Answer: a) [tex]\frac{x^{2} }{2352.25 }[/tex] + [tex]\frac{y^{2} }{529}[/tex] = 1
b) The distance of two foci is 85.4 feet
c) Area = 3502.67 square feet
Step-by-step explanation: a) An ellipse has the equation in the form of:
[tex]\frac{x^{2} }{a^{2} }[/tex]+[tex]\frac{y^{2} }{b^{2} }[/tex] = 1, where a is the horizontal axis and b is the vertical axis.
For the Statuary Hall, a = [tex]\frac{97}{2}[/tex] = 48.5 and b = [tex]\frac{46}{2}[/tex] = 23, so the equation will be
[tex]\frac{x^{2} }{2352.25 }[/tex] + [tex]\frac{y^{2} }{529}[/tex] = 1.
b) To determine the distance of the foci, we have to calculate 2c, where c is the distance between one focus and the center of the ellipse. To find c, as a, b and c create a triangle with a as hypotenuse:
[tex]a^{2}[/tex] = [tex]b^{2} + c^{2}[/tex]
[tex]c^{2} = a^{2} - b^{2}[/tex]
c = [tex]\sqrt{48.5^{2} - 23^{2} }[/tex]
c = 42.7
The distance is 2c, so 2·42.7 = 85.4 feet.
The two foci are 85.4 feet apart.
c)The area of an ellipse is given by:
A = a.b.π
A = 48.5 · 23 · 3.14
A = 3502.67 ft²
The area of the floor room is 3502.67ft².
On the kite, vertex A at the top, vertex B at the right, vertex C at the bottom, and vertex D at the left. Side A B is marked congruent to side A D. Side D C is marked congruent to side B C. Diagonal A C and B D are drawn.
Angle D A C is 39 degrees. Find m ∠ 1 and m ∠ 3 in the kite. The diagram is not drawn to scale.
Answer:
Part 1) [tex]m\angle 1=39^o[/tex]
Part 2) [tex]m\angle 3=51^o[/tex]
Step-by-step explanation:
The picture of the question in the attached figure
Part 1) Find the measure of angle 1
we know that
The longer diagonal of a kite bisects the kite into two equal parts
That means
[tex]m\angle 1=39^o[/tex]
In this problem the longer diagonal is the segment AC
Part 2) Find the measure of angle 3
we know that
The intersection of the diagonals of a kite form 90 degrees.
That means ----> The triangle ADO (O is the intersection point both diagonals) is a right triangle
so
[tex]39^o+m\angle 3=90^o[/tex] ----> by complementary angles in a right triangle
[tex]m\angle 3=90^o-39^o=51^o[/tex]
In article presents measures of penetration resistance for a certain fine-grained soil. fifteen measurements, expressed as a multiple of a standard quantity, had a mean of 2.64 and a standard deviation of 1.02. can you conclude that the mean penetration resistance is greater than 2.5? Find the p-value and state a conclusion.
Answer:
[tex]t=\frac{2.64-2.5}{\frac{1.02}{\sqrt{15}}}=0.532[/tex]
[tex]p_v =P(t_{(14)}>0.532)=0.302[/tex]
If we compare the p value and the significance level assumed [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis at 5% of significance
Step-by-step explanation:
Data given and notation
[tex]\bar X=2.64[/tex] represent the sample mean
[tex]s=1.02[/tex] represent the sample standard deviation for the sample
[tex]n=15[/tex] sample size
[tex]\mu_o =2.5[/tex] represent the value that we want to test
[tex]\alpha[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the mean is greater than 2.5, the system of hypothesis would be:
Null hypothesis:[tex]\mu \leq 2.5[/tex]
Alternative hypothesis:[tex]\mu > 2.5[/tex]
If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{2.64-2.5}{\frac{1.02}{\sqrt{15}}}=0.532[/tex]
P-value
The first step is calculate the degrees of freedom, on this case:
[tex]df=n-1=15-1=14[/tex]
Since is a one side right tailed test the p value would be:
[tex]p_v =P(t_{(14)}>0.532)=0.302[/tex]
Conclusion
If we compare the p value and the significance level assumed [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis at 5% of significance
Find the value of x. Round your answer to nearest tenth.
Answer: 27.3 degrees
cos x = 16/18
x = arccos(16/18)
x = 27.3 degrees
Value of x is 30°
Step-by-step explanation:
Step 1: Find value of x by using the trigonometric ratio cosine of x. Here, given that adjacent side is 16 and hypotenuse is 18cos x° = adjacent side/hypotenuse = 16/18 = 8/9
x° = cos inverse(8/9) = 27.12° = 30° (Rounded off to nearest ten)
Which of the following are ordered pairs for the equation y = 1/8x + 11?
(0,11) (5,12) (7,13)
(0,11) (8,12) (16,13)
(0,11) (7,13) (8,15)
(0,11) (6,12) (8,13)
Final answer:
The set of ordered pairs that correctly satisfies the equation y = 1/8x + 11 is (0,11) (8,12) (16,13). This is determined by substituting the x-values into the equation to check if the y-values correspond.
