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Question

A rectangular garden must have a perimeter of 155 feet and an area of at least 1400 square feet. Describe the possible lengths of the garden. Round your answers to the nearest foot. The approximate length of the garden is at least __ feet and at most __ feet.

Answer 1

L+W=155/2=77.5

L=77.5-W

LW>1400

(77,5-W)W > 1400

it's a quadratic inequality, use the quadratic formula

L and W both

greater than about 28.6, but less than 48.8

28.6<W, L < 48.8

largest area would be with L=W= 38.75 feet

Answer 2

The problem can be expressed with 2 equations:

a+b = 155 and ab >= 1400

Which leads to the following quadratic inequality substituting b in the second equation by 155-a:

a^2 - 155 a + 1400 <= 0

there are 2 solutions for a^2 - 155a + 1400 = 0

2*a1 = 155 - sqrt(155^2-1400*4) - a1~9.63 rounded to 10

2*a2 = 155 + sqrt(155^2-1400*4) - a2 ~145.36 rounded to 145

For (a - 10)(a-145) to be negative, the garden must be at least 10 ft and at most 145 ft