if the diameter of the sphere is increased by 60% by what percent is the volume increases

The volume \( V \) of a sphere with diameter \( d \) is given by the formula:

\[
V = \frac{\pi}{6} d^3
\]

If the diameter \( d \) is increased by 60%, then the new diameter \( d' \) will be:

\[
d' = d \times 1.6 = 1.6d
\]

Now, let’s calculate the new volume \( V' \) with the new diameter \( d' \):

\[
V' = \frac{\pi}{6} (d')^3 = \frac{\pi}{6} (1.6d)^3 = \frac{\pi}{6} \times 1.6^3 \times d^3
\]

Calculating \( 1.6^3 \):

\[
1.6^3 = 4.096
\]

So,

\[
V' = \frac{\pi}{6} \times 4.096 \times d^3 = 4.096 \times \frac{\pi}{6} d^3
\]

Now, the increase in volume is:

\[
\text{Increase in Volume} = V' - V = 4.096V - V = (4.096 - 1)V = 3.096V
\]

To find the percentage increase:

\[
\text{Percentage Increase} = \frac{\text{Increase in Volume}}{V} \times 100 = 3.096 \times 100 = 309.6\%
\]

### Answer:
If the diameter of the sphere is increased by 60%, the volume of the sphere increases by **309.6%**.

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### Related Questions:
1. What happens to the surface area of the sphere if the diameter is increased by 60%?
2. If the radius of the sphere is increased by 60%, what is the percentage increase in volume?
3. How does the formula for volume change if the shape were a cylinder instead of a sphere?
4. What percentage increase in diameter would double the volume of a sphere?
5. How would an increase in diameter affect the density of the sphere if mass remains constant?

### Tip:
Remember that volume scales with the cube of the diameter (or radius) for spheres, so even small increases in diameter result in large increases in volume.

If the diameter of the sphere is increased by 60%, by what percent is the volume increased?

Learn how to calculate the percentage increase in the volume of a sphere when its diameter is increased by 60%. This problem covers geometry concepts related to spheres and volume, with a detailed explanation suitable for students in Grades 9-11.

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