The name of the shape graphed by the function r ^ 2 = 9 cos (2 theta) is called the “lemniscate”. A lemniscate is a plane curve with a feature shape which consists of two loops that meet at a central point. The curve is also sometimes called as the lemniscate of Bernoulli.
Explanation:
The period of coskθ is 2π/k. In this case, k = 2 therefore the period is π.
r ^ 2 = 9 cos 2θ ≥0 → cos 2θ ≥0. So easily one period can be chosen as θ ∈ [0, π] wherein cos 2θ ≥0.
As cos(2(−θ)) = cos2θ, the graph is symmetrical about the initial line.
Also, as cos (2(pi-theta) = cos 2theta, the graph is symmetrical about the vertical θ = π/2
A Table for half period [0,π4/] is adequate for the shape in Quarter1
Use symmetry for the other three quarters:
(r, θ) : (0,3)(3/√√2,π/8)(3√2/2,π/6)(0,π/4)
Answer:
lemniscate
Step-by-step explanation:
a p e x
Rationalize the denominator.
10/√24x
write it in simplest form
7x1,000,000+
3x100,000+
5x10,000+
6x1.00+
2x100+3x10+7+1
The volume of a rectangular prism varies jointly with the length and width of the figure when the height remains constant. The volume of a rectangular prism is 672 cubic centimeters. The figure has a length of 8 centimeters and a width of 14 centimeters. A second prism has a length of 12 centimeters and a width of 8 centimeters. What is the volume of the second prism? 576 cubic centimeters 768 cubic centimeters 784 cubic centimeters 1,344 cubic centimeters
1st Prism = 8*14 = 112
672/112 = 6
2nd prism = 8*12=96
96*6 = 576 cubic cm
answer is 576 cubic centimeters
Answer:
The correct option is 1.
Step-by-step explanation:
The volume of a rectangular prism varies jointly with the length and width of the figure when the height remains constant.
Let the height of both rectangular prism be h cm.
The volume of a prism is
[tex]V=l\times b\times h[/tex]
Where, l is length, b is breadth or width and h is height.
The volume of a rectangular prism is 672 cubic centimeters. The figure has a length of 8 centimeters and a width of 14 centimeters.
[tex]672=8\times 14\times h[/tex]
[tex]672=112h[/tex]
Divide both sides by 112.
[tex]\frac{672}{112}=h[/tex]
[tex]6=h[/tex]
The value of h is 6 cm. It means the height of both prism is 6 cm.
A second prism has a length of 12 centimeters and a width of 8 centimeters. So, the volume of second prism is
[tex]V=12 \times 8\times 6[/tex]
[tex]V=576[/tex]
The volume of second prism is 576 cubic centimeters. Therefore the correct option is 1.
A scatter plot is made with the data shown.
Time (hr)
1
2
3
4
5
6
7
8
9
Distance from Destination (mi)
320
280
240
200
160
120
80
40
0
What type of association will the scatter plot for this data represent between the time, in hours, and the distance from the destination, in miles?
No association
Negative linear association
Positive nonlinear association
Positive linear association
This is a negative linear association because when ever the x (top line) is increasing and the y (bottom line) is decreasing, that shows that you are going farther to the end of the x axis and lower down the y axis.
Negative linear association is the type of association will the scatter plot for this data represent between the time, in hours, and the distance from the destination, in miles
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The scatter plot for this data will represent a negative linear association between the time, in hours, and the distance from the destination, in miles.
As the time in hours increases, the distance from the destination in miles decreases, and this relationship is a straight line that slopes downwards from left to right which indicates a negative linear association.
Therefore, Negative linear association is the type of association will the scatter plot for this data represent between the time, in hours, and the distance from the destination, in miles.
To learn more on Graph click:
https://brainly.com/question/17267403
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If 3a2 – 5ab – 2b2 is factored, one of the factors might be:
Write the ratio in lowest terms: 4415 feet to 22245 feet
since they both end with 5 divide each number by 5
4415 = 883
22245/5 = 4449
there is no number that can go into 883 evenly
so the lowest term
would be 883/4449
The correct ratio in lowest terms is [tex]\(\frac{883}{4449}\)[/tex].
To find the ratio in lowest terms, we first write the ratio of the two given lengths:
[tex]\[ \frac{4415 \text{ feet}}{22245 \text{ feet}} \][/tex]
Next, we divide both the numerator and the denominator by their greatest common divisor (GCD) to simplify the ratio. The GCD of 4415 and 22245 can be found by using the Euclidean algorithm or by inspection.
We can start by dividing both numbers by 5:
[tex]\[ \frac{4415 \div 5}{22245 \div 5} = \frac{883}{4449} \][/tex]
Now, we check if 883 and 4449 have any common divisors other than 1. Since 883 is a prime number and does not divide evenly into 4449, we have found the simplest form of the ratio.
