A backyard gardener wants to enclose a rectangular area for a new garden.He will use a portion
of the backyard fence as the fourth side and has acquired 100 feet of wire fencing to enclose
the other three sides.
1.If the sides of the fence perpendicular to the current fence have length L, provide a formula for
the area enclosed by the fence.
2.What size should the garden be to make the most of the contained space?
Solution:
a) L + W + L, or simply 2L + W, equals 100 because we are aware that the fence available is only
100 feet long. As a result, we are able to express the width, W, in terms of L.
W = 100 - 2L
We now have to create an equation for the area that the fence encloses. The area of a rectangle
is equal to its length multiplied by its width:
A = L(W)
A = L(100 - 2L)
A(L) = 100L - 2L²
The area of the fence is represented by this formula in terms of the length L. The function is to
be written in general form in order to accurately solve for the other equations we may encounter:
A(L) = -2L² + 100L
b) -The leading coefficient of the quadratic equation is negative, which means the graph will
open downward, with the vertex representing the area's maximum value. Because the equation
is not represented in the standard polynomial form with decreasing powers, we must be careful
when determining the vertex. For this reason, we modified the equation above in general form.
-Determine the values for a, b, and c:
