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.
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