The function f(x) = 5x^2 + 20x + 25 can be rewritten in vertex form as f(x) = 5(x + 2)^2 + 5, and its vertex is located at (-2, 5).
To rewrite the function f(x) = 5x^2 + 20x + 25 in vertex form, we complete the square. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Let's complete the square:
f(x) = 5(x^2 + 4x) + 25
To complete the square, we need to add and subtract (4/2)^2 = 4 to the expression inside the parentheses:
f(x) = 5(x^2 + 4x + 4 - 4) + 25
Rearranging the terms:
f(x) = 5((x^2 + 4x + 4) - 4) + 25
Now we can factor the perfect square inside the parentheses:
f(x) = 5((x + 2)^2 - 4) + 25
Expanding and simplifying further:
f(x) = 5(x + 2)^2 - 20 + 25
f(x) = 5(x + 2)^2 + 5
Therefore, the function f(x) = 5x^2 + 20x + 25 in vertex form is f(x) = 5(x + 2)^2 + 5. The vertex is given by the coordinates (-2, 5).
The correct question should be :
What is the vertex form of the function f(x) = 5x^2 + 20x + 25? Identify the coordinates of the vertex.
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The playground at a park is shaped like a trapezoid the dimensions what is the area of the playground in square feet
Answer:
[tex]Area = 1560ft^2[/tex]
Step-by-step explanation:
Given
See attachment for playground
Required
Determine the area
The playground is a trapezoid. So;
[tex]Area = \frac{1}{2}(Sum\ parallel\ sides) * Height[/tex]
From the attachment, the parallel sides are: 68ft and 36ft
The height is: 30ft
So, the area is:
[tex]Area = \frac{1}{2}(68ft + 36ft) * 30ft[/tex]
[tex]Area = \frac{1}{2}(104ft) * 30ft[/tex]
[tex]Area = 52ft * 30ft[/tex]
[tex]Area = 1560ft^2[/tex]
I WILL GIVE BRAINLIEST!!!!!!
Write in complete sentences to explain what a budget is, how to make one, and how to balance it.
find w such that 2u v − 3w = 0. u = (−6, 0, 0, 2), v = (−3, 5, 1, 0)
To find the value of w that satisfies the equation 2u v - 3w = 0, where u = (-6, 0, 0, 2) and v = (-3, 5, 1, 0), we can substitute the given values into the equation and solve for w.
Substituting the given values of u and v into the equation 2u v - 3w = 0, we have:
2(-6, 0, 0, 2)(-3, 5, 1, 0) - 3w = 0.
Expanding the scalar multiplication and performing the dot product, we get:
(-12, 0, 0, 4)(-3, 5, 1, 0) - 3w = 0,
(36 + 0 + 0 + 0) - 3w = 0,
36 - 3w = 0.
Simplifying the equation, we have:
36 = 3w,
w = 12.
Therefore, the value of w that satisfies the equation is 12. By substituting w = 12 into the equation 2u v - 3w = 0, we get:
2(-6, 0, 0, 2)(-3, 5, 1, 0) - 3(12) = 0,
(-12, 0, 0, 4)(-3, 5, 1, 0) - 36 = 0,
36 - 36 = 0,
0 = 0.
Hence, the value of w = 12 makes the equation true, satisfying the given condition.
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An insurance policy sells for $600. Based on past data, an average of 1 in 50 policyholders will file a $5,000 claim, and average of 1 in 100 policyholders will file a $10,000 claim, and an average of 1 in 200 policyholders will file a $30,000 claim. What is the expected value per policy sold?
Answer:
$250
Step-by-step explanation:
Calculation to determine the expected value per policy sold
Expected value per policy sold =$600-(1/50)*$5,000-(1/100)*$10,000-(1/200)*$30,000
Expected value per policy sold =$600-$100-$100-$150
Expected value per policy sold =$250
Therefore the expected value per policy sold will be $250
1 Which is an arithmetic sequence?
F)2, 5, 9, 14, ...
G)100, 50, 12.5, 1.6, ...
H)3, 10, 17, 24,...
j) -2,-1,-1/2,-1/4
Answer:
H) 3, 10, 17, 24, ...Step-by-step explanation:
An arithmetic sequence is when the difference of the terms is same
F)2, 5, 9, 14, ...
