Question

Shenelle has 100 meters of fencing to build a rectangular

garden.
The garden's area (in square meters) as a function of the
garden's width x (in meters) is modeled by:
A(x) = − (x − 25)² + 625
-
What side width will produce the maximum garden
area?
meters

Answer 1

The area of the rectangular garden is given by the function:

A(x) = −(x − 25)² + 625

where x is the width of the garden in meters. We want to find the width that produces the maximum area, so we need to find the maximum value of the function A(x).

To do this, we can take the derivative of A(x) with respect to x and set it equal to zero:

A'(x) = −2(x − 25) = 0

Solving for x, we get:

x = 25

This means that the maximum area is obtained when the width of the garden is 25 meters. We can verify that this is indeed a maximum by checking the second derivative of A(x):

A''(x) = −2

Since A''(25) is negative, we know that the function A(x) has a maximum at x = 25.

Therefore, the side width that produces the maximum garden area is 25 meters