Find the area of the shaded region using Pick's Theorem.



A. 25
B. 42
C. 18
D. 27

If the 0's are indeed zeroes, then it would appear that you're using degrees, so that the equation

r(θ) = 2 sin(0°) + 3 sin(90°)

describes a circle r = 3 (since sin(0°) = 0 and sin(90°) = 1). In this case, the area would simply be π r ² = 9π.

But I suspect you meant to use θ, so the curve has the far more interesting equation,

r(θ) = 2 sin(θ) + 3 sin(9θ)

Since sin(9θ) has a period of 2π/9 and sin(θ) has a period of 2π, their sum has a period of 2π. (That is, 2π times the LCM of 1 and 1/9, which is 1.) But we observe r(θ) is odd, since

r(-θ) = 2 sin(-θ) + 3 sin(-9θ)

… = -2 sin(θ) - 3 sin(9θ)

… = - r(θ)

which tells us that the curve closes itself over a half-period. So the area bounded by the curve is given by the integral,

[tex]\displaystyle\int_0^\pi\int_0^{r(\theta)}r\,\mathrm dr\,\mathrm d\theta[/tex]

If you're not familiar with double integrals yet, all you need to know is that this reduces to the area formula you may/should be familiar with,

[tex]\displaystyle\frac12\int_0^\pi r(\theta)^2\,\mathrm d\theta[/tex]

Compute the integral:

[tex]=\displaystyle\frac12\int_0^\pi(2\sin(\theta)+3\sin(9\theta))^2\,\mathrm d\theta[/tex]

[tex]=\displaystyle\frac12\int_0^\pi(4\sin^2(\theta)+12\sin(\theta)\sin(9\theta)+9\sin^2(9\theta))\,\mathrm d\theta[/tex]

Apply some identities:

sin²(θ) = (1 - cos(2θ)) / 2

sin(θ) sin(9θ) = (cos(9θ - θ) - cos(9θ + θ)) / 2 = (cos(8θ) - cos(10θ)) / 2

sin²(9θ) = (1 - cos(18θ)) / 2

[tex]=\displaystyle\frac14\int_0^\pi(4(1-\cos(2\theta))+12(\cos(8\theta)-\cos(10\theta))+9(1-\cos(18\theta))\,\mathrm d\theta[/tex]

[tex]=\displaystyle\frac14\int_0^\pi(13 - 4\cos(2\theta) + 12\cos(8\theta) - 12\cos(10\theta) - 9\cos(18\theta))\,\mathrm d\theta[/tex]

[tex]=\dfrac14 \left(13\theta-2\sin(2\theta)+\dfrac32\sin(8\theta)-\dfrac65\sin(10\theta)-\dfrac12\sin(18\theta)\right)\bigg|_0^\pi[/tex]

[tex]=\dfrac1{40} \left(130\theta-20\sin(2\theta)+15\sin(8\theta)-12\sin(10\theta)-5\sin(18\theta)\right)\bigg|_0^\pi[/tex]

[tex]=\dfrac{130\pi-20\sin(2\pi)+15\sin(8\pi)-12\sin(10\pi)-5\sin(18\pi)}{40}[/tex]

= 13π/4

Answer:

Step-by-step explanation:

If you add the lengths of two sides, 12 and 14, you will get 28. Using the above theorem, the third side cannot be greater than or equal to 28. Therefore, the greatest possible whole-number length of the unknown side is 27.

Answer:

Step-by-step explanation:

7 or C

Answer:

-3

Step-by-step explanation:

-11(-3)+7=40

33+7=40

Answer:

Angle 3 is = to angle 2 because:

Angle 3 & angle 7 are corresponding angles, meaning they have the same measure.

Angle 7 & angle 2 are vertical angles, meaning they also have the same measure.

Therefore, if angle 7 is equal to angle 3, then angle 2 must also be equal to angle 3.

Hope this helps! Have a great day!

Answer: 14.967

Step-by-step explanation:

Use Pythagorean Theorem to find the missing length.  

a² + b² = c²        

x² + 10² = 18²      

x² + 100 = 324         [Subtract 100 from both side]  

          x²= 224        

      √x² = √224     [solve the square root from both sides]

        x   ≈  14.967

Answer:

im guessing .30, .3, and 30/100

Step-by-step explanation:

im not sure how to explain it but fractions. yes.

Answer:

5 Probability

5.1 Probabilities

Probabilities are used to describe how likely or unlikely it is that something will happen.

Weather forecasters often talk about how likely it is to rain, or perhaps snow, in particular

parts of the country.

Worked Example 1

(a) When you roll a dice, which number are you most likely to get?

(b) If you rolled a dice 600 times how my sixes would you expect to get?

(c) Would you expect to get the same number of ones?

Solution

(a) You are equally likely to get any of the six numbers.

(b) You would expect to get a six in about 1

6

of the throws, so 100 sixes.

(c) Yes, in fact you would expect to get about 100 of each number.

Worked Example 2

Use one of the following to describe each one of the statements (a) to (d).