Thus, the ratio of 4415 feet to 22245 feet in lowest terms is [tex]\(\frac{883}{4449}\)[/tex].
Factor each expression 1)4a^2-16ab^3+8ab^2c 2) n^2+8n+15 3) g^2-9g+20 4) z^2-7z-30 5) 4y^3-36y
determine the value of x in the diagram where lines a and b are parallel
25°
15°
75°
55°
The population of a town is decreasing at a rate of 1.1% each year. If there are 3,000 people in the town right now, how many people will be living in the town in 10 years? Round your answer to the nearest whole number.
total = 3000*(1-0.011)^10
1-0.011 = 0.989
total = 3000*(0.989)^10
0.989^10 = 0.895288314
3000* 0.895288314 = 2685.86
rounded to nearest whole number = 2686
there will be 2686 people
Please help !
(cos Θ − cos Θ)^2 + (cos Θ + cos Θ)^2
Answer: The required simplified form is [tex]4\cos^2\theta.[/tex]
Step-by-step explanation: We are given to simplify the following trigonometric expression :
[tex]T=(\cos \theta-\cos\theta)^2+(\cos \theta+\cos \theta)^2.[/tex]
To simplify, first we need to evaluate the terms within the brackets.
The simplification is as follows :
[tex]T\\\\=(\cos \theta-\cos\theta)^2+(\cos \theta+\cos \theta)^2\\\\=0^2+(2\cos\theta)^2\\\\=0+4\cos^2\theta\\\\=4\cos^2\theta.[/tex]
Thus, the required simplified form is [tex]4\cos^2\theta.[/tex]
Celia is staring at the clock waiting for school to end so that she can go to track practice. She notices that the 4-inch-long minute hand is rotating around the clock and marking off time like degrees on a unit circle. Part 1: How many radians does the minute hand move from 1:25 to 1:50? (Hint: Find the number of degrees per minute first.) Part 2: How far does the tip of the minute hand travel during that time?
there are 60 minutes marks on a clock
360/60 = 6
6 degree between every minute
From 1:25 to 1:50 there are 25 minutes
25 x 6 = for a total of 150 degrees
150*pi/180 = 5pi/6 radians. ( 2.61799 radians)
Total distance the tip of minute hand has traveled in this time = 2pi*(4 in) * (5pi/6)/(2pi) = 10pi/3 inches (10.47 inches)
You are getting a line-up ready for a school kickball game. you have 55 girls and 55 boys. the rules state each child must kick the same number of times and alternate girl-boy or boy-girl. how many ways can a line-up be made for one round of kicking
To solve this problem,
we must first imagine out that the sequence of the children is either
GBGBGB.... or BGBGBG....
So there are 2
possible sequence all in all. Now to solve for the total arrangements per
sequence, the
girls can be arranged in n! ways in their alloted spots, and so can the boys n!
in their alternate spots, therefore:
Total arrangements = 2 * n! * n!
If n = 55
Total arrangements = 2 * 55! * 55!
Total arrangements = (The answer is very big ~almost infinite)
If n = 5
Total arrangements = 2 * 5! * 5!
Total arrangements = 28,800
So I believe the correct given is 5 boys and 5 girls and there are a total of 28,800 arrangements.
The lengths of three sides of a quadrilateral are shown below: Side 1: 1y2 + 3y − 6 Side 2: 4y − 7 + 2y2 Side 3: 3y2 − 8 + 5y The perimeter of the quadrilateral is 8y3 − 2y2 + 4y − 26. Part A: What is the total length of sides 1, 2, and 3 of the quadrilateral? (4 points) Part B: What is the length of the fourth side of the quadrilateral? (4 points) Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer.
Answer:
Part A: 12y^2+9y-21
Part B: 4y^2+6y^2+7y-5
Part C: A set of numbers is closed, or has closure, under a given operation if the result of the operation on any two numbers in the set is also in the set. For example, the set of real numbers is closed under addition, because adding any two real numbers results in another real number. Likewise, the real numbers are closed under subtraction, multiplication and division (by a nonzero real number), because performing these operations on two real numbers always yields another real number. Polynomials are closed under the same operations as integers.
Step-by-step explanation:
Hope this helps!!
In an upcoming race, the top 3 finishers will be recognized with the same award. Ryan is one of 12 people entered in the race.
If all racers are equal in skill, what is the probability that Ryan will be one of the top 3 racers?
Answer: The required probability is 25%.
Step-by-step explanation: Given that in an upcoming race, the top 3 finishers will be recognized with the same award and Ryan is one of 12 people entered in the race.
