A cylinder has a circular base with an area of 40 cm2 and a height of 60 cm. Determine the volume of a cone that has five times the radius and six times the height of the cylinder.

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.

Learn how to calculate the volume of a cone that has five times the radius and six times the height of a given cylinder with a circular base area of 40 cm² and a height of 60 cm. Detailed step-by-step solution provided.

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