Question

A glass cylinder with diameter 20cm has water to a height of 9cm. A metal cube of 8cm edge is immersed in it completely. Calculate the height by which water will rise in the cylinder.

Answer 1

Hint: In this question use the concept that the height of water that will rise is eventually due the immersion of a cube into water, so the volume of water that rises up is equal to the volume of the cube.

Complete step-by-step answer:
Given data
Diameter of the cylinder = 20 cm.
Height of water in the cylinder = 9 cm
As we know that the radius (r) is half of the diameter (d)
Therefore the radius of the cylinder (r) = $ \dfrac{{20}}{2} = 10 $ cm.

Now let us suppose that the water rises by h cm.
So as we know that the volume displaced is in the form of a cylinder with base radius (r) same as cylinder.
So the volume of water displaced = $ \pi {r^2}h $ cm³.
So the water displaced is after when we placed a 8 cm edge metal cube in the cylinder.
So as we know that the volume (V_C) of the cube is the cube of its side.
 $ \Rightarrow {V_C} = {8^3} = 512{\text{ c}}{{\text{m}}^3} $
So the volume of the cube should be equal to the water displaced.
 $ \Rightarrow \pi {r^2}h = 512 $

Now substitute the values we have,
 $ \Rightarrow \dfrac{{22}}{7} \times {\left( {10} \right)^2} \times h = 512 $
Now simplify the equation we have,
 $ \Rightarrow h = \dfrac{{512 \times 7}}{{22 \times 100}} = 1.629 $ cm
So the total height of water in the cylinder = 9 + 1.629 = 10.629.
So the height by which the water will rise = 1.629 cm.
So this is the required answer.

Note: While solving problems of this kind it is advised to remember the direct basic formula for volume of cylinder and cube. In this problem we have firstly taken out the radius and then found the volume of water displaced which is within a cylinder. Then the volume of the cube is calculated and both are kept equal as explained above to find the required height of water rise.