Question

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

Answer 1

Hint: Make use of the formula of volume of a cuboid and solve this.

Complete step-by-step answer:
Let us first construct a figure with the data given
 

Let us consider the length of the box to be= x metres and
The Breadth of the tank be= y metres
The height of the tank is given as =h metres
Volume of the tank is also given as 8cubic metres
The formula for the volume of the tank=$l\times b\times h$
$$\begin{gathered} \Rightarrow 8=2\times x\times y \\ \Rightarrow 4=xy \\ \Rightarrow y={\displaystyle \frac{4}{x}}--------(i) \end{gathered}$$
Given: Building a tank costs Rs.70 per sq.meter for base
Area of base=$l\times b$
Area of base=xy
Cost of base=70(xy)
Also given cost is Rs.45 per square metre
Area of closed sides=2(hl+hb)
                                 =2(2x+2y)
                          =4(x+y)
Cost of making sides =45[4(x+y)]=180(x+y)
Let C be the total cost of tank
C(x)=Cost of Base+Cost of sides
C(x)=70(xy)+180(x+y)
We got the value of $y={\displaystyle \frac{4}{x}}$ from eq (i)
Let’s substitute the value here, so we get
C(x)=70(4)+$180\left(x+{\displaystyle \frac{4}{x}}\right)$
C(x)=280+180$(x+4x^{-1})$
We need to minimise the cost of the tank
So, let's find out the minimum of this
So, we will differentiate C(x) with respect to x
So, we get $C^{\prime }(x)={\displaystyle \frac{d(280+180(x+4x^{-1}))}{dx}}$
$$\begin{gathered} C^{\prime }(x)=0+180(1+(-1)4x^{-1-1} \\ {C}^{\prime }(x)=180(1-4x^{-2}) \\ {C}^{\prime }(x)=180(1-{\displaystyle \frac{4}{x^2}}) \end{gathered}$$
Putting C’(x)=0,
(x-2)(x+2)=0
So, x=2 or x=-2
Since length cannot be negative , the value of x=2
Finding $C^{{}^{″}}(x)$ ,
Differentiating $C^{\prime }(x)$ with respect to x, we get
$C^{{}^{″}}(x)={\displaystyle \frac{1440}{x^3}}$
From this, we get $x=2$ is a point of minima
Thus, the least cost of construction
$C(x)=280+180\left(2+{\displaystyle \frac{4}{2}}\right)$
From this we get $280+720=1000$
So, from this we get the least cost of construction to be equal to Rs.1000.

Note: Whenever we have to find the minima, we have to find out the second order derivative and then find out the least cost. One should be careful while differentiating.