Question

Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs $1/ft2; the material for the base costs $2/ft2. Find the dimensions of the box of greatest possible volume if you are allowed to spend $144 for the material to make it. List largest to smallest.

Answer 1

Answer:

height = 14.5 ft

width = length = 3.2 ft

Step-by-step explanation:

Let the side of the square base is a and the height of the box is h.

The material required to make the box is equal to the total surface area of the box.

Area of base = a²

Cost of square base is $ 2 per ft²

So, the cost of making base = 2a²

Area of four walls + area of top = 2 (a² + ah + ah) + a² =  3a² + 2ah

Cost of making walls and the top is $ 1 per ft²

So, the cost of making walls and the top = 1 x ( 3a² + 2ah)

Total cost of making the box = 2a² + 3a² + 2ah = 5a² + 2ah

According to the question

144 = 5a² + 2ah

$h = \frac{144-5a^{2}}{2a}$     .... (1)

The volume of the box is V = length x width x height

V = a x a x h

V = 72 a - 2.5 a³     from equation (1)

Differentiate both sides with respect to a.

dV/da = 72 - 7.5 a²

Put, it equal to zero for maxima and minima

7.5 a² = 72

a = 3.2 ft

Put in equation (1)

$h = \frac{144-5\times 3.2\times 3.2}{2\times 3.2}$

h = 14.5 ft

So, the dimensions of the box are

height = 14.5 ft

width = length = 3.2 ft