About this video
A cone has volume 4 cubic meters. The instructor scales its radius by 2.5 and its height by 7, then shows the new volume can be found without recomputing from scratch: radius enters squared, so it multiplies volume by 6.25, and height enters linearly, multiplying by 7. The result is 4 times 6.25 times 7, which equals 175 cubic meters.
You'll be able to
- write the volume formula for a right circular cone.
- predict how scaling the radius and height changes volume.
- compute the new volume of a scaled cone.
Key words
- Right circular cone
- A cone whose base is a circle and whose apex sits directly above the base centre.
- Scale factor
- The number by which a dimension is multiplied when a shape is enlarged or reduced.
- Volume of a cone
- One third times pi times the radius squared times the height.
Transcript
- [Instructor] The volume of right circular cone A is four cubic meters. What is the volume in cubic meters of a right circular cone with 2.5 times the radius and seven times the height of cone A. Pause this video and think about this on your own before we work through this together.
All right. So let's think about the volume of cone A. Well, we know it's four cubic meters. We also know what the formula for this is. It's going to be 1/3 times pi times the radius of the circular base squared times the height and that's going to be, of course, four cubic meters, so that's what they told us.
Now, let's think about the volume of this new, large cone, larger cone. What's it going to be? Well, it's going to be 1/3 times pi. Now what's the radius going to be for this larger one? Well, it says that it is 2.5 times the radius, so if r is the radius of our original cone, now we're going to have 2.5r squared.
That's the radius of this larger cone, the radius of its base of the larger cone and then what's its height? Well, instead of a height of just h, we're now going to have a height of seven times h, so let me write this times seven h and now what is this going to give us?
Well, let's see. This is going to be equal to 1/3 pi times 2.5 squared is 6.25. So it's going to be 6.25r squared times 7h times 7h and now what I'm going to do is I'm gonna take out all of the parts of this expression that essentially make up the volume of A and then see what we're scaling it by.
So if we take, we have a 1/3 pi r squared h, so 1/3 pi r squared h. So this is equal to 1/3 pi r squared h times, what do we have leftover? We have 6.25 times seven. So times 6.25 times seven. Now we know what I have here in black. This is the volume of our original cone, which they told us is four cubic meters.
So this is equal to four right over here and so let's see what's four times 6.25? That's going to be 24 plus one, so it's 25 times seven is going to be equal to 175 cubic meters, which is exactly what we have here for choice C. Now another way that you could have thought about this is saying, all right, my volume is based on the square of the radius and the height, so if I increase my radius by 2.5 times, that's gonna increase my volume by the square of that by 6.25 times and if I increase my height by a factor of seven, well, I just have an h there, not an h squared, so that's going to increase my volume by a factor of seven.
So overall, we're going to increase our volume by a factor of seven times 6.25 and then you multiply that times the four cubic meters and you once again get to the 175 meters cubed.
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