Explanation:
The question is asking which set of ordered pairs satisfies the equation y = 1/8x + 11. To determine the correct set, we will plug in the x-values from each ordered pair into the equation and see if the resulting y-value matches the one provided in the ordered pair. Let's examine each set:
(0,11) - If we plug in x=0, we get y=1/8*0+11=11. This matches the ordered pair, so (0,11) is correct.
(5,12) - With x=5, y becomes 1/8*5+11=11.625. This does not match the ordered pair (5,12), so this set is incorrect.
(7,13) - With x=7, y is 1/8*7+11=11.875. This does not match the ordered pair (7,13), so this set is also incorrect.
(8,12) - With x=8, y is 1/8*8+11=12. This matches the ordered pair, so (8,12) is correct.
(16,13) - With x=16, y is 1/8*16+11=13. This matches the ordered pair, so (16,13) is correct.
(6,12) - With x=6, y is 1/8*6+11=11.75. This does not match the ordered pair (6,12), so this set is incorrect.
(8,13) - Again, with x=8, y should be 12, not 13, so the ordered pair (8,13) is incorrect.
The only set with all matching ordered pairs for the equation y = 1/8x + 11 is (0,11) (8,12) (16,13).
Parents wish to have 160,000 available for a child's education. If the child is now 8 years old, how much money must be set aside at 3 % compounded semiannually to meet their financial goal when the child is 18?
Answer:
$118,791.2985
Step-by-step explanation:
Given that,
A = 160,000
T = 18 - 8 = 10 years
Rate = 3%
since it is semi annual, n = 2
A = P ( 1 + R/2) ∧ 2T
160000 = P ( 1 + 3/2*100) ∧ 2 * 10
160000 = P ( 1 + 3/200) ∧20
160000 = P( 1 + 0.015) ∧20
160000 = P(1.015)∧20
P= 160000/ (1.015)∧20
P = 160000/ 1.3469
= $118,791.2985
Which of the following represents an example to calculate the sum of numbers (that is, an accumulator), given that the number is stored in the variable number and the total is stored in the variable total? A) total +number B) total +number =total C) total + number D) number+number
The accumulator pattern for calculating the sum of numbers using variables number and total in programming is correctly represented by 'total = total + number'. This demonstrates the commutative property of addition, indicating that 'total += number' updates the running total with the new number.
Explanation:The correct representation of an accumulator to calculate the sum of numbers in a programming context, when the current number is stored in a variable number and the total is stored in a variable total, would be to update the total by adding the number to it. This would look like total = total + number or in a more simplified form as total += number.
Addition in programming is similar to addition in mathematics; it's commutative and associative. This means that the order of adding numbers does not change the sum, as represented by A + B = B + A. Whether we are dealing with numbers, such as integers or real numbers, or other structures like vectors, the concept of addition remains fundamentally the same.
The accumulator pattern is commonly used to build a sum or total by updating a running total each time a new value is added, which is also illustrated by the expression total += number. This pattern is essential in various programming tasks, such as summing a series of numbers in a loop.
Jorge is setting up his tent. He is using two nylon ropes to pull the tent taut and stabilize it at each end. If the tent is 5 feet tall, and Jorge stakes the ropes into the ground 3 feet from the tent. What is the total length of nylon rope he will use,
Answer:
Total length of the Nylon rope will be 5.8 feet.
Step-by-step explanation:
Given:
Height of the tent = 5 ft
ground Distance from stake to tent = 3 ft
We need to find the Total length of the nylon rope.
Solution:
Now we can say that the total length of the nylon rope, the height of the tent, the ground distance from the stake to the tent, forms a right angle triangle.
From above we can see that;
the height of the tent, the ground distance from the stake to the tent are the two legs of the right angled triangle.
While the Total length of the nylon rope is the hypotenuse.
Now using Pythagoras theorem we get;
[tex]h^2=l_1^2+l_2^2[/tex]
[tex]l_1[/tex] ⇒ the height of the tent
[tex]l_2[/tex] ⇒ the ground distance from the stake to the tent
[tex]h[/tex] ⇒ the Total length of the nylon rope
substituting the values we get;
[tex]h^2=5^2+3^2\\\\h^2=25+9\\\\h^2=34[/tex]
Taking square root on both side we get;
[tex]\sqrt{h^2} =\sqrt{34} \\\\h=5.8\ ft[/tex]
Hence Total length of the Nylon rope will be 5.8 feet.
Answer:
Just find the hypotenuse by doing a^2 + B^2 = C^2
Step-by-step explanation:
: Logan is driving a boat that has a speed of 18 mph in standing water (no current). She drives the boat up and down a river to pick up people on tubing trips. The boat travels 4 miles each way and it takes half and hour to complete the round trip.How fast is the current that helps the boat one way and slows the boat the other way.
