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:
\(\[
V = \pi \times r^2 \times h
\]\)
Where \(\( r \)\) is the radius of the base, \(\( h \)\) is the height, and \(\( \pi \)\) (pi) is approximately 3.141592653589793.
3. Rearrange the Volume Formula to Solve for \(\( r^2 \)\):
By rearranging the volume formula, we get:
\(\[
r^2 = \frac{V}{\pi \times h}
\]\)
Plug in the given values for volume and height:
\(\[
r^2 = \frac{197.9}{3.141592653589793 \times 7}
\]\)
Simplifying this, we obtain:
\(\[
r^2 \approx 8.999075210824596
\]\)
4. Find the Radius \(\( r \)\):
To find the radius \(\( r \)\), we take the square root of \(\( r^2 \)\):
\(\[
r \approx \sqrt{8.999075210824596}
\]\)
Calculating this, we get:
\(\[
r \approx 2.9998458645111414 \, \text{cm}
\]\)
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:
\(\[
C = 2 \times \pi \times r
\]\)
Substituting the radius we found:
\(\[
C = 2 \times 3.141592653589793 \times 2.9998458645111414
\]\)
Simplifying this, we get:
\(\[
C \approx 18.848587459699846 \, \text{cm}
\]\)
Hence, the circumference of the base of the cylinder is approximately 18.85 cm.
