I cant figure out how to do this, please help.

Question

A rectangular box with volume 469 cubic feet is built with a square base and top. The cost is $1.50 per square foot for the top and the bottom and $2.00 per square foot for the sides. Let x represent the length of a side of the base. Express the cost as a function of x.

Answer 1

^Kirsten,

 

If you draw a picture for this question and label the base with x, you can see that the area of the base is x². If the base costs $1.50 per square foot to make, then the cost of the base is 1.50x². This is because the total cost is the cost per square foot times the area in square feet. The cost for the top will also be the same, 1.50x².

 

As for the four sides, let the height of them be h. The area of one side (remember that the box has a square base), must be hx (height times width). The total area for all four sides is 4hx. The cost of all four sides will then equal 2.00·4hx = 8hx.

 

The total cost to make the box will then equal 3x² + 8hx.

 

Where does the volume of 469 ft³ come into play here? The volume of the box is hx² (length times width times height). If you set this equal to 469, then solve for h, you will get

 

h = 469/x².

 

You can now plug this into the total cost formula to write the answer in terms of x only:

 

total cost = C(x) = 3x² + 8(469/x²)x = 3x² + 3752/x .

 

This should be the correct answer. Please reply back if you have any more questions.