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Sagot :
Certainly! Let's break down the problem step by step, starting with the given areas, and we'll make the conjecture about the similar solids.
### Step-by-Step Solution:
1. Given Information:
- The area of the circular base of the larger cylinder is [tex]\(81\pi\)[/tex].
- The area of the circular base of the smaller cylinder is [tex]\(9\pi\)[/tex].
2. Calculate the Radii:
We use the formula for the area of a circle, [tex]\(A = \pi r^2\)[/tex], to find the radii of both cylinders.
- For the larger cylinder:
[tex]\[ 81\pi = \pi r_{\text{large}}^2 \][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 81 = r_{\text{large}}^2 \][/tex]
Taking the square root of both sides:
[tex]\[ r_{\text{large}} = 9 \][/tex]
- For the smaller cylinder:
[tex]\[ 9\pi = \pi r_{\text{small}}^2 \][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 9 = r_{\text{small}}^2 \][/tex]
Taking the square root of both sides:
[tex]\[ r_{\text{small}} = 3 \][/tex]
3. Determine the Scale Factor:
The scale factor between the larger and smaller cylinders is the ratio of their radii:
[tex]\[ \text{Scale factor} = \frac{r_{\text{large}}}{r_{\text{small}}} = \frac{9}{3} = 3 \][/tex]
4. Surface Area Ratio:
When dealing with similar solids, the ratio of their surface areas is the square of the scale factor. Given the scale factor of 3:
[tex]\[ \text{Ratio of surface areas} = (\text{Scale factor})^2 = 3^2 = 9 \][/tex]
### Conclusions:
- The dimensions of the larger cylinder are 3 times the dimensions of the smaller cylinder. This is true since the scale factor calculated is 3.
- The surface area of the larger cylinder is [tex]\(3^2 = 9\)[/tex] times the surface area of the smaller cylinder. This is also true based on the ratio of the surface areas.
- If proportional dimensional changes are made to a solid figure, then the surface area will change by the square of the scale factor of similar solids. This statement is verified by our calculations and is true.
Therefore, the conjecture is supported by the given data, the calculations, and the relationships:
- The dimensions of the larger cylinder are 3 times the dimensions of the smaller cylinder.
- The surface area of the larger cylinder is [tex]\(9\)[/tex] times the surface area of the smaller cylinder.
- Proportional dimensional changes in a solid result in the surface area changing by the square of the scale factor.
Each statement provided is accurate and adheres to the relationships within similar geometric solids.
### Step-by-Step Solution:
1. Given Information:
- The area of the circular base of the larger cylinder is [tex]\(81\pi\)[/tex].
- The area of the circular base of the smaller cylinder is [tex]\(9\pi\)[/tex].
2. Calculate the Radii:
We use the formula for the area of a circle, [tex]\(A = \pi r^2\)[/tex], to find the radii of both cylinders.
- For the larger cylinder:
[tex]\[ 81\pi = \pi r_{\text{large}}^2 \][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 81 = r_{\text{large}}^2 \][/tex]
Taking the square root of both sides:
[tex]\[ r_{\text{large}} = 9 \][/tex]
- For the smaller cylinder:
[tex]\[ 9\pi = \pi r_{\text{small}}^2 \][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[ 9 = r_{\text{small}}^2 \][/tex]
Taking the square root of both sides:
[tex]\[ r_{\text{small}} = 3 \][/tex]
3. Determine the Scale Factor:
The scale factor between the larger and smaller cylinders is the ratio of their radii:
[tex]\[ \text{Scale factor} = \frac{r_{\text{large}}}{r_{\text{small}}} = \frac{9}{3} = 3 \][/tex]
4. Surface Area Ratio:
When dealing with similar solids, the ratio of their surface areas is the square of the scale factor. Given the scale factor of 3:
[tex]\[ \text{Ratio of surface areas} = (\text{Scale factor})^2 = 3^2 = 9 \][/tex]
### Conclusions:
- The dimensions of the larger cylinder are 3 times the dimensions of the smaller cylinder. This is true since the scale factor calculated is 3.
- The surface area of the larger cylinder is [tex]\(3^2 = 9\)[/tex] times the surface area of the smaller cylinder. This is also true based on the ratio of the surface areas.
- If proportional dimensional changes are made to a solid figure, then the surface area will change by the square of the scale factor of similar solids. This statement is verified by our calculations and is true.
Therefore, the conjecture is supported by the given data, the calculations, and the relationships:
- The dimensions of the larger cylinder are 3 times the dimensions of the smaller cylinder.
- The surface area of the larger cylinder is [tex]\(9\)[/tex] times the surface area of the smaller cylinder.
- Proportional dimensional changes in a solid result in the surface area changing by the square of the scale factor.
Each statement provided is accurate and adheres to the relationships within similar geometric solids.
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