Question

Look at this cylinder: 4 m 7 m If the radius and height are tripled, then which of the following statements about its surface area will be true?

Answer 1

To analyze the change in surface area of a cylinder when both its radius and height are tripled, let's first recall the formula for the surface area (SA) of a cylinder. The surface area of a cylinder can be calculated using the formula:

\[
SA = 2\pi r(h + r)
\]

where \( r \) is the radius and \( h \) is the height of the cylinder.

Given that the original cylinder has a radius of \( r = 4 \, \text{m} \) and a height of \( h = 7 \, \text{m} \), we can substitute these values into the formula to find the original surface area.

**Original Surface Area Calculation:**
- Radius \( r = 4 \, \text{m} \)
- Height \( h = 7 \, \text{m} \)

\[
SA_{\text{original}} = 2\pi (4)(7 + 4) = 2\pi (4)(11) = 88\pi \, \text{m}^2
\]

Now, if both the radius and height are tripled, the new dimensions will be:

- New radius \( r' = 3 \times 4 = 12 \, \text{m} \)
- New height \( h' = 3 \times 7 = 21 \, \text{m} \)

**New Surface Area Calculation:**
Now we can calculate the surface area with the new dimensions:

\[
SA_{\text{new}} = 2\pi (12)(21 + 12) = 2\pi (12)(33) = 792\pi \, \text{m}^2
\]

**Comparing the Surface Areas:**
Next, let's find the ratio of the new surface area to the original surface area.

\[
\text{Ratio} = \frac{SA_{\text{new}}}{SA_{\text{original}}} = \frac{792\pi}{88\pi} = \frac{792}{88} = 9
\]

So, the new surface area is 9 times the original surface area.

### Conclusion:
When both the radius and height of the cylinder are tripled, the surface area increases by a factor of 9.

Therefore, the statement that is true about the surface area of the cylinder after tripling its dimensions is that **the surface area becomes 9 times larger**.