A circular piece of sheet metal has a diameter of 20 in. The edges are to be cut off to form a rectangle of area 60 in. What are the dimensions of the rectangle?The tips of the rectangle touch the...

A circular piece of sheet metal has a diameter of 20 in. The edges are to be cut off to form a rectangle of area 60 in. What are the dimensions of the rectangle?

The tips of the rectangle touch the circle.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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We are given a circle with diameter 20in, and we want to cut a rectangle out of it; the vertices of the rectangle lie on the circle and the area of the rectangle is 60 sq in.

(1) Note that if a rectangle is inscribed in a circle, opposite vertices lie on a diagonal. (If a right angle is inscribed in a circle, the sides of the angle intercept the circle at the endpoints of a diameter)

So the diagonal of the rectangle has length 20 in

(2) `l*w=60 => w=60/l`

(3) From the pythagorean theorem we also have `60^2=l^2+w^2 => 3600=l^2+(60/l)^2`

Thus:

`l^2+3600/l^2=400`

`l^4-400l^2+3600=0`

This is a quadratic in `l^2` ; using the quadratic formula we get

`l^2=(400+-sqrt(400^2-4(1)(3600)))/2`

`=(400+-sqrt(145600))/2`

`~~390.7878,9.2122`

So `l~~19.78,3.04` .

Checking we see that (19.78)(3.04)=60.13 and `19.78^2+3.04^2=400.49=(20.01)^2` .

Thus the dimensions are approximately 19.78in by 3.04in.

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