Question

A rectangular storage container with an open top is to have avolume

20m3 . The length of its base is twice the width.The
material for the base costs $5 per square meter. The materialfor
the sides costs $9 per square meter. Find the minimum cost tobuild
such container.

Answer 1

Dimensions of the container:

x = 3 m

y = 6 m

h = 1.1 m

C(min) = 270 $

Explanation:

Volume of storage container

V = 20 m³

Let "y" be the length and "x" the width then y = 2*x

V = x*y*h ⇒ V = 2*x²*h ⇒ 20 = 2*x²+h ⇒ h = 10/ x²

Costs:

Total cost = cost of base ( 5*2*x² ) + cost of side with base x ( 2*9*x*h) +

cost of side witn base y =2x (2*9*2x*h)

C(t) = 10*x² + 18*x*h + 36*x*h

C(x) = 10x² + 54*x*10/x² ⇒ C(x) 10*x² + 540 /x

Taking derivatives on both sides of the equation we get:

C´(x) = 20*x - 540/x²

C´(x) = 0 ⇒ 20*x - 540/x² = 0 ⇒ 2x - 54/x² = 0

2x³ - 54 = 0

x³ = 27 x = 3 m

Then y = 2*x ⇒ y = 2*3 y = 6 and h = 10 / x² h = 1.1 m

And the minimum cost is

C (min) = 10*x² + 540/x ⇒ C (min) = 90 + 180

C(min) = 270 $