Answer:
For a unique solution, there must be as many consistent, independent equations as there are variables.
Step-by-step explanation:
Each equation lets you write an expression for one variable in terms of the others in that equation. That expression can be substituted throughout the system, reducing the number of equations and variables by one each.
After n-1 such substitutions in a system of n equations, there will be one equation left. If that is an equation in one variable, then the solution to the system can be easily found. If more than one variable remains (> n variables in n equations), then the system cannot be solved for definitive values of each of the variables.
___
If the equations are inconsistent, there is no solution. If the equations are dependent, then there is no unique solution.
To solve a system of linear equations, one generally requires the same number of equations as there are variables for a unique solution, otherwise the system may have either an infinite number of solutions or none.
To solve a system of linear equations, you generally need as many equations as there are variables. If you have two variables, you should have two linear equations to find a unique solution. For example, the equations 3x - 2y = 24 and 3x + 9y = 30 form a system that can be solved because there are two equations with two variables (x and y).
If there are more variables than equations, then you may either have infinitely many solutions or no solution, depending on the equations' consistency. If you have fewer equations than variables, as in the case of three variables with only two equations, you typically have an undetermined system where you can choose one variable as a parameter and express the others in terms of it, leading to an infinite number of solutions.
For homogeneous equations where the vector b is zero (Ax = 0), the solutions form a subspace of the vector space, and nontrivial solutions exist if the system is not of full rank.
Solve the equation. Round to the nearest hundredth. Show work.
[tex]2e^{8x} = 1 - e^{4x}[/tex]
Answer:
The value of x = -0.17
Step-by-step explanation:
∵ [tex]2e^{8x}=1-e^{4x}[/tex]
Let [tex]e^{4x}=y[/tex]
∴ [tex]e^{8x}=y^{2}[/tex]
∴ 2y² = 1 - y
∴ 2y² + y - 1 =0 ⇒ factorize
∴ (2y - 1)(y + 1) = 0
∴ 2y - 1 = 0 ⇒ 2y = 1 ⇒ y = 1/2
∴ y + 1 = 0 ⇒ y = -1
∵ [tex]y=e^{4x}[/tex]
Note: [tex]e^{4x}=-1[/tex] ⇒ refused
([tex]e^{ax}[/tex] never gives -ve values)
∴ [tex]e^{4x}= 1/2[/tex] ⇒ insert ln in both sides
∵ [tex]ln(e)^{ax}=axln(e)=ax[/tex] ⇒ ln(e) = 1
∴ 4xln(e) = ln(1/2) ⇒ 4x = ln(1/2)
∴ x = [ln(1/2)]/4 = -0.17
A circle has a circumference of 615.44615. point, 44 units. What is the radius of the circle?
Answer:
The radius is 98
Step-by-step explanation:
The circumference of a circle is the distance around the edge of the circle. To find the circumference, we use the formula C = 2πr. You have the circumference so substitute C = 615.44. Then solve for r.
C = 2πr
615.44 = 2πr
98 = r
Answer: 98 units
Step-by-step explanation:
A container is in the shape of a rectangular prism with a square base. It has a volume of 99 cubic inches and a height of 11 inches. How many softballs with a diameter of 3.8 inches will fit into the container? Use the drop-down menus to explain your answer.
Answer:
A total of zero softballs will fit into the container
Step-by-step explanation:
step 1
Find the dimensions of the base of the prism
we know that
The volume of the prism is equal to
[tex]V=Bh[/tex]
where
B is the area of the base
h is the height of the prism
In this problem we have
[tex]V=99\ in^{3}[/tex]
[tex]h=11\ in[/tex]
substitute in the formula and find the area of the base B
[tex]99=B(11)[/tex]
[tex]B=99/11=9\ in^{2}[/tex]
the length side of the square base is the square root of the area
so
[tex]\sqrt{9}=3\ in[/tex]
we have that
The diameter of the softball 3.8 inches will fit (11/3.8=2.89 ) 2 times in the length of the container
The diameter of the softball 3.8 inches will fit 0 times in the width of the container
so
A total of 0 times of softballs will fit in the width of the container
therefore
A total of zero softballs will fit into the container
Answer:
Zero softballs with a diameter of 3.8 inches will fit into the container as length of the container is less the diameter of the softball.
