# Math Help - Rectangle Inscribed inside a Semicircle (w/ picture)

1. ## Rectangle Inscribed inside a Semicircle (w/ picture)

I am not good with these certain types of problems. It has 3 parts to the question:

A) Verify that A(theta)=25sin2(theta) models the area of the rectangle. Justify your verification and determine the domain for theta.

B) Use your calc. to find the largest possible area for such a rectangle.

C) What are the dimensions of the inscribed rectangle with the largest possible area?

Picture for the question: Image - TinyPic - Free Image Hosting, Photo Sharing & Video Hosting

2. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

area of a triangle = (1/2)(base)(height) ... the rectangle is made up of 4 congruent triangles.

Don't forget the double angle identity for sine ... you'll need it.

3. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Im not to sure where to start to verify that it models the rectangle.

4. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Originally Posted by Binary
Im not to sure where to start to verify that it models the rectangle.
look at the previously attached sketch ...

area of the rectangle = 4(area of the green shaded triangle)

5. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

I dont get what Im solving for. Can you please explain what should go through my head when I look at it. Im really struggling right now

6. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Originally Posted by Binary
I dont get what Im solving for. Can you please explain what should go through my head when I look at it. Im really struggling right now
area of the triangle = $\frac{1}{2}bh = \frac{1}{2} \cdot 5\cos{\theta} \cdot 5\sin{\theta}$

area of the rectangle = 4(area of the triangle) = $4 \cdot \frac{1}{2} \cdot 5\cos{\theta} \cdot 5\sin{\theta}$

you take it from here ... show that the above expression is the same as $25\sin(2\theta)$, I already told you that the double angle identity for sine would be required.

7. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

This is what i got: (t = theta)

25sin(2t) = 25(2sin(t)cos(t)) = 50sin(t)cos(t)

area of rectangle: (4)(1/2)(5cos(t))(5sin(t)) = 2(25cos(t)sin(t)) = 50sin(t)cos(t).

Is this correct?

NOTE: ALSO ON THE PAPER IT SAYS THE SEMI CIRCLE HAS A RADIUS OF 5cm. Forgot to add that in the beginning.

8. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Originally Posted by Binary
Is this correct?
are you convinced it's correct?

NOTE: ALSO ON THE PAPER IT SAYS THE SEMI CIRCLE HAS A RADIUS OF 5cm. Forgot to add that in the beginning.
... note the prominence of the number "5" in your solution.

9. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

I think it is haha.

Originally Posted by skeeter

... note the prominence of the number "5" in your solution.
What do you mean?

10. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Please can anyone help me solve this problem? I dont have any examples to look at and cant find anything online. Anyonne?

11. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Originally Posted by Binary
Please can anyone help me solve this problem? I dont have any examples to look at and cant find anything online. Anyonne?
$A = 25\sin(2\theta)$

what is the problem?

the instructions say to find the greatest possible area using your calculator ... so, have you graphed the area equation?

12. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Yes I graphed the equation. It is just a line that I am getting. I dont know how to calculate the area with just a line.

13. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Originally Posted by Binary
Yes I graphed the equation. It is just a line that I am getting. I dont know how to calculate the area with just a line.
check and make sure your calculator is in radian mode ... the domain of the graph should be between 0 and pi/2

14. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

Okay. After I graph it in radian mode, what do I do to find the area. I havent ever used my calculator to calculate max area yet. This is why im struggling.

15. ## Re: Rectangle Inscribed inside a Semicircle (w/ picture)

What i can tell from what I see in my calculator. It looks like the Area = 25cm^2, But as you stated there are 4 congruent sides, so Im guessing the area = 100cm^2?

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