Answer: the speed of the current is 6 mph
Step-by-step explanation:
Let x represent the speed of the current.
Logan is driving a boat that has a speed of 18 mph in standing water.
Assuming the current slowed down the boat while she was going up(upstream), it means that his total speed was (18 - x) mph
Also, if the current helped the boat while she was going down(downstream), it means that his total speed was (18 + x) mph
Time = distance/speed
The boat travels 4 miles each way. The time taken to travel upstream is
4/(18 - x)
Time taken to travel downstream is
4/(18 + x)
The round trip took 0.5 hour. It means that
4/(18 - x) + 4/(18 + x) = 0.5
Multiplying through by (18 - x)(18 + x), it becomes
4/(18 - x) + 4/(18 + x) = 0.5(18 - x)(18 + x)
4(18 + x) + 4(18 - x) = 0.5(18 - x)(18 + x)
72 + 4x + 72 - 4x = 0.5(324 + 18x -
18x - x²)
144 = 0.5(324 - x²)
144 = 162 - 0.5x²
0.5x² = 162 - 144
0.5x² = 18
x² = 18/0.5 = 36
x = √36
x = 6
A sociologist surveyed 300 people about their level of anxiety on a scale of 1 to 100. Unfortunately, the person inputting the data into the computer accidentally transposed six of the numbers causing the statistics to have errors.What type of error is this?1. Sampling error 2. Non sampling error
Answer:
sampling error i think
Step-by-step explanation:
What is the solution of the system of equations shown in the graph?
I Think it's c. 0,4 but I could be wrong
Option b: The solution to the system of equations is (2,0)
Explanation:
Given that the graph that contains the system of equations.
We need to determine the solution to the system of equations.
The solution to the system of equations is the points of intersection of these two lines.
The two lines intersect at x - axis at 2 and y - axis 0.
This can be written in coordinates as (2,0)
Thus, the point of intersection of the two lines is the point (2,0)
Hence, Option b is the correct answer.
The time that it takes a randomly selected job applicant to perform a certain task has a distribution that can be approximated by a normal distribution with a mean value of 115 seconds and a standard deviation of 20 seconds. The fastest 10% are to be given advanced training. What task times qualify individuals for such training
The time for the fastest 10 % is less than 89.4 seconds
Here's how to find the qualifying task times:
Calculate the z-score for the 10th percentile:
A z-score represents the number of standard deviations a specific point is away from the mean. In this case, we want the z-score for the lower 10th percentile, which can be found using a z-score table or online calculators. The approximate z-score for the 10th percentile is -1.28.
Translate the z-score to task time:
We know the z-score for the 10th percentile (-1.28) and the standard deviation (20 seconds). We can use the formula to find the corresponding task time (t):
t = mean + (z-score) * standard deviation
t = 115 seconds + (-1.28) * 20 seconds
t ≈ 89.4 seconds
Therefore, task times less than 89.4 seconds qualify individuals for advanced training, as they fall within the lower 10th percentile of the normal distribution.
Complete question:
The time that it takes a randomly selected job applicant to perform a certain task has a distribution that can be approximated by a normal distribution with a mean value of 115 sec and a standard deviation of 20 sec. The fastest 10% are to be given advanced training. What task times qualify individuals for such training? (Round the answer to one decimal place.)
Determine whether each of these sets is the power set of aset,wherea and b are distinct elements. a) ∅ c) {∅,{a},{∅,a}} b) {∅,{a}} d) {∅,{a},{b},{a,b}}
Among the sets given, ∅ is not a power set while {∅,{a}}, {∅,{a},{∅,a}}, and {∅,{a},{b},{a,b}} can be considered power sets of the sets {a}, {a}, and {a,b} respectively according to the definition of power set.
Explanation:In mathematics, a power set of any set S is the set of all subsets of S, including the empty set and S itself. We can use this definition to examine the four sets provided and determine if they qualify as power sets.
a) ∅ is not a power set because a power set must at least contain the empty set and the set itself.b) {∅,{a}} is the power set of the set {a}, because it includes the empty set and the set {a} itself.c) {∅,{a},{∅,a}} is the power set of the set {a}, again, because it includes the empty set, the element a and the set {a} itself.d) {∅,{a},{b},{a,b}} is the power set of the set {a,b}, as it includes the empty set, single element sets {a} and {b}, and the set itself {a,b}.Learn more about Power Sets here:https://brainly.com/question/35520738
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A certain vibrating system satisfies the equation . Find the value of the damping coefficient for which the quasi period of the damped motion is greater than the period of the corresponding undamped motion.
Question:
A certain vibrating system satisfies the equation u''+γu'+u=0. Find the value of the damping coefficientγfor which the quasi period of the damped motion is 50% greater than the period of the corresponding undamped motion.
