Question

a triangular prism has a base area of 20 cm² and a height of 13 cm. if the prism is dilated with a scale factor of k = 0.5, what is the new volume? 260 cm³ 130 cm³ 65 cm³ 32.5 cm³

Explanation:

Step1: Find the original volume

The volume formula of a prism is \(V = B\times h\), where \(B\) is the base area and \(h\) is the height.
Given \(B = 20\space cm^{2}\) and \(h=13\space cm\), the original volume \(V_{original}=20\times13 = 260\space cm^{3}\).

Step2: Use the scale - factor for volume

When a three - dimensional object is dilated with a scale factor \(k\), the new volume \(V_{new}=V_{original}\times k^{3}\).
Here \(k = 0.5\), so \(V_{new}=260\times(0.5)^{3}\).
Calculate \((0.5)^{3}=0.5\times0.5\times0.5 = 0.125\).
Then \(V_{new}=260\times0.125\).
\(260\times0.125=\frac{260}{8}=32.5\space cm^{3}\).

Answer:

\(32.5\space cm^{3}\)