Question

A firm receives an order for a square-base rectangular storage container with a lid. The container has a volume of 20 cubic meters. Material for the base costs 20 dollar per square meter. Material for the sides and the lid costs 10 dollars per square meter. What is the lowest cost of materials for making such a container?

Answer 1

Answer:$ 506.05

Explanation:

Given

volume of container $=20 m^3$

Let  L be the length of square-base and h be the height of Rectangular box

Cost of base$=20 \ /m^2$

Cost of side and lid$=10 \ /m^2$

Cost of base $c_1=L^2\times 20$

$h=\frac{20}{L^2}$

cost of lid and side $c_2=10\times 4L\cdot h+10\times L^2$

Total cost $C=c_1+c_2$

$C=20L^2+10L^2+40L\cdot h$

$C=30L^2+\frac{800}{L}$

differentiate C w.r.t to L to get minimum cost

$\frac{\mathrm{d} C}{\mathrm{d} L}=60L-\frac{800}{L^2}=0$

$L^3=\frac{80}{6}$

$L=2.37 m$

thus $h=\frac{20}{5.613}=3.56 m$

Thus Lowest cost is $C=30\times 5.617+\frac{800}{2.37}= 506.05$