Question

Lori wants to build a rectangular garden. She plans to use a side of a river for oneside of the garden, so she will not place fencing along this side of the garden. Shehas 36 yards of fencing material.What is the maximum area in square yards that will be enclosed?

Answer 1

Maximum Area that will be enclosed = 162 square yards

Step-by-step explanation

Let the side of the rectangular garden along the the side of the river have a width of y yards.

Then let the other side of the rectangle be x yards.

Perimeter of this rectangular garden will be the total sum of fencing material available.

x + x + y = 36

2x + y = 36

Then, the area of a rectangle is given as

Area = Length × Width

Area = x × y

Area = xy

We can substitute for y using the equation of the perimeter.

2x + y = 36

y = 36 - 2x

A = xy

A = x (36 - 2x)

A = (36x - 2x²)

We are then asked to find the maximum area of this rectangular garden.

Since we have managed to express the Area of the garden as a function of one of the sides of the rectangle, we will find the maximum like we would for any function.

At maximum point, the first derivative of any function is 0, that is,

(dA/dx) = 0

A = 36x - 2x²

(dA/dx) = 36 - 4x = 0

4x = 36

Divide both sides by 4

(4x/4) = (36/4)

x = 9 yards

This value corresponds to the length of the rectangle that will give the maximum area of the rectangular garden.

y = 36 - 2x = 36 - 2 (9) = 36 - 18 = 18 yards

Maximum Area = Length × Width

Maximum Area = 9 × 18

Maximum Area = 162 square yards

Hope this Helps!!!