Question

An enclosed rectangular box of height h with square base of side b has volume of 4 cubic meters. The two square bases are made of material costing $40 per square meter. The remaining sides cost $20 per square meter. Which values of b and h minimize the cost of the box? Round your answer to three decimal places, if necessary. Enter your answer for h here.

Answer 1

To minimize the cost of the box, we need to determine the values of b and h that minimize the surface area of the box. The surface area of the box is given by Surface Area = 2b² + 4bh. To minimize the surface area, we can take the derivative of the surface area equation with respect to h and set it equal to zero. Solving for h, we find that h = sqrt(2b).

Step-by-step explanation:

The surface area of the box is given by:

Surface Area = 2b² + 4bh

Since we know that the volume of the box is 4 cubic meters, we can use this information to eliminate one variable. The volume of the box is given by:

Volume = b²h = 4

Substituting this value of b²h into the surface area equation, we get:

Surface Area = 2(4/h) + 4bh

To minimize the surface area, we can take the derivative of the surface area equation with respect to h and set it equal to zero:

d(Surface Area)/dh = -8/h² + 4b

= 0

Simplifying this equation, we get:

4b = 8/h²

Solving for h, we find that:

h = sqrt(2b)

Therefore, to minimize the cost of the box, the value of h should be equal to the square root of 2 times the value of b.