To solve the problem of determining the maximum percentage change observed in a metal sphere when it is heated, we will analyze the changes in radius, surface area, volume, and mass step by step.
### Step 1: Understand the Physical Changes
When a metal sphere is heated, it expands. This expansion affects its radius, surface area, volume, and mass.
### Step 2: Analyze the Change in Radius
Let the initial radius of the sphere be \( r \). When heated, the radius changes by a small amount \( \Delta r \). The percentage change in radius can be expressed as:
\[
\text{Percentage change in radius} = \frac{\Delta r}{r} \times 100
\]
### Step 3: Analyze the Change in Surface Area
The surface area \( A \) of a sphere is given by the formula:
\[
A = 4\pi r^2
\]
When the radius changes, the new surface area becomes:
\[
A' = 4\pi (r + \Delta r)^2
\]
Using the binomial expansion, we can approximate the change in surface area. The change in surface area can be expressed as:
\[
\Delta A = A' - A \approx 4\pi (2r\Delta r) = 8\pi r \Delta r
\]
Thus, the percentage change in surface area is:
\[
\text{Percentage change in area} = \frac{\Delta A}{A} \times 100 = \frac{8\pi r \Delta r}{4\pi r^2} \times 100 = \frac{2\Delta r}{r} \times 100
\]
### Step 4: Analyze the Change in Volume
The volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3}\pi r^3
\]
When the radius changes, the new volume becomes:
\[
V' = \frac{4}{3}\pi (r + \Delta r)^3
\]
Again using the binomial expansion, we can approximate the change in volume:
\[
\Delta V = V' - V \approx 4\pi r^2 \Delta r
\]
Thus, the percentage change in volume is:
\[
\text{Percentage change in volume} = \frac{\Delta V}{V} \times 100 = \frac{4\pi r^2 \Delta r}{\frac{4}{3}\pi r^3} \times 100 = \frac{3\Delta r}{r} \times 100
\]
### Step 5: Analyze the Change in Mass
The mass of the sphere remains constant during heating, as mass is conserved. Therefore, the percentage change in mass is:
\[
\text{Percentage change in mass} = 0\%
\]
### Step 6: Compare the Percentage Changes
- Percentage change in radius: \( \frac{\Delta r}{r} \times 100 \)
- Percentage change in surface area: \( \frac{2\Delta r}{r} \times 100 \)
- Percentage change in volume: \( \frac{3\Delta r}{r} \times 100 \)
- Percentage change in mass: \( 0\% \)
From the analysis, we can see that the maximum percentage change occurs in the volume, followed by surface area, then radius, and lastly mass which does not change at all.
### Conclusion
The maximum percentage change will be observed in the **volume** of the metal sphere.
