The cross-section of a piece of conduit is an annulus. It has an outer diameter of 60 mm and an inner diameter of 50 mm.
a Find the area of the cross-section to the
nearest square millimetre.
b What is the increase in cross-sectional area if the
outer diameter is increased to 65 mm? Answer correct to the nearest square
millimetre.

To solve this problem, we'll use the formula for the area of an annulus. An annulus is a ring-shaped object, and the area of an annulus is given by the difference between the areas of the outer and inner circles:

\[
A = \pi \left( R^2 - r^2 \right)
\]
where \(R\) is the outer radius and \(r\) is the inner radius.

### Part (a): Find the area of the cross-section to the nearest square millimetre.

Given:
- Outer diameter = 60 mm → Outer radius, \(R = \frac{60}{2} = 30\) mm
- Inner diameter = 50 mm → Inner radius, \(r = \frac{50}{2} = 25\) mm

Using the formula for the area of an annulus:

\[
A = \pi \left( R^2 - r^2 \right) = \pi \left( 30^2 - 25^2 \right)
\]

Let's calculate this area.

### Part (b): Find the increase in cross-sectional area if the outer diameter is increased to 65 mm.

The new outer radius is:

\[
R_{new} = \frac{65}{2} = 32.5 \, \text{mm}
\]

The new area of the annulus is:

\[
A_{new} = \pi \left( R_{new}^2 - r^2 \right) = \pi \left( 32.5^2 - 25^2 \right)
\]

The increase in the area is the difference between the new and the original areas.

Let me compute both results.### Part (a): 
The area of the cross-section is approximately **864 square millimetres**.

### Part (b): 
The increase in the cross-sectional area, when the outer diameter is increased to 65 mm, is approximately **491 square millimetres**.

Would you like further details or have any questions?

Here are five related questions to expand on the problem:
1. How would the area change if both the inner and outer diameters increased proportionally?
2. How does the annular area compare to the area of a solid circle with the same outer diameter?
3. What is the ratio of the inner area to the outer area?
4. How does increasing the thickness of the conduit affect the area?
5. How would you calculate the volume if the conduit had a certain length?

**Tip:** Always check the units when solving area problems, especially when converting between diameters and radii!

The cross-section of a piece of conduit is an annulus. It has an outer diameter of 60 mm and an inner diameter of 50 mm. a) Find the area of the cross-section to the nearest square millimetre. b) What is the increase in cross-sectional area if the outer diameter is increased to 65 mm? Answer correct to the nearest square millimetre.

Learn how to calculate the area of an annulus (a ring-shaped cross-section) with outer and inner diameters of 60 mm and 50 mm, respectively. This problem also explores the increase in cross-sectional area when the outer diameter is increased to 65 mm. Detailed steps include applying the annulus area formula and solving for the area to the nearest square millimetre. Suitable for Grades 9-11.

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