Question

two similar cylinders have diameters as shown below. (note: the figures are not drawn to scale.) (a) if the surface area of the cylinder on the left is 125 cm², find the surface area of the cylinder on the right. (b) if the volume of the cylinder on the left is 375 cm³, find the volume of the cylinder on the right. (c) if the height of the cylinder on the left is 30 cm, find the height of the cylinder on the right.

Explanation:

Step1: Recall similarity - ratio formula for similar solids

For two similar cylinders, if the ratio of their corresponding linear dimensions (such as radii or diameters) is \(k\), the ratio of their surface - areas is \(k^{2}\), the ratio of their volumes is \(k^{3}\), and the ratio of their heights is \(k\). The ratio of the diameters of the two cylinders is \(\frac{d_1}{d_2}=\frac{4}{5}\), so \(k = \frac{4}{5}\).

Step2: Solve part (a)

The formula for the surface - area of a cylinder is \(S = 2\pi r^{2}+2\pi rh\). For two similar cylinders, \(\frac{S_1}{S_2}=k^{2}\). Given \(S_1 = 125\ cm^{2}\) and \(k=\frac{4}{5}\), we have \(\frac{S_1}{S_2}=(\frac{4}{5})^{2}=\frac{16}{25}\). Then \(S_2=\frac{125\times25}{16}=\frac{3125}{16}=195.3125\ cm^{2}\).

Step3: Solve part (b)

The formula for the volume of a cylinder is \(V=\pi r^{2}h\). For two similar cylinders, \(\frac{V_1}{V_2}=k^{3}\). Given \(V_1 = 375\ cm^{3}\) and \(k = \frac{4}{5}\), we have \(\frac{V_1}{V_2}=(\frac{4}{5})^{3}=\frac{64}{125}\). Then \(V_2=\frac{375\times125}{64}=\frac{46875}{64}=732.421875\ cm^{3}\).

Step4: Solve part (c)

For two similar cylinders, \(\frac{h_1}{h_2}=k\). Given \(h_1 = 30\ cm\) and \(k=\frac{4}{5}\), we can solve for \(h_2\) by cross - multiplying. \(\frac{h_1}{h_2}=\frac{4}{5}\), so \(h_2=\frac{30\times5}{4}=37.5\ cm\).

Answer:

(a) \(195.3125\ cm^{2}\)
(b) \(732.421875\ cm^{3}\)
(c) \(37.5\ cm\)