Answer: y = √(20/9) = √20/3 = 1.49071
Step-by-step explanation:
u''+γu'+u=0
m =1, k =1, w• = √ (k/m) = 1
The period of undamped motion T, is given by T = 2π/w•, T = 2π/1 = 2π
The quasi period Tq = 2π/quasi frequency
Quasi frequency = ((4km - y^2)^1/2)/2m
Therefore the quasi period Tq = 4πm/((4km - y^2)^1/2)
From the question the quasi period is 50% greater than the period of undamped motion
Therefore Tq = T + (1/2)T = (3/2)T
Thus,
4πm/((4km - y^2)^1/2) = (3/2)(2π)
Where, k =1, m=1,
4π/((4 - y^2)^1/2) = 3π,
(4 - y^2)^1/2 = 4π/3π,
(4 - y^2) = (4/3)^2,
4 - y^2 = 16/9,
y^2 =4 - 16/9,
y^2 = 20/9,
y = √(20/9)
Answer:
Answer is 1.49071
Step-by-step explanation:
See the picture for the complete details
There were 88 vendors at the craft fair. They needed to set up an equal number in each of the rows and needed 4 flags to mark each row. How many rows and flags were needed?
Answer:
22 rows and 22 flags are needed
Step-by-step explanation:
Total number of vendors = 88
Existing number of rows and flags= 4
Number of rows and flags needed= 88/4 =22
Answer:
4rows and 16flags
Step-by-step explanation:
Since there were 88 vendors at the craft fair and 4flags on each rows. To set up equal number of vendors on each row, we will use the expression;
Number of vendors per row = Total number of vendors/total number of flags per row = 88/4 = 22 vendors
If there are 22 vendors in a rows and there are 88vendors in total, the total of rows will be;
Total number of vendors/number of vendors per row
= 88/22
= 4 rows
If there are four rows in total and 4flags in each row, the total of flags needed will be;
Total number of row × total flag per row
= 4×4
= 16flags
This shows that there are 4rows and 16flags were needed.
A puzzle in the newspaper presents a matching problem. The names of 10 U.S. presidents are listed in one column, and their vice presidents are listed in random order in the second column. The puzzle asks the reader to match each president with his vice president.
(1) If you make the matches randomly, how many matches are possible?
Number of possible matches
(2) What is the probability all 10 of your matches are correct? (Round your answer to 8 decimal places.)
Answer:
(1) 3628800
(2) 0.00000028
Step-by-step explanation:
We are given that a puzzle in the newspaper presents a matching problem. The names of 10 U.S. presidents are listed in one column, and their vice presidents are listed in random order in the second column.
(1) If we make the matches randomly, number of possible matches are given by = 10!
Because after making each match the number will decrease so,
Number of possible matches = 10! = 10*9*8*7*6*5*4*3*2*1 = 3628800 .
(2) The probability all 10 of your matches are correct is given by;
Number of outcomes in favor ÷ Total number of matches
So, there will be only 1 case when all 10 of the matches are correct.
Therefore, required probability = [tex]\frac{1}{10!}[/tex] = [tex]\frac{1}{3628800}[/tex] = 0.00000028 .
A right triangle has legs with the lengths of 2 and 5, find the length of the hypotenuse
Answer Choices:
√21
√3
√7
√29
Answer:
[tex]\sqrt{29}\ units[/tex]
Step-by-step explanation:
we know that
In a right triangle
Applying the Pythagorean Theorem
[tex]c^2=a^2+b^2[/tex]
where
c is the hypotenuse (the greater side)
a and b are the legs (perpendicular sides)
In this problem we have
[tex]a=2\ units\\b=5\ units[/tex]
substitute
[tex]c^2=2^2+5^2[/tex]
[tex]c^2=29[/tex]
[tex]c=\sqrt{29}\ units[/tex]
The number of defects in the first five cars to come through a new production line are 9, 7, 10, 4, and 6, respectively. If the sixth car through the production line has either 3, 7, or 12 defects, for which of theses values does the mean number of defects per car for the first six cars equal the median?
I. 3
II. 7
III. 12
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
Answer:
D
Step-by-step explanation:
Depending on the number of defects in 6th car, we have 3 cases:
3,4,6,7,9,10 (median 13/2 =6.5) 4,6,7,7,9,10 (median 14/2 = 7) 4,6,7,9,10,12 (median 16/2 = 8)To find mean we can use the sum of numbers given (= 36) and add either 3, 7, or 12 to it and then divide by 6 (number of cars).
(36 + 3) /6 = 39/6 = 6.5 (== median above)
(36 + 7) /6 = 43/6 = 7.1 (does not equal median above)
(36 + 12) /6 = 48/6 = 8 (== median above)
Answer: D (I and III only).
Why do we use two supply curves in the aggregate goods and services market? What is the difference between them, and why do they have different slopes?
Answer:
Supply curve vs Aggregate Supply
Step-by-step explanation:
- Supply express a direct relationship between what producers supply and how that relationship affects the price of a specific product or service. Hence, expresses a linear increase - or a constant slope
- Aggregate supply are the total supply in an economy at a particular period of time and a particular price threshold. Aggregate supply is an economy's gross domestic product (GDP), the total amount a nation produces and sells.
