Question

Gerry Anium is designing a rectangular garden. It will sit next to a long, straight rock wall, thus leaving only three sides to be fenced. Gerry has bought 150 feet of fencing in one-foot sections. Subdivision into shorter pieces is not possible. The garden is to be rectangular and the fencing (all of which must be used) will go along three of the sides

(a) If each of the two sides attached to the wall were 40 ft long, what would the length of the third side be?
(b) Let x be the length of the side parallel to the wall. Find the lengths of the other two sides, in terms of x.
(c) Express the area of the garden as a function of x, and graph this function. For what values of x does this graph have meaning?

Answer 1

See below. The garden is 40' x 70.' for 2800 ft^2 total area.

Explanation:

a) Two 40 foot sections would require 80 feet of fencing. Since one of the remaining sides is the rock wall, the fencing left for the fourth side, x, would be: 150' - 2*(40') = 70 feet for the fourth wall, x. x = 70 feet.

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b) The total fencing available is 150 feet. The side opposite the stone wall is length x. (150' - x) represents the remaining fencing available for the two identical sides, which we'll say are y feet in length.

We know that x + 2y = 150'

Rearrange to find y:

x + 2y = 150

2y = 150 - x

y = (150 - x)/2

Since x is 70 feet, each side would be 40 feet.

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c) Area of the garden is width, x, times length. y.

Area = x*y

Area = (70')*(40')

Area = 2800 ft^2

To express the area only as a function of x, we can substitute the definition of y:

Area = x*y

Area = x*((150-x)/2)

Area = (150x - x^2)/2

Check: Area = (150*70 - (70^2))/2 = 2800 ft^2

x has meaning for the range of 0 - 70 feet.