Question

You are tasked with constructing a rectangular box with a square base, an open top, and a volume of 184 in3. Determine what the dimensions of the box should be to minimize the surface area of the box. What is the minimum surface area? Keep your answer in radical form and omit units.

Answer 1

Answer:

7.17 by 7.17 by 3.58

Step-by-step explanation:

Since the rectangular box is square based, the total surface area is expressed as;

S = x²+2xh+2xh (the top is opened)

S = x² + 4xh ..... 1

x is the side length of the square

h is the height of the box

Also the volume V = x²h

Given V = 184

184 = x²h

h = 184/x² .... 2

Substitute 2 into 1;

S = x² + 4xh

S = x² + 4x(184/x²)

S = x² + 4(184/x)

S = x² + 736/x

Since we need a dimension that will minimize the surface area of the box then, dS/dx = 0

dS/dx = 2x -736/x²

0 = 2x -736/x²

multiply through by x²

0 = 2x³- 736

2x³ = 736

x³ = 736/2

x³ = 368

x = ∛368

x= 7.17 in

Since V = x²h

184 = 7.17²h

h = 184/7.17²

h = 184/51.4089

h = 3.58 in

Hence the dimensions of the box should be to minimize the surface area of the box are 7.17 by 7.17 by 3.58