I've tried this multiple times and cant seem to get the right answer could someone help solve this?
Answers
Answer:
[tex]\dfrac{6\pi -9\sqrt{3} }{16}[/tex]
Step-by-step explanation:
You need to rewrite [tex]\cos^2 \theta[/tex] using cos double angle identity:
[tex]\cos 2\theta= \cos^2\theta-\sin^2\theta[/tex]
[tex]\implies \cos 2\theta= \cos^2\theta-(1-\cos^2\theta)[/tex]
[tex]\implies \cos 2\theta=2 \cos^2\theta-1[/tex]
[tex]\implies \cos^2\theta=\dfrac12(\cos 2\theta+1)[/tex]
Then substitute this into the integration.
Please see the attachment for the full integration (it was clearer for me to type this in MS word than use the equation editor here)
Recall the double angle identity for cosines
[tex]{\boxed{\bf{2\cos^{2}(\theta)=1+\cos (2\theta)}}}[/tex]Using this the integral becomes :
[tex]{:\implies \quad \displaystyle \sf \dfrac{1}{2}\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\{3\cos (\theta)\}^{2}d\theta}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{1}{2}\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}9\cos^{2}(\theta)d\theta}[/tex]
As we can take constant out of the integrand so we have
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{2}\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\cos^{2}(\theta)d\theta}[/tex]
Using the double angle identity for cosines ;
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{2}\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\bigg\{\dfrac{1+\cos (2\theta)}{2}\bigg\}d\theta}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\{1+\cos (2\theta)\}d\theta}[/tex]
Now , as integrals follow distributive property, so we now have
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg(\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}d\theta+\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\cos (2\theta)d\theta \bigg)}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg\{\bigg(\theta +\dfrac{\sin (2\theta)}{2}\bigg) \bigg|_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\bigg\}}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg[\dfrac{\pi}{2}-\dfrac{\pi}{3}+\dfrac{1}{2}\bigg\{\sin (\pi)-\sin \left(\dfrac{2\pi}{3}\right)\bigg\}\bigg]}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg\{\dfrac{3\pi -2\pi}{6}+\dfrac{1}{2}\bigg(-\dfrac{\sqrt{3}}{2}\bigg)\bigg\}}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg(\dfrac{\pi}{6}-\dfrac{\sqrt{3}}{4}\bigg)}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{9}{4}\bigg(\dfrac{2\pi -3\sqrt{3}}{12}\bigg)}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{3(2\pi -3\sqrt{3})}{4\times 4}}[/tex]
[tex]{:\implies \quad \displaystyle \sf \dfrac{6\pi -9\sqrt{3}}{16}}[/tex]
[tex]{:\implies \quad \displaystyle \bf \therefore \quad \underline{\underline{\int_{\tiny \dfrac{\pi}{3}}^{\tiny \dfrac{\pi}{2}}\{3\cos (\theta)\}^{2}d\theta =\dfrac{6\pi -9\sqrt{3}}{16}}}}[/tex]
Used Concepts :- [tex]{\boxed{\displaystyle \bf \int dx=x+C}}[/tex] [tex]{\boxed{\displaystyle \bf \int \cos (nx)dx=\dfrac{\sin (nx)}{n}+C}}[/tex]Related Questions
Solve the equation 3m – 10 = 5. Find the value of m
Answers
Answer:
So I don’t know if this is right cause yeha but I think that m=5
Step-by-step explanation:
we can substitute different numbers starting with one
Or this is the way I did we can do 10+5 which = 15
So that means 15/3=5
So m=5
also I don’t know if this is correct….:/
Answer:
m=5
Step-by-step explanation:
step 1:letters to the left side of the =, numbers to the right (If there are numbers attached to the letter, keep it on the left) if a number is being taken to the right, change the sign to the opposite of what it was
3m= 5+10
step 2: add the numbers on the right
3m=15
step 2: divide both sides by 3 to get m alone
m=5
please answer this question
Answers
Answer:
½ i + ¾ j = (½ , ¾)
Step-by-step explanation:
The gradient of a scalar function is a vector. Which is calculated as written in the image.
A laundry basket contains 8 socks, of which 2 are blue.
Answers
Answer:
They are probably a pair.
Step-by-step explanation:
If the two of those are blue, they probably go together. Maybe the other 6 are also pairs???
Assessment Practice
16. Marique is making a large table in the shape of a trapezoid. She needs
to calculate the area of the table. The longest side of the table is twice
as long as the table's width. Find the area of the table by decomposing
the trapezoid into familiar shapes. Show your work
Answers
Given that the length of the table is twice the width which is 1.25 m. and
the difference in the parallel side is 1 m., the area of the table is 2.5 m².
