Answers To Your Questions About Square Foot Gardening Plants
Charles asks…
math optimization question?
4.) Rancher Roy wants to make a rectangular garden that he will plant during November with winter vegetables. The outer fencing (the fencing around the garden) will cost $45 per foot. He also wants to use interior fencing to make 8 rectangular subplots. The interior fencing will cost $16 per foot. Rancher Roy wants his garden to have an area of 850 square feet. What are the dimensions of the garden that will have the minimum fencing cost? What is the minimum of the fencing cost?
Green Thumb answers:
Rectangular garden of 850 square feet:
XY=850
outer fencing length = 2X + 2Y
outer fencing cost = 45 ( 2X + 2Y)
8 rectangular subplots (assuming all 8 subplots are of the same size and shape and fencing can be purchased by any length): two possible situations, either 1x 8 or 2×4
Situation 1×8:
interior fencing length = 7Y,
interior fencing cost = 16(7Y)
total fencing cost f(X)
= 45 ( 2X + 2Y) + 16(7Y)
= 90X + 202Y
= 90X + 202( 850 / X )
= 90X + 171700 / X
to find minimum, f’(X) = 0 = 90 – 171700 / X^2
therefore, X=sqrt(171700 / 90) = 43.678, Y=19.461
total fencing cost f(X) = 7862.06
Situation 2×4:
interior fencing length = X + 3Y
interior fencing cost = 16 (X+3Y)
total fencing cost f(X)
= 45 ( 2X + 2Y) + 16(X+3Y)
= 106X + 138Y
= 106X + 138( 850 / X )
= 106X + 117300 / X
to find minimum, f’(X) = 0 = 106 – 117300 / X^2
therefore, X=sqrt(117300 / 106) = 33.265, Y= 25.552
total fencing cost f(X) = 7052.32
Situation 2 has the minimum fencing cost of $7052.32, the dimensions of the garden is 33.265 ft x 25.552 ft
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