Optimization Calculus Problem (Please help)

Question

A landscape architect is designing an 5130 square foot, rectangular garden. The perimeter of the garden is to be lined with shrubbery on three sides, which costs $48 per foot, whereas the fourth side of the garden is to be enclosed with fencing, at a cost of $47 per foot. Determine the dimensions of such a garden with a minimal cost. What is the minimum cost?

x = ______

y = ______

Minimum Cost = $ ______

Answer 1

A = hw where width is the side with fencing

Cost (C) = (2h+w)48 + w(47)

Substitute in h = A/w

C = (2A/w + w)48 + 47w = 96A/w + 95w

Set dC/dw = 0 -96A/w² + 95 = 0 , so w² = (96/95)(5130) w = 72

I'll leave the rest to you. Note that 2nd derivative is 192A/w³ .> 0, so it is a minimum.