How can the area of the table be calculated?The possible table dimensions are;
Width of the table = 1.25 m
Width of the table = The height of the trapezoid
The difference in length between the parallel sides = 2 × 0.5 m = 1 m
Longest side of the table is a parallel side of the trapezoid
Given;
Length of the longest side of the table = 2 × The width
Which gives;
Length of the longest side of the table = 2 × 1.25 m = 2.5 m
Length of other parallel side = 2.5 m - 1 m = 1.5 m
[tex]Area \ of \ a \ trapezoid = \mathbf{\dfrac{Sum \ of \ length \ of \ parallel \ sides}{2} \times Height}[/tex]
Which gives;
[tex]\mathbf{Area \ of}\ the \ \mathbf{table} = \dfrac{2.5 + 1.5}{2} \times 1.25 = 2.5[/tex]
The area of the = 2.5 m²
Learn more about the properties of a trapezoid here:
https://brainly.com/question/4758162
— length of the longest side of the table = 2 × width which gives you 'length of the longest side of the table = 2 × 1.25 meters' which equals 2.5 meters. to find length of other parallel side use the equation 2.5 meters minus 1 meter, which is 1.5 meters. lastly to find the area, do [tex]\frac{sum of length of parallel sides}{2}[/tex] multiplied by the height, which afer you fill it out turns into [tex]\frac{2.5+1.5}{2} *1.25[/tex] , which equals 2.5 meters. so the area is 2.5 meters squared.
P.S — PFP DUPEEE LOL
Roger can type 1530 words in 1/2 hour. How many words can he type per
minute?
PLS HELP ME
Answers
Answer:
25.5
Step-by-step explanation:
1530 divided by 60 is 25.5 and 25.5 multiplied by 60 is 1530.
This graph represents the relationship between the x and y.
What is an equation showing the relationship between x and y?
Enter your answer in the box.
PLEASE ASAP!!! MARKING BRAINLIEST!! FIRST ANSWER!!
Answers
Answer:
Therefore, x is 1 and 2.
Step-by-step explanation:
As you plot both equations on the same graph, you will get something like this, shown in the graph.
Then, you have to find the x solutions where they intersect.
So, both equations intersect at x = 1 and 2.
Consider the line 7x+5y=−2.
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?
Answers
Answer: slope of a line perpendicular is [tex]\frac{5}{7}[/tex]; slope of a line parallel to this line is [tex]-\frac{7}{5}[/tex].
Step-by-step explanation:
Rewrite the line in y = mx + b form, where m is slope and b is y-intercept.
7x + 5y = -2
5y = -7x -2
y = [tex]-\frac{7}{5}[/tex]x - [tex]\frac{2}{5}[/tex]
The product of two slope of perpendicular line is -1.
Let x be the other line that perpendicular line with the line 7x + 5y = -2.
（[tex]-\frac{7}{5}[/tex])x = -1
x = [tex]\frac{5}{7}[/tex]
The slope of a line parallel to this line is the same, which is [tex]-\frac{7}{5}[/tex].
Which answers describe the shape below? Check all that apply. A. Parallelogram B. Quadrilateral O C. Trapezoid D. Rhombus E. Rectangle F. Square
Answers
Answer:
Step-by-step explanation:
A, B, E
100 POINTS!
Victoria will borrow $8,711 at 14.5% APR. She will pay it back over 5 years. What will her monthly payments be? (Use the table)
Show work for 100 POINTS!
Answers
Answer:
$204.95
Step-by-step explanation:
[tex]\mathsf{monthly \ payment=\dfrac{\frac{r}{12}\times P}{(1-(1+\frac{r}{12})^{-n}}}[/tex]
where:
P = total amount borrowedr = APR in decimal formn = total number of months of term of loanGiven:
P = 8711r = 14.5% = 0.145n = 5 × 12 = 60 monthsSubstituting given values into formula:
[tex]\implies \mathsf{monthly \ payment=\dfrac{\frac{0.145}{12}\times 8711}{(1-(1+\frac{0.145}{12})^{-60}}}[/tex]
[tex]\implies \mathsf{monthly \ payment=\$204.95}[/tex]
To the nearest millimeter, a cell phone is 86 mm long and 44 mm wide. What is the ratio of the width to the length?
Answers
Answer:
1
Step-by-step explanation:
44 ÷ 86
Simplify 44/86
22/43
Find the closest integer to 22/43
Answer:
44 mm : 86 mm
Step-by-step explanation:
A cell phone is 86 mm long and 44 mm wide
Long = length
Wide = width
Tasha is planning an expansion of a square flower garden in a city park. If each side of the original garden is increased by 10 m, the new total area of the garden will be 169 m2. Find the length of each side of the original garden
Answers
Answer:
6m
Step-by-step explanation:
To find the original length you must take away the add-ons. To do so we know that they added 7 and A= l*l (or l squared). They added the 7 to the length so multiply 7*7 (49) subtract your answer from 169 (120) and that's your new area. To find the original length divide by 4 (I believe) and get 30m2
Suppose a car is initially worth $20,000, and suppose that after one year, the car is now worth $18,673. Write an exponential function that models this scenario, and then estimate the value of the car after five years
Answers
y=-3(x + 5)^2 - 4
Convert the equation from vertex formi to standard form.
