The statement "the length is 9 inches shorter than twice the width" can be expressed as
[tex]l=2w-9[/tex]
We know the height is 4 inches long, and the volume is 1,064 cubic inches.
The volume of a rectangular box is
[tex]V=w\cdot l\cdot h[/tex]
Where w is x. Let's replace each given information.
[tex]1064=x\cdot(2x-9)\cdot4[/tex]
Now, we solve for x.
[tex]\begin{gathered} 1064=(2x^2-9x)\cdot4 \\ \frac{1064}{4}=2x^2-9x \\ 266=2x^2-9x \\ 2x^2-9x-266=0 \end{gathered}[/tex]
Where a = 2, b = -9, and c = -266. We use the quadratic formula to find the solutions.
[tex]\begin{gathered} x_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_{1,2}=\frac{-(-9)\pm\sqrt[]{(-9)^2-4(2)(-266)}}{2(2)} \\ x_{1,2}=\frac{9\pm\sqrt[]{81+2128}}{4}=\frac{9\pm\sqrt[]{2209}}{4}=\frac{9\pm47}{4} \\ x_1=\frac{9+47}{4}=\frac{56}{4}=14 \\ x_2=\frac{9-47}{4}=\frac{-38}{4}=-9.5 \end{gathered}[/tex]
However, the positive number is the only one that makes sense to the problem since distances can't be negative.
Therefore, the right answer is 14 inches.
