Answer
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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=$l \times b \times h$
$
\Rightarrow 8 = 2 \times x \times y \\
\Rightarrow 4 = xy \\
\Rightarrow y = \dfrac{4}{x} - - - - - - - - (i) \\
$
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 $y = \dfrac{4}{x}$ from eq (i)
Let’s substitute the value here, so we get
C(x)=70(4)+$180\left( {x + \dfrac{4}{x}} \right)$
C(x)=280+180$(x + 4{x^{ - 1}})$
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 $C'(x) = \dfrac{{d(280 + 180(x + 4{x^{ - 1}}))}}{{dx}}$
\[\
C'(x) = 0 + 180(1 + ( - 1)4{x^{ - 1 - 1}} \\
C'(x) = 180(1 - 4{x^{ - 2}}) \\
C'(x) = 180(1 - \dfrac{4}{{{x^2}}}) \\
\]
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 $C'(x)$ with respect to x, we get
${C^{''}}(x) = \dfrac{{1440}}{{{x^3}}}$
From this, we get $x=2$ is a point of minima
Thus, the least cost of construction
$C(x)=280+180\left( {2 + \dfrac{4}{2}} \right)$
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.
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=$l \times b \times h$
$
\Rightarrow 8 = 2 \times x \times y \\
\Rightarrow 4 = xy \\
\Rightarrow y = \dfrac{4}{x} - - - - - - - - (i) \\
$
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 $y = \dfrac{4}{x}$ from eq (i)
Let’s substitute the value here, so we get
C(x)=70(4)+$180\left( {x + \dfrac{4}{x}} \right)$
C(x)=280+180$(x + 4{x^{ - 1}})$
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 $C'(x) = \dfrac{{d(280 + 180(x + 4{x^{ - 1}}))}}{{dx}}$
\[\
C'(x) = 0 + 180(1 + ( - 1)4{x^{ - 1 - 1}} \\
C'(x) = 180(1 - 4{x^{ - 2}}) \\
C'(x) = 180(1 - \dfrac{4}{{{x^2}}}) \\
\]
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 $C'(x)$ with respect to x, we get
${C^{''}}(x) = \dfrac{{1440}}{{{x^3}}}$
From this, we get $x=2$ is a point of minima
Thus, the least cost of construction
$C(x)=280+180\left( {2 + \dfrac{4}{2}} \right)$
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.
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