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# Solution for A Tank with Rectangular Base and Rectangular Sides, Open at the Top is to the Constructed So that Its Depth is 2 M and Volume is 8 M3. If Building of Tank Cost 70 - CBSE (Science) Class 12 - Mathematics

#### Question

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

#### Solution

Let lb and h be the length, breadth and height of the tank, respectively.

Height, h = 2 m

Volume of the tank = 8 m3

Volume of the tank = l × b × h

∴  × b × 2 = 8

$\Rightarrow lb = 4$

$\Rightarrow b = \frac{4}{l}$

Area of the base = lb = 4 m2

Area of the 4 walls, A= 2h (l + b)

$\therefore A = 4\left( l + \frac{4}{l} \right)$

$\Rightarrow \frac{dA}{dl} = 4\left( 1 - \frac{4}{l^2} \right)$

$\text { For maximum or minimum values of A, we must have }$

$\frac{dA}{dl} = 0$

$\Rightarrow 4\left( 1 - \frac{4}{l^2} \right) = 0$

$\Rightarrow l = \pm 2$

However, the length cannot be negative.

Thus,
l = 2 m

$\therefore b = \frac{4}{2} = 2 m$

$\text { Now,}$

$\frac{d^2 A}{d l^2} = \frac{32}{l^3}$

$\text { At }l = 2:$

$\frac{d^2 A}{d l^2} = \frac{32}{8} = 4 > 0$

Thus, the area is the minimum when l = 2 m

We have
l = b = h = 2 m

∴ Cost of building the base = Rs 70 × (lb) = Rs 70 × 4 = Rs 280

Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90 (2) (2 + 2)= Rs 8 (90) = Rs 720

Total cost = Rs (280 + 720) = Rs 1000

Hence, the total cost of the tank will be Rs 1000.

Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.

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Solution A Tank with Rectangular Base and Rectangular Sides, Open at the Top is to the Constructed So that Its Depth is 2 M and Volume is 8 M3. If Building of Tank Cost 70 Concept: Graph of Maxima and Minima.
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