14 - 9 = 5, 9 - 5 = 4. 5-2 = 35 ≠ 4 ≠ 3, no
G)100, 50, 12.5, 1.6, ...
1.6 - 12.5 = -10.912.5 - 50 = -37.550 - 100 = -50-10.9 ≠ -37.5 ≠ -50, no
H)3, 10, 17, 24,...
24-17 = 717 - 10 = 710 - 3 = 77 is the common difference, yes
j) -2,-1,-1/2,-1/4
-1/4 - (-1/2) = 1/4-1/2 - (-1) = 1/2-1 - (-2) = 11/4 ≠ 1/2 ≠ 1, no
The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, shown by the dashed line around the window. Each square in the window has an area of 169 in2. Round your answers to the nearest whole number.
Answer:
[tex](a)\ Area = 3765.32[/tex]
[tex](b)\ Area = 4773[/tex]
Step-by-step explanation:
Given
[tex]A_1 = 169in^2[/tex] --- area of each square
[tex]Shade = 4in[/tex]
See attachment for window
Solving (a): Area of the window
First, we calculate the dimension of each square
Let the length be L;
So:
[tex]L^2 = A_1[/tex]
[tex]L^2 = 169[/tex]
[tex]L = \sqrt{169[/tex]
[tex]L=13[/tex]
The length of two squares make up the radius of the semicircle.
So:
[tex]r = 2 * L[/tex]
[tex]r = 2*13[/tex]
[tex]r = 26[/tex]
The window is made up of a larger square and a semi-circle
Next, calculate the area of the larger square.
16 small squares made up the larger square.
So, the area is:
[tex]A_2 = 16 * 169[/tex]
[tex]A_2 = 2704[/tex]
The area of the semicircle is:
[tex]A_3 = \frac{\pi r^2}{2}[/tex]
[tex]A_3 = \frac{3.14 * 26^2}{2}[/tex]
[tex]A_3 = 1061.32[/tex]
So, the area of the window is:
[tex]Area = A_2 + A_3[/tex]
[tex]Area = 2704 + 1061.32[/tex]
[tex]Area = 3765.32[/tex]
Solving (b): Area of the shade
The shade extends 4 inches beyond the window.
This means that;
The bottom length is now; Initial length + 8
And the height is: Initial height + 4
In (a), the length of each square is calculated as: 13in
4 squares make up the length and the height.
So, the new dimension is:
[tex]Length = 4 * 13 + 8[/tex]
[tex]Length = 60[/tex]
[tex]Height = 4*13 + 4[/tex]
[tex]Height = 56[/tex]
The area is:
[tex]A_1 = 60 * 56 = 3360[/tex]
The radius of the semicircle becomes initial radius + 4
[tex]r = 26 + 4 = 30[/tex]
The area is:
[tex]A_2 = \frac{3.14 * 30^2}{2} = 1413[/tex]
The area of the shade is:
[tex]Area = A_1 + A_2[/tex]
[tex]Area = 3360 + 1413[/tex]
[tex]Area = 4773[/tex]
please help, tysm for your assistance if you do :)
Answer:
27/49
plz mark me as brainliest
Can anyone please help asap I keep getting those links bots
if i do something to the numerator of a fraction, am i supposed to do the same to the denominator too? and if yes,why?
for example i want to multiply 2/2 over 6/2, is it necessary to multiply 2/2 or can I just multiply 2?
Step-by-step explanation:
When performing operations on fractions, it is important to maintain the relationship between the numerator and the denominator. In general, if you do something to the numerator, you should also do the same to the denominator.
In your example, if you want to multiply the fraction 2/2 by 6/2, it is necessary to multiply both the numerator and the denominator by the same value. Here's why:
When you multiply fractions, you multiply the numerators together and the denominators together. So, in this case, the multiplication would be:
(2/2) * (6/2) = (2 * 6) / (2 * 2) = 12/4
If you had only multiplied the numerator (2) by 6, the result would have been:
(2 * 6) / 2 = 12/2
As you can see, these two results are different. The correct result is 12/4, which simplifies to 3/1 or simply 3. If you only multiplied the numerator, you would have obtained 12/2, which simplifies to 6.
So, it's necessary to apply the same operation (in this case, multiplication by 2) to both the numerator and the denominator in order to maintain the value of the fraction.
