Question

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container. Be sure to justify that the cost you find is the cheapest cost for this container

Answer 1

Answer:163.54$

Step-by-step explanation:

Given data

Volume of Storage(V)=${10m^3}$

Length=2breadth

Let Length be L,Breadth be & height be H

therefore

10=LBH

Now substitutes the values

10=2${B^2}H$

5=${B^2}H$

Now cost for base is ${C_1}=2{B^2}\times10$

Cost for side walls is${C_2}={2LH}\times6+2BH}\times6$

Now total cost(C)=${C_1}+{C_2}$

C=20${B^2}H$+${2LH}\times6$+$2BH}\times6$

C=20${B^2}H$+24BH+$12BH$

C=$20{B^2}+36B\times\frac{5}{B^{2}}$

Now Differentiating With respect to Breadth to get minimum cost

$\frac{\mathrm{d} C}{\mathrm{d} B}=0$

$we\ get\ B=\sqrt[3]{4.5}=1.65m$

$L=3.30m$

$H=1.836m$

and mimimum cost C

${C=163.54\ }$