Question

A company is constructing an​ open-top, square-based, rectangular metal tank that will have a volume of 49 cubic feet. What dimensions yield the minimum surface​ area? Round to the nearest tenth.

Answer 1

Answer:

b = 4.6 ft

h = 2.3 ft

Step-by-step explanation:

The volume of the tank is given by:

$b^2*h=49$

Where 'b' is the length of the each side of the square base, and 'h' is the height of the tank.

The surface area can be written as:

$A=b^2+4bh\\A=b^2+4b*({\frac{49}{b^2}})\\A=b^2+\frac{196}{b}$

The value of b for which the derivate of the expression above is zero is the value that yields the minimum surface area:

$\frac{dA}{db} =0=2b-\frac{196}{b^2}\\2b^3=196\\b=4.61\ ft$

The value of h is then:

$h=\frac{49}{4.61^2}\\h=2.31\ ft$

Rounded to the nearest tenth, the dimensions are b = 4.6 ft and h = 2.3 ft.