Zero softballs can fit in length and zero softballs will fit in width.
Step-by-step explanation:
Length of the square base in rectangular pyramid = s
Breadth of the square base in rectangular pyramid = s
Height of the square base in rectangular pyramid ,l = 11 inches
Volume of the square base in rectangular pyramid ,V=[tex]99 inches^3[/tex]
Volume of the cuboid = l × b × w
V= s × s × l
[tex]99 inches^3=s^2\times 11 inches[/tex]
s = 3 inches
Softballs with a diameter of 3.8 inches.
But the length of the container is less the diameter of the softball which means not even single ball will not be able to get into the container. So zero softballs can fit in length and zero softballs will fit in width.
Look at the tally chart or line plot from Exercise 2 above. What was the total distance ridden by all of the children?
The senior class of a high school needs to elect a new class president. Ten people want to run for office, but only four are allowed to so the class needs to determine which four people can run.
They randomly select 5 people from each of the ten homerooms and have them select two people out of the ten that they would like to run for office. The table shows the results.
Which four people should be on the ballot to run for office from the results of the survey?
A) George Smith, Jamal Brown, Lucas Hernandez, and Lucy Coen
B) Walter Heyes, Laurie Langston, George Smith, and Jamal Brown
C) Michael Mast, Joel Friedburg, Alice King, and Laurie Langston
D) George Smith, Laurie Langston, Jamal Brown, and Lucas Hernandez
Answer:
A
Step-by-step explanation:
Add up the votes for each person and order them greatest to least and pick the top 4.
The data and summary given in the represents a random sample of twelfth-grade girls and bots in a school district. Is there an overlap between the two data sets?
Answer:
it is A) No, there is not an overlap between the two data setsStep-by-step explanation:
i did it on usatestprep
Answer:
A-No, there is not an overlap between the two data sets.
Step-by-step explanation:
A father is four times as old as his son. In 4 years, the father will be three times as old as the son. How old is each now?
Answer:
M = 4S
In three years
M + 3 = 3(S + 3)
So you put the 4S in to substitute for the M.
4S + 3 = 3(S + 3)
4S + 3 = 3S + 9
S = 6
If the son is 6, the father must be 24.
We can check this by adding three to both ages. Then, the son will be 9 and the father will be 27, which is three times 9.
Step-by-step explanation:
Need help ASAP!!
2. The Geo Air pilot is looking at SCCA from the plane. From the aircraft the angle of depression is 17 degrees. If the plane is at an altitude of 10,000 feet, approximately how far is the plane to SCCA? Round your answer to the nearest tenth. The image is not drawn to scale.
Answer:
The horizontal distance from the plane to SCCA is [tex]32,708.5\ ft[/tex]
Step-by-step explanation:
Let
x-----> the horizontal distance from the plane to SCCA
we know that
see the attached figure to better understand the problem
[tex]tan(17\°)=\frac{10,000}{x}[/tex]
[tex]x=\frac{10,000}{tan(17\°)}[/tex]
[tex]x=32,708.5\ ft[/tex]
please help me out with this.....
In a parallelogram, adjacent angles sum to 180. Since the labeled angle is adjacent to the 124° angle, we have
[tex] 2z+16 +124 = 180 \iff 2z = 180-124-16 \iff 2z= 40 \iff z = 20[/tex]
A two-way frequency table is shown below displaying the relationship between age and preferred cola beverage. We took a sample of 100 people and recorded the following results:
Cola Rootbeer Dr. Fizz TOTAL
10-25 10 5 20 35
26-40 15 10 10 35
41-55 20 10 0 30
TOTAL 45 25 30 100
What is the probability (rounded to the nearest whole percent) that a randomly selected person is 41-55 in age or prefers drinking Dr. Fizz?
55%
35%
60%
0%
The probability that a randomly selected person is 41-55 in age or prefers drinking Dr. Fizz is approximately 60%.
