The radius of a sphere is `(4.5+-0.2)cm`. What is the percentage error in volume?

Explanation

To find the percentage error in the volume of a sphere given the radius and its uncertainty, we can follow these steps:

### Step 1: Understand the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Determine the relationship between the radius and volume
To find the percentage error in volume due to the uncertainty in the radius, we can use the formula for the relative error in volume:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
where:
- \( \Delta V \) is the uncertainty in volume,
- \( \Delta r \) is the uncertainty in radius,
- \( r \) is the radius.

### Step 3: Substitute the values
From the question, we have:
- \( r = 4.5 \, \text{cm} \)
- \( \Delta r = 0.2 \, \text{cm} \)

Now we can substitute these values into the formula:
\[ \frac{\Delta V}{V} = 3 \frac{0.2}{4.5} \]

### Step 4: Calculate the relative error
Now we calculate the right-hand side:
\[ \frac{\Delta V}{V} = 3 \times \frac{0.2}{4.5} = 3 \times 0.044444 \approx 0.13333 \]

### Step 5: Convert to percentage error
To find the percentage error, we multiply by 100:
\[ \text{Percentage Error} = 0.13333 \times 100 \approx 13.33\% \]

### Final Answer
The percentage error in the volume of the sphere is approximately:
\[ \text{Percentage Error} \approx 13.3\% \]

---