Certain

Very likely

Likely

Unlikely

Very unlikely

Impossible

(a) It will snow tomorrow.

(b) It will rain tomorrow.

(c) You win a car in a competition tomorrow.

(d) You are late for school tomorrow.

Solution

(a) Very unlikely for most places in Britain, especially in summer.

(b) Likely or Very likely in Britain in winter.

(c) Very unlikely if you have entered the competition. Impossible if you have not

entered the competition.

(d) Very unlikely, unless the school bus breaks down.

Answer:

91m + 13n + 52

Step-by-step explanation:

13(n + 4 + 7m)

= (13) (n+4+7m)

= (13) (n) + (13) (4) + (13) (7m)

= 13n + 52 + 91m

= 91m + 13n + 52

hope this helps and is right :)

p.s. i really need brainliest

Answer:

13n + 52 +91m

Step-by-step explanation:

As far as I can tell, all you are doing is distributing the 13 into the numbers. You multiply 13* 1n, 13*4, and 13*7m to give you 13n +52+91m.

Answer:

it is 3 inches

3 x 4 = 12

Answer:

sin b

Step-by-step explanation:

Answer:

cos A

Step-by-step explanation:

sine = [tex]\frac{opposite}{hypotenuse}[/tex]

sine B = [tex]\frac{3}{5}[/tex]

cos = [tex]\frac{adjacent}{hypotenuse}[/tex]

cos A = [tex]\frac{3}{5}[/tex]

Answer:

[tex]3x^5+6x^4+12x^3[/tex]

Step-by-step explanation:

Given that,

[tex]f(x) = 3x^3+ 6x^2+ 12x[/tex]

and

[tex]g(x)=x^2[/tex]

We need to find the product of f(x) and g(x). SO,

[tex]f(x)\times g(x)=(3x^3+ 6x^2+ 12x)\times (x^2)\\\\=3x^3\times x^2+6x^2\times x^2+12x\times x^2\\\\=3x^5+6x^4+12x^3[/tex]

Hence, the correct option is (d).

Answer:

trust your instincts

Step-by-step explanation:

I can not interfere with your destiny you will need to decide your destiny

Answer:

Perimeter = 12.246 units

Area = 11.314 units²

Step-by-step explanation:

A=2√2r²=11.314

P=2(8)(√2-√2)=12.246

Answer:

The developer bought 4.9 acres of land, increasing its property to 29.4 acres.

Step-by-step explanation:

Given that a developer has 24.5 acres of land, and she buys an adjacent plot of land and increases the total acreage she owns by 20%, to determine the amount of acres she bought and the total acres she now owns, the following calculations must be done:

24.5 x 1.2 = 29.4

24.5 x 0.2 = 4.9

Thus, the developer bought 4.9 acres of land, increasing its property to 29.4 acres.

Answer:

The slope of the line would be 3/6 or 1/2

Step-by-step explanation:

To get from point (1,2) to point (7,5) you have to rise 3 places and run (move to the right) 6 times. Rise/run=3/6

hope this helps. ; )

Answer:

B

Step-by-step explanation:

Answer:

Its E

Step-by-step explanation:

Answer:

798

Step-by-step explanation:

first you do 40x20=800 then 800-6=794 then 794+4=798

Answer:

2

Step-by-step explanation:

Answer:

What do you want us to find?

Step-by-step explanation:

Answer:

10/1

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

and the answer is c

Answer: Just look it up on goo gle

Step-by-step explanation:

The rate per annum compound interest is approximately 11.5%.

To find the rate per annum compound interest, we can use the formula for compound interest:

[tex]A = P(1 + r/n)^{nt}[/tex]

Where:

A = the final amount (N5353 in this case)

P = the principal amount (N4000 in this case)

r = the annual interest rate (what we want to find)

n = the number of times interest is compounded per year (assuming once per year)

t = the number of years (5 years in this case)

Now, we can plug in the values:

[tex]N5353 = N4000(1 + r/1)^{1*5}[/tex]

Divide both sides by N4000:

[tex]1.33825 = (1 + r)^5[/tex]

Now, isolate (1 + r) by taking the fifth root of both sides:

[tex]1 + r = (1.33825)^{1/3}\\1 + r \approx 1.115[/tex]

Now, subtract 1 from both sides to find the interest rate (r):

r ≈ 1.115 - 1

r ≈ 0.115

Finally, convert the interest rate to a percentage:

r ≈ 0.115 * 100

r ≈ 11.5%

To know more about rate:

https://brainly.com/question/32398403

#SPJ2

Answer:

4

Step-by-step explanation:

1) Simplify 2⁴ to 16.

[tex]20 - 16[/tex]

2) Simplify.

[tex]4[/tex]

Therefor, the answer is 4.

Answer:

20 - 2^4 =4

Step-by-step explanation:

Answer:

x-6=6

Step-by-step explanation:

6+6=12

Answer:

c

Step-by-step explanation:

because 6+6=12 and the missing number would be 12 because 12-6=6

Answer:I don’t understand

Step-by-step explanation:

Answer:

1 6 3b 0ny 5i bhk

Step-by-step explanation: