Question

If a marble of radius 2.1 cm is put into a cylindrical cup full of water of radius 5cm and height 6cm, then how much water flows out of the cylindrical cup
(a) $38.8c{{m}^{3}}$
(b) $55.4c{{m}^{3}}$
(c) $19.4c{{m}^{3}}$
(d) $471.4c{{m}^{3}}$

Answer 1

Hint: First, we draw both the figures of marble and cylindrical cup with given dimensions. Then, we use the condition that the volume of the water that spills out is nearly equal to the volume of the marble which is actually a sphere of radius 2.1 cm. Then, we use the formula of the volume V of the sphere in which r is radius of that sphere as $V=\dfrac{4}{3}\pi {{r}^{3}}$.Then, after substituting the value of radius we get the amount of water that flows out of the cylindrical cup.

Complete step by step answer:
In this question, we are supposed to find the volume of the marble with the radius 2.1 cm and volume of the cylindrical cup of radius 5cm and height 6cm.
So, firstly we draw both the figures of marble and cylindrical cup with dimensions as:

Now, from the condition given in the question that the marble is dropped in the cylindrical cup filled with water and we need to find the amount of water flows out.
So, the volume of the water that spills out is nearly equal to the volume of the marble which is actually a sphere of radius 2.1 cm.
Now, by using the formula of the volume V of the sphere in which r is radius of that sphere is given by:
$V=\dfrac{4}{3}\pi {{r}^{3}}$
Now, by substituting the value of radius of marble given as 2.1cm in the above formula as:
$V=\dfrac{4}{3}\pi {{\left( 2.1 \right)}^{3}}$
Now, solve the above expression to get the value of the water that flows out as:
$\begin{align}
  & V=\dfrac{4}{3}\pi \left( 9.261 \right) \\
 & \Rightarrow V=\dfrac{116.318}{3} \\
 & \Rightarrow V=38.8c{{m}^{3}} \\
\end{align}$
So, the amount of water that flows out of the cylindrical cup is $38.8c{{m}^{3}}$.
Hence, option (a) is correct.

Note: The issue with these kinds of questions is that the extra data is given in the question which is of no use to confuse us that we should go for the formula of the cylindrical volume that is not required to get the amount of the water flowing out. Here , the volume of the water that spills out is nearly equal to the volume of the marble which is actually a sphere of radius 2.1 cm. This can be very easily calculated by using the formula of the volume of the sphere which is a marble in this case as $V=\dfrac{4}{3}\pi {{r}^{3}}$.