- Aggregate supply convey how much firms are willing to produce at a specific price point.
- Aggregate supply is a response to increasing prices that drive firms to utilize more inputs to produce more output. The incentive is that if the price of inputs remains the same but if the price of outputs increase, the firm will generate larger profits and margins by producing and selling more. The aggregate supply curve is represented by a curve that slopes upward, which indicates that as the price per unit goes up, a firm will supply more. Increasing slope - Not constant.
- The supply curve eventually becomes vertical, indicating that at a certain price point a firm cannot produce anymore, as they are limited by certain inputs, e.g. number of employees and number of factories.
fewer students can afford to attend college. Some commentators have suggested that a heightened sense of patriotism may increase military enlistments, while others think that the existence of actual hostilities may deter young people from choosing a military path. A polling organization wants to investigate what this year’s high school seniors are planning to do after they graduate.
1) During the ‘90s about 63% of high school graduates enrolled in college. The pollsters hope to estimate the percentage of this year’s seniors planning to attend college with a margin of error no greater than 4%. What size sample would suffice if they want to have 90% confidence in the estimate?
The pollsters randomly select 5 cities in Upstate New York and then randomly selected on high school in each city. The guidance office at each of the chosen schools is instructed to ask 100 randomly selected seniors what their current plans are, and to report the results back to the pollsters. The data collected from the 5 schools are summarized in the following table.
Plans
Count
College
289
Employment
112
Military
26
Other (travel, parenting, etc.)
51
Undecided / No response
22
2) Determine a 90% confidence interval for the percentage of seniors planning to go to college this year. Explain in context what your interval means. Make sure you include your checks that the conditions for inference have been met.
3) During the 90’s about 4.5% of high school seniors enlisted in the military. Do these data suggest that the percentage who enlist is different this year? Test an appropriate hypothesis and state your conclusion. Make sure you state your null and alternative hypothesis, the test statistic, the P-value, your alpha level, and your conclusion.
4) A few of the seniors did not respond to the guidance queries, and others said they were undecided. Some of the people might eventually decide to enlist in the military. Suppose that half of this small group also enlist. Would that cause you to change your conclusion in Question 3? Make sure you perform another hypothesis test and make sure you state your null, alternative hypothesis, the test statistic, the P-value, your alpha level, and your conclusion.
To estimate the percentage of seniors attending college with a 4% margin of error and 90% confidence, a sample size of 424 is needed. For constructing a confidence interval and hypothesis testing about military enlistment, various statistical formulas are applied, including adjustments for undecided responses.
Explanation:To estimate the percentage of this year’s seniors planning to attend college with a margin of error no greater than 4% and 90% confidence, we use the formula for determining sample size for a proportion, which is n = (Z^2 * p * (1-p)) / E^2, where Z is the Z-score corresponding to the confidence level, p is the estimated proportion, and E is the margin of error. With a 90% confidence level, the Z-score is approximately 1.645, and assuming we don't have a preliminary estimate, we use p = 0.5 for maximum variability, thus maximizing the required sample size.
To calculate: n = (1.645^2 * 0.5 * 0.5) / 0.04^2 = 423.3. Therefore, a sample size of 424 is required to achieve the desired margin of error and confidence level.
For the second question regarding the confidence interval for the percentage of seniors planning to go to college this year, the total sample size is the sum of students from all categories, which is 500. The proportion planning to go to college is 289/500. Using a formula for constructing a confidence interval for a proportion, C.I. = p ± (Z*sqrt(p(1-p)/n)), we can find our interval.
Testing the hypothesis regarding military enlistment involves setting up a null hypothesis (H0: p = 0.045) and an alternative hypothesis (H1: p ≠ 0.045), where p is the proportion of high school seniors enlisting in the military. The test statistic is calculated using a formula, and the P-value associated with this statistic is compared against the alpha level to decide whether to reject H0.
If considering undecided or no response students as potential enlistees changes this calculation significantly, we reassess by including an adjusted number of potential enlistees.
