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
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
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.