Answers:
- Length = 13 inches
- Width = 9 inches
- Height = 3 inches
=========================================================
Explanation:
- x = width, any positive real number
- x+4 = length, because it's 4 more than the width
- x/3 = one third the width = height
Dividing by 3 is the same as multiplying by 1/3.
Multiply those expressions to get the volume.
Length*width*height = volume
(x+4)*(x)*(x/3) = 351
(1/3)*x^2*(x+4) = 351
x^2(x+4) = 3*351
x^3+4x^2 = 1053
x^3+4x^2-1053 = 0
From here, we need to use a graphing calculator to find the x intercept(s).
We could use the rational root theorem to check each possible rational root (assuming there are any to begin with), but such a task is a bit of tedious busywork in my opinion. I prefer to stick to technology whenever possible.
On your graphing calculator, use the "root" or "x intercept" feature. If you typed y = x^3+4x^2-1053 into Desmos (a free online graphing tool), then you can simply click on the x intercept to have its coordinates show up. You may have to adjust the window.
In this case, there's only one real number root and it's when x = 9. The other two remaining roots are nonreal complex numbers, which we'll ignore.
So again, the only practical value is x = 9 which means the width is 9 inches.
Use that value of x to find the other dimensions
length = x+4 = 9+4 = 13 inches
height = x/3 = 9/3 = 3 inches
-----------------------
Check:
length*width*height = 13*9*3 = 351
This confirms the correct dimensions.
