To calculate the cross-sectional area of a hollow shaft, we need to follow a series of steps involving basic geometry and mathematics. Let's break it down step-by-step:
1. Determine the Radii:
- The outer diameter of the shaft is given as 3.25 cm. The radius is half of the diameter.
[tex]\[
\text{Outer radius} = \frac{3.25 \text{ cm}}{2} = 1.625 \text{ cm}
\][/tex]
- The inner diameter of the shaft is given as 2.5 cm. Similarly, the inner radius is half of the inner diameter.
[tex]\[
\text{Inner radius} = \frac{2.5 \text{ cm}}{2} = 1.25 \text{ cm}
\][/tex]
2. Calculate the Outer Cross-Sectional Area:
- The formula for the area of a circle is given by [tex]\( A = \pi r^2 \)[/tex] where [tex]\( r \)[/tex] is the radius.
[tex]\[
\text{Area of the outer circle} = \pi \times (1.625 \text{ cm})^2 \approx 8.295768100885548 \text{ cm}^2
\][/tex]
3. Calculate the Inner Cross-Sectional Area:
- Using the same formula for the area of a circle:
[tex]\[
\text{Area of the inner circle} = \pi \times (1.25 \text{ cm})^2 \approx 4.908738521234052 \text{ cm}^2
\][/tex]
4. Calculate the Cross-Sectional Area of the Hollow Shaft:
- The cross-sectional area of the hollow shaft is the difference between the outer area and the inner area.
[tex]\[
\text{Cross-sectional area} = \text{Area of the outer circle} - \text{Area of the inner circle}
\][/tex]
[tex]\[
\text{Cross-sectional area} \approx 8.295768100885548 \text{ cm}^2 - 4.908738521234052 \text{ cm}^2 \approx 3.3870295796514958 \text{ cm}^2
\][/tex]
Therefore, the cross-sectional area of the hollow shaft is approximately [tex]\( 3.387 \)[/tex] cm².