To find the probability that a randomly selected person is 41-55 in age or prefers drinking Dr. Fizz, we need to sum the frequencies for the 41-55 age group and the frequency for Dr. Fizz, and then divide by the total number of people.
From the table:
- Frequency of 41-55 age group = 30
- Frequency of Dr. Fizz preference = 30
Total number of people = 100
Therefore, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Frequency of 41-55 age group} + \text{Frequency of Dr. Fizz}}{\text{Total number of people}} \][/tex]
[tex]\[ = \frac{30 + 30}{100} \][/tex]
[tex]\[ = \frac{60}{100} \][/tex]
[tex]\[ = 0.60 \][/tex]
Converting to a percentage, rounded to the nearest whole percent:
[tex]\[ \text{Probability} \approx 60\% \][/tex]
So, the correct answer is: 60%.
An angle represents 1/12 of a circle. Use the numbers to show how to find the measure of the angle in degrees 1/12=1/12x?/?=360
Answer:
The measure of the angle is [tex]30\°[/tex]
Step-by-step explanation:
Let
x-----> the measure of the angle in degrees
we know that
The measure of a complete circle is [tex]360\°[/tex]
so
by proportion
[tex]\frac{360}{1}\frac{degrees}{circle}=\frac{x}{(1/12)}\frac{degrees}{circle}[/tex]
[tex]x=\frac{1}{12}(360\°)[/tex]
[tex]x=\frac{360\°}{12}[/tex]
[tex]x=30\°[/tex]
An angle that represents 1/12 of a circle measures 30 degrees. We found this by understanding that a full circle is 360 degrees and then multiplied 1/12 by 360.
Explanation:The question is asking for the measure in degrees of an angle that represents 1/12 of a circle. As we know, a circle consists of 360 degrees next we measure the angle just like when we are measuring the angle in the sky. To find the angle, we can use a simple proportion. Since the whole circle is 360 degrees, we know that 1/12 of the circle will be equal to 1/12 of 360 degrees.
So, the calculation will be like: 1/12 x 360 = 30 degrees. Therefore, an angle that represents 1/12 of a circle measures 30 degrees.
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Two numbers are between 10 and 20. Their greatest common factor is 3. Which two numbers could they be?
Answer:
12 and 15 or
12 and 18 or
15 or 18
Step-by-step explanation:
The two numbers have to both be divisible by 3, since their greatest common factor is 3. This only leaves..
12, 15, or 18 (no other numbers between 10 and 20 are divisible by 3)
The factors of 12 are: 2, 3, 4, 6, and 12
The factors of 15 are: 3, 5, and 15
The factors of 18 are: 2, 3, 6, 9, and 18
All three numbers have 3 as a greatest common factor, so we have 3 pairs of numbers they could be...
12 and 15 or
12 and 18 or
15 or 18
Answer:
15 and 18
Step-by-step explanation:
Because The number 3 is the greatest common factor of 15 and 18, and both numbers are between 10 and 20.
You can spend no more than $20. You have already spent $6. Let s be the amount of money (in dollars) you can spend now. Which inequality and solution represent this situation?
Please answer this question! Will give brainliest!
Step-by-step answer:
Angle ECF = 110 = angle of intercepted (minor) arc EF.
The inscribed angles (angles EDF and EHF) are equal to half the angle of the inscribed arc, namely 110/2 = 55 degrees.
For your information, an inscribed angle is an angle with its vertex on (circumference of) the circle, formed by two intersecting chords, with a base on the inscribed arc.
Answer:
m∠EHF = m∠EDF = 55°Step-by-step explanation:
Look at the pictures.
(the picture 1)
Inscribed angle and central angle.
(the picture 2)
In a circle, central angle is two times larger than inscribed angle that intercept the same arc.
In a circle, inscribed angles that intercept the same arc are congruent.
We have the central angle C = 110°. The inscribed angle EHF is two times smaller than ∠C. Therefore m∠EHF = 110° : 2 = 55°.
The inscribed angles EHF and EDF that intercept the same arc. Therefore are congruent. m∠EHF = m∠EDF.
