# Re: Pre-cal: find a function that models the area of a rectangle in terms of its height

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Posted by Soroban on September 17, 2002 at 02:03:15:

In Reply to: Pre-cal: find a function that models the area of a rectangle in terms of its height posted by Brandi Jenison on September 16, 2002 at 15:02:02:

: I have a question. There is a semi-circle with a rectangle in it. The radius of the semi-circle is 10. I am supposed to find a function that models the area A of the rectangle in terms of its height h. The answer is A(h) = 2h(square root 100-h squared) when 0: Can you help?!
: Brandi

First of all, Brandi, asking four times won't get you the answer faster.

Did you drew a sketch?
Do you know the equation of a circle? (No, not the area.)
Do you know the formula for the area of a rectangle?

I'll try to explain this without a diagram - tricky!

You have your semicircle centered at the origin, extending left to right
from -10 to 10. It extends upward from 0 to 10.

About halfway up the semicircle, draw a horizontal line.
It cuts the Y-axis at h. That's the height of the rectangle.

Where the line cuts the semicircle (left and right) is the length.
Draw lines straight down to the X-axis. There is
our rectangle!

Since the area of a rectangle is Length x Width
(or Length x Height),
we need to find the Length.

The equation of that circle is: x^2 + y^2 = 100.
For y = h, we solve x^2 + h^2 = 100, and get x = sqrt(100 - h^2)
But that's the distance from 0 to the right side of the rectangle.
The Length is twice that. So, Length = 2sqrt(100-h^2)

We've got it!
Area = Length x Height = 2h sqrt(100-h^2)

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