Question

1d. if the polygon shown below is rotated about the x - axis, what will be the volume of the 3 - dimensional figure that is created? round your answer to the nearest tenth of a unit.

Explanation:

Step1: Determine the shape of the 3 - D figure

When the rectangle is rotated about the \(x\) - axis, it forms a cylinder. The formula for the volume of a cylinder is \(V=\pi r^{2}h\).

Step2: Find the radius \(r\) and height \(h\)

The distance from the \(x\) - axis to the side \(AB\) (or \(CD\)) gives the radius \(r\). The \(y\) - coordinate of point \(B\) is \(- 2\), so \(r = 2\). The length of \(AD\) gives the height \(h\). The \(x\) - coordinate of \(A\) is \(-4\) and of \(D\) is \(4\), so \(h=\vert4-(-4)\vert=8\).

Step3: Calculate the volume

Substitute \(r = 2\) and \(h = 8\) into the formula \(V=\pi r^{2}h\).

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Answer:

\(100.5\)