Which shows the correct way to evaluate 10 × 4 – (5 – 3) + 2? 10 × 4 – (5 – 3) + 2 10 × 4 – (2 + 2) 10 × 4 – 4 10 × 0 0 10 × 4 – (5 – 3) + 2 10 × 4 – (2 + 2) 10 × 4 – 4 40 – 4 36 10 × 4 – (5 – 3) + 2 10 × 4 – 2 + 2 10 × 2 + 2 20 + 2 22 10 × 4 – (5 – 3) + 2 10 × 4 – 2 + 2 40 – 2 + 2 38 + 2 40
Answer:
10 × 4 – (5 – 3) + 2
10 × 4 – 2 + 2
40 – 2 + 2
38 + 2
40
Step-by-step explanation:
To solve;
10 × 4 – (5 – 3) + 2
We use BODMAS
We start by removing brackets, to get
10 × 4 – 2 + 2
Then we proceed to multiplication to get;
40 – 2 + 2
Then subtraction to get;
38 + 2
The answer is 40
The correct answer to evaluate is 10 × 4 – (5 – 3) + 2 is 40
What is the evaluationThe correct way to evaluate the expression 10 × 4 – (5 – 3) + 2 is as follows:
10 × 4 – (5 – 3) + 2
= 40 – (5 – 3) + 2
= 40 – 2 + 2
= 38 + 2
= 40
Therefore, the correct evaluation is 40.
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Describe a real world scenario and which it would be important to apply trigonometric ratios or special right triangles.
Trigonometric ratios are vital in real-world applications such as construction for measuring building heights, in navigation or aviation for determining distances, and in physics for predictions and descriptions of natural phenomena.
Explanation:Trigonometry and special right triangles are fundamental in various real-world scenarios, especially in fields such as engineering, astronomy, and construction. One example where trigonometric ratios are crucial is in construction when determining the height of a building using a measured baseline and the angles of elevation. If a surveyor knows the distance from his point of observation to the base of the building (the adjacent side in a right triangle) and the angle of elevation to the top of the building, they can use the tangent ratio (opposite over adjacent) to calculate the building's height.
Another scenario involves navigation or aviation, where knowing the distance between two locations is necessary. By measuring the angle from two different points to a third point, one can apply the basic problem of trigonometry to find the distance to the third point using a known baseline and the measured angles, a process often referred to as triangulation.
Lastly, in physics, the principles of special right triangles like the Pythagorean theorem are used to predict certain outcomes. Whether a calculation is made using trigonometry or some other physics principle, the predictions must agree and accurately describe natural phenomena. The Pythagorean theorem is always reliable as long as the algebra and arithmetic are correctly done, illustrating the logic and interconnectedness of mathematical postulates.
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10% for infomercials in a 24 got day, how many hours of infomercials are there
Answer:
40 yes it is i got 100 on this
Step-by-step explanation:
There are 2.4 hours of infomercials in a 24-hour day if they account for 10% of the daily broadcast.
Explanation:The question asks us to calculate the amount of time designated for infomercials in a 24-hour day if they make up 10% of the day's broadcast. To find the answer, we can use the basic percentage calculation.
To calculate 10% of a day, we need to know that a day has 24 hours. So 10% of 24 hours is calculated as follows:
(10/100) × 24 = 2.4
Therefore, there are 2.4 hours of infomercials in a 24-hour day.
Please help me out :)
79-y=2y+22
Add y on both sides
79=3y+22
Subtract 22 from both sides
57=3y
Divide by 3 on both sides
19=y
Plz help me
WILL GIVE BRAINLIEST
Factor 3x^3−12x
3x^3−12x
=3x(x+2)(x−2)
Answer:
3x(x+2)(x−2)
3x^3 - 12x //Common factor: 3x
3x (x^2 - 4)
3x (x - 2) (x + 2)
Answer: C
//Hope this helps.
Select the correct answer from each drop-down menu. Determine the dependence between the quantities for the given graph.
The cost per package depends on the weight of the package.
Step-by-step explanation:We know that a dependent variable is one which depends on some other variable or the value of the variable is calculated corresponding to the independent variable.
