Question

these solids are similar. find the volume of the larger solid. r = 3 cm v = ? cm³ r = 1 cm v = 9 cm³

Explanation:

Step1: Find the scale factor of the radii

The scale factor \(k\) of the radii is \(\frac{3}{1}=3\).

Step2: Use the volume - scale factor relationship

For similar solids, if the scale factor of corresponding linear dimensions (like radius here) is \(k\), the scale factor of volumes is \(k^{3}\).
The volume of the smaller solid \(V_{s}=9\space cm^{3}\). Let the volume of the larger solid be \(V_{l}\).
We know that \(\frac{V_{l}}{V_{s}}=k^{3}\). Substituting \(k = 3\) and \(V_{s}=9\), we get \(\frac{V_{l}}{9}=3^{3}\).
Since \(3^{3}=27\), then \(V_{l}=9\times27\).

Step3: Calculate the volume of the larger solid

\(V_{l}=9\times27 = 243\space cm^{3}\)

Answer:

\(243\space cm^{3}\)