A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8m³. If building of tank costs ₹ 70 per sq. meters for the base and ₹ 45 per square metres for sides. What is the cost of least expensive tank?

Solution:

Let l , b and h represent the length, breadth, and height of the tank respectively

Then, we have height, h = 2m and volume of the tank, V = 8m³

Volume of the tank

V = lbh

⇒ 8 = l x b x 2

⇒ lb = 4

⇒ b = 4 / l

Now, area of the base, lb = 4

Area of the 4 walls,

A = 2h (l + b)

= 4 (l + 4/l)

Hence,

dA/dl = 4 (l + 4 / l²)

Now,

dA / dl = 0

⇒ 4 (l + 4 / l²) = 0

⇒ l² = 4

⇒ l = ± 2

However, the length cannot be negative.

Therefore, we have l = 4

Hence,

b = 4 / l

= 4/2 = 2

Now,

⇒ d²A/dl² = 32/l³

When, l = 2

Then,

⇒ d²A/dl² = 32/8

= 4 > 0

Thus, by second derivative test, the area is the minimum when l = 2

We have l = b = h = 2

Therefore,

Cost of building the base in ₹ is

70 x (lb) = 70(4)

= 280

Cost of building the walls in ₹ is

2h (l + b) x 45 = 2 x 2 (2 + 2) x 45

= 720

Required total cost is ₹ is

280 + 720 = 1000

Thus, the total cost of the tank will be ₹ 1000

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 9

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 8m³. If building of tank costs ₹ 70 per sq. meters for the base and ₹ 45 per square metres for sides. What is the cost of least expensive tank?

Summary:

Given that a  tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8m³. Hnece the cost of least expensive tank will be ₹ 1000