Answers
The points K(5,-5), L(9,-5), and M (7,3)
Answers
Answer:
i plotted the points, just connect them
Could I get some help? QUESTION: This table shows the profit for a company (in millions of dollars) in different years.
The quadratic regression equation that models these data is y=-0.34x^2+4.43x+3.46. Using the quadratic regression equation, what was the predicted profit in year 8?
A. $17.14 million
B. $16.8 million
C. $17 million
D. $16 million
Answers
Answer:
17 million
Step-by-step explanation:
-0.34([tex]8^{2}[/tex]) + 4.43(8) + 3.46
-21.76 + 35.44 + 3.46
13.68 + 3.46
17.14
Answer: C 17 million
Have an amazing day!
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Each of 5 gift bags contains x pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
Write an equation for the story. (use "X")
what is the reason for multiplication in the story?
Answers
Explanation:
Each of the five bags contain x pencils, where x is a placeholder for a positive whole number.
This means Tyler starts off with 5x pencils
He then adds 3 pencils per bag, so 3*5 = 15 extra pencils, to get the expression 5x+15. We're told the new grand total is 20 pencils.
Therefore, the equation for this story problem is 5x+15 = 20
The reason for the multiplication is because we have the same number of pencils per bag. So we're adding multiple groups together, or using multiplication to do a shortcut.
eg: instead of 5+5+5 = 15 we can do 5*3 = 15
Variables y and x have a proportional relationship, where y = 18 when x = 2.
What is the value of x when y = 36?
Enter your answer in the box.
X=
Answers
Answer:
x = 324Step-by-step explanation:
y=Kxmake k the subject of formula=> Kx = y=> k = y/xadd the value of y = 18 when x = 2.=> k = 18/2divide 18 by 2=> k = 9What is the value of x when y = 36?
=> x = Kxsubstitute k equals to 9 and from the question y equals to 36 into this equation=> x = 9(36)multiply 36 by 9=> x = 324PLEASE HELP i WILL MaRK BRAINLY plsssssss
A "light" popcom has 120 Calories per serving. This is 25% fewer Calories than a serving of regular popcorn. How many
Calories does each serving of regular popcom have?
175
b. 250
480
d. 160
C.
a
Answers
We're given:
The popcorn is 120 calories per serving120 calories is 25% fewer calories than original popcornWe must find the calorie count per serving of the original popcornDetermine VariablesLet x equal the calorie count per serving of the original popcorn.
Create an EquationThe light popcorn has a calorie count of 120, which is 25% fewer calories than regular popcorn. This means that 75% of the calorie count of regular popcorn is equal to that of the light popcorn:
[tex]x*0.75=120[/tex]
Solve the Equation[tex]x*0.75=120[/tex]
⇒ Divide both sides by 0.75:
[tex]\dfrac{x*0.75}{0.75}=\dfrac{120}{0.75}\\\\x=160[/tex]
AnswerTherefore, each serving of regular popcorn has 160 calories.
How do I create a teacher account?
Answers
Answer:
At the top of the screen (Under the search) tap For Teachers.
Then just do what it wants.
(I'm taking a guess at this cuz I'm not a teacher.)
The Box is being packed without gaps or overlaps with unit cubes
Answers
The volume of a box is the product of its dimensions
The volume of the box is 11 unit cube
How to determine the volumeFrom the question, we have the following highlights
The cubes are unit cubesThe boxes are packed without gapsThe volume of a unit cube is:
Volume = 1 unit cube
There are 11 cubes in the box. So, the volume of the box is:
Volume = 11 *1 unit cube
Volume = 11 unit cube
Hence, the volume of the box is 11 unit cube
Read more about volumes at:
https://brainly.com/question/1972490
What is the total weight of 23 baseballs that weigh 5.25 ounces each?
Answers
Answer:
120.75 ounces
Step-by-step explanation:
It's just a multiplication you will multiply 23 ×5.25 will get you your answer
Aiko works in the fish department of a pet store. She is asked to drain, clean, and refill two reef tanks.
The first tank holds 175 gallons of water and the second tank holds 200 gallons of water. The hoses that
she uses drain the tanks at a rate of 25 gallons of water per hour.
1. write an equation for each tank that represents the total amount of water in the gallons in the tank in terms of the number of hours that the tanks are draining
tank one equation ________
tank two equation ________
2. how much water is in each tank after 3 hours?
tank one ___
tank two___
Answers
The equation for the first tank will be 175 - 25h and the equation for the second tank is 200 - 25h.
