Question

(Ball = radius 5, height 10)

image.png
When calculating the volume of this ball, Phil used the volume formula V= πr²h.

Here is his work:

image.png


Did Phil find the correct volume for the ball?


If yes, explain what is happening in each step of his work.

If no, identify his mistake and find the correct volume.
Make sure to show your steps and calculations.

Answer 1

To determine whether Phil calculated the volume of the ball correctly, we need to first check his volume formula and the dimensions of the ball he used.

The formula Phil used is \( V = \pi r^2 h \). However, this formula is used for calculating the volume of a cylinder, not a sphere (or "ball").

For a sphere, the correct volume formula is:
\[
V = \frac{4}{3} \pi r^3
\]

### Given:
- Radius \( r = 5 \)
- Height (which is irrelevant for the calculation of the sphere's volume) \( h = 10 \)

### Correct Steps to Calculate the Volume of the Ball:
1. **Identify the radius**: The radius of the ball is \( 5 \).
2. **Use the correct formula**:
\[
V = \frac{4}{3} \pi r^3
\]
3. **Substitute the radius into the formula**:
\[
V = \frac{4}{3} \pi (5)^3
\]
4. **Calculate \( (5)^3 \)**:
\[
(5)^3 = 125
\]
5. **Substitute \( 125 \) back into the formula**:
\[
V = \frac{4}{3} \pi (125)
\]
6. **Multiply**:
\[
V = \frac{500}{3} \pi
\]

### Final Volume Calculation:
Now we can simplify this to get a numerical approximation if necessary:
\[
V \approx 523.6 \quad \text{(using } \pi \approx 3.14\text{)}
\]

### Conclusion:
Phil made a mistake by using the formula for the volume of a cylinder instead of the formula for a sphere. The correct volume of the ball (sphere) with a radius of 5 is:
\[
V = \frac{500}{3} \pi \quad \text{or approximately } 523.6 \text{ cubic units.}
\]