The height of container is 8 feet
Solution:
Let "w" be the width of container
Let "l" be the length of container
Let "h" be the height of container
The width of a container is 5 feet less than its height
Therefore,
width = height - 5
w = h - 5 ------ eqn 1
Its length is 1 foot longer than its height
length = 1 + height
l = 1 + h ---------- eqn 2
The volume of container is given as:
[tex]v = length \times width \times height[/tex]
Given that volume of the container is 216 cubic feet
[tex]216 = l \times w \times h[/tex]
Substitute eqn 1 and eqn 2 in above formula
[tex]216 = (1 + h) \times (h-5) \times h\\\\216 = (h+h^2)(h-5)\\\\216 = h^2-5h+h^3-5h^2\\\\216 = h^3-4h^2-5h\\\\h^3-4h^2-5h-216 = 0[/tex]
Solve by factoring
[tex](h-8)(h^2+4h+27) = 0[/tex]
Use the zero factor principle
If ab = 0 then a = 0 or b = 0 ( or both a = 0 and b = 0)
Therefore,
[tex]h - 8 = 0\\\\h = 8[/tex]
Also,
[tex]h^2+4h+27 = 0[/tex]
Solve by quadratic equation formula
[tex]\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}[/tex]
[tex]\mathrm{For\:} a=1,\:b=4,\:c=27:\quad h=\frac{-4\pm \sqrt{4^2-4\cdot \:1\cdot \:27}}{2\cdot \:1}[/tex]
[tex]h = \frac{-4+\sqrt{4^2-4\cdot \:1\cdot \:27}}{2}=\frac{-4+\sqrt{92}i}{2}[/tex]
Therefore, on solving we get,
[tex]h=-2+\sqrt{23}i,\:h=-2-\sqrt{23}i[/tex]
Thus solutions of "h" are:
h = 8
[tex]h=-2+\sqrt{23}i,\:h=-2-\sqrt{23}i[/tex]
"h" cannot be a imaginary value
Thus the solution is h = 8
Thus the height of container is 8 feet