x2 + y2 – 8x + 12y- 8 = 4
Answer:
3x-7y=-6
Step-by-step explanation:
Solve for all the missing angles for triangle ABC: a= 10cm, b=15cm, c= 20cm. State the angles in order (Angle A,B,C) and round answers to the nearest hundredth
Answer:
Part 1) [tex]A=28.96^o[/tex]
Part 2) [tex]B=46.57^o[/tex]
Part 3) [tex]C=104.47^o[/tex]
Step-by-step explanation:
step 1
Find the measure of angle A
Applying the law of cosines
[tex]a^2=b^2+c^2-2(b)(c)cos(A)[/tex]
we have
[tex]a=10\ cm\\b=15\ cm\\c=20\ cm[/tex]
substitute
[tex]10^2=15^2+20^2-2(15)(20)cos(A)[/tex]
Solve for A
[tex]2(15)(20)cos(A)=15^2+20^2-10^2[/tex]
[tex]600cos(A)=525[/tex]
[tex]cos(A)=(525/600)[/tex]
using a calculator
[tex]A=cos^{-1}(525/600)=28.96^o[/tex]
step 2
Find the measure of angle B
Applying the law of cosines
[tex]b^2=a^2+c^2-2(a)(c)cos(B)[/tex]
we have
[tex]a=10\ cm\\b=15\ cm\\c=20\ cm[/tex]
substitute
[tex]15^2=10^2+20^2-2(10)(20)cos(B)[/tex]
Solve for A
[tex]2(10)(20)cos(B)=10^2+20^2-15^2[/tex]
[tex]400cos(B)=275[/tex]
[tex]cos(B)=(275/400)[/tex]
using a calculator
[tex]B=cos^{-1}(275/400)=46.57^o[/tex]
step 3
Find the measure of angle C
we know that
The sum of the interior angles in any triangle must be equal to 180 degrees
so
[tex]A+B+C=180^o[/tex]
we have
[tex]A=28.96^o[/tex]
[tex]B=46.57^o[/tex]
substitute
[tex]28.96^o+46.57^o+C=180^o[/tex]
[tex]C=180^o-75.53^o[/tex]
[tex]C=104.47^o[/tex]
A rectangle that is x feet wide is inscribed in a circle of radius 25 feet. Express the area of the rectangle as a function of x. Give the function and state its domain.
Answer:
[tex]A(x)=x(\sqrt{2500-x^{2} } )[/tex] is the area expressed as function of x,
and x: >0, <50 is the domain of the function.
Step-by-step explanation:
Draw an appropriate figure and then express the area of the rectangle as a function of x.
We know that the diagonal of a rectangle inscribed in a circle, will be equal to the diameter of the circle. 50 feet
let w = the width of the rectangle
therefore
[tex]x^2 + w^2 = 50^2\\w^2 = 2500 - x^2\\w = \sqrt{2500-x^{2} }[/tex]
recall
Area = x*w
replace w
[tex]A(x)=x(\sqrt{2500-x^{2} } )[/tex] is the area expressed as function of x
b)state the domain of the function
x: >0, <50
To find the area of the rectangle inscribed in a circle, we can use the Pythagorean theorem to find the length of the rectangle. Then, we can calculate the area by multiplying the width and length. The function that represents the area of the rectangle as a function of x is A(x) = x * sqrt(2500 - x^2), and the domain is x >= 0 and x <= 50.
Explanation:To find the area of the rectangle, we need to consider that the diagonal of the rectangle is equal to the diameter of the circle, which is twice the radius. Therefore, the diagonal measures 50 feet.
We can use the Pythagorean theorem to find the length of the rectangle. Let's denote the length as y. We have:
x^2 + y^2 = 50^2
This equation represents the relationship between the width and length of the rectangle. By solving for y, we can express the length as a function of x.
Once we have the width and length of the rectangle, we can calculate the area by multiplying them:
A = x * y = x * sqrt(50^2 - x^2)
The function that represents the area of the rectangle as a function of x is A(x) = x * sqrt(2500 - x^2). The domain of this function is x >= 0 and x <= 50.
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According to the Fisher effect, if a lender and a borrower would agree on an interest rate of 8 percent when no inflation is expected, they should set a rate of _______ when an inflation rate of 3 percent is expected.
Answer:
5%
Step-by-step explanation:
According to fisher equation
Nominal rate = Real rate + Inflation
N = R + I
N is calculated when there is no inflation and for the current year = 8%
The Real rate is calculated from the base year
Real rate is consider "Inflation factor" and R is unknown
Inflation rate (I) = 3%
Hence, N = R + I
8 = R + 3
R = 8 - 3
R = 5%
A local pizza restaurant surveyed a random sample of 150 people that live in their town about their favorite type of pizza. Of the people surveyed, 40 said that pepperoni pizza was their favorite type of pizza. There are 2,800 residents that live in the town. Based on the data, is 750 a reasonable estimate for the number of residents in the town whose favorite pizza is p
Answer:
756 is a reasonable estimate because the proportion of the sample whose favorite pizza is pepperoni is about 27 %
Step-by-step explanation:
given data
random sample = 150 people
favorite pepperoni pizza = 40
total of residents = 2800
number of residents favorite pizza = 750
solution
we get here first Proportion favorite pizza that is
Proportion = [tex]\frac{40}{150}[/tex]
Proportion = 0.27 = 27%
and now we get 27% of total of residents that is = 27% × 2800
= 756
so we can say 756 is a reasonable estimate because here proportion of an sample whose favorite pizza are pepperoni pizza about 27 %
If using the method of completing the square to solve the quadratic equation x^2+3x-13=0x 2 +3x−13=0, which number would have to be added to "complete the square"?
