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Sagot :
Absolutely! Let's break down the problem and solve it step by step.
### Given:
- Curved Surface Area (CSA) of the cylinder, [tex]\( \text{CSA} = 19,400 \, \text{cm}^2 \)[/tex]
- Ratio of radius ([tex]\( r \)[/tex]) to height ([tex]\( h \)[/tex]) is [tex]\( 7:9 \)[/tex]
We need to find:
1. Radius and height
2. Total Surface Area (TSA)
3. Volume
### Key Formulas:
1. Curved Surface Area:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
2. Total Surface Area:
[tex]\[ \text{TSA} = 2\pi r (h + r) \][/tex]
3. Volume:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
### Step-by-Step Solution:
#### Part (a): Finding Radius and Height
Given the ratio of radius to height is 7:9, let:
[tex]\[ r = 7k \][/tex]
[tex]\[ h = 9k \][/tex]
From the given CSA formula:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ 19,400 = 2 \pi (7k) (9k) \][/tex]
[tex]\[ 19,400 = 2 \pi \times 63k^2 \][/tex]
[tex]\[ 19,400 = 126 \pi k^2 \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k^2 = \frac{19,400}{126 \pi} \][/tex]
Taking the square root:
[tex]\[ k \approx 7.000686637\\ \][/tex]
So, the radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] are:
[tex]\[ r = 7 \times 7.000686637 \][/tex]
[tex]\[ r \approx 49.004808462344215 \, \text{cm} \][/tex]
[tex]\[ h = 9 \times 7.000686637 \][/tex]
[tex]\[ h \approx 63.00618230872827 \, \text{cm} \][/tex]
#### Part (b): Finding Total Surface Area (TSA)
Using the formula for TSA:
[tex]\[ \text{TSA} = 2 \pi r (h + r) \][/tex]
Substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ \text{TSA} = 2 \pi \times 49.004808462344215 \times (63.00618230872827 + 49.004808462344215) \][/tex]
[tex]\[ \text{TSA} \approx 34,488.88888888889 \, \text{cm}^2 \][/tex]
#### Part (c): Finding Volume
Using the formula for volume:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
Substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ \text{Volume} \approx \pi \times (49.004808462344215)^2 \times 63.00618230872827 \][/tex]
[tex]\[ \text{Volume} \approx 475,346.6420847388 \, \text{cm}^3 \][/tex]
### Summary of Results:
1. Radius (r): [tex]\( 49.004808462344215 \, \text{cm} \)[/tex]
2. Height (h): [tex]\( 63.00618230872827 \, \text{cm} \)[/tex]
3. Total Surface Area (TSA): [tex]\( 34,488.88888888889 \, \text{cm}^2 \)[/tex]
4. Volume: [tex]\( 475,346.6420847388 \, \text{cm}^3 \)[/tex]
This step-by-step solution comprehensively tackles parts (a), (b), and (c) of the problem effectively.
### Given:
- Curved Surface Area (CSA) of the cylinder, [tex]\( \text{CSA} = 19,400 \, \text{cm}^2 \)[/tex]
- Ratio of radius ([tex]\( r \)[/tex]) to height ([tex]\( h \)[/tex]) is [tex]\( 7:9 \)[/tex]
We need to find:
1. Radius and height
2. Total Surface Area (TSA)
3. Volume
### Key Formulas:
1. Curved Surface Area:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
2. Total Surface Area:
[tex]\[ \text{TSA} = 2\pi r (h + r) \][/tex]
3. Volume:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
### Step-by-Step Solution:
#### Part (a): Finding Radius and Height
Given the ratio of radius to height is 7:9, let:
[tex]\[ r = 7k \][/tex]
[tex]\[ h = 9k \][/tex]
From the given CSA formula:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ 19,400 = 2 \pi (7k) (9k) \][/tex]
[tex]\[ 19,400 = 2 \pi \times 63k^2 \][/tex]
[tex]\[ 19,400 = 126 \pi k^2 \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k^2 = \frac{19,400}{126 \pi} \][/tex]
Taking the square root:
[tex]\[ k \approx 7.000686637\\ \][/tex]
So, the radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] are:
[tex]\[ r = 7 \times 7.000686637 \][/tex]
[tex]\[ r \approx 49.004808462344215 \, \text{cm} \][/tex]
[tex]\[ h = 9 \times 7.000686637 \][/tex]
[tex]\[ h \approx 63.00618230872827 \, \text{cm} \][/tex]
#### Part (b): Finding Total Surface Area (TSA)
Using the formula for TSA:
[tex]\[ \text{TSA} = 2 \pi r (h + r) \][/tex]
Substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ \text{TSA} = 2 \pi \times 49.004808462344215 \times (63.00618230872827 + 49.004808462344215) \][/tex]
[tex]\[ \text{TSA} \approx 34,488.88888888889 \, \text{cm}^2 \][/tex]
#### Part (c): Finding Volume
Using the formula for volume:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
Substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ \text{Volume} \approx \pi \times (49.004808462344215)^2 \times 63.00618230872827 \][/tex]
[tex]\[ \text{Volume} \approx 475,346.6420847388 \, \text{cm}^3 \][/tex]
### Summary of Results:
1. Radius (r): [tex]\( 49.004808462344215 \, \text{cm} \)[/tex]
2. Height (h): [tex]\( 63.00618230872827 \, \text{cm} \)[/tex]
3. Total Surface Area (TSA): [tex]\( 34,488.88888888889 \, \text{cm}^2 \)[/tex]
4. Volume: [tex]\( 475,346.6420847388 \, \text{cm}^3 \)[/tex]
This step-by-step solution comprehensively tackles parts (a), (b), and (c) of the problem effectively.
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