Question

what is the volume of the object generated by rotating the rectangle about the vertical axis?
give your answer in terms of \\(\pi\\).
2 in.
5 in.
axis
enter your answer in the box.
\\(\square\\) in\\(^3\\)

Explanation:

Step1: Identify the shape after rotation

When a rectangle is rotated about a vertical axis, it forms a cylinder. The radius of the cylinder ($r$) is the horizontal distance from the axis to the rectangle, which is 5 in, and the height of the cylinder ($h$) is the vertical side of the rectangle, which is 2 in.

Step2: Use the formula for the volume of a cylinder

The formula for the volume $V$ of a cylinder is $V = \pi r^2 h$. Substitute $r = 5$ and $h = 2$ into the formula.

Step3: Calculate the volume

First, calculate $r^2$: $5^2 = 25$. Then multiply by $h$: $25\times2 = 50$. Then multiply by $\pi$: $V=\pi\times25\times2 = 50\pi$.

Answer:

$50\pi$