Re: If radius of a right circular cylinder is increased by
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25 May 2025, 04:15
We have a right circular cylinder where:
- The radius is increased by $\(\mathbf{2 0 \%}\)$.
- The height remains the same.
We need to find the percentage change in the volume of the cylinder after this increase in radius.
Formula for Volume of a Cylinder
The volume $V$ of a right circular cylinder is given by:
$$
\(V=\pi r^2 h\)
$$
where:
- $r$ is the radius,
- $h$ is the height.
Initial Volume
Let's denote:
- Original radius $\(=r\)$,
- Original height $\(=h\)$,
- Original volume $\(=V_{\text {original }}=\pi r^2 h\)$.
After Increasing the Radius
The radius is increased by $\(20 \%\)$. So, the new radius $\(r_{\text {new }}\)$ is:
$$
\(r_{\text {new }}=r+0.20 r=1.20 r\)
$$
The height remains the same: $h$.
New Volume
The new volume $\(V_{\text {new }}\)$ is:
$$
\(V_{\text {new }}=\pi\left(r_{\text {new }}\right)^2 h=\pi(1.20 r)^2 h=\pi\left(1.44 r^2\right) h=1.44 \pi r^2 h\)
$$
Calculating the Change in Volume
The original volume was $\pi r^2 h$, and the new volume is $1.44 \pi r^2 h$.
The increase in volume is:
$$
\(V_{\text {new }}-V_{\text {original }}=1.44 \pi r^2 h-\pi r^2 h=0.44 \pi r^2 h\)
$$
Percentage Increase in Volume
To find the percentage increase:
$$
\(\text { Percentage Increase }=\left(\frac{V_{\text {new }}-V_{\text {original }}}{V_{\text {original }}}\right) \times 100 \%=\left(\frac{0.44 \pi r^2 h}{\pi r^2 h}\right) \times 100 \%=0.44 \times 100 \%\)
$$
Verification
Let's verify with an example. Suppose:
- Original radius $\(r=5$\) units,
- Height $\(h=10\)$ units.
Original volume:
$$
\(V=\pi(5)^2 \times 10=250 \pi\)
$$
New radius:
$$
\(r_{\text {new }}=5+0.20 \times 5=6 \text { units }\)
$$
New volume:
$$
\(V_{\text {new }}=\pi(6)^2 \times 10=360 \pi\)
$$
Increase in volume:
$$
\(360 \pi-250 \pi=110 \pi\)
$$
Percentage increase:
$$
\(\left(\frac{110 \pi}{250 \pi}\right) \times 100 \%=\left(\frac{110}{250}\right) \times 100 \%=0.44 \times 100 \%=44 \%\)
$$
This confirms our earlier calculation.
Why Not Other Options?
- 20\%: This would be the increase if the volume depended linearly on the radius, but it depends on $\(r^2\)$.
- 37\%: Doesn't match our calculation.
- $\(\mathbf{7 2 . 8 \%}\)$ : This would be the case if both radius and height increased by $\(20 \%\)$, leading to $\((1.20)^3=1.728$ or $72.8 \%\)$ increase, but height is constant here.
- 64.8\%: Doesn't correspond to any straightforward calculation in this context.
Final Answer
The percentage change in the volume of the cylinder when the radius is increased by $20 \%$ (with height constant) is $\mathbf{4 4 \%}$.
Correct Option: \(44\%\)