Generally we consider the values on the y-axis or the vertical axis as the dependent values because they are dependent upon the x-value or the value on the horizontal axis.
Here from the graph we may observe that the cost of the package depends on the weight of the package.
The cost per package depends on the weight of the package.
We know that a dependent variable is one which depends on some other variable or the value of the variable is calculated corresponding to the independent variable.
Generally we consider the values on the y-axis or the vertical axis as the dependent values because they are dependent upon the x-value or the value on the horizontal axis.
Here from the graph we may observe that the cost of the package depends on the weight of the package.
Mia has 7/8 ponds of bird food. She puts an equal portion into 4 bird feeders how much bird food in pounds does she put into each bird feeder
Answer:
B
Step-by-step explanation:
first you wanna multiply and sehow many mutiplesof 4 will go into 8.
The equal amount of bird food put into the 4 feeders is 7 / 32 pounds.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that Mia has 7/8 pounds of bird food. She puts an equal portion into 4 bird feeders.
The amount of bird food for each feeder will be calculated as,
Bird food = (7/8) / (4)
Bird food = [ 7/ (8 x 4 )]
Bird food = 7 / 32 pounds
Therefore, the equal amount of bird food put into the 4 feeders is 7 / 32 pounds.
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Based on this data what is a reasonable estimate of the probability that the next tucker family vacation lasts less than 3 days
A. 0.24
B. 0.44
C. 0.56
D. 0.89
Answer:
I am assuming it is the same table as mine. So it would be 0.56
Step-by-step explanation:
Find the area of the regular polygon below. leave your answer in simplest radical form.
Answer:
384√3 in²
Step-by-step explanation:
Given in the question a regular 6 sided polygon
To find it's area you have to use the following formula
1/2 x perimeter x apothemPerimeter = the sum of the lengths of all the sides
Suppose length of one side = x
Apothem = a segment that joins the polygon's centre to the midpoint of any side that is perpendicular to that side = 8√3
Since the polygon have 6 sides so
perimeter = 6x
x = 2(8√3)/√3x = 2(8)
x = 16
perimeter = 6(16) = 96 in
plug values in the formula of area
1/2 x 96 x 8√3384√3 in²
Answer:
= 384√3 In²
Step-by-step explanation:
The polygon is a hexagon ; thus the angle subtended by each side at the center will be given by;
θ = 360/6
= 60°
Therefore; we can calculate the length of each side;
Tan θ = opp/Adj
Tan θ = x /8√3
Tan 30 = x /8√3
Therefore; 1/√3 =x /8√3
x = 8√3× 1/√3
= 8
The length of each side is 8 × 2 = 16 In
The area of the polygon will be;
Area of one triangle multiplied by the number of a triangle;
= 1/2 × 16 × 8√3 ×6
= 384√3 In²Determine whether the graphs of the given equations are parallel, perpendicular, or neither
y= 2x+13
Y= -2x+2
A.Parallel
B.Perpendicular
C.Neither
Answer: the answer should be A= Parallel
Step-by-step explanation:
Answer:
C. Neither
Step-by-step explanation:
The first equation is [tex]y=2x+13[/tex]
This equation is already in the slope-intercept form; [tex]y=mx+c[/tex]
The slope of this equation is 2.
The second equation is [tex]y=-2x+2[/tex].
This equation is also already in the slope-intercept form.
The slope of this equation is [tex]-2[/tex].
Since the two slopes are not the same, the two lines are not parallel.
If these two lines are perpendicular, then the product of their slopes is -1.
But [tex]2\times -2=-4[/tex] which is not equal to -1.
Therefore the two lines are also not perpendicular.
The correct choice is C.
according to the graph, what is the value of the constant in the equation below? apex
Answer:
Option B. 36
Step-by-step explanation:
we know that
A relationship between two variables, x, and y, represent an inverse variation if it can be expressed in the form [tex]y*x=k[/tex] or [tex]y=k/x[/tex]
The graph of the figure represent an inverse variation
so
In this problem
Constant=Height*Width
Take any point in the graph
example -----> the point (4,9)
Constant=4*9=36
The graph shows a probability distribution. What is P(2.5≤X≤4)?
