Question

A rectangular box is to have a square base and a volume of 12 ft3. If the material for the base costs $0.17/ft2, the material for the sides costs $0.10/ft2, and the material for the top costs $0.13/ft2, determine the dimensions (in ft) of the box that can be constructed at minimum cost.
a.length ft

b.width ft

c.height ft

Answer 1

Answer:

(a)Length =2 feet

(b)Width =2 feet

(c)Height=3 feet

Step-by-step explanation:

Let the dimensions of the box be x, y and z

The rectangular box has a square base.

Therefore, Volume of the box$V=x^2z$

Volume of the box$=12 ft^3$

$Therefore, x^2z=12\\z=\frac{12}{x^2}$

The material for the base costs $\ 0.17/ft^2$, the material for the sides costs $\ 0.10/ft^2$, and the material for the top costs $\ 0.13/ft^2$.

Area of the base $=x^2$

Cost of the Base $=\ 0.17x^2$

Area of the sides $=4xz$

Cost of the sides=$=\ 0.10(4xz)$

Area of the Top $=x^2$

Cost of the Base $=\ 0.13x^2$

Total Cost, $C(x,z) =0.17x^2+0.13x^2+0.10(4xz)$

Substituting $z=\frac{12}{x^2}$

$C(x) =0.17x^2+0.13x^2+0.10(4x)(\frac{12}{x^2})\\C(x)=0.3x^2+\frac{4.8}{x} \\C(x)=\dfrac{0.3x^3+4.8}{x}$

To minimize C(x), we solve for the derivative and obtain its critical point

$C'(x)=\dfrac{0.6x^3-4.8}{x^2}\\Setting \:C'(x)=0\\0.6x^3-4.8=0\\0.6x^3=4.8\\x^3=4.8\div 0.6\\x^3=8\\x=\sqrt[3]{8}=2$

Recall: $z=\frac{12}{x^2}=\frac{12}{2^2}=3$

Therefore, the dimensions that minimizes the cost of the box are:

(a)Length =2 feet

(b)Width =2 feet

(c)Height=3 feet