We are to find the probability that Ryan will be one of the top 3 racers, if all racers are equal in skills.
Let S denote the sample space of the experiment for selecting a racer and A denote the event of selecting the top 3 finishers.
Then, according to the given information, we have
n(S) = 12 and n(A) = 3.
So, the probability of event A is given by
[tex]P(A)=\dfrac{n(A)}{n(S)}=\dfrac{3}{12}=\dfrac{1}{4}\times100\%=25\%.[/tex]
Thus, the required probability is 25%.
Points A and B lie on a circle centered at point O. If OA = 5 and length of ABowncircumference=14, what is the area of sector AOB? Use the value π = 3.14, and choose the closest answer.
This question can simply be answered directly. To solve this, we should recall that the formula of a circle is:
Area of circle = π r^2 where r is the radius of the circle
Now we are given that segment OA is equivalent to 5 units. Segment OA is also the diameter of that circle therefore d = 5.
Now let us convert the formula knowing that radius is one half the diameter:
r = d / 2
Area of circle = π (d / 2)^2
Area of circle = π d^2 / 4
Substituting:
Area of circle = π (5)^2 / 4
Area of circle = 3.14 * 25 / 4
Area of circle = 19.625 = 19.6 square units
Answer:
The answer would be 19.6 for plato Users
Step-by-step explanation:
If runners in a long distance race were to run straight from the starting line to the finish line they would run 13 kilometers. However, the road they run makes them travel longer than that. They must run 5 kilometers south and then head west "x" kilometers for the remainder of the race. How far do the runners travel?
5^2 +x^2 = 13^2
25 + x^2 = 169
x^2 = 144
x = sqrt(144) = 12
they run west for 12 KM
12+5 = 17 total km
I don't know the answer can someone pls help me
line QP =
11.5*(11.5 +24) = 11.5 * 35.5 = 408.25
SqRT(408.25) = 20.205
round answer to 20 units
If the perimeter of the adult pinball machine is 172 inches, what is the length, in inches of ? Type the numeric answer only in the box below.
What is the following product?
The scale of a map is 1 1/4 inches = 100 miles. On that map, 2 cities are 4 1/8 inches short. What is the actual distance between the cities?
need to divide 4 1/8 by 1 1/4
4 1/8 = 33/8
1 1/4 = 5/4
33/8 / 5/4 = 33/8 * 4/5 = 132/40 reduces to 33/10 = 3 3/10
100* 3 3/10 = 330 miles
Final answer:
Using the map scale of 1 1/4 inches equals 100 miles, and the measured map distance of 4 1/8 inches, the actual distance between the two cities is calculated to be 330 miles.
Explanation:
To find the actual distance between two cities on a map, we can set up a proportion based on the scale of the map. In this case, the scale is 1 1/4 inches = 100 miles. The measured distance on the map between the two cities is 4 1/8 inches.
We'll convert these measurements to an easier-to-calculate form by changing the mixed numbers to improper fractions.
First, we convert 1 1/4 inches to an improper fraction: 1 1/4 = 5/4 inches. Likewise, we convert 4 1/8 inches to an improper fraction: 4 1/8 = 33/8 inches. Now we set up the proportion using the scale.
(5/4 inches) / (100 miles) = (33/8 inches) / (x miles)
To solve for x, cross-multiply and divide:
(5/4) * x = (33/8) * 100
x = (33/8 * 100) / (5/4)
x = (33 * 100 * 4) / (8 * 5)
x = (33 * 4) / (2)
x = 66 miles
Therefore, the actual distance between the two cities is 330 miles.
literal equations, please help! this one is confusing to me.
help. Find mBAC in circle O. (The figure is not drawn to scale.)
A. 170
B. 95
C. 47.5
D. 42.5
Answer: The answer is (C) 47.5.
Step-by-step explanation: In the given figure, O is the centre of a circle, where AC is the diameter and OB is the radius. We are to find the measure of ∠BAC.
We have
[tex]m\angle AOB+m\angle BOC=180^\circ\\\\\Rightarrow m\angle BOC=180^\circ-85^\circ\\\\\Rightarrow m\angle BOC =95^\circ.[/tex]
∠BOC and ∠BAC are angles at the centre and at the circumference subtended by the arc BC, so
[tex]m\angle BAC=\dfrac{1}{2}\times m\angle BOC=\dfrac{1}{2}\times 95^\circ=47.5^\circ.[/tex]
Thus, (C) is the correct option.
PLZ HEEEEEEEELPPPPPPPP!!!!!!!!!
