A heap of rice is in the form of a cone of base diameter 24 metres and height three and a half metres. Find the volume of the rice. How much canvas cloth is required to exactly cover the heap?
Here is our cone shape. The diameter of the circular base is 24 metres. That’s the distance from one side to the other through the centre. And the height of the heap is three and a half metres. We know that volume is the area of the base times height.
And specifically, when we’re working with cones, we multiply the base times the height by one-third. Multiplying this piece by one-third is really important. If we only multiply the area of the base times the height, we’re finding the volume of a cylinder with the same base and height as this cone.
Back to the value of our cone, the area of the base is found by multiplying 𝜋 times the radius squared. Then we have our height of ℎ and our one-third. We can simplify this expression to say volume equals 𝜋𝑟 squared ℎ over three. From here, we’ll need to plug in all the missing values.
Starting off, we’ll leave 𝜋 as 𝜋 until the very last step. We need to know what the radius squared is. To find the radius, you take the diameter and divide by two. The radius is half of the diameter. Our diameter is 24 metres, and that makes the radius 12 metres.
In the formula, we need to square the radius, 12 squared. We multiply that amount by the height. And we’ve already been given a height of three and a half metres. We take all of this and divide it by three. Again, we’ll leave 𝜋 as 𝜋. 12 squared equals 144. We need to multiply that by three and a half, all over three. 144 times three and a half equals 504. Keep our 𝜋 value, divided by three. 504 is divisible by three.
We can say that our volume is 168𝜋 because 504 divided by three equals 168. If we wanted to find an approximate answer, we can substitute the value 22 over seven for 𝜋, which would look like this: 168 times 22 over seven. This gives us an estimated volume of 528. Volume is a measure of the units cubed. We’re dealing in metres, and so our final answer will be 528 metres cubed.
On to part two of the question, we want to cover the rice heap with a canvas cloth. What kind of measure are we talking about here?
This must be a measure of surface area. It’s about covering the outside of our cone. This surface area is found by multiplying 𝜋 times the radius times the slant height. The slant height of a cone is the distance from the top of the cone down the side to a point on the edge of the base.
We can use a right triangle to calculate the slant height here. We know that the height is three and a half metres and the radius is 12 metres. The length of our slant height squared is equal to three and a half squared plus 12 squared. Three and a half squared equals 12.25. 12 squared equals 144. When we add those together, we get 156.25.
Again, this is the square of our slant height, and that means we need to take the square root on both sides. And then we find the slant height is equal to 12 and a half metres. And then we’re ready to plug in the information back into our surface area formula: 𝜋 times the radius times 12 times the slant height of 12 and a half.
12 and a half times 12 equals 150. We’ll bring down 𝜋, and our surface area is equal to 150 times 𝜋. Just like in the volume, we’ll use 22 over seven for an approximation of 𝜋. Now our problem looks like this: 150 times 22 over seven. 150 times 22 equals 3300. We need to divide that value by seven. When you do that division and round to the nearest hundredth, you get 471.43. Surface area is a measure of your units squared, so our final answer 471.43 metres squared.