Question

A closed box with a square base is to have a volume of 256 comma 000 cm cubed. The material for the top and bottom of the box costs ​$10.00 per square​ centimeter, while the material for the sides costs ​$2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total​ cost? Write an equation for​ C(x), the cost of the box as a function of​ x, the length of a side of the base.

Answer 1

Explanation:

Let the side of the square base be a cm and height be h cm.

volume = a²h = 256000

area of base + lid = 2a²

area of side wall = 4ah

Total cost C = 10 x 2a² + 2.5 x 4ah

C = 20 a² + 10 ah

= 20 a² + 10 a x
`(256000)/(a^2)`

= 20 a² +
`(2560000)/(a)`

differentiating

`(dC)/(da) = 40a -(2560000)/(a^2)`

for minimum cost

`(dC)/(da) = 40a -(2560000)/(a^2) = 0`

a³ = 64000

a = 40 cm

length of side base = 40 cm

height h =

40 x 40 x h = 256000

h = 160 cm

cost = 20 a² + 10 ah

= 20 x 40² + 10 x 40 x 160

= 32000+64000

= 96000 .