To solve the problem, let's break it down step-by-step. We need to find the volume of a cone with specific dimensions relative to a given cylinder.

Step 1: Analyze the cylinder

We know:

• The area of the circular base of the cylinder $A_{\text{base}}$ is $40 \, \text{cm}^2$.
• The height of the cylinder $h_{\text{cylinder}}$ is $60 \, \text{cm}$.

Step 2: Find the radius of the cylinder

The area of the circular base of a cylinder is given by the formula: $A_{\text{base}} = \pi r^2$

Given $A_{\text{base}} = 40 \, \text{cm}^2$: $\pi r^2 = 40$ $r^2 = \frac{40}{\pi}$ $r = \sqrt{\frac{40}{\pi}}$

Step 3: Determine the dimensions of the cone

The problem states that the cone has:

• Five times the radius of the cylinder.
• Six times the height of the cylinder.

Let's calculate these: $r_{\text{cone}} = 5r$ $h_{\text{cone}} = 6h_{\text{cylinder}}$

First, compute the radius of the cone: $r_{\text{cone}} = 5 \times \sqrt{\frac{40}{\pi}}$ $r_{\text{cone}} = 5 \sqrt{\frac{40}{\pi}}$

Next, compute the height of the cone: $h_{\text{cone}} = 6 \times 60 \, \text{cm} = 360 \, \text{cm}$

Step 4: Find the volume of the cone

The volume $V_{\text{cone}}$ of a cone is given by the formula: $V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}$

Substitute the values for $r_{\text{cone}}$ and $h_{\text{cone}}$: $V_{\text{cone}} = \frac{1}{3} \pi \left(5 \sqrt{\frac{40}{\pi}}\right)^2 \times 360$

Simplify inside the square: $\left(5 \sqrt{\frac{40}{\pi}}\right)^2 = 25 \left(\frac{40}{\pi}\right) = \frac{1000}{\pi}$

Now substitute back: $V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{1000}{\pi}\right) \times 360$

Simplify further: $V_{\text{cone}} = \frac{1}{3} \times 1000 \times 360$ $V_{\text{cone}} = \frac{360000}{3}$ $V_{\text{cone}} = 120000 \, \text{cm}^3$

Thus, the volume of the cone is 120,000 cubic centimeters (cm³).

Would you like more details on any specific part of the calculation, or do you have any other questions?

Here are some related questions to expand your understanding:

1. How do you find the surface area of a cone with the same dimensions?
2. What is the volume of the original cylinder?
3. How does changing the height of a cone affect its volume?
4. How do you calculate the lateral surface area of a cone?
5. What is the formula for the volume of a cylinder and how is it derived?
6. How does the volume of a cylinder compare to that of a cone with the same base and height?
7. What are the properties of similar geometric shapes?
8. How do you derive the formula for the volume of a cone?

Tip

When working with composite shapes or transforming one shape into another (like a cylinder to a cone), always pay attention to the relationship between their dimensions and how it affects volume or surface area calculations.