Question

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2m$ and volume is $8m^3$.If building of tank costs Rs 70per sq meters for the base and Rs 45per square metre for sides.What is the cost of least expensive tank?

Answer 1

The correct answer is $Rs\,1000$
Let $l, b,$ and $h$ represent the length, breadth, and height of the tank respectively.
Then, we have height $(h) = 2 m$
Volume of the tank$=8m^3$
Volume of the tank$=l\times b\times h$
$∴8=l\times b\times 2$
$⇒lb=4$
$⇒b=\frac{l}{4}$
Now,area of the base$=lb=4$
Area of the 4 walls$(A)=2h(l+b)$
$∴A=4(l+\frac{4}{l})$
$⇒\frac{dA}{dl}=4(1-\frac{4}{l^2})$
Now,$\frac{dA}{dl}=0$
$⇒1-\frac{4}{l^2}=0$
$⇒l^2=4$
$⇒l=±2$
However, the length cannot be negative.
Therefore, we have $l=4.$
$∴b=\frac{4}{l}=\frac{4}{2}=2$
Now,$\frac{d^2A}{dl^2}=\frac{32}{l^3}$
When $l=2,\frac{d^2A}{dl^2}=\frac{32}{8}=4>0.$
Thus, by second derivative test, the area is the minimum when $l=2.$
We have $l=b=h=2.$
∴Cost of building the base$=Rs70\times (lb)=Rs70(4)=Rs280$
Cost of building the walls$=Rs\,2h(l+b)\times 45=Rs\,90(2)(2+2)=Rs8(90)=Rs720$
Required total cost$=Rs(280+720)=Rs\,1000$
Hence, the total cost of the tank will be $Rs1000.$