Question

The width of a container is 5 feet less than its height. Its length is 1 foot longer than its height. The volume of the container is 216 cubic feet. How tall is the container?

Answer 1

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:

`v = length * width * height`

Given that volume of the container is 216 cubic feet

`216 = l * w * h`

Substitute eqn 1 and eqn 2 in above formula

`216 = (1 + h) * (h-5) * 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`

Solve by factoring

`(h-8)(h^2+4h+27) = 0`

Use the zero factor principle

If ab = 0 then a = 0 or b = 0 ( or both a = 0 and b = 0)

Therefore,

`h - 8 = 0\\\\h = 8`

Also,

`h^2+4h+27 = 0`

Solve by quadratic equation formula

`\text {For a quadratic equation } a x^(2)+b x+c=0, \text { where } a \\eq 0\\\\x=\frac{-b \pm \sqrt{b^(2)-4 a c}}{2 a}`

`\mathrm{For\:} a=1,\:b=4,\:c=27:\quad h=(-4\pm √(4^2-4\cdot \:1\cdot \:27))/(2\cdot \:1)`

`h = (-4+√(4^2-4\cdot \:1\cdot \:27))/(2)=(-4+√(92)i)/(2)`

Therefore, on solving we get,

`h=-2+√(23)i,\:h=-2-√(23)i`

Thus solutions of "h" are:

h = 8

`h=-2+√(23)i,\:h=-2-√(23)i`

"h" cannot be a imaginary value

Thus the solution is h = 8

Thus the height of container is 8 feet