If the radius of the base of a conical container is increased by 20% and its height is increased by 10%, by what percentage will its total volume increase?

The correct option is 58.4%.

Step-by-step explanation:

1. Formula for Volume of a Cone:
The volume ($V$) of a conical container with base radius $r$ and height $h$ is given by the formula:

$V=\frac{1}{3}\pi r^{2}h$

2. Express Initial Volume:
Let the original base radius be $r$ and the original height be $h$.
The initial volume ($V_1$) is:

$V_1=\frac{1}{3}\pi r^{2}h$

3. Determine the New Dimensions:
The radius is increased by 20%:

$r^{\prime }=r+0.20r=1.2r$

The height is increased by 10%:

$h^{\prime }=h+0.10h=1.1h$

4. Calculate the New Volume:
Substitute the new dimensions $r^{\prime }$ and $h^{\prime }$ into the volume formula to find the new volume ($V_2$):

$V_2=\frac{1}{3}\pi {\left(1.2r\right)}^2\left(1.1h\right)$

$V_2=\frac{1}{3}\pi \left(1.44r^2\right)\left(1.1h\right)$

$V_2=(1.44\times 1.1)\times \left(\frac{1}{3}\pi r^{2}h\right)$

$V_2=1.584V_1$

5. Calculate Percentage Increase in Volume:
The percentage change in volume is calculated using:

$\text{Percentage Increase}=\frac{V_2-V_1}{V_1}\times 100\%$

$\text{Percentage Increase}=\frac{1.584V_1-V_1}{V_1}\times 100\%$

$\text{Percentage Increase}=(1.584-1)\times 100\%$

$\text{Percentage Increase}=0.584\times 100\%=58.4\%$

Thus, the total volume of the container increases by 58.4%.