Question

if a polyhedron has a volume of 8 cm³ and is dilated by a factor of 1/4, what will be the volume of the dilated polyhedron? round your answer to the nearest thousandth if necessary. volume ≈ ______ cm³

Explanation:

Step1: Recall the volume - dilation formula

If a solid is dilated by a scale factor \(k\), the volume of the new solid \(V_{new}\) is related to the volume of the original solid \(V_{old}\) by the formula \(V_{new}=V_{old}\times k^{3}\). Here, \(V_{old} = 8\space cm^{3}\) and \(k=\frac{1}{4}\).

Step2: Substitute the values into the formula

Substitute \(V_{old}=8\) and \(k = \frac{1}{4}\) into \(V_{new}=V_{old}\times k^{3}\). Then \(V_{new}=8\times(\frac{1}{4})^{3}\).
First, calculate \((\frac{1}{4})^{3}=\frac{1^{3}}{4^{3}}=\frac{1}{64}\).
Then \(V_{new}=8\times\frac{1}{64}\).
Since \(8\times\frac{1}{64}=\frac{8}{64}\), and simplifying \(\frac{8}{64}\) by dividing both the numerator and denominator by 8 gives \(\frac{1}{8}=0.125\).

Answer:

\(0.125\space cm^{3}\)