Word Problem

Question

Carla is a carpenter who has been hired to make a closed box with a square base and volume of 250 cubic meters. The material for the top and bottom of the box costs $2 per square meter, and the material for the sides costs $1 per square meter. Can Carla construct the box for less than $300?

Answer 1

Some parts of the box has a price per area.  Since we have a box with a square base, we need to utilize the surface area and the volume.

 

Let x = length = width     ----->  square base

Let y = height

 

Set up two equations using the variables.  One for volume and the other for cost.  The cost is affected by the surface area. If we draw the box and label the sides, we can write these equations.

 

x²y = 250              eq1

 

C = 2(2x²) + 4xy

 

C = 4x² + 4xy                 eq2

 

 

Now the question here is can she construct a box for less than $300?  If the dimensions come out to be positive, then she can.  If  she gets a negative or zero dimension, then she cannot.  We use eq2 to solve this problem.

 

First, we need to get eq2 in terms of one variable.  Let have it in terms of x, since we have more x variables.

 

Substituting eq1 into eq2,

 

C = 4x² + 4x(250 / x²)

 

C = 4x² + 1000/x

 

C = (4x³ + 1000) / x

 

 

Set C=299

 

299 = (4x³ + 1000) / x

 

 

Multiply both sides of the equation by x.

 

299x = 4x³ + 1000

 

 

Subtract 299x on both sides of the equation.

 

0 = 4x³ - 299x + 1000

 

 

Solve for x from the bolded equation.  If the value of x is positive, then she can construct a box.