The radius of a right circular cylinder is decreased by 20% and its height is increased by 25%. What is the absolute value of the percent change in the volume of the cylinder?
+2441 Hello, Guest!
A formula that will be important for this problem is the volume formula for a cylinder. It is the following:
\(V_{\text{cylinder}}=\pi r^2h\)
Let's compute the volume of the original right circular cylinder:
| \(V_{\text{original}}=\pi r_{\text{old}}^2 h_{\text{old}}\) | The original volume is the one where radius and height remained unchanged. |
Now, let's consider a volume wherein the variables are tweaked somewhat.
| \(r_{\text{new}}=r_{\text{old}}-20\%*r_{\text{old}}\) | As a decimal, 20%=0.2 |
| \(r_{\text{new}}=r_{\text{old}}-0.2r_{\text{old}}=0.8r_{\text{old}}\) | Now, let's find how the height was affected. |
| \(h_{\text{new}}=h_{\text{old}}+25\%*h_{old}\) | As a decimal, 25%=0.25 |
| \(h_{\text{new}}=h_{\text{old}}+0.25h_{\text{old}}=1.25h_{\text{old}}\) | Now, we have tweaked both variables to fit the description in the original problem. |
Now, let's find the volume of the new right cylinder:
| \(V_{\text{new}}=\pi r_{\text{new}}^2h_{\text{new}}\) | Plug in the known values for the radius and height. |
| \(V_{\text{new}}=\pi (0.8r_{\text{old}})^2*1.25h_{\text{old}}\) | The only thing left to do is simplify. |
| \(V_{\text{new}}=\pi *0.64r_{\text{old}}^2*1.25h_{\text{old}}\) | |
| \(V_{\text{new}}=0.8\pi r_{\text{old}}^2h_{\text{old}}\) | |
Now the only thing left to do is to calculate the percent change. I want you to try to do that. See what you can do. Check in with me if you would like.
+2441 You are correct in stating that the new volume of the cylinder is 80% of the original right cylinder. Although this is a true observation, it does not answer what the value of the percent is.
However, what 80% tells you is that this is certainly a percent decrease.
| \(V_{\text{new}}=V_{old}-xV_{old}\) | I am subtraction a portion of the old volume to see what the decimal will be. Use substitution here. |
| \(V_{\text{new}}=0.8\pi r_{\text{old}}^2h_{\text{old}}\\ V_{\text{original}}=\pi r_{\text{old}}^2 h_{\text{old}}\) | |
| \(0.8\pi r_{\text{old}}^2 h_{\text{old}}=\pi r_{\text{old}}^2 h_{\text{old}}-x\pi r_{\text{old}}^2 h_{\text{old}}\) | Now, solve. I will factor. |
| \(0.8\pi r_{\text{old}}^2 h_{\text{old}}=\pi r_{\text{old}}^2 h_{\text{old}}(1-x)\) | I will divide both sides by \(\pi r^2 h\). This get's rid of a lot of the variables. |
| \(0.8=1-x\) | |
| \(-0.2=-x\) | |
| \(x=0.2\Rightarrow 20\%\text{ change}\) | The question asks for the percent change, so I convert from a decimal to a percent. |
