Please Help! 2 Optimization Calculus Questions

Question

1. The volume of a storage facility must be 800m3. The shape of this building must be a rectangular prism with length double the width. If the floor costs $50/m2, the walls cost $100/m2, and the roof (which is flat) costs $250/m2, what dimensions will create the lowest total cost?



2. What is the largest area possible for a rectangle contained between the x-axis and the quadratic with y-intercept of (0,10) and x-intercepts of (-5,0) and (5,0)

Answer 1

Part A

The rectangular prism has dimensions w, 2w, and h where volume = length x width x height = (2w)(w)(h) = 2w²h=800m³

We need h in terms of w. h = 800/(2w²) = (400)/(w²)

Floor cost = (50)(2w²) = 100w²

Wall cost = (100)(6wh) = (100)(6w)(400)/(w²) = (240,000)/(w)

Roof cost = (250)(2w²) = (500)(w²)

Cost = C = (100)(w²) + (240,000)/(w) + (500)(w²) = (600)(w²) + (240,000)(w^-1)

Differentiating, C' = (1200)(w) - (240,000)/(w²)

We want to minimize cost so we set C' = 0

(1200)(w) - (240,000)/(w²) = 0

w - (200)/(w²) = 0

w = (200)/(w²)

w³ = 200

w = 5.85 = width

Hence, length = 11.7, and height = (400)/(5.85)² = 11.7

For part B, I'm not entirely sure of what quadratic, but with a y-intercept of 10 and x-intercepts of -5 and 5 it appears to be a parabola. The area of the rectangle is 10x10 = 100. No calculus needed.

Incidentally, the area contained within the parabola bounded by a rectangle is 2/3 the area of the rectangle.