Question

level 3/4

1. how much material is required to make a pipe that is 300 cm long whose outer diameter is 8 cm with an inner diameter of 5 cm? show your thinking.

total volum= vol large cylinder - vol inner cylinder
r=4 h=300 h=300 r=2.5

Explanation:

Step1: Recall Cylinder Volume Formula

The volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length, in this case).

Step2: Find Outer and Inner Radii

• Outer diameter \( D_{outer} = 8 \, \text{cm} \), so outer radius \( r_{outer} = \frac{8}{2} = 4 \, \text{cm} \).
• Inner diameter \( D_{inner} = 5 \, \text{cm} \), so inner radius \( r_{inner} = \frac{5}{2} = 2.5 \, \text{cm} \).
• Length of the pipe (height for both cylinders) \( h = 300 \, \text{cm} \).

Step3: Calculate Volume of Outer Cylinder

Using the formula \( V_{outer} = \pi r_{outer}^2 h \), substitute \( r_{outer} = 4 \) and \( h = 300 \):
\( V_{outer} = \pi \times 4^2 \times 300 = \pi \times 16 \times 300 = 4800\pi \, \text{cm}^3 \).

Step4: Calculate Volume of Inner Cylinder

Using the formula \( V_{inner} = \pi r_{inner}^2 h \), substitute \( r_{inner} = 2.5 \) and \( h = 300 \):
\( V_{inner} = \pi \times (2.5)^2 \times 300 = \pi \times 6.25 \times 300 = 1875\pi \, \text{cm}^3 \).

Step5: Calculate Volume of Material (Difference)

The volume of the material is \( V_{material} = V_{outer} - V_{inner} \).
Substitute the values:
\( V_{material} = 4800\pi - 1875\pi = (4800 - 1875)\pi = 2925\pi \, \text{cm}^3 \).
If we use \( \pi \approx 3.14 \), then \( V_{material} \approx 2925 \times 3.14 = 9184.5 \, \text{cm}^3 \).

Answer:

The volume of material required is \( 2925\pi \, \text{cm}^3 \) (or approximately \( 9184.5 \, \text{cm}^3 \)).