A rectangle is inscribed in a semicircle of radius 5 cm.
One side of the rectangle is on the diameter of the semicircle
and two of the vertices of the rectangle are on the semicircle).
1) Draw a model of this situation
The equation of P is: .
* * *
* | * P
* - - - + - - - o
*| | . |*
| | . W|
* | | . | *
-5 : - - - L - - - : 5
The area of the rectangle is: .
If 2) Suppose the side of the rectangle which is on the diameter is 2 cm long.
What will be the area of the rectangle?
Therefore: . cm²
Given , find 3) Create a rule which can be used to find the length of the side of rectangle
which is on the diameter of the semicircle when the area of the rectangle is known.
We have: .
. . Then: .
Square both sides: .
Quadratic Formula: .
And the positive real root is: .
I'd let my calculator graph it . . . 4) Graph the function from #3 and use it
to find possible lengths which will give an area of 24.
Hard? . . . I don't think so! 5) Why would #4 be hard with algebra alone?
With , we have: .