Question

select the correct answer.
an online retailer is shipping a shower curtain pole that is 13 feet long. their largest box is in the shape of a rectangular prism that is
$10\text{ ft} \times 9\text{ ft} \times 8\text{ ft}$ will the pole fit in the box?
a. no, because the longest diagonal of the box is 12.1 feet long.
b. no, because the longest diagonal of the box is 12.8 feet long.
c. yes, because the longest diagonal of the box is 15.7 feet long.
d. yes, because the longest diagonal of the box is 13.5 feet long.

Explanation:

Step1: Find base diagonal first

First, calculate the diagonal of the base (10ft × 9ft face) using the Pythagorean theorem.
$\text{Base diagonal} = \sqrt{10^2 + 9^2} = \sqrt{100 + 81} = \sqrt{181}$

Step2: Calculate space diagonal

Use the base diagonal and height (8ft) to find the 3D space diagonal of the box.
$\text{Space diagonal} = \sqrt{(\sqrt{181})^2 + 8^2} = \sqrt{181 + 64} = \sqrt{245} \approx 15.7$

Step3: Compare to pole length

The pole is 13ft, which is shorter than 15.7ft.

Answer:

C. Yes, because the longest diagonal of the box is 15.7 feet long.