Question

2d. if the polygon shown below is rotated about the y - 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: Identify the shape after rotation

When the right - triangle \(ABC\) (\(A=(0,0)\), \(B=(0,5)\), \(C=(3,0)\)) is rotated about the \(y\) - axis, it forms a cone.
The formula for the volume of a cone is \(V=\frac{1}{3}\pi r^{2}h\).

Step2: Determine the radius \(r\) and height \(h\) of the cone

For the cone formed by rotating the triangle about the \(y\) - axis:

• The radius \(r\) of the base of the cone is the \(x\) - coordinate of point \(C\), so \(r = 3\).
• The height \(h\) of the cone is the \(y\) - coordinate of point \(B\), so \(h=5\).

Step3: Calculate the volume of the cone

Substitute \(r = 3\) and \(h = 5\) into the volume formula \(V=\frac{1}{3}\pi r^{2}h\).

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Using \(\pi\approx3.14\), we get \(V\approx15\times3.14 = 47.1\)

Answer:

\(47.1\)