Answer:
[tex]\frac{9}{4}[/tex]
Step-by-step explanation:
Given
x² + 3x - 13 = 0 ( add 13 to both sides )
x² + 3x = 13
To complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2([tex]\frac{3}{2}[/tex] )x + ([tex]\frac{3}{2}[/tex] )² = 13 + ([tex]\frac{3}{2}[/tex] )², that is
x² + 2([tex]\frac{3}{2}[/tex] )x + [tex]\frac{9}{4}[/tex] = 13 + [tex]\frac{9}{4}[/tex]
(x + [tex]\frac{3}{2}[/tex] )² = [tex]\frac{61}{4}[/tex]
The required number to be added to complete the square is [tex]\frac{9}{4}[/tex]
Hence, required number to be added to complete the square is 9/4
What is Quadratic Equation?A quadratic equation is any equation that can be rewritten in standard form as ax2+bx+c=0 in algebra. When x is an unknown and a, b, and c are known numbers, and an is less than 0. Because there is no ax2 term when a = 0, the equation is linear rather than quadratic.
How to solve?Given equation =x² + 3x - 13 = 0 ( add 13 to both sides )
=x² + 3x = 13
using complete the square and add ( half the coefficient of the x- term )² to both sides
=x² + 2(3/2 )x + ( 3/2)² = 13 + (3/2 )², that is
=x² + 2(3/2 )x + = 13 + 9/4
=(x + 3/2 )² = 61/4
The required number to be added to complete the square is 9/4
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As a business person you need to visit each of the following cities once. You need to start in Atlanta and end in Atlanta. Use the "Nearest Neighbor" Algorithm to find one of the cheaper routes to visit each city and return home again.
Group of answer choices
A
B
C
D
Answer:
it was B
Step-by-step explanation:
The algorithm employing "Nearest Neighbour" to determine one of the cheaper routes to visit every city and get back home again would be:
B). ATL-CHI-BOS-DEN-ATL
Algorithm is described as the compilation of sequenced steps that assist in resolving a mathematical problem. This involves a step-by-step organization that assists in determining the final value post following the steps. In the given scenario, the cheaper route to visit all the given cities once and returning home again would be beginning from Atlanta via Chicago, Boston, Denver, and returning back to Atlanta.
Thus, option B is the correct answer.
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The questions below deal with the Gizmo Company, which has the following production function. If the real wage is equal to 8 widgets and only an integer number of workers can be hired the Gizmo company should hire 3 workers. 5 workers. 4 workers. 2 workers.
Answer:
2 workers
Step-by-step explanation:
Here is additional information for your question:
The questions below deal with the Gizmo Company, which has the following production function.
# Workers # Produce
0 0
1 10
2 19
3 26
4 31
5 34
If the real wage is equal to 8 widgets and only an integer number of workers can be hired the Gizmo company should hire?
My answer:
2 workers. (where MPL is less than real wage)
pamela is 10 years younger than jiri the sum of their age is 70
Answer:
Pamela = 30
Jiri = 40
Step-by-step explanation:
Two equations needed:
J-10 = P
P+J = 70
Plug and solve:
(J-10) + J = 70
2J - 10 =70
2J = 80
J = 40
40 - 10 = P
P=30
Answer:
30
Step-by-step explanation:
Let Jiri's age be x
Let Pamela's age be (x - 10)
The sum of their ages becomes
x + (x - 10) = 70
x + x - 10 = 70
2x - 10 = 70
2x =70+10
2x = 80
x = 40
Therefore, it means that Jiri's age is 40
Pamela's age is 40-10= 30
In 2010, the area's population tallied at 2.13 million. Since then, the population has grown at a rate of 2.4% per year. Write an equation that you can use to predict the population for the number of years after 2010.
Answer:
Step-by-step explanation:
We would apply the formula for exponential growth which is expressed as
y = b(1 + r)^t
Where
y represents the population, t years after 2010.
t represents the number of years.
b represents the initial population.
r represents rate of growth.
From the information given,
b = 2.13 × 10^6
r = 2.4% = 2.4/100 = 0.024
Therefore, the equation that you can use to predict the population for the number of years after 2010 is
y = 2.13 × 10^6(1 + 0.024)^t
y = 2.13 × 10^6(1.024)^t
We have two fair three-sided dice, indexed by i = 1, 2. Each die has sides labeled 1, 2, and 3. We roll the two dice independently, one roll for each die. For i = 1, 2, let the random variable Xi represent the result of the i-th die, so that Xi is uniformly distributed over the set {1, 2, 3}. Define X = X2 − X1. 1. Calculate the numerical values of following probabilities:____________. (a) P(X = 0) = (b) P(X = 1) = (c) P(X = −2) = (d) P(X = 3) = Let Y = X2 . Calculate the following probabilities:_________. (a) P(Y = 0) =(b) P(Y = 1) = (c) P(Y = 2) =
Answer:
(a) P(X = 0) = 1/3
(b) P(X = 1) = 2/9
(c) P(X = −2) = 1/9
(d) P(X = 3) = 0
(a) P(Y = 0) = 0
(b) P(Y = 1) = 1/3
(c) P(Y = 2) = 1/3
Step-by-step explanation:
Given:
- Two 3-sided fair die.
