Increasing the Radius and Height of a Cone

To find out how much the volume of a cone increases when both the radius and height are increased by 10%, we can follow these steps:

1. Determine the formula for the volume of a cone:
The volume of a cone can be calculated using the formula V = (1/3)πr²h, where V represents the volume, π is a mathematical constant (approximately 3.14159), r is the radius, and h is the height of the cone.

2. Calculate the initial volume of the cone:
Let's assume the initial radius of the cone is r and the initial height is h. Using the volume formula, we can calculate the initial volume as V1 = (1/3)πr²h.

3. Calculate the increased radius and height:
Since both the radius and height are increased by 10%, we can calculate the new values as follows:
- Increased radius: r + 10% of r = r + 0.1r = 1.1r
- Increased height: h + 10% of h = h + 0.1h = 1.1h

4. Calculate the increased volume of the cone:
After obtaining the increased radius (1.1r) and height (1.1h), we can calculate the new volume using the volume formula:
V2 = (1/3)π(1.1r)²(1.1h)

5. Calculate the percentage increase in volume:
To find the percentage increase in volume, we need to compare the initial volume (V1) with the increased volume (V2). The percentage increase can be calculated using the formula:
Percentage Increase = ((V2 - V1) / V1) * 100

Example:
Let's consider an example to illustrate the calculation. Suppose the initial radius (r) of a cone is 5 units and the initial height (h) is 8 units.

1. Calculate the initial volume:
V1 = (1/3)π(5)²(8) = (1/3)π(25)(8) ≈ 209.44 cubic units

2. Calculate the increased radius and height:
Increased radius = 5 + 0.1(5) = 5 + 0.5 = 5.5 units
Increased height = 8 + 0.1(8) = 8 + 0.8 = 8.8 units

3. Calculate the increased volume:
V2 = (1/3)π(5.5)²(8.8) ≈ 266.71 cubic units

4. Calculate the percentage increase in volume:
Percentage Increase = ((266.71 - 209.44) / 209.44) * 100 ≈ 27.37%

Conclusion:
When both the radius and height of a cone are increased by 10%, the volume of the cone increases by approximately 27.37%.