Video: CBSE Class X • Pack 4 • 2015 • Question 18

CBSE Class X • Pack 4 • 2015 • Question 18

05:07

Video Transcript

A hemispherical bowl with an internal diameter of 36 centimetres was completely full with a liquid. This liquid was poured into 72 cylindrical bottles of diameter six centimetres. All of the bottles are completely filled, but 10 percent of the liquid was spilled during the transfer. Find the height of each bottle.

The volume of a hemisphere is equal to two-thirds multiplied by 𝜋𝑟 cubed. We’re told in the question that the diameter is equal to 36 centimetres. This means that the radius must be equal to 18 centimetres as the radius is half of the diameter. The volume of the hemisphere is, therefore, given by two-thirds multiplied by 𝜋 multiplied by 18 cubed.

We will look at simplifying this later. The liquid was poured into cylindrical bottles. And the volume of a cylinder is equal to 𝜋𝑟 squared multiplied by ℎ. Once again, we were told the diameter. The diameter of each bottle was equal to six centimetres. This means that the radius of each bottle was equal to three centimetres. The height of the cylinder was unknown. We will call this ℎ.

Therefore, the volume of each cylinder is 𝜋 multiplied by three squared multiplied by ℎ. There were 72 of these bottles in total. Therefore, we need to multiply this volume by 72. The total volume of the 72 bottles is 𝜋 multiplied by three squared multiplied by ℎ multiplied by 72. Both of these volumes are in centimetres cubed. However, it is sometimes easier to leave the unit out of our calculation. We must make sure that we put the units back in at the end.

We were told that 10 percent of the liquid was spilled in transfer. This means that the volume of liquid in the 72 bottles is 90 percent of the volume of liquid in the bowl. 100 percent minus 10 percent is equal to 90 percent. 90 percent can be written as a fraction as nine tenths or nine over 10. We can, therefore, say that the volume in the hemisphere multiplied by nine tenths is equal to the volume in the cylinders.

Substituting in our two expressions gives us two-thirds multiplied by 𝜋 multiplied by 18 cubed multiplied by nine tenths is equal to 𝜋 multiplied by three squared multiplied by ℎ multiplied by 72.

Our first step here is to divide both sides of the equation by 𝜋. We now need to try and cancel the numbers to try and work out our value of ℎ. Two-thirds multiplied by 18 multiplied by 18 multiplied by 18 multiplied by nine tenths is equal to three multiplied by three multiplied by ℎ multiplied by 72.

We could start by cancelling the two fractions on the left-hand side. Two is a factor of two and 10. And three is a factor of three and nine. The first fraction becomes one and the second fraction becomes three-fifths. 18 can be written as nine multiplied by two. And 72 can be written as nine multiplied by two multiplied by two multiplied by two as 72 is equal to nine multiplied by eight.

Our next step is to cancel the like terms on both sides. We can divide both sides by nine. We can also divide both sides by eight, two multiplied by two multiplied by two. Both sides are also divisible by three. This leaves us with nine multiplied by nine multiplied by one-fifth is equal to three multiplied by ℎ.

Dividing both sides by three gives us three multiplied by nine multiplied by one-fifth is equal to ℎ. Three multiplied by nine is equal to 27 and 27 multiplied by one-fifth is twenty-seven fifths or 27 over five. Twenty-seven fifths is equal to five and two-fifths as 27 divided by five is five remainder two. This is equal to 5.4.

We can, therefore, say that the height of each cylindrical bottle is 5.4 centimetres.

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