If the radius of the base of a right circular cylinder is halved, keep

1.	If the radius of the base of a right circular cylinder is halved, keeping the height the same, then find the ratio of the volume of the cylinder thus obtained to the volume of original cylinder?
Answer» Let the radius of the base of original cylinder is r. And the height of the original cylinder is h. ∴ The volume of the original cylinder is \(\pi\)r²ℎ. Given that radius of base of reduced cylinder is half of radius of original cylinder and height will be same. ∴ The radius of reduced cylinder is \(\frac{r}2\) And the height of the reduced cylinder is h. ∴ The volume of the reduced cylinder is \(\pi\)\((\frac{r}{2})^2\) ℎ. Now, the ratio of the volume of reduced cylinder and the volume of original cylinder is \(\frac{Volume\,of\,reduced\,cylinder}{Volume\,of\,original\,cylinder}\) = \(\frac{\pi(\frac{r}2)^2h}{\pi{r}^2h}\) = \(\frac{\pi{r}^2h}{4\pi{r}^2h}\) = \(\frac{1}4\). Hence, the required ratio is 1 : 4.

If the radius of the base of a right circular cylinder is halved, keeping the height the same, then find the ratio of the volume of the cylinder thus obtained to the volume of original cylinder?

Answer»

Let the radius of the base of original cylinder is r.

And the height of the original cylinder is h.

∴ The volume of the original cylinder is \(\pi\)r²ℎ.

Given that radius of base of reduced cylinder is half of radius of original cylinder and height will be same.

∴ The radius of reduced cylinder is \(\frac{r}2\)

And the height of the reduced cylinder is h.

∴ The volume of the reduced cylinder is \(\pi\)\((\frac{r}{2})^2\) ℎ.

Now, the ratio of the volume of reduced cylinder and the volume of original cylinder is

\(\frac{Volume\,of\,reduced\,cylinder}{Volume\,of\,original\,cylinder}\) = \(\frac{\pi(\frac{r}2)^2h}{\pi{r}^2h}\) = \(\frac{\pi{r}^2h}{4\pi{r}^2h}\) = \(\frac{1}4\).

Hence, the required ratio is 1 : 4.

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