Part 2- Could anybody help me answer this (Math) Geometry problem?

Question

Problem:
The volume of a rectangular box is 12cm^3. It has a square base and its height is 1 cm more than its width. Find the dimension of the box.

Assumptions:
let x=measurement of one side of the square base.
and x+1cm= the height

Since the volume of a rectangular box is LWH. L=W because it has a square base. My solution would be:
LWH=12cm^3
x*x(x+1cm)=12cm^3

Then,
x^3 + (x^2)cm =12cm^3
x^3 + (x^2)cm = 12cm^3
x^3 + (x^2)cm - 12cm^3=0

I don't know what will be the next step....

Are my assumptions and solution wrong?
Could anybody give me any alternative way on solving this problem. Thanks alot again. =D

Answer 1

Hi,

Your equation is correct. To solve it, if you graph this equation on a graphing calculator like the TI83, you will see it crosses the x axis when x = 2. So the square base is 2 cm on a side and the height is x + 1 = 2 + 1 = 3 cm.

You could also verify this by synthetic division if you don't have a calculator.

In synthetic division, put coefficients inside including a 0 for the +0x term. Divide by possible solution x = 2.

......_____________
..2..| 1...1...0..-12
.......__________ Copy the 1 below the line.

......_____________
..2..| 1...1...0..-12
.......__________
........1
Multiply that number at the bottom times the 2 out front. Write the answer above the line in the next column.

......_____________
..2..| 1...1...0..-12
.......___2______
.........1

......_____________
..2..| 1...1...0..-12
.......___2______ Add this column.
.........1...3
Multiply the number at the bottom (3) times the 2 out front. Write the answer above the line in the next column.

......_____________
..2..| 1...1...0..-12
.......___2_6____ Add this column.
.........1...3..6

Multiply the number at the bottom (6) times the 2 out front. Write the answer above the line in the next column.

......_____________
..2..| 1...1...0..-12
.......___2_6__12_ Add this column.
.........1...3..6......0

The zero remainder verifies that x = 2 worked and is the length of the base. Therefore the height is 2 + 1 or 3 cm.

The third way to find it from your equation is just by trial and error. If the base edge was 3, then the base has an area of 9 sq. cm. When you multiply this times a height of 4 cm (3 + 1), the volume is 108 cm³, which is far to large, so x must be less than 3.

If x = 2, then the area of the base is 4 cm². Multiplying this by its height of 3 (2 + 1) gives a volume of 12 cu. cm. This is correct, so the base is 2 cm on a side and the height is 3 cm.

I hope this helps!! :-)

Answer 2

You're totally going on the right way. It seems kinda hard since it doesn't look like you can factor anything right away (which would help to solve for x). But another method you can use to solve for x is by using the Fundamental Theorem of Algebra. Basically, you can find the root in a function that has an n degree of at least 1.

To find such root, you look at the leading coefficient, in this case 1 (because it's x^3), and also at the coefficient with n degree of zero, your constant, in this case -12. Next you list the factors of each one of them

1: +- 1 ----we will call 1 q
-12: +-1, +-2, +-3, +-4, +-6, +-12 ----we will call -12 p

Next you plug in the possible solutions p/q into the function
x^3 + x^2 - 12 , and find which one =0. Notice your possible solutions are

p/q = +-1, +-2, +-3, +-4, +-6, +-12

If you plug them in, you 'll find that 2 is your answer.

Answer 3

I can't remember how to solve it algebraically, but you can solve it using guess and check by getting to x^3+x^2=12, then substituting in numbers for x, x has to be smaller then 3 because 3^3=27 and larger then 1 because 1^3=1 if you try 2, 2^3+2^2=8+4=12 I know there's a way to find it algebraically, but I'm blanking on how. Sorry.

Answer 4

You did it right.
Now you must solve the cubic equation
x³ + x² - 12 = 0

There is no easy way to solve such equations.
However by trial-and-error method you can notice that x = 2 is the root. The polynomial can then be rewritten as
(x³ + x² - 12) = (x² + 3x + 6)(x-2)

Since quadratic equation
x² + 3x + 6 = 0
has no real roots, then


Answer: unique solution is L=W=2, H = 3.

Answer 5

your assumptions are right...I'm sure there's a way to do it where you complete the square or something but if you go to the step with x^3 + x^2 = 12 just think of a number whose square and cube would add up to 12. I would set up a table with numbers going down the left side and x^2 and x^3 on top and find the squares and cubes of the numbers and see which ones add up to 12

Answer 6

Be careful with your units.
You should have:
x^3 cm^3 + x^2 cm^3 = 12cm^3.

If you substitute 2 for x, you will see that it is the correct answer:

8cm^3 + 4cm^3 = 12cm^3.

So, the dimensions are 2cm x 2cm x 3cm.

Answer 7

X=2. This means the l and w are 2 and the h =3. 2X2X3=12.

Answer 8

you are right and if you look this equation has x=2 as a root so
can factor it as (x-2)*(x^2+3x+6) =0and the 2nd factor has no real roots so x=2

Answer 9

First of all you you find the root of it hte root time 2 x1 will equal to the answer