Question

A baker is building a rectangular solid box from cardboard to be able to safely deliver a birthday cake. The baker wants the volume of the delivery box to be 540 cubic inches. If the width of the delivery box is 3 inches longer than the length and the height is 4 inches longer than the length, what must the length of the delivery box be?
10 inches
9 inches
6 inches
3 inches

Answer 1

Answer:

C

Step-by-step explanation:

The volume of a box (rectangular prism) is given by:

$\displaystyle V = \ell wh$

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:

$\displaystyle (540) = \ell(\ell + 3)(\ell + 4)$

Solve for the length. Expand:

$\displaystyle \begin{aligned} 540 &= \ell (\ell^2 + 7\ell +12) \\ 540&= \ell ^3 + 7\ell^2 +12\ell \\ \ell^3 + 7\ell ^2 +12\ell -540 &= 0\end{aligned}$

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:

$\displaystyle (6)^3 +7(6)^2 + 12(6) -540 \stackrel{\checkmark}{=} 0$

Hence, one factor is (x - 6).

By synthetic division (shown below), we can see that:

$\displaystyle \ell^3 + 7\ell^2 +12\ell -540 =(\ell -6)(\ell ^2 + 13\ell +90)$

The second factor has no real solutions. Hence, our only solution is that l = 6.

In conclusion, our answer is C.

Answer 2

Answer:

6 inches

Step-by-step explanation:

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