Question

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 cost Rs 70 per square meter for the base and Rs 45 per square meter for sides, what is the cost of least expensive tank?

Answer 1

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

Also, 

Assume volume of tank as V 

h = 2 m (given) 

V = 8 cm³ 

⇒ lbh = 8 

⇒ 2lb = 8 (given) 

⇒ lb =4 

⇒ b = \(\frac{4}{1}\)…(1) 

Cost for building base = Rs 70/m² 

Cost for building sides = Rs 45/m² 

Cost for building the tank, C = Cost for base + cost for sides 

⇒ C = lb × 70 + 2(l + b) h × 45 

⇒ C = l × \(\frac{4}{I}\) × 70 + 2(l + \(\frac{4}{I}\)) × 2 × 45 

⇒ C = 280 + 180(l + \(\frac{4}{I}\)) …(2) 

Condition for maxima and minima,

⇒ \(\frac{dC}{dI}\) = 0

⇒ 180(1 - \(\frac{4}{I^2}\)) = 0

⇒ \(\frac{4}{I^2}\) = 1

⇒ l² = 4

⇒ l = ±2 cm

Since, 

l cannot be negative 

So, 

l = 2 cm

\(\frac{d^2c}{dI^2}\) = 180\((\frac{8}{I^3})\)

For l = 2\(\frac{d^2c}{dI^2}\) = 180

⇒ \(\frac{d^2c}{dI^2}\) > 0

Therefore,

Cost will be minimum for l = 2

From equation 2,

C = 280 + 180(l +\(\frac{4}{I}\))

⇒ C = Rs 1000