Question

Martha is fencing off a small herb garden in her backyard. If she has 20 feet of fencing and wants a rectangular garden, what length and width will give her the largest area? Draw the graph of the function A(w) for area in terms of width and determine the maximum.

Answer 1

Options for the fencing
1, 9
2, 8
3, 7
4, 6

the one with the highest area is the 4, 6

Answer 2

A = lw

P = 2l +2w

20 = 2l + 2w.

We express the length using (1).

2l = 20 - 2w

l = 10 - w.

We write the area using the width and length expression using width.

A = w(10 - w).

A = 10w - w2 .

The relationship of the area is quadratic. It has a max value that is where the vertex of the parabola is. The vertex is the average of the two zeros of the equation. We have

w1 = 0

and

w2 = 10.

1/2 (0 + 10) = 5 feet.

Martha will have largest area garden if (l = w = 5 feet).

A = 5x5 = 25 ft2

Note, if l = 4 and w =6,

A = 4x6 =24 ft2.