Question

cylinders a and b are similar solids. the base of cylinder a has a circumference of 4π units. the base of cylinder b has an area of 9π units. the dimensions of cylinder a are multiplied by what factor to produce the corresponding dimensions of cylinder b? options: \\(\frac{4}{9}\\), \\(\frac{2}{3}\\), \\(\frac{3}{2}\\), \\(\frac{9}{4}\\)

Explanation:

Step1: Find the radius of cylinder A

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(C = 4\pi\), then \(2\pi r=4\pi\). Dividing both sides by \(2\pi\), we get \(r = 2\).

Step2: Find the radius of cylinder B

The formula for the area of a circle is \(A=\pi R^{2}\). Given \(A = 9\pi\), then \(\pi R^{2}=9\pi\). Dividing both sides by \(\pi\) gives \(R^{2}=9\), so \(R = 3\).

Step3: Find the scale factor

Since the cylinders are similar, the scale factor \(k\) is the ratio of the radii (or any corresponding linear dimensions). The scale factor \(k=\frac{R}{r}\). Substituting \(R = 3\) and \(r = 2\), we get \(k=\frac{3}{2}\).

Answer:

\(\frac{3}{2}\) (the second option)