Video Transcript
By modeling the trunk of the tree
as a cylinder and the head of the tree as a sphere, ignoring any air between the
leaves and branches, work out an estimate for the volume of the tree seen in the
given figure. Give your answer in terms of
𝜋.
In this question, we’re working out
an estimate for the volume of the tree because in reality it won’t be a perfect
sphere and a perfect cylinder, but these are reasonable models. We need to recall two key
formulae. Firstly, the volume of a sphere is
calculated using the formula four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius of the
sphere. And secondly, the volume of a
cylinder is 𝜋𝑟 squared ℎ, where 𝑟 is the radius of the circular base and ℎ is the
height of the cylinder.
Let’s consider the volume of the
leaves first of all, which we’re modeling as a sphere. From the diagram, we can see that
the diameter of this sphere is nine feet. The radius will be half of
this. So, the radius of this sphere is
4.5 or nine over two feet. We have then that the volume of the
leaves modeled as a sphere is four-thirds multiplied by 𝜋 multiplied by nine over
two cubed. To cube nine over two, we can cube
the numerator and denominator separately. Nine cubed is 729 and two cubed is
eight. We can then cancel a factor of four
from the numerator and denominator and also a factor of three, which leaves 243 over
two 𝜋. Notice that the question asked us
to give our answer in terms of 𝜋, so we won’t evaluate this.
Next, we consider the volume of the
tree trunk, which we’re modeling as a cylinder. We can see that the height of the
cylinder is eight feet and the diameter of its base is 1.5 feet. If we write this as a fraction,
it’s three over two. And so, the radius will be half of
this. We can double the denominator to
give a radius of three over four feet. The volume of the trunk is,
therefore, estimated as 𝜋 multiplied by three-quarters squared multiplied by
eight.
Again, we can square the numerator
and denominator of our fraction separately to give nine over 16. And then, we can cancel a factor of
eight from the numerator and denominator to give nine 𝜋 over two as the volume of
the tree trunk. The total volume of the tree then
can be found by adding these two values together, 243 over two 𝜋 plus nine over two
𝜋. 243 over two plus nine over two is
252 over two. And 252 divided by two is 126. So, by modeling this tree as a
composite solid composed of a cylinder and a sphere, we found an estimate for the
total volume of the tree to be 126𝜋 cubic feet.
