# Math Help - Finding area of a rectangle as a function of x when...

1. ## Finding area of a rectangle as a function of x when...

A rectangle has a perimeter of 100 inches and one side has length x. Then, use the function found in the first part to find the dimension of the rectangle with perimeter of 100 inces and the largest possible area.

2. Hi

The other side of the rectangle is 50-x
Its area is therefore x(50-x)
The largest area is obtained when x=25 (the rectangle is a square)

3. Originally Posted by Incursion
A rectangle has a perimeter of 100 inches and one side has length x. Then, use the function found in the first part to find the dimension of the rectangle with perimeter of 100 inces and the largest possible area.
Hi Incursion,

if the perimeter is 100, then half the perimeter is 50,
which is x + other side "y", so x+y is 50, hence y=50-x

Thus the rectangle area is xy=x(50-x) square units in terms of x.

To find the maximum area of this rectangle,
(since the minimum area will be zero as the opposite sides eventually touch)
the area function is differentiated and the value of x causing it to be zero found, as the derivative gives the tangent slope,
a tangent of slope zero balances on the maximum point of the area curve,
which is a quadratic.

$x(x-50)=x^2-50x$

$\frac{d}{dx}\left(x^2-50x\right)=2x-50$

This is zero when x=25.
This value of x gives the rectangle of maximum area.

$y=50-x_{max}=50-25=25$

All sides are 25 inches long.