Question 718556
For a box (right rectangular prism, in Mathematical terms), the volume is found using: V = l*w*h (<u>l</u>ength times <u>w</u>idth times <u>h</u>eight).<br>
We've been given that the volume is {{{120m^3}}}. We know the width, 4m. Let's call the length "x". Then, since the height is 1m less, the height would be x-1.<br>
Substituting these numbers and expressions into the volume formula we get:
{{{(120) = (x)*(4)*(x-1)}}}
Note the use of parentheses. It is a very good habit to use parentheses like this when making substitutions. Sometimes they are not necessary but sometimes they are critical. The parentheses around the 120, the x and the 4 are not necessary (but they don't hurt anything, either). The parentheses around x-1 are critical!! If we wrote {{{120 = x*4*x-1}}} we would have the wrong equation!<br>
Simplifying...
{{{120 = (4x)*(x-1)}}}
Using the Distributive Property:
{{{120 = 4x^2-4x}}}<br>
Now we solve for x. This is a quadratic equation so we want one side to be zero. Subtracting 120 from each side:
{{{0 = 4x^2-4x-120}}}
Factor. First the GCF:
{{{0 = 4(x^2-x-30)}}}
Then the trinomial:
{{{0 = 4(x-6)(x+5)}}}
Zero Product Property:
4 = 0 or x-6 = 0 or x+6 = 0
Solve these. The first equation is simply false. There are no solutions (numbers that make it true) for it. Solving the other two we get:
x = 6 or x = -6<br>
Since x represents the length of the box and since negative lengths make no sense, we reject the negative solution. So the only solution is x = 6. In words, the length is 6m. And since the height is x-1, the height is 5.