Video Transcript
Determine to the nearest tenth the
volume of a right cone having a height of 106 centimeters, given that the perimeter
of its base is 318 centimeters. Use 𝜋 equal to 22 over seven.
As we are dealing with a right
cone, we know that the apex lies directly above the centroid of the circular
base. This means that the height and
radius are at right angles or perpendicular to one another. We are told that this perpendicular
distance or height is equal to 106 centimeters. We are also told that the perimeter
or circumference of the base is equal to 318 centimeters. We recall that the volume of a cone
𝑉 is equal to one-third 𝜋𝑟 squared multiplied by ℎ. In order to calculate this value,
we firstly need to work out the radius of the circular base.
The circumference of any circle 𝐶
is equal to two 𝜋𝑟. In this question, we know that the
circumference is 318 centimeters. Therefore, 318 is equal to two
𝜋𝑟. Dividing both sides of this
equation by two, we get 159 is equal to 𝜋 multiplied by 𝑟. At this stage, we could substitute
22 over seven for 𝜋. However, we will simply divide both
sides of the equation by 𝜋, giving us 𝑟 is equal to 159 over 𝜋. The radius of the circular base of
our cone is 159 over 𝜋 centimeters.
We can now substitute the values of
ℎ and 𝑟 into our formula for the volume. 𝑉 is equal to one-third multiplied
by 𝜋 multiplied by 159 over 𝜋 all squared multiplied by 106. Substituting in our value for 𝜋
and typing this into the calculator gives us an answer of 284219.727 and so on. We are asked to round our answer to
the nearest tenth. This is the same as rounding to one
decimal place, giving us a volume of 284219.7 cubic centimeters. Note that our units for volume will
always be cubic units.