A Store owners offers a discount of 20% off the regular price of all jackets. Jessica has a coupon that gives her an additional 5% off the discount price. The original price of jacket Jessica buys is $84. What is the price of the jacket after the discount and Jessica coupon?
Answer:
$63
Step-by-step explanation:
The store is 20% off, Jessica has a coupon that is 5% off add that together and it's 25% off. $84 - 25% = $63
pls help i’ll give brainliest
Answer:
The answer is 56.33 in decimal form but in fraction form the answer is 5633/
100
Step-by-step explanation:
6.55 x 8.6
An after school music program has 15 out 50 students practicing. Write 15/50 (15 over 50) as a decimal and as a percent.
Decimal -
Percent -
Answer:
Percent- 30
decimal-0.3
Step-by-step explanation:
Hope this helps and have a wonderful day!!!
The answers are:
5
10
35
55
please help
Answer:
35.
Step-by-step explanation:
can i get brainliesttt
jus solved it.
Which statements are correct? Check all that apply. 5 students study both French and Spanish. 63 students study French. 2 students study neither French nor Spanish. 30 students study French, but not Spanish. 63 students study Spanish.
Answer:
The correct answers are:
• Option 1,
• Option 3 and
• Option 4
Step-by-step explanation:
From the two-way table, it is true that;
1.) 5 students study both French and Spanish.
2.) 30 students study French but not Spanish.
3.) 2 students study neither French nor Spanish
4.) 35 students study French
5.) 68 students study Spanish
6.) 63 students study Spanish but not French
This makes the correct answers options 1, 3 and 4
P.S: I believe your question is from the attached image
Answer:
A. 5 students study both French and Spanish.
C. 2 students study neither French nor Spanish.
D. 30 students study French, but not Spanish.
:)
Students deliver catalogues and leaflet to houses. One day they have to deliver 384 catalogues and 1890 leaflets. Each student can deliver either 16 catalogues or 90 leaflets in hour. Each student can only work for 1 hour. All students hired are paid £51.30 per day, even if they don't work a full day. If the minimum number of wages are hired, how much will the wage bill be
Answer:
£2308.5 per day
Step-by-step explanation:
Since in one day they have to deliver 384 catalogues and 1890 leaflets and each student can deliver either 16 catalogues or 90 leaflets in hour, the amount of students required to deliver 384 catalogues in one hour is 384/16 = 24 students.
Also, the number of students required to deliver 1890 leaflets in one hour is 1890/90 = 21 students.
So the total number of students required to make the delivery is thus 24 + 21 = 45 students. This is the minimum number of students required for the delivery.
Since all students hired are paid £51.30 per day, even if they don't work a full day, so the amount of wage paid for this minimum amount of students is thus minimum amount × wage = 45 × £51.30 per day = £2308.5 per day
Answer:
£307.80
Step-by-step explanation:
Wow seems the verified answer is wrong.
Who would've thought?
16*384=24
90*1890=21
21+24=45
45/8=5.62500
5.625 rounds to 6
51.30*6=£307.80
Thats your working out
You have to divide 45 by 8 because there are 8 hours in a day in which they can work.
Then round the number as you cant have a fraction of a person
Which you would then multiply by 51.30
Brainliest would be appreciated <33
Hope it helps!
In a large population of adults, the mean IQ is 111 with a standard deviation of 22. Suppose 55 adults are randomly selected for a market research campaign. (Round all answers to 4 decimal places, if needed.)
(a) The distribution of IQ is approximately normal is exactly normal may or may not be normal is certainly skewed.
(b) The distribution of the sample mean IQ is approximately normal exactly normal not normal left-skewed right-skewed with a mean of ? and a standard deviation of .
(c) The probability that the sample mean IQ is less than 107 is .
(d) The probability that the sample mean IQ is greater than 107 is .
(e) The probability that the sample mean IQ is between 107 and 117 is
(a) The distribution of IQ is approximately normal.
(b) The distribution of the sample mean IQ is approximately normal with a mean of 111 and a standard deviation of 3.0410.
(c) The probability that the sample mean IQ is less than 107 is 0.1056.
(d) The probability that the sample mean IQ is greater than 107 is 0.8944.
(e) The probability that the sample mean IQ is between 107 and 117 is 0.7881.
