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How many times smaller is the volume of a cylinder if the diameter is multiplied by $\frac{1}{2}$ ?

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Solve proportions: word problems

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Q. How many times smaller is the volume of a cylinder if the diameter is multiplied by

\frac{1}{2}

?

Volume Formula: The volume of a cylinder is given by the formula $V = \pi r^2 h$ , where $r$ is the radius and $h$ is the height of the cylinder.
Halving Diameter: If the diameter is halved, then the new diameter is $d/2$ , and the new radius is $r/2$ , since the radius is half of the diameter.
Volume Comparison: We need to compare the volume of the original cylinder with the volume of the new cylinder with the halved diameter. Let's call the original volume $V_1$ and the new volume $V_2$ .
Original Volume: The original volume $V_1$ is $\pi r^2 h$ .
New Volume Calculation: The new volume $V_2$ , with the halved diameter (and thus halved radius), is $\pi(\frac{r}{2})^2h = \pi(\frac{r^2}{4})h = (\frac{1}{4})\pi r^2h$ .
Volume Ratio Calculation: To find out how many times smaller the new volume is compared to the original volume, we divide $V_1$ by $V_2$ : $\frac{V_1}{V_2} = \frac{\pi r^2 h}{(\frac{1}{4})\pi r^2 h}$ .
Simplified Ratio: Simplify the expression: $V_1/V_2 = (\pi r^2 h) / ((1/4)\pi r^2 h) = 4/1 = 4$ .
Final Conclusion: The new volume is $4$ times smaller than the original volume when the diameter of the cylinder is halved.

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