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

Question

Philoid · Accepted Answer

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

then, we have h = 2m

Volume of the tank = 8 m³

Volume of the tank = l × b × h

⇒ 8 = l × b × 2

⇒ lb = 4

⇒ b =

Now, area of the base = lb = 4

Area of the 4 walls (A) = 2h(l + b)

Now,

⇒ = 0

⇒ l² = 4

⇒ l = ±2

Since, length cannot be negative therefore l =2.

⇒ b = 2

Now,

When l =2,

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

We have, l =b=h=2

Therefore, 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.

Required total cost = Rs(280 + 720) = Rs 1000.

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

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