Find two positive numbers a and b (witha≤b) whose sum is 88 and whose product is maximized.
a + b = 88
ab = y
a = 88 - b
y = (88 - b)*b
y = -b^2 + 88b
Take the derivative and set equal to 0
y' = -2b + 88 = 0
2b = 88
b = 44
a=44
both numbers are 44
To find two positive numbers a and b whose sum is 88 and whose product is maximized, we can use the concept of quadratic equations. By taking the derivative of the product function and setting it equal to zero, we can find the maximum value. Plugging that value back into the product function will give us the maximum product.
Explanation:To find two positive numbers a and b whose sum is 88 and whose product is maximized, we need to use the concept of quadratic equations. Let's assume a as x and b as 88-x. The product of the two numbers can be expressed as the quadratic equation P(x) = x(88-x). To maximize the product, we need to find the maximum value of P(x). We can use calculus to find the maximum value by taking the derivative of P(x) and setting it equal to zero. Solving that equation will give us the value of x, and plugging it back into P(x) will give us the maximum product.
Let's go through the steps:
Write the quadratic equation: P(x) = x(88-x).Take the derivative of P(x) and set it equal to zero: P'(x) = 88-2x = 0.Solve for x: x = 44.Plug x back into P(x) to find the maximum product: P(44) = 44(88-44) = 1936.So, the two positive numbers a and b are 44 and 88-44, which is 44 as well. Their sum is 88 and their product is maximized at 1936.
A "Local" train leaves a station and runs at an average rate of 35 mph. An hour and a half later an "Express" train leaves the station and travels at an average rate of 56 mph on a parallel track. How many hours after the Express train starts will the it overtake the Local?
Answer:
2.5 hrs.
Hope this helps <3
Part 1: what are the conditions for using the standard deviation formula when conducting a significance test? be specific about p versus p-hat. part 2: what are the conditions for approximating with a normal distribution?'
To use the standard deviation formula in a significance test, the sample must be random and the population standard deviation should typically be unknown, while approximation with a normal distribution requires a sufficiently large sample size and the success-failure condition for proportions. Choosing the correct distribution depends on whether the standard deviation is known and the sample size.
Explanation:Conditions for Using the Standard Deviation Formula and Approximating with a Normal Distribution
For conducting a significance test using the standard deviation formula, certain conditions must be met:
The sample must be randomly selected.If surveying a proportion, we use \( \hat{p} \) for samples and \( p \) for populations.If calculating a sample standard deviation, the population standard deviation should be unknown.The sample size should be sufficiently large if the population distribution is not normal (usually n > 30).To approximate the sample distribution with a normal distribution, particularly when conducting hypothesis testing:
The sample size must be large enough (typically n > 30).The sample should be randomly selected and should represent the population.For proportions, the sample should meet the success-failure condition where \( np \geq 10 \) and \( n(1-p) \geq 10 \).The conditions for a hypothesis test often include:
Stating the null and alternative hypotheses.Deciding on a significance level (e.g., \( \alpha = 0.05 \)).Knowing whether population parameters are known, which determines the choice of test statistic.Finding the p-value and comparing it with the significance level to make a decision.Examples of Hypothesis Testing with Different Conditions
If you know the population standard deviation, you might use the z-distribution for hypothesis testing.If the population standard deviation is unknown but the sample size is large, the t-distribution might be the appropriate choice.The F-statistic is used when comparing two variances, such as two standard deviations of test scores.Select the inequality that corresponds to the given graph. graph of an inequality with a dashed line through the points negative 3 comma 0 and 0 comma 4 and shading below the line
A. 4x-3y>-12
B. x+4y>4
C. 4x-2y<-8
D. 2x+4y=>-16
Answer:
b
Step-by-step explanation:
i took the test
How to subtract a negative fraction from a positive fraction?
Someone please help me out!
try it with a couple different numbers
try 4: 1^2 +2^2 +3^2 +4^2 = 1 +4 +9 +16 =30
formula: 4(4+1)(2*4+1)/6 = 4*5*9 = 180/6=30
this one works
now try 5
1^2 + 2^2+3^2+4^2+5^2 = 55
formula: 5(5+1)(2*5+1)/6 = 5*6*11 = 330/6=55
I tried a larger number ( 14) as well and it worked
so you can say that this is true
Factor out the greatest common factor of 5ab^2+10ab
Answer: 5ab
Step-by-step explanation:
The given polynomial : [tex]5ab^2+10ab[/tex]
The prime factorization of [tex]5ab^2= 5\times a\times b\times b[/tex]
The prime factorization of [tex]10ab= 5\times2\times a\times b[/tex]
We can see that the greatest common factor of [tex]5ab^2\text{ and }10ab[/tex] is [tex]5\times a\times b[/tex]
Hence, the greatest common factor of [tex]5ab^2+10ab[/tex] = 5ab