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 \(Rs1000.\)