Determine, to the nearest tenth,
the volume of a right cone having a height of 106 centimetres, given that the
perimeter of its base is 318 centimetres. Use 𝜋 equal to 22 over seven.
The cone in this question has a
height of 106 centimetres. This is the distance from the apex
to the centre of the circular base. The perimeter of the base is 318
centimetres. As the base is a circle, this is
the circumference of the circle. We’re asked to calculate the volume
of the cone. This is equal to one-third 𝜋𝑟
squared multiplied by the height.
At present, we know the height of
the cone, but we don’t know its radius. We recall that the circumference of
a circle is equal to two 𝜋𝑟. This means that 318 is equal to two
𝜋𝑟. As we’re told to use 𝜋 as 22 over
seven, 318 is equal to two multiplied by 22 over seven multiplied by 𝑟. The right-hand side of our equation
simplifies to 44 over seven 𝑟. Dividing both sides by 44 over
seven gives us 𝑟 is equal to 1113 over 22. We can now substitute this value
along with the height into our formula for the volume of a cone. The volume is equal to one-third
multiplied by 22 over seven multiplied by 1113 over 22 squared multiplied by
106. Typing this into the calculator
gives us 284219.727 and so on.
We’re asked to give our answer to
the nearest tenth. Therefore, the deciding number is
the two in the hundredths column. As this is less than five, we round
down. The volume of a right cone with
height 106 centimetres and base circumference of 318 centimetres is 284219.7 cubic
centimetres. Note that our units here are cubed
as we’re dealing with volume.