Question

A farmer needs to enclose three sides of a garden with a fence (the fourth side is a river). The farmer has

30 feet of fence and wants the garden to have an area of 108 sq-feet. What should the dimensions of the
garden be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides);
the length with be the longer dimežision (needing one side). Additionally, the length should be as long as
possible.)

Answer 1

length×width = 108

length + 2×width = 30

length = 30 - 2×width

using that in the first equation

(30 - 2×width)×width = 108

-2×width² + 30×width - 108 = 0

the general solution to such a quadratic equation like

ax² + bx + c = 0

is

x = (-b ± sqrt(b² - 4ac))/(2a)

so,

width = (-30 ± sqrt(30² - 4×-2×-108))/(2×-2) =

= (-30 ± sqrt(900 - 864))/-4 =

= (-30 ± sqrt(36))/-4 = (-30 ± 6)/-4

width1 = (-30 + 6)/-4 = -24/-4 = 6 ft

width2 = (-30 - 6)/-4 = -36/-4 = 9 ft

for width = 6 ft

length = 30 - 2×width = 30 - 12 = 18 ft.

for width = 9 ft

length = 30 - 2×width = 30 - 18 = 12 ft

given the request that the length sound be as long as possible,

width = 6 ft

length = 18 ft

is our solution.