Hi! Your answer is 125/3 π
Please see an explanation for a clear understanding to the problem.
Any questions about the answer/explanation can be asked through comments! :)
Step-by-step explanation:
For this problem, I'll be demonstrating two methods to solve your problem which are Integration method and Formula method. Let's start with Formula method first!
Formula Method
<u>Volume of Cone Formula</u>
<u /><u />
r is a radius. Therefore, r² is a diameter.
h is height of cone.
Our r is 5. Therefore, 5² = 25.
Our h is 5.
Substitute these values in the equation.
Hence, the volume of a cone is 125/3 pi. If you want in Mixed Number form then It'd be 41 and 2/3 pi [ 41 2/3 pi]
Integration Method - Proof
Let r':r = x:h
Therefore,
r and h both are same (r is still radius while h is still height)
The reason why we add pi because the surface is a circle. If the surface is not a circle, we don't add pi.
The range starts from 0 ≤ x ≤ h
Substitute S(x) in.
Integrate, separate everything in the integrand out except for x^2
As you notice, Integration method is pretty useful, but it's mostly for an advanced volume lesson only.
Integration Method - Problem Solving
Let r':r = x:h
r is given as 5
h is given as 5 as well.
Therefore, r':5 = x:5
The range of volume starts from surface to height. Therefore the range is 0 ≤ x ≤ 5
Notice that both methods have same answer. Hence, we can conclude that the volume of a cone is 125/3 pi units²