The image shows a cylinder with radius $r = 4$, height $h$, and the following conditions:

• $P_b = P_p$: This indicates that the perimeter (circumference) of the base is equal to the perimeter (circumference) of the side. We'll calculate both.
• We are asked to find the volume $V$ and the lateral surface area $P_c$ of the cylinder.

Step-by-step solution:

1. Determine the height $h$:

   From the condition $P_b = P_p$, the circumference of the base equals the perimeter of the side of the cylinder: $P_b = 2\pi r$ The lateral perimeter (side) of the cylinder is: $P_p = h$ Since $P_b = P_p$, we set: $2\pi r = h$ Substituting $r = 4$: $2\pi(4) = h$ $h = 8\pi$ So, the height $h$ of the cylinder is $8\pi$ units.

2. Volume of the cylinder $V$:

   The volume of a cylinder is given by the formula: $V = \pi r^2 h$ Substituting $r = 4$ and $h = 8\pi$: $V = \pi (4)^2 (8\pi)$ $V = \pi (16)(8\pi)$ $V = 128\pi^2$ The volume of the cylinder is $128\pi^2$ cubic units.

3. Lateral surface area $P_c$:

   The lateral surface area of the cylinder is given by: $P_c = 2\pi r h$ Substituting $r = 4$ and $h = 8\pi$: $P_c = 2\pi(4)(8\pi)$ $P_c = 64\pi^2$ So, the lateral surface area $P_c$ is $64\pi^2$ square units.

Summary:

• Height $h = 8\pi$ units
• Volume $V = 128\pi^2$ cubic units
• Lateral surface area $P_c = 64\pi^2$ square units

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

1. What is the total surface area of the cylinder (including both bases)?
2. How would the volume change if the radius were doubled but the height remained the same?
3. Can you derive the formula for the height from the perimeter condition $P_b = P_p$?
4. How does the volume of a cylinder compare to the volume of a cone with the same radius and height?
5. What is the lateral surface area in terms of decimal approximations (without using $\pi$)?

Tip: Always check units and dimensions carefully to ensure your formula fits the situation.