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# help pleease

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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?

Jun 6, 2018

#1
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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.

Jun 6, 2018
#2
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Jun 6, 2018
#3
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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.
TheXSquaredFactor  Jun 6, 2018