Question

the two cylinders shown below are similar. cylinder 1 has radius ( r_1 ) and height ( h_1 ), while cylinder 2 has radii ( r_2 ) and height ( h_2 ).
if ( \frac{r_1}{r_2}=\frac{h_1}{h_2}=\frac{1}{ka},a > 1 ), and ( k > 1 ), what is the formula for finding the difference in the volumes of these two cylinders?
( v_d=pi r_1^2h_1(k^3 - a^3) )
( v_d=pi r_1^2h_1(k^3a^3 - 1) )
( v_d=pi r^2h(1 - k^3a^3) )
( v_d=pi r_1^2h_1(1 - k^3a^3) )

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is \(V = \pi r^{2}h\). For Cylinder 1, \(V_1=\pi r_{1}^{2}h_{1}\). Since \(\frac{r_{1}}{r_{2}}=\frac{h_{1}}{h_{2}}=\frac{1}{ka}\), we can express \(r_{2}=r_{1}ka\) and \(h_{2}=h_{1}ka\).

Step2: Calculate the volume of Cylinder 2

Using the volume formula \(V = \pi r^{2}h\), for Cylinder 2, \(V_2=\pi r_{2}^{2}h_{2}\). Substitute \(r_{2}=r_{1}ka\) and \(h_{2}=h_{1}ka\) into the formula: \(V_2=\pi(r_{1}ka)^{2}(h_{1}ka)=\pi r_{1}^{2}h_{1}(k^{3}a^{3})\).

Step3: Calculate the difference in volumes

The difference in volumes \(V_d=V_2 - V_1\) (assuming \(V_2>V_1\) as \(ka > 1\)). Substitute \(V_1=\pi r_{1}^{2}h_{1}\) and \(V_2=\pi r_{1}^{2}h_{1}(k^{3}a^{3})\) into the formula: \(V_d=\pi r_{1}^{2}h_{1}(k^{3}a^{3}- 1)\).

Answer:

\(V_d=\pi r_{1}^{2}h_{1}(k^{3}a^{3}- 1)\) (the second option)