Question

The dimensions of a pool: height = x - 2 meters, length = 2x + 5 meters, and width = 2x - 1 meters. If the volume of the pool is 182 cubic meters, what is the value of x?​

Answer 1

x = 4

Explanation:

volume = height * width * length==>

(x-2)*(2x+5)*(2x-1) = 182 = 2*7*13 ==>

x = 4

Answer 2

x = 4

Explanation:

The pool can be modeled as a rectangular prism.

Given values:

• height = (x - 2) meters
• length = (2x + 5) meters
• width = (2x - 1) meters
• volume = 182 cubic meters

Volume of a rectangular prism:

`V=h\:l\:w`

(where h is the height, l is the length and w is the width).

Substitute the given values into the formula, expand and equal to zero:

`\begin{aligned}V & = h \: l \: w\\\implies 182 & = (x-2)(2x+5)(2x-1)\\182 & = (x-2)(4x^2+8x-5)\\182 & = 4x^3+8x^2-5x-8x^2-16x+10\\182 & = 4x^3-21x+10\\0 & = 4x^3-21x-172 \end{aligned}`

To find the value(s) of x that make the above true, factor the found cubic function using the Factor Theorem.

Factor Theorem

If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x).

`f(x)=4x^3-21x-172`

Try substituting different values of x into the function:

`f(2)=4(2)^3-21(2)-172=-182`

`f(4)=4(4)^3-21(4)-172=0`

Therefore, since f(4) = 0 when x = 4, (x - 4) is a factor of f(x).

Substitute the found value of x into the expressions for height, length and width to check it does not give any negative values:

⇒ height = (4) - 2 = 2 m

⇒ length = 2(4) + 5 = 13 m

⇒ width = 2(4) - 1 = 7 m

As x = 4 gives a positive value for all three dimensions, we do not need to factor the cubic polynomial any further to find any additional values of x.

As a final check, multiply the three found dimensions to find the volume:

⇒ 2 m × 13 m × 7 m = 182 m³

This confirms that x = 4.