To find P(2.5≤X≤4), calculate the length of the interval between 2.5 and 4, and divide by the total length of the distribution's support.
For a uniform distribution U(0.5, 4), this would result in a probability of ¾ or 0.75.
Explanation:To calculate the probability P(2.5≤X≤4) for a random variable X, given the graph of the probability distribution, you would typically integrate the probability density function (pdf) from 2.5 to 4 (in the case of a continuous distribution) or sum the probabilities for each whole number value of X between 2.5 and 4 (in the case of a discrete distribution).
For a uniform distribution U(0.5, 4), the probability is uniform (constant) across the interval.
Since the total area under the distribution is equal to 1, the probability of any interval can be found by calculating the length of the interval divided by the total length of the distribution's support (4 - 0.5).
For P(2.5≤X≤4), this would be ¾ or 0.75 since the interval length from 2.5 to 4 is 1.5 and the total length of distribution's support is 3.5 (4-0.5).
Answer:
3/8
Step-by-step explanation:
You want the probability P(2.5 ≤ x ≤ 4), given X has a uniform distribution between 0 and 4.
ProbabilityThe probability can be found by integrating the PDF over the interval [2.5, 4]:
[tex]\displaystyle\int_{2.5}^4{\dfrac{1}{4}}\,dx=\dfrac{1}{4}(4-2.5)=\dfrac{1}{4}\cdot\dfrac{3}{2}=\dfrac{3}{8}[/tex]
The probability is 3/8.
Suppose a firm that produces light bulbs wants to know whether it can say that its light bulbs typically last more than 1500 hours. Hoping to find support for their claim, the firm collects a random sample of n = 25 light bulbs and records the lifetime (in hours) of each bulb. The information related to the hypothesis test is presented below. Test of H0: f$mu leq f$ 1500 versus H1: f$ mu f$ > 1500 Sample mean 1509.5 Sample standard deviation 24.27 Assuming the life length of this type of lightbulb is normally distributed, what is the p-value associated with this test? Place your answer, rounded to 3 decimal places in the blank. For example, .123 would be a legitimate entry.
Answer:
p score = 0.031
Step-by-step explanation:
We will be running a hypothesis test to find the p-value. See attached photo for the work needed and the running of the test.
Our hypothesis are:
H0: µ = 1500
HA: µ > 1500 (claim)
They say that the life of the light bulbs are more than 1,500 hours, so that is the alternate hypothesis since it's strictly more than, not equal to or greater.
we have a sample mean of: 1509.5
we have a sample standard deviation of: 24.27
We just need to find the p-value, we don't need to make a conclusion about the test results.
The p-value of the hypothesis test that the light bulbs last more than 1500 hours is estimated to be less than 0.05, supporting the company's claim. The calculation involved computing a t-statistic from the sample data and then finding the probability of getting a t-statistic larger than the computed value.
Explanation:The question involves conducting a hypothesis test for the claim that the company's light bulbs last more than 1500 hours. The null hypothesis H0 for this test would indicate that the population mean longevity is less than or equal to 1500 hours (H0: μ ≤ 1500), while the alternative hypothesis H1 posits that the mean exceeds 1500 hours (H1: μ > 1500). The company collected a sample (n=25) and computed the sample mean ( = 1509.5 hours) and the sample standard deviation (s = 24.27 hours). To calculate the p-value for this test, we need to first calculate the test statistic (z or t) by using the given sample data and then find the area to the right of this test statistic in the relevant distribution.
Using the formula for calculating the test statistic in t-tests: t = ( - μ0)/(s / √n), where μ0 is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Here, t = (1509.5 - 1500)/(24.27 / √25) = approximately 1.96.
Since the alternative hypothesis is looking for values greater than 1500, we seek the probability that a test statistic is greater than what we observed (i.e., t > 1.96). This probability is equal to the p-value. To obtain it, we use the t-distribution with n-1 = 24 degrees of freedom. Since exact p-values can be challenging to retrieve without a statistical software or detailed tables, it's typically adhered to note if the p-value is less than or greater than the significance level, which is 0.05 in this case. Due to the calculated t-statistic, our p-value is approximately less than 0.05. Hence, this result supports rejecting the null hypothesis and lends credibility to the company's claim that its light bulbs typically last more than 1500 hours.
