Question

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

• A. 1 : 4
• B. 2 : 1
• C. 1 : 2
• D. 4 : 1

Answer 1

The Correct Option is A

Given: A right-circular cylinder with:

• Original radius = \( r \)
• New radius = \( \frac{r}{2} \) (halved) 
• Height = \( h \) (same)

Step 1: Formula for Volume of a Cylinder

\[ V = \pi r^2 h \]

Step 2: Compute the New Volume

\[ V' = \pi \left(\frac{r}{2}\right)^2 h \] \[ V' = \pi \frac{r^2}{4} h = \frac{1}{4} \pi r^2 h \]

Step 3: Find the Ratio

\[ \frac{V'}{V} = \frac{\frac{1}{4} \pi r^2 h}{\pi r^2 h} = \frac{1}{4} \]

Final Answer: 1:4

Answer 2

To solve the problem of finding the ratio of the volumes of a new cylinder to an original cylinder when the radius is halved, we use the formula for the volume of a cylinder: \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height.

Let's denote:

• \(V_1\) as the volume of the original cylinder with radius \(r\).
• \(V_2\) as the volume of the new cylinder with radius \(\frac{r}{2}\).

The volume of the original cylinder \((V_1)\) is:

\(V_1 = \pi r^2 h\)

The volume of the new cylinder \((V_2)\) is:

\(V_2 = \pi\left(\frac{r}{2}\right)^2 h = \pi \cdot \frac{r^2}{4} \cdot h = \frac{\pi r^2 h}{4}\)

Now, the ratio of the volume of the new cylinder to the original cylinder is:

\(\text{Ratio} = \frac{V_2}{V_1} = \frac{\frac{\pi r^2 h}{4}}{\pi r^2 h} = \frac{1}{4}\)

Therefore, the ratio of the volume of the new cylinder to the volume of the original cylinder is 1:4.