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Answer :
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### Volume of Cylinders
The volume [tex]\( V \)[/tex] of a cylinder can be calculated using the formula:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
#### Part A:
- Given: [tex]\( r = 12 \)[/tex] inches, [tex]\( h = 4 \)[/tex] inches
- Calculation:
[tex]\[ V = \pi \times 12^2 \times 4 \][/tex]
[tex]\[ V \approx 1809.6 \, \text{cubic inches} \][/tex]
#### Part B:
- Given: [tex]\( r = 6 \)[/tex] feet, height uses the result from Part A as [tex]\( h = 1809.6 \)[/tex]
- Calculation:
[tex]\[ V = \pi \times 6^2 \times 1809.6 \][/tex]
[tex]\[ V = 204660.9 \, \text{cubic feet} \][/tex]
#### Part C:
- Given: [tex]\( r = 3 \)[/tex] cm, [tex]\( h = 13 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 3^2 \times 13 \][/tex]
[tex]\[ V \approx 367.6 \, \text{cubic centimeters} \][/tex]
#### Part D:
- Given: [tex]\( r = 9 \)[/tex] m, [tex]\( h = 11 \)[/tex] m
- Calculation:
[tex]\[ V = \pi \times 9^2 \times 11 \][/tex]
[tex]\[ V \approx 2799.2 \, \text{cubic meters} \][/tex]
#### Part E:
- Given: [tex]\( r = 8 \)[/tex] ft, [tex]\( h = 15 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 8^2 \times 15 \][/tex]
[tex]\[ V \approx 3015.9 \, \text{cubic feet} \][/tex]
#### Part F:
- Given: [tex]\( d = 10 \)[/tex] cm, [tex]\( h = 7 \)[/tex] cm
- Calculate radius [tex]\( r = \frac{d}{2} = 5 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 5^2 \times 7 \][/tex]
[tex]\[ V \approx 549.8 \, \text{cubic centimeters} \][/tex]
#### Part G:
- Given: [tex]\( d = 3 \)[/tex] cm, [tex]\( h = 9 \)[/tex] cm
- Calculate radius [tex]\( r = \frac{d}{2} = 1.5 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 1.5^2 \times 9 \][/tex]
[tex]\[ V \approx 63.6 \, \text{cubic centimeters} \][/tex]
#### Part H:
- Given: [tex]\( d = 8 \)[/tex] ft, [tex]\( h = 15 \)[/tex] ft
- Calculate radius [tex]\( r = \frac{d}{2} = 4 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 4^2 \times 15 \][/tex]
[tex]\[ V \approx 754.0 \, \text{cubic feet} \][/tex]
#### Part I:
- Given: [tex]\( d = 14 \)[/tex] m, [tex]\( h = 15 \)[/tex] m
- Calculate radius [tex]\( r = \frac{d}{2} = 7 \)[/tex] m
- Calculation:
[tex]\[ V = \pi \times 7^2 \times 15 \][/tex]
[tex]\[ V \approx 2309.1 \, \text{cubic meters} \][/tex]
#### Part J:
- Given: [tex]\( d = 6 \)[/tex] ft, [tex]\( h = 21 \)[/tex] ft
- Calculate radius [tex]\( r = \frac{d}{2} = 3 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 3^2 \times 21 \][/tex]
[tex]\[ V \approx 593.8 \, \text{cubic feet} \][/tex]
### Finding Missing Dimensions
To find a missing height or diameter, rearrange the volume formula to solve for the unknown.
#### Part K:
- Given: [tex]\( d = 3 \)[/tex] in, [tex]\( V = 7.1 \)[/tex] cubic inches
- Calculate radius [tex]\( r = \frac{d}{2} = 1.5 \)[/tex] in
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{7.1}{\pi \times 1.5^2} \][/tex]
[tex]\[ h \approx 1 \, \text{inch} \][/tex]
#### Part L:
- Given: [tex]\( d = 11 \)[/tex] ft, [tex]\( V = 190.1 \)[/tex] cubic feet
- Calculate radius [tex]\( r = \frac{d}{2} = 5.5 \)[/tex] ft
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{190.1}{\pi \times 5.5^2} \][/tex]
[tex]\[ h \approx 2 \, \text{feet} \][/tex]
#### Part M:
- Given: [tex]\( r = 5 \)[/tex] in, [tex]\( V = 1727.9 \)[/tex] cubic inches
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{1727.9}{\pi \times 5^2} \][/tex]
[tex]\[ h \approx 22 \, \text{inches} \][/tex]
#### Part N:
- Given: [tex]\( h = 19 \)[/tex] cm, [tex]\( V = 731.2 \)[/tex] cubic centimeters
- Using radius squared formula
[tex]\[ \pi r^2 = \frac{731.2}{19} \][/tex]
- Finding diameter [tex]\( d \)[/tex]:
[tex]\[ d = 2 \sqrt{\frac{731.2}{\pi \times 19}} \][/tex]
[tex]\[ d \approx 7 \, \text{centimeters} \][/tex]
These steps should help you solve problems involving the calculation of volumes and dimensions of cylinders.
