1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Sladkaya [172]
2 years ago
15

Find the area that the curve encloses and then sketch it. r = 3 + 8 sin(6)

Mathematics
1 answer:
Rudiy272 years ago
7 0

Answer:

A=41\pi\: \text{units}^2\approxA\approx128.8053\:\text{units}^2

Step-by-step explanation:

I assume you mean r=3+8\sin\theta:

Use the formula \displaystyle A=\int\limits^a_b \frac{1}{2} {r(\theta)^2} \, d\theta where a and b are the lower and upper bounds and r(\theta) is the equation of the polar curve.

Since the graph is symmetrical about the line \displaystyle \theta=\frac{\pi}{2}, let the bounds of integration be \displaystyle \biggr(-\frac{\pi}{2},\frac{\pi}{2}\biggr) to find half the area of the curve, and then find twice of that area:

\displaystyle A=\int\limits^a_b \frac{1}{2} {r(\theta)^2} \, d\theta\\\\A=2\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \frac{1}{2} {(3+8\sin\theta)^2} \, d\theta\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} 9+48\sin\theta+64\sin^2\theta \, d\theta\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} 9+48\sin\theta+64\biggr(\frac{1-\cos2\theta}{2} \biggr) \, d\theta\\\\\\A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} (9+48\sin\theta+32-32\cos2\theta) \, d\theta

\displaystyle A=\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} (41+48\sin\theta-32\cos2\theta) \, d\theta\\\\A=41\theta-48\cos\theta-16\sin2\theta\biggr|^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\\\\

A=\biggr[41\biggr(\frac{\pi}{2}\biggr)-48\cos\biggr(\frac{\pi}{2}\biggr)-16\sin2\biggr(\frac{\pi}{2}\biggr)\biggr]-\biggr[41\biggr(-\frac{\pi}{2}\biggr)-48\cos\biggr(-\frac{\pi}{2}\biggr)-16\sin2\biggr(-\frac{\pi}{2}\biggr)\biggr]\\\\A=\biggr[\frac{41\pi}{2}-24\sqrt{2}\biggr]-\biggr[-\frac{41\pi}{2}+24\sqrt{2}\biggr]\\ \\A=41\pi\\\\A\approx128.8053

Thus, the area of the curve is 41π square units. See below for a graph of the curve and its shaded area.

You might be interested in
Find the missing ratio and explain how you found it
ankoles [38]

Answer:

5:4

Step-by-step explanation

If your shoes to flip-flops are 10:8 then the starting ratio will be 10:8 but 10 and 8 are both divisible to 2 so you can simplify 10:8 to 5:4

7 0
3 years ago
Read 2 more answers
WHAT IS THE LINE OF BEST FIT FOR THE FOLLOWING DATA (1,3) (2,0) (5,-8) (9, -25) . WRITE YOUR ANSWERS IN THE FORM OF Y= AX+ B​
dimulka [17.4K]

Answer:

3.471 + 7.2516 x

Step-by-step explanation:

Given the data:

(1,3) (2,0) (5,-8) (9, -25) .

X:

1

2

5

9

Y:

3

0

-8

-25

The line of best fit for the data given can be obtained using an online regression calculator :

The equation of best fit for the data is thus;

Best fit equation; y = -3.471 + 7.2516x

Intercept = - 3.471 ;

Slope = 7.2516

5 0
3 years ago
PLS- I NEED HELPPPP PLS HELP MEEEEEEEE.
Elis [28]

Answer:

with what?

Step-by-step explanation:

8 0
3 years ago
What is the length of segment RS?
Evgesh-ka [11]

Answer:

3

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Mr .Cruz is the director of an after school program. He says that 170 or 85%of the students went on to attend collage. A parent
Nimfa-mama [501]

Answer:

  200 students were in the program

Step-by-step explanation:

The parent should know how to figure ...

  170/85% = total/100%

This is the same as ...

  total = 170/0.85 = 200

200 students were in the program.

_____

<em>Critical Thinking</em>

There are three declarative statements here. <em>There is no question</em>. We have made up a question to answer.

Another possible answer to the non-question could be, "The parent can ask his student how many students were in the program."

3 0
3 years ago
Other questions:
  • (0.6)2, (-0.3) + (-0.06), |-3 + 2.64|, -(-0.36) ,2x - 0.64x which is equivalent to 036
    10·1 answer
  • Lris has a family photograph with a width of8 in and a height of10 in. she wants to enlarge the photograph so that its width is
    15·1 answer
  • Simplify (-5x^3)^-2 using only positive exponents
    15·2 answers
  • What is −2 1/4 ⋅ (−2 13) ?<br><br><br> A −5 1/4<br> B −4 1/12 <br> C 4 1/12 <br> or D 5 1/4
    14·1 answer
  • Draw a box plot for the following data. {35, 50, 50, 48, 48, 32, 38, 41, 48, 34, 29, 23, 41, 34, 43}
    14·1 answer
  • Which of the following correctly uses absolute value to show the distance between −50 and 12? (10 points)
    11·2 answers
  • 4% more than 150 is, can yall explain me each step
    9·1 answer
  • (5 + 42) ÷ 3 ● 6<br> I need the answer and Steps Plz
    10·1 answer
  • Write and solve an equation to find x. Then find the measure of angle YXZ.
    8·1 answer
  • A solid metal cone has a radius of 4cm and a vertical height of 12cm.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!