How to solve the research campaign?(a) The given information does not indicate any significant departure from normality, and with a large population, the Central Limit Theorem suggests that the distribution of IQ will be approximately normal.
(b) The mean of the sample mean IQ will be equal to the population mean, which is 111.
The standard deviation of the sample mean can be calculated by dividing the population standard deviation (22) by the square root of the sample size (55).
Therefore, the standard deviation of the sample mean IQ is 22 / √(55) = 3.0410.
(c) To calculate this probability, standardize the sample mean IQ value using the formula z = (x - μ) / (σ / √(n)),
where x = value to find the probability for, μ = population mean, σ = population standard deviation, and n = sample size.
In this case, find the probability for x = 107.
By plugging in the values, calculate the z-score and then use a standard normal distribution table or calculator to find the corresponding probability, which is 0.1056.
(d) Similar to part (c), use the same formula to standardize the value of 107 and calculate the z-score.
Then, find the probability of the sample mean IQ being greater than 107 by subtracting the probability found in part (c) from 1, which is 0.8944.
(e) To calculate this probability, find the individual probabilities for both values and then subtract the probability found in part (d) from the probability found in part (c).
The probability of the sample mean IQ being less than 107 is 0.1056, and the probability of the sample mean IQ being greater than 117 is 0.2119 (which can be found by subtracting the probability of being less than 117 from 1).
Therefore, the probability of the sample mean IQ being between 107 and 117 is 0.1056 + (1 - 0.2119) = 0.7881.
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. In a volleyball game, Alexis scored 4 points more than twice the
number of points Jessica scored. Jessica scored 3 points. How many
points did Alexis score?
F. 1 point G. 7 points H. 10 points I. 12 points
Answer: 10
Step-by-step explanation:
Alexis Scored 4 more than twice the number of points Jessica scored.
Jessica scored 3
twice the number of 3 would be 3 x 2 which equals six
4 more than twice the number which is 6 would be 10, 4+6=10
How many solutions does this equation have? 8 + 10z = 3 + 9z
-no solution
-one solution
-infinitely many solutions
help I will give brainiest if you can atleast do three
1.) A=pi(r)^2
2.) V=Bh
3.) 3.1
4.) 20
5.)36
6.) 10
7.)18
Answer:
Step-by-step explanation:
1.) formula of circle =pi times r^2
2.)volume of cylinder =pi times r^2 times h
3.)value of pi rounded = 3.14
4.) diameter of can A =2r=2(10) = 20
diameter of Can A is 20
5.)diameter of can B =2r =2(18) =36
diameter of Can B is 36
6.)radius = 10
7.)radius =18
I did all of them.
For a random variable X where X ~ N(p, p(1-p)/k) and 0<=p<=1, find the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9
The value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}
Given a random variable X where X ~ N(p, p(1-p)/k) and 0<=p<=1, we need to find the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9.
In general, if X ~ N(μ,σ²), then
P[|X-μ| < a] = 2Φ(a/σ) - 1
where Φ(z) is the standard normal cumulative distribution function.
Therefore, we can say that
P[|X-p| < 0.2] = 2Φ(0.2/√(p(1-p)/k)) - 1 ≥ 0.9
or 2Φ(0.2/√(p(1-p)/k)) ≥ 1.9
or Φ(0.2/√(p(1-p)/k)) ≥ 0.95
or 0.2/√(p(1-p)/k) ≥ Φ^(-1)(0.95)
where Φ^(-1)(z) is the inverse of the standard normal cumulative distribution function.
Therefore, Φ^(-1)(0.95) = 1.6450.2/√(p(1-p)/k) ≥ 1.645
or k ≤ 0.2²p(1-p)/1.645²
From the above inequality, we get the maximum value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is given by the formula:
k ≤{0.2^2 p(1-p)}/{1.645^2}
Therefore, the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}
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Equation in slope intercept form that represents their shown
Answer:
I think the answer would be Y= -2X+5 .
Hope it helps u ^^♥️
A recipe uses 6 tablespoons of butter for every 8 oz of cheese. the rate is __ tablespoons for every 1 oz. the raze is __ oz for every 1 tablespoon.
1/4 or .75
6 divided by 4 equal 1/4 or .75
The table below lists the number of games played in a yearly best-of-seven baseball championship series, along with the expected proportions for the number of games played with teams of equal abilities. Use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions.
