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 m 3 . If building of tank costs Rs 70 per sq meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
Consider x and y as the length and breadth of the base of the tank. Height of the tank is given as 2 m and volume of the tank is given as 8 m 3 .
The formula for the volume of tank is length × breadth × height .
Write the equation for the volume of the tank.
x ⋅ y ⋅ 2 = 8 x y = 4 y = 4 x
The cost of building base of the tank at the rate of 70 per sq. meter is 70 x y .
Write the equation for the cost of building of four walls of the tank at a rate of 45 per sq. meter.
45 ( x ⋅ 2 + x ⋅ 2 + y ⋅ 2 + y ⋅ 2 ) = 180 x + 180 y
Consider z as the total cost of building the tank.
Write the equation for the total cost of building the tank.
z = 70 x y + 180 x + 180 y
Differentiate both sides of the equation,
d z d x = 0 + 180 − 720 x 2 d 2 z d x 2 = 1440 x 3
Now,
d z d x = 0 180 − 720 x 2 = 0 180 x 2 = 720 x = ± 2
Length cannot be negative.
At x = 2 ,
d 2 z d x 2 = 1440 8 = 180 > 0
Thus, z is minimum at x = 2 .
Write the equation for the minimum cost.
280 + 180 × 2 + 720 2 = 280 + 360 + 360 = 1000
Thus, the minimum cost is 1000 .
Consider x and y as the length and breadth of the base of the tank. Height of the tank is given as
The formula for the volume of tank is
Write the equation for the volume of the tank.
The cost of building base of the tank at the rate of
Write the equation for the cost of building of four walls of the tank at a rate of
Consider
Write the equation for the total cost of building the tank.
Differentiate both sides of the equation,
Now,
Length cannot be negative.
At
Thus,
Write the equation for the minimum cost.
Thus, the minimum cost is
