Penalty: 1 off tion ar models the volume of a popcorn box as a right rectangular prism and the box can hold 69 c inches of popcorn when it is full. Its width is 3 in and its height is \( 5 \frac{3}{4} \) in. Find the length of popcorn box in inches. Round your answer to the nearest tenth if necessary.

To find the length of the popcorn box, we start with the volume formula for a rectangular prism, which is: \[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \] We know the volume is 69 cubic inches, the width is 3 inches, and the height is \(5 \frac{3}{4}\) inches. First, convert \(5 \frac{3}{4}\) into an improper fraction: \[ 5 \frac{3}{4} = \frac{23}{4} \text{ inches} \] Now, substitute the known values into the volume formula: \[ 69 = \text{Length} \times 3 \times \frac{23}{4} \] To isolate the length, we first calculate \(3 \times \frac{23}{4}\): \[ 3 \times \frac{23}{4} = \frac{69}{4} \] Now the equation becomes: \[ 69 = \text{Length} \times \frac{69}{4} \] To solve for Length, multiply both sides by the reciprocal of \(\frac{69}{4}\): \[ \text{Length} = 69 \times \frac{4}{69} \] The 69 cancels out: \[ \text{Length} = 4 \text{ inches} \] The length of the popcorn box is \(4\) inches. If you like to remember this formula, just think of it as the "box filling equation" where you plug in the numbers that hold such delicious popcorn! Enjoy your next movie night with perfectly portioned snacks! For a real-world application, if you ever decide to make your own popcorn boxes for a party or event, knowing the relationship between volume, length, width, and height can help you design boxes that fit specific quantities of popcorn! You can create a fun assembly line and let friends decorate their own custom popcorn boxes, turning snack time into a crafty and interactive event.