Chapter 7.8, Problem 36E

To determine

To find: Dimensions of the given gift bag.

Answer to Problem 36E

The required dimensions are $16,6$ and $12inches$ .

Explanation of Solution

Given information: Length of gift bag is $2w+4$ , width is $w$ , height is $(18-w)inches$ . Volume is $1152$ $cubic$ $inches$ .

Calculation:

Volume of gift bag $=(length)(width)(height)$

  $(2w+4)(w)(18-w)=1152$

  $(2w^2+4w)(18-w)=1152$ (Distributive Property)

  $36w^2-2w^3+72w-4w^2=1152$ (Using FOIL method)

  $-2w^3+32w^2+72w-1152=0$ (Combining like terms)

  $-2(w^3-16w^2-36w+576)=0$ (Factoring out 2)

  $w^3-16w^2-36w+576=0$

Writing left hand side factored and simplifying,

  $\begin{array}{l}{w}^2(w-16)-36(w-16)=0\\ (w^2-36)(w-16)=0\end{array}$

  $w=-6,6,16$

As width should be a positive integer, $\therefore w=6,16$.

If $width=16$ then height $(18-16=2)$ will be less than width, which should not be as such as per given statement, $\therefore width=6$ .

Putting $w=16$ in given equations,

Length of gift bag $=2(6)+4$

  $=16inches$

Width of gift bag $=6inches$

Height of gift bag $=18-6$

  $=12inches$

Thus, the required dimensions are $16,6$ and $12inches$ .