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A point is on a circle if the difference from the center of the circle to the point is equal to the
it should be the radius
What are the values of the variables in the triangle below? if the answer is not an integer, leave it in simplest radical form. the diagram is not drawn to scale
Answer:
x = 69 and y = [tex]23\sqrt{3}[/tex]
Step-by-step explanation:
Firstly the hypotenuse is the side opposite the 90 degree angle. So hypotenuse is [tex]46\sqrt{3}[/tex]
Since the angle given is 30 degree, with respect to this angle, the side length y is opposite and the side length x is adjacent.
Now, we can use trigonometric ratios to solve for x and y. Sine is defined as [tex]sin\theta=\frac{Opposite}{Hypotenuse}[/tex] and Cos is defined as [tex]Cos\theta=\frac{Adjacent}{Hypotenuse}[/tex]
Hence, we can write:
[tex]Sin(30)=\frac{y}{46\sqrt{3} }\\y=46\sqrt{3}*Sin30 \\y=46\sqrt{3}*\frac{1}{2}\\y=23\sqrt{3}[/tex]
Also, we can figure out:
[tex]Cos(30)=\frac{x}{46\sqrt{3} }\\Cos(30)*46\sqrt{3}=x\\ x=\frac{\sqrt{3} }{2}*46\sqrt{3} \\x=\frac{46*3}{2}\\x=69[/tex]
2nd answer choice is right.
ANSWER
[tex]x = 69,y = 23 \sqrt{3} [/tex]
EXPLANATION
Recall and use the mnemonics SOH CAH TOA.
We use the cosine ratio to find x.
[tex] \cos(30 \degree) = \frac{adjacent}{hypotenuse} [/tex]
[tex] \cos(30 \degree) = \frac{x}{46 \sqrt{3} } [/tex]
[tex] \frac{ \sqrt{3} }{2} = \frac{x}{46 \sqrt{3} } [/tex]
Cross multiply,
[tex]2x = 46 \sqrt{3} \times \sqrt{3} [/tex]
[tex]2x = 46(3)[/tex]
[tex]x = 23(3)[/tex]
[tex]x = 69[/tex]
We use the sine ratio, to find y.
[tex] \sin(30 \degree) = \frac{opposite}{hypotenuse} [/tex]
[tex]\sin(30 \degree) = \frac{y}{46 \sqrt{3} } [/tex]
[tex] \frac{1}{2} = \frac{y}{46 \sqrt{3} } [/tex]
Solve for y.
[tex] \frac{1}{2} \times 46 \sqrt{3} = y[/tex]
[tex]23 \sqrt{3} = y[/tex]
Therefore,
[tex]x = 69,y = 23 \sqrt{3} [/tex]
(10CQ) The series 1/49+1/64+1/81+ ... is divergent.
Answer:
False
Step-by-step explanation:
We have the serie:
[tex]\frac{1}{49}+ \frac{1}{64} + \frac{1}{81}+...[/tex]
To test whether the series converges or diverges first we must find the rule of the series
Note that:
[tex]7^2 = 49\\\\8^2 = 64\\\\9^2 = 81[/tex]
Then we can write the series as:
[tex]\frac{1}{7^2}+ \frac{1}{8^2} + \frac{1}{9^2}+...[/tex]
Then:
[tex]\frac{1}{7^2}+ \frac{1}{8^2} + \frac{1}{9^2}+... = \sum_{n=7}^{\infty}\frac{1}{n^2}\\\\\sum_{n=7}^{\infty}\frac{1}{n^2} = \sum_{n=1}^{\infty}\frac{1}{(n+6)^2}[/tex]
The series that have the form:
[tex]\sum_{n=1}^{\infty}\frac{1}{n^p}[/tex]
are known as "p-series". This type of series converges whenever [tex]p > 1[/tex].
In this case, [tex]p = 2[/tex] and [tex]2 > 1[/tex]. Then the series converges