Question

Prove that the twice of the volume of a cylinder is equal to the product of its radius of base and curved surface.

Hint:

When working with cylinders, remember the formulas for volume and curved surface area: \( V = \pi r^2 h \) and \( A_{\text{curved}} = 2 \pi r h \).

Answer 1

The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h, \] where \( r \) is the radius of the base and \( h \) is the height of the cylinder.
The curved surface area \( A_{\text{curved}} \) of the cylinder is given by:
\[ A_{\text{curved}} = 2\pi r h. \] Now, we need to prove that twice the volume is equal to the product of the radius and the curved surface area:
\[ 2V = r \cdot A_{\text{curved}}. \] Substitute the expressions for \( V \) and \( A_{\text{curved}} \):
\[ 2 \times \pi r^2 h = r \times 2 \pi r h. \] Simplify both sides:
\[ 2 \pi r^2 h = 2 \pi r^2 h. \] Since both sides are equal, we have proved that:
\[ 2V = r \cdot A_{\text{curved}}. \]
Conclusion:
Twice the volume of a cylinder is indeed equal to the product of its radius of base and the curved surface area.