Question

A carpenter wants to build a rectangular box with square sides in which to put round things. the material for the bottom costs $20/ft^2, material for the sides costs $10/ft^2 and the material for the top costs $50/ft^2

If the volume of the box must be 5 ft^3, then find the dimensions that will minimize the cost (and find the minimum cost).

Answer 1

Answer:
`1.627* 1.627* 1.88\ ft^3`

Explanation:

Given

Suppose side face have a dimension of l\times l

and width of h

volume
`V=l^2\cdot h`

`h=(V)/(l^2)`

volume
`V=5 ft^3`

Area of side wall is
`A_s=l^2`

Area of top Wall
`A_t=l* h`

Area of bottom
`A_b=l* h`

Cost of bottom wall
`c_b=20* l* h=20lh`

Cost of top wall
`c_t=50* l* h=50lh`

Cost of side walls
`c_s=4* l^2* 10=40l^2`

total cost
`C=c_s+c_t+c_b=20lh+50lh+40l^2`

`C=70lh+40l^2`

`C=70* l* (5)/(l^2)+40l^2`

differentiate C w.r.t l to get minima or maxima

`\frac{\mathrm{d} C}{\mathrm{d} l}=0`

`-(350)/(l)+80l^2=0`

`l^3=(350)/(80)`

`l=1.627 ft`

`h=(5)/(2.648)`

`h=1.88 ft`

Dimension of Box is
`1.627* 1.627* 1.88\ ft^3`