Answer:
C
Step-by-step explanation:
The volume of a box (rectangular prism) is given by:
[tex]\displaystyle V = \ell wh[/tex]
We are given that the desired volume is 540 cubic inches. The width is three inches longer than the length and the height is four inches longer than the length. Substitute:
[tex]\displaystyle (540) = \ell(\ell + 3)(\ell + 4)[/tex]
Solve for the length. Expand:
[tex]\displaystyle \begin{aligned} 540 &= \ell (\ell^2 + 7\ell +12) \\ 540&= \ell ^3 + 7\ell^2 +12\ell \\ \ell^3 + 7\ell ^2 +12\ell -540 &= 0\end{aligned}[/tex]
We cannot solve by grouping, so we can consider using the Rational Root Theorem. Our possible roots are:
±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±27, ±30, ±36, ±45, ±54, ±60, ±90, ±108, ±135, ±180, ±270, and/or ±540.
(If you are allowed a graphing calculator, this is not necessary.)
Testing values, we see that:
[tex]\displaystyle (6)^3 +7(6)^2 + 12(6) -540 \stackrel{\checkmark}{=} 0[/tex]
Hence, one factor is (x - 6).
By synthetic division (shown below), we can see that:
[tex]\displaystyle \ell^3 + 7\ell^2 +12\ell -540 =(\ell -6)(\ell ^2 + 13\ell +90)[/tex]
The second factor has no real solutions. Hence, our only solution is that l = 6.
In conclusion, our answer is C.
