We are given a cylinder with height 19 cm and radius 4.75 cm. This cylinder is placed inside a rectangular box with dimensions 25 cm, 15 cm, and 10 cm. We need to find the volume of the packing material needed to fill the remaining space in the box. This is the volume of the box minus the volume of the cylinder.

1. Problem Description

We are given a cylinder with height 19 cm and radius 4.75 cm. This cylinder is placed inside a rectangular box with dimensions 25 cm, 15 cm, and 10 cm. We need to find the volume of the packing material needed to fill the remaining space in the box. This is the volume of the box minus the volume of the cylinder.

2. Solution Steps

First, calculate the volume of the rectangular box.

The formula for the volume of a rectangular box is:

$V_{box} = length \times width \times height$

$V_{box} = 25 \times 15 \times 10 = 3750 \text{ cm}^3$

Next, calculate the volume of the cylinder.

The formula for the volume of a cylinder is:

$V_{cylinder} = \pi r^2 h$

where $r$ is the radius and $h$ is the height.

$r = 4.75 \text{ cm}$

$h = 19 \text{ cm}$

$V_{cylinder} = \pi (4.75)^2 (19)$

$V_{cylinder} = \pi (22.5625) (19)$

$V_{cylinder} = \pi (428.6875)$

Using $\pi \approx 3.14159$:

$V_{cylinder} \approx 3.14159 \times 428.6875 \approx 1346.53 \text{ cm}^3$

Finally, calculate the volume of the packing material by subtracting the volume of the cylinder from the volume of the box:

$V_{packing} = V_{box} - V_{cylinder}$

$V_{packing} = 3750 - 1346.53$

$V_{packing} = 2403.47$

Rounding to the nearest cubic centimeter, we get

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3. Final Answer

2403 cubic centimeters