Question

A conical cup 36 cm high has a diameter of base 28 cm. It is full of water. The water was poured into a cylindrical jar of a radius of base 10 cm. The height of water in the vessel is
(A) 23.52 cm
(B) 16.92 cm
(C) 11.76 cm
(D) 13.65 cm

Answer 1

Hint: The height of the conical cup is 36 cm and has a diameter of base 28 cm. Calculate the volume of the conical cup using the formula, \[Volume=\dfrac{1}{3}\pi {{\left( radius \right)}^{2}}\left( height \right)\] . Since the cone is filled with water, the volume of the water in the conical cup is exactly equal to the volume of the conical cup. Let us assume that the height of water in the cylindrical vessel is x cm. The radius of the circular base of the conical cup is equal to 10 cm. Calculate the volume of the cylindrical vessel using the formula, \[Volume=\pi {{\left( radius \right)}^{2}}\left( height \right)\] . Since the water from the conical cup is poured into the cylindrical vessel. So, the volume of the water in the conical cup is equal to the volume of the water in the cylindrical vessel. Now, compare the volume of the cylindrical vessel and the volume of the conical cup. Then, get the value of x.

Complete step-by-step answer:
According to the question, it is given that a conical cup 36 cm high has a diameter of base 28 cm. It is full of water. The water was poured into a cylindrical jar of a radius of base 10 cm.

The diameter of the base of the conical cup = 28 cm …………………………………(1)
We know that the radius is half of the diameter.
The radius of the circular base of the conical cup = \[\dfrac{28}{2}\] cm = 14 cm ………………………………….(2)
The height of the conical cup = 36 cm …………………………………(3)
We know the formula, \[Volume=\dfrac{1}{3}\pi {{\left( radius \right)}^{2}}\left( height \right)\] …………………………(4)
From equation (2), equation (3), and equation (4), we get
The volume of the conical cup = \[\dfrac{1}{3}\pi {{\left( 14cm \right)}^{2}}\left( 36cm \right)=\pi {{\left( 14cm \right)}^{2}}\left( 12cm \right)\] …………………………………(5)
Since the cone is filled with water, the volume of the water in the conical cup is exactly equal to the volume of the conical cup.
From equation (5), we have the volume of the water.
The volume of the water in the conical cup = \[\pi {{\left( 14 \right)}^{2}}\left( 12 \right)\,c{{m}^{3}}\] …………………………………..(6)

Let us assume that the height of water in the cylindrical vessel is x cm.
The height of the cylindrical vessel = x cm ………………………………………(7)
The radius of the circular base of the conical cup = 10 cm ……………………………….(8)
We know the formula, \[Volume=\pi {{\left( radius \right)}^{2}}\left( height \right)\] …………………………………………..(9)
From equation (7), equation (8), and equation (9), we get
The volume of the water in the cylindrical vessel = \[\pi {{\left( 10cm \right)}^{2}}\left( xcm \right)\] ………………………………….(10)
Since the water from the conical cup is poured into the cylindrical vessel. So, the volume of the water in the conical cup is equal to the volume of the water in the cylindrical vessel.
On comparing equation (6) and equation (10), we get
\[\begin{align}
  & \Rightarrow \pi {{\left( 14 \right)}^{2}}\left( 12 \right)\,c{{m}^{3}}=\pi {{\left( 10cm \right)}^{2}}\left( xcm \right) \\
 & \Rightarrow 196\times 12=100\times x \\
 & \Rightarrow \dfrac{196\times 12}{100}=x \\
 & \Rightarrow \dfrac{2352}{100}=x \\
 & \Rightarrow 23.52=x \\
\end{align}\]
Therefore, the height of the water in the cylindrical vessel is 23.52 cm.
Hence, the correct option is (A).



Note:In this question, one might make a silly mistake while calculating the volume of the conical cup. One might put radius equal to 28 cm in the formula \[Volume=\dfrac{1}{3}\pi {{\left( radius \right)}^{2}}\left( height \right)\] . This is wrong because the diameter is equal to 28 cm. We know that the radius is half of the diameter.
The radius of the circular base of the conical cup is equal to \[\dfrac{28}{2}\] cm.