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Answer :
We are given that the volume of a cylinder varies jointly with the square of its radius and its height. This relationship is expressed 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, and [tex]$k$[/tex] is a constant.
Step 1. Find the constant [tex]$k$[/tex] using Cylinder A
For Cylinder A, we have:
- Volume: [tex]$V_A = 254.34$[/tex] cubic inches,
- Radius: [tex]$r_A = 3$[/tex] inches,
- Height: [tex]$h_A = 9$[/tex] inches.
Substitute these values into the formula:
[tex]$$
254.34 = k \cdot 3^2 \cdot 9.
$$[/tex]
Calculate [tex]$3^2$[/tex]:
[tex]$$
3^2 = 9.
$$[/tex]
Then the equation becomes:
[tex]$$
254.34 = k \cdot 9 \cdot 9.
$$[/tex]
Multiply [tex]$9 \cdot 9$[/tex]:
[tex]$$
254.34 = k \cdot 81.
$$[/tex]
Now, solve for [tex]$k$[/tex]:
[tex]$$
k = \frac{254.34}{81} \approx 3.14.
$$[/tex]
Step 2. Compute the volume of Cylinder B
For Cylinder B, the given dimensions are:
- Radius: [tex]$r_B = 4$[/tex] inches,
- Height: [tex]$h_B = 5$[/tex] inches.
Now substitute the values into the formula along with the computed [tex]$k$[/tex]:
[tex]$$
V_B = k \cdot r_B^2 \cdot h_B.
$$[/tex]
First, find [tex]$r_B^2$[/tex]:
[tex]$$
r_B^2 = 4^2 = 16.
$$[/tex]
Now substitute all the values:
[tex]$$
V_B = 3.14 \cdot 16 \cdot 5.
$$[/tex]
Multiply [tex]$16 \cdot 5$[/tex]:
[tex]$$
16 \cdot 5 = 80.
$$[/tex]
Then:
[tex]$$
V_B = 3.14 \cdot 80 = 251.2.
$$[/tex]
Thus, the volume of Cylinder B is
[tex]$$
\boxed{251.2 \text{ cubic inches}}.
$$[/tex]
[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, and [tex]$k$[/tex] is a constant.
Step 1. Find the constant [tex]$k$[/tex] using Cylinder A
For Cylinder A, we have:
- Volume: [tex]$V_A = 254.34$[/tex] cubic inches,
- Radius: [tex]$r_A = 3$[/tex] inches,
- Height: [tex]$h_A = 9$[/tex] inches.
Substitute these values into the formula:
[tex]$$
254.34 = k \cdot 3^2 \cdot 9.
$$[/tex]
Calculate [tex]$3^2$[/tex]:
[tex]$$
3^2 = 9.
$$[/tex]
Then the equation becomes:
[tex]$$
254.34 = k \cdot 9 \cdot 9.
$$[/tex]
Multiply [tex]$9 \cdot 9$[/tex]:
[tex]$$
254.34 = k \cdot 81.
$$[/tex]
Now, solve for [tex]$k$[/tex]:
[tex]$$
k = \frac{254.34}{81} \approx 3.14.
$$[/tex]
Step 2. Compute the volume of Cylinder B
For Cylinder B, the given dimensions are:
- Radius: [tex]$r_B = 4$[/tex] inches,
- Height: [tex]$h_B = 5$[/tex] inches.
Now substitute the values into the formula along with the computed [tex]$k$[/tex]:
[tex]$$
V_B = k \cdot r_B^2 \cdot h_B.
$$[/tex]
First, find [tex]$r_B^2$[/tex]:
[tex]$$
r_B^2 = 4^2 = 16.
$$[/tex]
Now substitute all the values:
[tex]$$
V_B = 3.14 \cdot 16 \cdot 5.
$$[/tex]
Multiply [tex]$16 \cdot 5$[/tex]:
[tex]$$
16 \cdot 5 = 80.
$$[/tex]
Then:
[tex]$$
V_B = 3.14 \cdot 80 = 251.2.
$$[/tex]
Thus, the volume of Cylinder B is
[tex]$$
\boxed{251.2 \text{ cubic inches}}.
$$[/tex]
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