Games_Played Actual_contests Expected_proportion
4 16 0.125
5 21 0.25
6 21 0.3125
7 38 0.3125
determine the null hypotheses
what is the t statistics
what is the p value
what is the conclusion for the test statistic
The null hypothesis is that the actual numbers of games fit the distribution indicated by the expected proportions.
The following is the calculation of the t-statistics for the given data.
[tex]T=\frac{(O_i-E_i)} {\sqrt{E_i}} [/tex]where [tex]O_i[/tex] represents the observed frequency, and [tex]E_i[/tex] represents the expected frequency.t statistics for
4: [tex]\frac {(16-25)} {\sqrt {25(0.125)}} [/tex] = -3.2t statistics for
5: [tex]\frac {(21-25)} {\sqrt {25(0.25)}} [/tex] = -1.8t statistics for
6: [tex]\frac {(21-31)} {\sqrt {25(0.3125)}} [/tex] = -3.2t statistics for
7: [tex]\frac {(38-31)} {\sqrt {25(0.3125)}} [/tex] = 3.2
The critical value of t at the 0.05 level of significance is ± 2.132. Since the t-statistics of 3.2 > 2.132, we reject the null hypothesis. So, there is a significant difference between the actual number of games played and the expected number of games played in the baseball championship series at the 0.05 significance level. p-value = P (|t| > 3.2) = 0.002
The conclusion for the test statistic: Since the p-value (0.002) is less than the level of significance (0.05), we reject the null hypothesis and conclude that there is a significant difference between the actual numbers of games played and the distribution indicated by the expected proportions.
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simplify this answer pls
Answer:
D
Step-by-step explanation:
when it's a power of the power we multiply the powers to get a single value for the power.
(6^(1/4))^4=6^(4*(1/4)) (4*(1/4)=1)
=6^1=6
so the answer is D
There’s a picture of my question plz help :)
Answer:
1,534 inches squared
Step-by-step explanation:
To find surface area we just solve for the area of all the sides and add those together. A rectangular prism (a box like above) has 6 sides. There are...
2 sides each of the following dimensions:
2(13×26)=
2(338)=676
2(13×11)=
2(143)=286
2(26×11)=
2(286)=572
Add the area of all 6 sides...
676+286+572=1,534
Remember it is squared not cubed.
let $p$ and $q$ be the two distinct solutions to the equation$$\frac{4x-12}{x^2 2x-15}=x 2.$$if $p > q$, what is the value of $p - q$?
let $p$ and $q$ be the two distinct solutions to the equation$$\frac{4x-12}{x^2 2x-15}=x 2.$$if $p > q$. The value of $p - q$ is 4.
To find the value of $p - q$, we first need to solve the given equation and determine the values of $p$ and $q$.
The equation is:
$$\frac{4x-12}{x^2 - 2x - 15} = x^2.$$
Step 1: Factorize the denominator:
The denominator can be factored as $(x - 5)(x + 3)$.
Step 2: Simplify the equation:
$$\frac{4x-12}{(x - 5)(x + 3)} = x^2.$$
Step 3: Multiply both sides of the equation by $(x - 5)(x + 3)$ to eliminate the denominator:
$$(4x - 12) = x^2(x - 5)(x + 3).$$
Step 4: Expand and rearrange the equation:
$$4x - 12 = x^4 - 2x^3 - 15x^2 + 25x.$$
Step 5: Rearrange the equation and combine like terms:
$$x^4 - 2x^3 - 15x^2 + 21x - 12 = 0.$$
Step 6: Factorize the equation:
$$(x - 3)(x + 1)(x - 2)(x + 2) = 0.$$
From this, we get four possible solutions: $x = 3$, $x = -1$, $x = 2$, and $x = -2$.
However, we are interested in the two distinct solutions $p$ and $q$, where $p > q$. Therefore, the values of $p$ and $q$ are $p = 3$ and $q = -1$.
Finally, we can find the value of $p - q$:
$$p - q = 3 - (-1) = 3 + 1 = 4.$$
Hence, the value of $p - q$ is 4.