### Volume of Cylinders
The volume [tex]\( V \)[/tex] of a cylinder can be calculated using the formula:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
#### Part A:
- Given: [tex]\( r = 12 \)[/tex] inches, [tex]\( h = 4 \)[/tex] inches
- Calculation:
[tex]\[ V = \pi \times 12^2 \times 4 \][/tex]
[tex]\[ V \approx 1809.6 \, \text{cubic inches} \][/tex]
#### Part B:
- Given: [tex]\( r = 6 \)[/tex] feet, height uses the result from Part A as [tex]\( h = 1809.6 \)[/tex]
- Calculation:
[tex]\[ V = \pi \times 6^2 \times 1809.6 \][/tex]
[tex]\[ V = 204660.9 \, \text{cubic feet} \][/tex]
#### Part C:
- Given: [tex]\( r = 3 \)[/tex] cm, [tex]\( h = 13 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 3^2 \times 13 \][/tex]
[tex]\[ V \approx 367.6 \, \text{cubic centimeters} \][/tex]
#### Part D:
- Given: [tex]\( r = 9 \)[/tex] m, [tex]\( h = 11 \)[/tex] m
- Calculation:
[tex]\[ V = \pi \times 9^2 \times 11 \][/tex]
[tex]\[ V \approx 2799.2 \, \text{cubic meters} \][/tex]
#### Part E:
- Given: [tex]\( r = 8 \)[/tex] ft, [tex]\( h = 15 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 8^2 \times 15 \][/tex]
[tex]\[ V \approx 3015.9 \, \text{cubic feet} \][/tex]
#### Part F:
- Given: [tex]\( d = 10 \)[/tex] cm, [tex]\( h = 7 \)[/tex] cm
- Calculate radius [tex]\( r = \frac{d}{2} = 5 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 5^2 \times 7 \][/tex]
[tex]\[ V \approx 549.8 \, \text{cubic centimeters} \][/tex]
#### Part G:
- Given: [tex]\( d = 3 \)[/tex] cm, [tex]\( h = 9 \)[/tex] cm
- Calculate radius [tex]\( r = \frac{d}{2} = 1.5 \)[/tex] cm
- Calculation:
[tex]\[ V = \pi \times 1.5^2 \times 9 \][/tex]
[tex]\[ V \approx 63.6 \, \text{cubic centimeters} \][/tex]
#### Part H:
- Given: [tex]\( d = 8 \)[/tex] ft, [tex]\( h = 15 \)[/tex] ft
- Calculate radius [tex]\( r = \frac{d}{2} = 4 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 4^2 \times 15 \][/tex]
[tex]\[ V \approx 754.0 \, \text{cubic feet} \][/tex]
#### Part I:
- Given: [tex]\( d = 14 \)[/tex] m, [tex]\( h = 15 \)[/tex] m
- Calculate radius [tex]\( r = \frac{d}{2} = 7 \)[/tex] m
- Calculation:
[tex]\[ V = \pi \times 7^2 \times 15 \][/tex]
[tex]\[ V \approx 2309.1 \, \text{cubic meters} \][/tex]
#### Part J:
- Given: [tex]\( d = 6 \)[/tex] ft, [tex]\( h = 21 \)[/tex] ft
- Calculate radius [tex]\( r = \frac{d}{2} = 3 \)[/tex] ft
- Calculation:
[tex]\[ V = \pi \times 3^2 \times 21 \][/tex]
[tex]\[ V \approx 593.8 \, \text{cubic feet} \][/tex]
### Finding Missing Dimensions
To find a missing height or diameter, rearrange the volume formula to solve for the unknown.
#### Part K:
- Given: [tex]\( d = 3 \)[/tex] in, [tex]\( V = 7.1 \)[/tex] cubic inches
- Calculate radius [tex]\( r = \frac{d}{2} = 1.5 \)[/tex] in
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{7.1}{\pi \times 1.5^2} \][/tex]
[tex]\[ h \approx 1 \, \text{inch} \][/tex]
#### Part L:
- Given: [tex]\( d = 11 \)[/tex] ft, [tex]\( V = 190.1 \)[/tex] cubic feet
- Calculate radius [tex]\( r = \frac{d}{2} = 5.5 \)[/tex] ft
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{190.1}{\pi \times 5.5^2} \][/tex]
[tex]\[ h \approx 2 \, \text{feet} \][/tex]
#### Part M:
- Given: [tex]\( r = 5 \)[/tex] in, [tex]\( V = 1727.9 \)[/tex] cubic inches
- Finding height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{1727.9}{\pi \times 5^2} \][/tex]
[tex]\[ h \approx 22 \, \text{inches} \][/tex]
#### Part N:
- Given: [tex]\( h = 19 \)[/tex] cm, [tex]\( V = 731.2 \)[/tex] cubic centimeters
- Using radius squared formula
[tex]\[ \pi r^2 = \frac{731.2}{19} \][/tex]
- Finding diameter [tex]\( d \)[/tex]:
[tex]\[ d = 2 \sqrt{\frac{731.2}{\pi \times 19}} \][/tex]
[tex]\[ d \approx 7 \, \text{centimeters} \][/tex]
These steps should help you solve problems involving the calculation of volumes and dimensions of cylinders.
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