Question

A hemispherical bowl is filled to the brim with a beverage. The contents of the bowl are transferred into a cylindrical vessel whose radius is 50% more than its height. If the diameter is the same for both the bowl and the cylinder, then the amount of the beverage that can be poured from the bowl into the cylindrical vessel is _________.
A. $66{\displaystyle \frac{2}{3}}\mathrm{\%}$
B. $78{\displaystyle \frac{1}{2}}\mathrm{\%}$
C. 100%
D. None of these

Answer 1

Hint: For solving this question we will need to find the volumes of the two vessels. The volume of a hemisphere is given by ${\displaystyle \frac{2}{3}}\pi r^3$ where ‘r’ is the radius of the hemisphere and volume of a cylinder is given by $\pi r^{2}h$ where ‘r’ is the radius of the cylinder and ‘h’ is the height of the cylinder. Volume of the bowl will be equal to the volume of the beverage and then we can compare the volume of the beverage and that of the vessel. After comparing them, we can find out how much of the beverage can be poured into the cylindrical vessel and this will give us our answer.

Complete step by step answer:
Now, it is mentioned in the question that the hemispherical bowl is filled to the brim with the beverage.
Therefore, the volume of the hemispherical bowl is equal to the volume of the beverage.
Let the radius of the hemispherical bowl be ‘r’

Thus, the volume of the beverage will be:
$volume={\displaystyle \frac{2}{3}}\pi r^3$
Now, we will find the volume of the cylindrical vessel.
It is given in the question that the diameter of both the hemispherical bowl and cylindrical vessel are equal. Hence, their radii are also equal.
Thus, the radius of the cylindrical vessel is ‘r’.

Now, it has been given to us that the radius of the cylindrical bowl is 50% more than the height of the vessel. Thus if the height of the vessel is ‘h’ then it is given as:
$\begin{array}{rl}& \Rightarrow r=h+{\displaystyle \frac{50}{100}}h\\ & \Rightarrow r=h+{\displaystyle \frac{1}{2}}h\\ & \Rightarrow r={\displaystyle \frac{3}{2}}h\\ & \Rightarrow h={\displaystyle \frac{2}{3}}r\end{array}$
Now, we have both the radius and height of the cylinder in terms of ‘r’. Now we will calculate the volume of this vessel.
Volume of a cylindrical vessel is given as:
$\begin{array}{rl}& \Rightarrow volume=\pi r^{2}h\\ & h={\displaystyle \frac{2}{3}}r\\ & \Rightarrow volume=\pi r^2\left({\displaystyle \frac{2}{3}}r\right)\\ & \Rightarrow volume={\displaystyle \frac{2}{3}}\pi r^3\end{array}$
Now, we can see that the volume of the beverage is exactly equal to the volume of the cylindrical vessel. Thus, the beverage can be completely transferred from the hemispherical bowl to the cylindrical vessel and the cylindrical vessel will be filled up to the brim.
Thus, 100% of the beverage can be transferred from the bowl to the vessel.

So, the correct answer is “Option C”.

Note: Always take one reference variable in these types of questions like we have taken ‘r’ in this question. Comparison can only be done if the variables in both the quantities to be compared are the same and it is easy to compare one variable. Also, don’t put the value of $\pi$ in these solutions and leave the symbol as it is.