Answer :
Final Answer:
The volume of the right circular cylinder is [tex]\( \frac{44}{3} \)[/tex] cubic centimeters.
Explanation:
Let r be the radius of the base and h be the height of the right circular cylinder. According to the given information, [tex]\( r = \frac{h}{7} \)[/tex]. The formula for the curved surface area (CSA) of a cylinder is [tex]\( 2\pi rh \)[/tex]. Given that the CSA is 176 sq. cm, we can substitute [tex]\( r = \frac{h}{7} \)[/tex] into the formula:
[tex]\[ 2\pi \left(\frac{h}{7}\right)h = 176 \][/tex]
Solving for h in the above equation gives h = 14 cm. Substituting this value of h into [tex]\( r = \frac{h}{7} \)[/tex] gives r = 2 cm. Now, using the formula for the volume of a cylinder [tex]\( V = \pi r^2 h \)[/tex], we can calculate the volume:
[tex]\[ V = \pi \left(2\right)^2 \cdot 14 = \frac{44}{3} \pi \][/tex]
Thus, the volume of the right circular cylinder is [tex]\( \frac{44}{3} \)[/tex] cubic centimeters. This demonstrates the relationship between the radius, height, and volume of the cylinder, given the information about the curved surface area.
