Question

Guadalupe models the volume of a popcorn box as a right rectangular prism and the box can hold 46 cubic inches of popcorn when it is full. Its length is
2, start fraction, 3, divided by, 4, end fraction
2 3/4
​
in and its height is
7, one half
7 1/2
​
in. Find the width of the popcorn box in inches. Round your answer to the nearest tenth if necessary.

Answer 1

The volume \( V \) of a right rectangular prism is given by the formula:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

In this case, we know the volume \( V \) is 46 cubic inches, the length is \( 2 \frac{3}{4} \) inches, and the height is \( 7 \frac{1}{2} \) inches. First, we need to convert these mixed numbers into improper fractions or decimals.

1. Convert the length \( 2 \frac{3}{4} \):

\[
2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \text{ inches}
\]

2. Convert the height \( 7 \frac{1}{2} \):

\[
7 \frac{1}{2} = 7 + \frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{15}{2} \text{ inches}
\]

Now we can substitute the values into the volume formula:

\[
46 = \left(\frac{11}{4}\right) \times \text{width} \times \left(\frac{15}{2}\right)
\]

Next, we can solve for the width. First, calculate the product of the length and height:

\[
\left(\frac{11}{4}\right) \times \left(\frac{15}{2}\right) = \frac{11 \times 15}{4 \times 2} = \frac{165}{8}
\]

Now, we rewrite the equation for the volume:

\[
46 = \frac{165}{8} \times \text{width}
\]

To isolate the width, multiply both sides by the reciprocal of \( \frac{165}{8} \):

\[
\text{width} = 46 \times \frac{8}{165}
\]

Now, calculate this step-by-step:

\[
46 \times \frac{8}{165} = \frac{368}{165}
\]

Next, we can perform the division to find the decimal value:

\[
\frac{368}{165} \approx 2.226
\]

Rounding to the nearest tenth gives us:

\[
\text{width} \approx 2.2 \text{ inches}
\]

Thus, the width of the popcorn box is approximately \( \boxed{2.2} \) inches.