Volume Increase of a Sphere When Radius Increases by 20%

The relationship between a sphere's volume and its radius is a fundamental concept in geometry and calculus. This article explores how the volume of a sphere changes when its radius is increased by a specific percentage. We will use mathematical reasoning to derive the percent increase in volume and verify the result using different methods.

Understanding the Volume Formula for a Sphere

The formula for the volume of a sphere is given by:

V (frac{4}{3} pi r^3)

where V is the volume and r is the radius of the sphere.

Original and New Radius

Let the original radius of the sphere be r. If the radius is increased by 20%, the new radius becomes:

r_{new} r 0.2r 1.2r

Original and New Volume

Using the volume formula, we can calculate the original volume:

V_{original} (frac{4}{3} pi r^3)

With the new radius, the volume becomes:

V_{new} (frac{4}{3} pi (1.2r)^3)

Expanding the expression for the new volume:

V_{new} (frac{4}{3} pi (1.2)^3 r^3 frac{4}{3} pi cdot 1.728 r^3)

Thus, the new volume is:

V_{new} 1.728 (frac{4}{3} pi r^3)

Calculating the Increase in Volume

The increase in volume is given by:

(Delta V V_{new} - V_{original} 1.728 (frac{4}{3} pi r^3) - (frac{4}{3} pi r^3) (1.728 - 1) (frac{4}{3} pi r^3))

Simplifying the expression:

(Delta V 0.728 (frac{4}{3} pi r^3))

Therefore, the increase in volume is 0.728 times the original volume.

Percent Increase in Volume

To find the percent increase in volume:

%text{ Increase} (frac{0.728 (frac{4}{3} pi r^3)}{(frac{4}{3} pi r^3)} cdot 100 0.728 cdot 100 72.8%)

Hence, when the radius of a sphere is increased by 20%, its volume increases by 72.8%.

Verifying the Result Using Different Methods

1. **Using an Extended Radius:**

Let’s assume the original radius R. When the radius is increased by 20%, the new radius is:

R_{new} 1.2R

Volume increase:

[V_{new} frac{4}{3} pi (1.2R)^3 frac{4}{3} pi 1.728R^3]

[Delta V 1.728 (frac{4}{3} pi R^3) - (frac{4}{3} pi R^3) 0.728 (frac{4}{3} pi R^3)

Percentage increase:

[%text{ Increase} (frac{0.728 (frac{4}{3} pi R^3)}{(frac{4}{3} pi R^3)} cdot 100 0.728 cdot 100 72.8%)

2. **Using a New Radius with a Fraction:**

Let R be the original radius. The new radius is:

R_{new} 1.1R)

Volume increase:

[V_{new} frac{4}{3} pi (1.1R)^3 frac{4}{3} pi 1.331R^3)

[Delta V 1.331 (frac{4}{3} pi R^3) - (frac{4}{3} pi R^3) 0.331 (frac{4}{3} pi R^3)

Percentage increase:

[%text{ Increase} (frac{0.331 (frac{4}{3} pi R^3)}{(frac{4}{3} pi R^3)} cdot 100 0.331 cdot 100 33.1%)

Note that this example uses a 10% increase in radius (1.1 times the original radius) and does not directly relate to the 20% increase discussed in the main calculation.

3. **General Discussion on Radius Increase:**

When the radius of a sphere is increased by a factor of (k) (where (k > 1) in most cases), the volume increases by (k^3) times. If (k 1.2) for a 20% increase:

(text{Increase in volume} (1.2)^3 - 1 1.728 - 1 0.728)

This confirms that the volume increases by 72.8% when the radius is increased by 20%.

Conclusion

In summary, when the radius of a sphere is increased by 20%, the volume of the sphere increases by 72.8%. This result is derived using the volume formula and verified through different methods of calculation. Understanding these concepts is crucial for various applications in geometry, physics, and engineering.

Frequently Asked Questions (FAQ)

Q: What is the formula for the volume of a sphere?

A: The volume of a sphere is given by V (frac{4}{3} pi r^3), where r is the radius of the sphere.

Q: How does the volume of a sphere change when the radius is increased?

A: The volume increases by (k^3) times if the radius is increased by a factor of (k). For a 20% increase, the volume increases by 72.8%.

Q: Can this method be applied to other shapes?

A: This method can be applied to other three-dimensional shapes whose volume is a function of the radius, such as a cylinder or a cone, but the formula and factors will differ based on the shape.

Keywords: sphere volume increase, volume formula, radius increase