Claudia works as a manufacturer of decorative jewelry boxes. A customer would like her to create some boxes with the following conditions: The sum of the length and width must be 30 centimeters. The height must be 3 centimeters less than the length. The box should have the greatest possible volume.

1) Write a function f (x) that gives the volume of a box of length x in polynomial form.

2) Write an inequality that describes the possible values of x for our box problem. Remember that the y value is the volume, and that none of the dimensions can be 0 or negative.

3) Use your calculator to find the value of x that gives the maximum volume. Use 2nd TRACE (CALC) and choose the maximum command. Round your answer to the nearest tenth.

4) Find the length, width, and height of the box Claudia should build.

Step-by-step explanation:

Presuming represents the length of the box (which you should have specified), your volume function:

is correct.

V (x) = -x^3+33x^2-90x=0

Obviously, in order to have a box, all three dimensions must be positive numbers.

Since the height is 3 less than the length, the length must be greater than 3 so that the height will be greater than zero. Since the sum of length and width is 30, the length must be less than 30. Hence, the possible range of values for is

If you know enough calculus to be able to take the derivative of this function, set the first derivative equal to zero and solve. You will obtain two possible extrema. Only one of these extrema will lie in the the acceptable range of values,

If you don't know the Calculus, then the best you can do is to graph the function and eyeball it.