To solve this problem, we need to find the circumference of the base of a cylinder given the height and the volume. Here is the step-by-step approach:
1. Understand the Problem:
- We are provided with the height (h) of the cylinder, which is 7 cm.
- The volume (V) of the cylinder is given as 197.9 cubic cm.
- We need to find the circumference (C) of the base of the cylinder.
2. Recall the Volume Formula of a Cylinder:
The volume (V) of a cylinder is given by the formula:
[tex]\[
V = \pi \times r^2 \times h
\][/tex]
Where [tex]\( r \)[/tex] is the radius of the base, [tex]\( h \)[/tex] is the height, and [tex]\( \pi \)[/tex] (pi) is approximately 3.141592653589793.
3. Rearrange the Volume Formula to Solve for [tex]\( r^2 \)[/tex]:
By rearranging the volume formula, we get:
[tex]\[
r^2 = \frac{V}{\pi \times h}
\][/tex]
Plug in the given values for volume and height:
[tex]\[
r^2 = \frac{197.9}{3.141592653589793 \times 7}
\][/tex]
Simplifying this, we obtain:
[tex]\[
r^2 \approx 8.999075210824596
\][/tex]
4. Find the Radius [tex]\( r \)[/tex]:
To find the radius [tex]\( r \)[/tex], we take the square root of [tex]\( r^2 \)[/tex]:
[tex]\[
r \approx \sqrt{8.999075210824596}
\][/tex]
Calculating this, we get:
[tex]\[
r \approx 2.9998458645111414 \, \text{cm}
\][/tex]
5. Find the Circumference (C):
The circumference of the base of the cylinder can be found using the formula for the circumference of a circle:
[tex]\[
C = 2 \times \pi \times r
\][/tex]
Substituting the radius we found:
[tex]\[
C = 2 \times 3.141592653589793 \times 2.9998458645111414
\][/tex]
Simplifying this, we get:
[tex]\[
C \approx 18.848587459699846 \, \text{cm}
\][/tex]
Hence, the circumference of the base of the cylinder is approximately 18.85 cm.