How to solve the equation?From the information given, the first tank holds 175 gallons of water and the second tank holds 200 gallons of water while the hoses that she uses drain the tanks at a rate of 25 gallons of water per hour.
Therefore, the equation for the first tank will be:
= 175 - (25 × h)
= 175 - 25h
The equation for the second tank will be:
= 200 - (25 × h)
= 200 - 25h
The quantity in the first tank after three hours will be:
= 175 - 25h
= 175 - (25 × 3)
= 175 - 75
= 100 gallons.
The quantity in the second tank after three hours will be:
= 200 - 25h
= 200 - (25 × 3)
= 200 - 75
= 125 gallons.
Learn more about equations on:
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If an exponential model has a 9% decay rate,
what multiplier would be used in the function
model?
0.9 1.09
1.9 0.91
Answers
=====================================================
Explanation:
If you lose 9% of a substance, then you keep the remaining 91% because 100% - 9% = 91%. Or you can think of it as 91% + 9% = 100%
Then we convert 91% to decimal form to get 0.91
-----------
Or you can do the steps like this:
9% = 0.09
1 - 0.09 = 0.91
Let a and b be real numbers where a b 0. Which of the following functions could represent the graph below?
f(x) = x(x -a)2(x - b)4
f(x) = x(x - a)3(x - b)2
f(x) = 4(x-a)(x - b)2
f(x) = x2 (x-a)5 (x-b)
Answers
Answer:
f(x) = x2 (x-a)5 (x-b)
Step-by-step explanation:
because if you graph the plots correctly you should get this
The equation of the graph of the polynomials is f ( x ) = x² ( x - a )⁵ ( x - b )
What is Equation of Graph of Polynomials?Graphs behave differently at various x-intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.
Identify the even and odd multiplicities of the polynomial functions' zeros.
Using end behavior, turning points, intercepts, and the Intermediate Value Theorem, plot the graph of a polynomial function.
The graphs cross or are tangent to the x-axis at these x-values for zeros with even multiplicities. The graphs cross or intersect the x-axis at these x-values for zeros with odd multiplicities
Given data ,
Let the function be represented as f ( x )
Now , the value of f ( x ) is
f ( x ) = x² ( x - a )⁵ ( x - b )
From the graph of the polynomial , we can see that
The critical points of the function f(x) = x² ( x - a )⁵ ( x - b ) are x = 0, x = a, x = b, and x = (7a + 5b)/3
Hence , the function is f ( x ) = x² ( x - a )⁵ ( x - b )
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the percentenge of 2:1 ratio
Answers
Answer: 200%
Step-by-step explanation: Seeing as 1:1 is a 100% and we all understand that just add 100% more for every number on the left if the other side is = to 1
Pls answer question b tyyy also pls simplify if possible
Answers
Answer:
9/5 + 9/4 = 81/20
= 4 1/20
= 4.05
Step-by-step explanation:
81/20 is the smallest it gets as a fraction I believe
and A. should be 5/4 if i'm not mistaken
Answer:
a) 1 1/4 or b) 4 1/20
Step-by-step explanation:
a) This one is the easier of the two. 7/8+3/8= 10/8 ---» 1 2/8 and 2/8 is 1/4. So 1 1/4.
b) This one is a bit more tricky. You need to find a common denominator between the two, in this case we can use 20.
9/5 ---» 36/20 and 9/4 ----» 45/20.
Do some adding and you get 81/20 ---» 4 1/20.
Hope this helped!
whats is the value of 5(7x+ 5), when x = 10
Answers
First you times 7 with 10
5(70+5)
Then you use distributive property, so now your problem should look like this
350+25
Combine like terms, to get 375 as your answer
5(7times10+5)
=5 times 70+5
=5 times 75
=375!
So 375 is your answer
what is the equation for Penny had some money. After she pays $5 for her new kitten, she will have $27. How much money did she start with? you will get 20 points
Answers
Answer:
X + 5 = 27 or 27 - 5 = X
the answer is $22
Step-by-step explanation:
Answer:
x-5=27 or 27=x-5 or x=27+5
Step-by-step explanation:
let x be the total amount of money Penny has.
after she deducts 5 dollars to pay for her bunny she is left with 27
therefore
x-5=27 (which is the same as representing it in any other way shown above)
if they ask for X, then solve your equation
x=27+5
x=$32
1 1 A recipe calls for cup of ingredient A for every 15 cups of ingredient B. You use 3 cups of ingredient A How many cups of ingredient B do you need? You need cups of ingredient B.
Answers
Answer:
45
Step-by-step explanation
Its actually quite simple all you have to do is multiply 15 by 3 and then you will get 45.
Tammy runs 8 miles in 45 minutes. How many minutes does she take per mile? If necessary, round your answer to the nearest hundredth.
Answers
Answer:
5.63
Step-by-step explanation: