Answer:
x = 4
Step-by-step 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:
[tex]V=h\:l\:w[/tex]
(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:
[tex]\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}[/tex]
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).
[tex]f(x)=4x^3-21x-172[/tex]
Try substituting different values of x into the function:
[tex]f(2)=4(2)^3-21(2)-172=-182[/tex]
[tex]f(4)=4(4)^3-21(4)-172=0[/tex]
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.
