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Answer :
Sure! Let's solve the problem step-by-step.
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula [tex]\( V = k r^2 h \)[/tex], where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( h \)[/tex] is the height,
- [tex]\( k \)[/tex] is a constant of proportionality.
### Step 1: Find the Constant [tex]\( k \)[/tex] Using Cylinder A
We know the volume, radius, and height of Cylinder A:
- Volume [tex]\( V_A = 254.34 \)[/tex] cubic inches,
- Radius [tex]\( r_A = 3 \)[/tex] inches,
- Height [tex]\( h_A = 9 \)[/tex] inches.
Plug these values into the formula to solve for [tex]\( k \)[/tex]:
[tex]\[ 254.34 = k \times 3^2 \times 9 \][/tex]
[tex]\[ 254.34 = k \times 9 \times 9 \][/tex]
Since [tex]\( 3^2 = 9 \)[/tex]:
[tex]\[ 254.34 = k \times 81 \][/tex]
Divide both sides by 81 to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{254.34}{81} \][/tex]
[tex]\[ k = 3.14 \][/tex]
### Step 2: Use the Constant [tex]\( k \)[/tex] to Find the Volume of Cylinder B
Now, we know [tex]\( k = 3.14 \)[/tex]. For Cylinder B, we have:
- Radius [tex]\( r_B = 4 \)[/tex] inches,
- Height [tex]\( h_B = 5 \)[/tex] inches.
Plug these values into the volume formula:
[tex]\[ V_B = 3.14 \times 4^2 \times 5 \][/tex]
First, calculate [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Now multiply the values together:
[tex]\[ V_B = 3.14 \times 16 \times 5 \][/tex]
[tex]\[ V_B = 3.14 \times 80 \][/tex]
[tex]\[ V_B = 251.2 \][/tex]
So, the volume of Cylinder B is [tex]\( 251.2 \)[/tex] cubic inches.
### Answer
The volume of Cylinder B is [tex]\( 251.2 \)[/tex] cubic inches.
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula [tex]\( V = k r^2 h \)[/tex], where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( h \)[/tex] is the height,
- [tex]\( k \)[/tex] is a constant of proportionality.
### Step 1: Find the Constant [tex]\( k \)[/tex] Using Cylinder A
We know the volume, radius, and height of Cylinder A:
- Volume [tex]\( V_A = 254.34 \)[/tex] cubic inches,
- Radius [tex]\( r_A = 3 \)[/tex] inches,
- Height [tex]\( h_A = 9 \)[/tex] inches.
Plug these values into the formula to solve for [tex]\( k \)[/tex]:
[tex]\[ 254.34 = k \times 3^2 \times 9 \][/tex]
[tex]\[ 254.34 = k \times 9 \times 9 \][/tex]
Since [tex]\( 3^2 = 9 \)[/tex]:
[tex]\[ 254.34 = k \times 81 \][/tex]
Divide both sides by 81 to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{254.34}{81} \][/tex]
[tex]\[ k = 3.14 \][/tex]
### Step 2: Use the Constant [tex]\( k \)[/tex] to Find the Volume of Cylinder B
Now, we know [tex]\( k = 3.14 \)[/tex]. For Cylinder B, we have:
- Radius [tex]\( r_B = 4 \)[/tex] inches,
- Height [tex]\( h_B = 5 \)[/tex] inches.
Plug these values into the volume formula:
[tex]\[ V_B = 3.14 \times 4^2 \times 5 \][/tex]
First, calculate [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Now multiply the values together:
[tex]\[ V_B = 3.14 \times 16 \times 5 \][/tex]
[tex]\[ V_B = 3.14 \times 80 \][/tex]
[tex]\[ V_B = 251.2 \][/tex]
So, the volume of Cylinder B is [tex]\( 251.2 \)[/tex] cubic inches.
### Answer
The volume of Cylinder B is [tex]\( 251.2 \)[/tex] cubic inches.
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