Question

A rectangular garden is to be constructed using a rock wall as one side and fencing for the other three sides. There are 28 yards of fencing available. Determine the dimensions that would create the garden of maximum area. What is the maximum area? Enter only the maximum area. Do not include units in your answer.

Answer 1

To maximize the area of a rectangular garden with one side already bordered by a rock wall and 28 yards of fencing for the other three sides, we use the principle that the optimal rectangle resembles a semi-square with the length double the width. This leads to dimensions of 14 yards by 7 yards, resulting in a maximum area of 98 square yards.

Step-by-step explanation:

To find the dimensions that would create the maximum area for a rectangular garden with one side against a rock wall and 28 yards of fencing for the other three sides, we can use calculus or understand that the maximum area for a given perimeter is a square. However, as one side of the rectangle is the rock wall, this forms a semi-square, or when the length is double the width. If we call the width of the rectangle w and the length l, we know that l + 2w = 28 yards (since only three sides need fencing).

To maximize the area, we set l = 2w, and

so we have 2w + 2w = 28,

leading us to w = 7 yards.

Hence, l = 14 yards.

The maximum area A is l × w = 14 yards × 7 yards = 98 square yards.

Therefore, the maximum area for the given amount of fencing is 98 square yards.

Answer 2

Maximum area = 98

Step-by-step explanation:

Fencing available = 28 yards

Let l be the length and w be the width of rectangular garden,

We have fencing in 3 sides 3 sides

That is

Fencing needed = 2l + w = 28

w = 28 - 2l

Area of rectangle = l x w = l x (28 -2l) = 28l-2l²

For maximum area we have,

`(dA)/(dl)=0\\\\(d)/(dl)\left ( 28l-2l^2\right )=0\\\\28-4l=0\\\\l=7yards`

We have

2l + w = 28

2 x 7 + w = 28

w = 14 yards

Maximum area = 7 x 14 = 98 yard²

Maximum area = 98