- Random Variable X_1 : Result on 1st die.
- Random Variable X_2: Result on 2nd die.
- Random Variable X = X_2 - X_1.
Solution:
- Possible outcomes of X : { - 2 , -1 , 0 ,1 , 2 }
- The corresponding probabilities for each outcome are:
( X = -2 ): { X_2 = 1 , X_1 = 3 }
P ( X = -2 ): P ( X_2 = 1 ) * P ( X_1 = 3 )
: ( 1 / 3 ) * ( 1 / 3 )
: ( 1 / 9 )
( X = -1 ): { X_2 = 1 , X_1 = 2 } + { X_2 = 2 , X_1 = 3 }
P ( X = -1 ): P ( X_2 = 1 ) * P ( X_1 = 3 ) + P ( X_2 = 2 ) * P ( X_1 = 3)
: ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 )
: ( 2 / 9 )
( X = 0 ): { X_2 = 1 , X_1 = 1 } + { X_2 = 2 , X_1 = 2 } + { X_2 = 3 , X_1 = 3 }
P ( X = -1 ):3*P ( X_2 = 1 )*P ( X_1 = 1 )
: 3*( 1 / 3 ) * ( 1 / 3 )
: ( 3 / 9 ) = ( 1 / 3 )
( X = 1 ): { X_2 = 2 , X_1 = 1 } + { X_2 = 3 , X_1 = 2 }
P ( X = 1 ): 2* P ( X_2 = 2 ) * P ( X_1 = 1 )
: 2* ( 1 / 3 ) * ( 1 / 3 )
: ( 2 / 9 )
( X = 2 ): { X_2 = 1 , X_1 = 3 }
P ( X = 2 ): P ( X_2 = 3 ) * P ( X_1 = 1 )
: ( 1 / 3 ) * ( 1 / 3 )
: ( 1 / 9 )
- The distribution Y = X_2,
P(Y=0) = 0
P(Y=1) = 1/3
P(Y=2) = 1/ 3
- The probability for each number of 3 sided die is same = 1 / 3.
In this exercise we have to use the knowledge of probability to calculate the chance of an event to occur, so:
A) P(X = 0) = 1/3
B) P(X = 1) = 2/9
C) P(X = −2) = 1/9
D) P(X = 3) = 0
A) P(Y = 0) = 0
B) P(Y = 1) = 1/3
C) P(Y = 2) = 1/3
organizing the following information given in the text we have that:
Two 3-sided fair die.Random Variable X_1 : Result on 1st dieRandom Variable X_2: Result on 2nd die.Random Variable X = X_2 - X_1.Possible outcomes of X : { - 2 , -1 , 0 ,1 , 2 }Then calculating the probability we find that:
A) For P(X = 0) we have
[tex]( X = -2 ): { X_2 = 1 , X_1 = 3 } \\P ( X = -2 ): P ( X_2 = 1 ) * P ( X_1 = 3 ) \\ : ( 1 / 3 ) * ( 1 / 3 ) \\ : ( 1 / 9 )[/tex]
B) For P(X = 1) er have:
[tex]( X = -1 ): { X_2 = 1 , X_1 = 2 } + { X_2 = 2 , X_1 = 3 }\\ P ( X = -1 ): P ( X_2 = 1 ) * P ( X_1 = 3 ) + P ( X_2 = 2 ) * P ( X_1 = 3)\\ : ( 1 / 3 ) * ( 1 / 3 ) + ( 1 / 3 ) * ( 1 / 3 )\\ : ( 2 / 9 )[/tex]
C) For P(X = −2) we have:
[tex]( X = 0 ): { X_2 = 1 , X_1 = 1 } + { X_2 = 2 , X_1 = 2 } + { X_2 = 3 , X_1 = 3 }\\ P ( X = -1 ):3*P ( X_2 = 1 )*P ( X_1 = 1 )\\ : 3*( 1 / 3 ) * ( 1 / 3 )\\ : ( 3 / 9 ) = ( 1 / 3 )[/tex]
D) For P(X = 3), we have:
[tex]( X = 1 ): { X_2 = 2 , X_1 = 1 } + { X_2 = 3 , X_1 = 2 }\\ P ( X = 1 ): 2* P ( X_2 = 2 ) * P ( X_1 = 1 )\\ : 2* ( 1 / 3 ) * ( 1 / 3 )\\ : ( 2 / 9 )\\ ( X = 2 ): { X_2 = 1 , X_1 = 3 }\\ P ( X = 2 ): P ( X_2 = 3 ) * P ( X_1 = 1 ) \\ : ( 1 / 3 ) * ( 1 / 3 ) \\ : ( 1 / 9 )[/tex]
In this next case we have to take into account that the number will be the probability of falling 3, that is:
A) P(Y=0) = 0
B) P(Y=1) = 1/3
C) P(Y=2) = 1/ 3
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