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find the missing side x
Answer:
[tex]\sqrt{968}[/tex]
Step-by-step explanation:
Since this is a right triangle, we are able to use pythagorean theorem, a^2+b^2=c^2. In this case x would be the "c", so 22^2+22^2=x^2. Isolate the variable and solve for x. 484+484=x^2
968=x^2
[tex]\sqrt{968\\}[/tex]=x
A soccer field is 100 meters long. What could be its length in yards? A. 33.3 B. 91 C. 100 D. 109
Answer:
D. 109
Step-by-step explanation:
100 meters into yards is 109.361
the ratio is 1.094 so 109
How many students are enrolled in a course either in calculus, discrete mathematics, data structures, 7. or programming languages at a school if there are 507, 292, 312, and 344 students in these courses, respectively; 14 in both calculus and data structures; 213 in both calculus and programming languages; 211 in both discrete mathematics 558 and data structures; 43 in both discrete mathematics and programming languages; and no student may take calculus and discrete mathematics, or data structures and programming languages, concurrently
Answer:
974
Step-by-step explanation:
Let assume that:
The set of student that took part in Calculus be = C
Those that took part in discrete mathematics be = D
Let those that took part in data structures be = DS; &
Those that took part in Programming language be = P
Thus;
{C} = 507
{D} = 292
{DS} = 312
{P} = 344
For intersections:
{C ∩ DS} = 14
{C ∩ P} = 213
{D ∩ DS} = 211
{D ∩ P} =43
{C ∩ D} = 0
{DS ∩ P} = 0
{C ∩ D ∩ DS ∩ P} = 0
According to principle of inclusion-exclusion;
{C ∪ D ∪ DS ∪ P} = {C} + {D} + {DS} + {P} - {C ∩ D} - {C ∩ DS} - {C ∩ P} - {D ∩ DS} - {D ∩ P} - {DS ∩ P}
{C ∪ D ∪ DS ∪ P} = 507 + 292 + 312 + 344 - 14 - 213 - 211 - 43 - 0
{C ∪ D ∪ DS ∪ P} = 974
Hence, the no of students that took part in either course = 974
Find the area of the region that lies inside both the curves.
r = sin 2θ , r = sin θ
The area of the region that lies inside both the curves r = sin 2θ and r = sin θ is π/3 + (1/16)√3.
To find the area of the region that lies inside both the curves, we need to determine the limits of integration for the angle θ.
The curves r = sin 2θ and r = sin θ intersect at certain values of θ. To find these points of intersection, we can set the two equations equal to each other and solve for θ:
sin 2θ = sin θ
Using the trigonometric identity sin 2θ = 2sin θ cos θ, we can rewrite the equation as:
2sin θ cos θ = sin θ
Dividing both sides by sin θ (assuming sin θ ≠ 0), we have:
2cos θ = 1
cos θ = 1/2
θ = π/3, 5π/3
Now we have the limits of integration for θ, which are π/3 and 5π/3.
The formula for calculating the area in polar coordinates is given by:
A = (1/2) ∫[θ₁,θ₂] (r(θ))² dθ
In this case, the function r(θ) is given by r = sin 2θ. Therefore, the area is:
A = (1/2) ∫[π/3,5π/3] (sin 2θ)² dθ
To evaluate this integral, we can simplify the expression (sin 2θ)²:
(sin 2θ)² = sin² 2θ = (1/2)(1 - cos 4θ)
Now, the area formula becomes:
A = (1/2) ∫[π/3,5π/3] (1/2)(1 - cos 4θ) dθ
We can integrate term by term:
A = (1/4) ∫[π/3,5π/3] (1 - cos 4θ) dθ
Integrating, we get:
A = (1/4) [θ - (1/4)sin 4θ] |[π/3,5π/3]
Evaluating the integral limits:
A = (1/4) [(5π/3 - (1/4)sin (20π/3)) - (π/3 - (1/4)sin (4π/3))]
Simplifying the trigonometric terms:
A = (1/4) [(5π/3 + (1/4)sin (2π/3)) - (π/3 + (1/4)sin (4π/3))]
Finally, simplifying further:
A = (1/4) [(5π/3 + (1/4)√3) - (π/3 - (1/4)√3)]
A = (1/4) [(4π/3 + (1/4)√3)]
A = π/3 + (1/16)√3
Therefore, the area of the region that lies inside both the curves r = sin 2θ and r = sin θ is π/3 + (1/16)√3.
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