Question

now it’s time to practice on your own
what is the volume of the object generated by rotating the square about the horizontal axis?
image of a square with side length 7 in. and a horizontal axis above it
express your answer in terms of \\(\pi\\).
enter your answer in the box.
volume = \\(\square\\) in\\(^3\\)

Explanation:

Step1: Identify the solid formed

When a square is rotated about a horizontal axis (which is along its top side), the resulting solid is a cylinder. The side length of the square (7 in) becomes the radius \( r \) of the base of the cylinder, and also the height \( h \) of the cylinder (since the vertical side of the square is 7 in, which becomes the height when rotated).

Step2: Recall the volume formula for a cylinder

The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \).

Step3: Substitute the values of \( r \) and \( h \)

Here, \( r = 7 \) in and \( h = 7 \) in. Substituting these into the formula:

$$ V = \pi \times (7)^2 \times 7 $$

Step4: Calculate the value

First, calculate \( 7^2 = 49 \), then multiply by 7: \( 49 \times 7 = 343 \). So the volume is \( 343\pi \) cubic inches.

Answer:

\( 343\pi \)