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

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    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
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    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.
    Attached Thumbnails Attached Thumbnails Rectangle Inscribed inside a Semicircle (w/ picture)-rectcircle.jpg  
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Im not to sure where to start to verify that it models the rectangle.
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Quote Originally Posted by Binary View Post
    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)
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    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
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Quote Originally Posted by Binary View Post
    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.
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    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.
    Last edited by Binary; April 14th 2012 at 08:56 AM.
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Quote Originally Posted by Binary View Post
    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.
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    I think it is haha.

    Quote Originally Posted by skeeter View Post

    ... note the prominence of the number "5" in your solution.
    What do you mean?
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    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?
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Quote Originally Posted by Binary View Post
    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?
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    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.
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    Re: Rectangle Inscribed inside a Semicircle (w/ picture)

    Quote Originally Posted by Binary View Post
    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
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    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.
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    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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