Question

A reinforcing bar, better known as rebar, is a cylindrical steel bar used to reinforce concrete construction. A
96
96-inch piece of rebar has a diameter of
2.25
2.25 inches and weighs
108.8
108.8 pounds. What is the density of the steel in pounds per cubic inch to the nearest thousandth of a cubic inch?



pounds per cubic inch

Answer 1

To find the density of the steel in the rebar, we need to calculate the volume of the rebar and then use the weight to find the density.

1. **Calculate the Volume of the Cylinder (Rebar)**:
The volume \( V \) of a cylinder can be calculated using the formula:

\[
V = \pi r^2 h
\]

where:
- \( r \) is the radius of the cylinder (half of the diameter)
- \( h \) is the height (or length) of the cylinder

Given:
- Diameter \( = 2.25 \) inches, so \( r = \frac{2.25}{2} = 1.125 \) inches
- Height \( h = 96 \) inches

Plugging in the values:

\[
V = \pi (1.125)^2 (96)
\]

Calculate the radius squared:

\[
1.125^2 = 1.265625
\]

Now plug it into the volume formula:

\[
V = \pi (1.265625) (96) \approx 3.141592653589793 \times 1.265625 \times 96 \approx 381.056116441301 \text{ cubic inches}
\]

2. **Calculate Density**:
Density \( \rho \) is defined as mass (or weight) divided by volume:

\[
\rho = \frac{W}{V}
\]

where:
- \( W = 108.8 \) pounds
- \( V \approx 381.056116441301 \) cubic inches

\[
\rho = \frac{108.8}{381.056116441301} \approx 0.2856 \text{ pounds per cubic inch}
\]

3. **Round to the Nearest Thousandth**:
Rounding \( 0.2856 \) to the nearest thousandth gives:

\[
\rho \approx 0.286 \text{ pounds per cubic inch}
\]

Therefore, the density of the steel in the rebar is approximately **0.286 pounds per cubic inch**.