SOLUTION: The width of a box is 200cm less than the length. The height is 100 cm less than the length. The volume of the box is 20m^3. Write a polynomial function to describe the volume.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The width of a box is 200cm less than the length. The height is 100 cm less than the length. The volume of the box is 20m^3. Write a polynomial function to describe the volume.       Log On


   

Question 813337: The width of a box is 200cm less than the length. The height is 100 cm less than the length. The volume of the box is 20m^3. Write a polynomial function to describe the volume. Find the dimensions of the box.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = the length of the box. Then "The width of a box is 200cm less than the length." translates into x - 0.2. (We use 0.2 meters instead of 200 cm since the volume is given in cubic meters.) And "The height is 100 cm less than the length." translates into x - 0.1. (We use 0.1 meters instead of 100 cm since the volume is given in cubic meters.)

Since the volume of a box is length times width times height, our polynomial will come from:
V%28x%29+=+x%2A%28x-0.2%29%28x-0.1%29
All we need to do now is multiply this out. Using the Distributive Property to multiply the first two factors:
V%28x%29+=+%28x%5E2-0.2x%29%28x-0.1%29
And now we use FOIL to multiply what remains:
V%28x%29+=+x%5E3-0.1x%5E2-0.2x%5E2%2B0.02x
which simplifies to:
V%28x%29+=+x%5E3-0.3x%5E2%2B0.02x
This is a polynomial for the volume of any box whose width is 200 less than the length and whose height is 100 less than the length.

To find the dimensions of the box if the volume is 20 cubic meters, we just replace the volume with 20 and solve:
20+=+x%5E3-0.3x%5E2%2B0.02x
Make one side zero (by subtracting 20):
0+=+x%5E3-0.3x%5E2%2B0.02x-20

Now we run into some problems. Normally we would try to factor the expression on the right using whatever methods we can. But as far as I can tell there is no way to factor this. If I am correct then there are no rational solutions to this equation. At this point we have a couple of options:
  • Double check what you've posted and the work I've done. If an error is found, correct it and try what I've done with the correct information. Maybe you will end up with an expression that will factor.
  • Use a graphing calculator to find an approximation for an irrational solution:
    1. Enter the function y=x%5E3-0.3x%5E2%2B0.02x-20 into the calculator.
    2. Look at the graph of the function.
    3. Use the trace facility to find where the graph comes closest to the x-axis.
    4. The x-coordinate of the point which is closest to the x-axis will be a decimal approximation for the irrational solution. (I got about 2.8 for the solution for x (which is the length). This would make the width approximately 2.8 - 0.2 or 2.6 meters and the height approximately 2.8 - 0.1 or 2.7 meters.