let thev in terms of slant and radius
Explanation
the volume of a cone is given by
[tex]\text{Volume}_{cone}=\frac{1}{3}\pi r^2h[/tex]where r is the radius and h is the heigth( yellow length)
so, we need to find the heigth in order to use the formula
Step 1
we have then, this triangle
this is a rigth triangle, then let
hypotenuse= slant height
side1 = radius
side2= heigth
we can apply the Pythagorean theorema to find the heigth, the P. T: tell us that the square of the hypotenuse side is equal to the sum of squares of the other two sides,so
[tex]\text{Slant}^2=height^2+radius^2[/tex]isolate heigth
[tex]\begin{gathered} \text{Slant}^2=height^2+radius^2 \\ \text{subtract }radius^2\text{ in both sides} \\ \text{Slant}^2-radius^2=height^2+radius^2-radius^2 \\ \text{Slant}^2-radius^2=height^2 \\ \text{take the square root in both sides} \\ \sqrt{\text{Slant}^2-radius^2}=\sqrt{height^2} \\ \sqrt[]{\text{Slant}^2-radius^2}=\text{heigth} \end{gathered}[/tex]therefore, the heigth is
[tex]heigth=\sqrt[]{\text{Slant}^2-radius^2}[/tex]Step 2
now, replace in the formula
[tex]\begin{gathered} \text{Volume}_{cone}=\frac{1}{3}\pi r^2h \\ \text{Volume}_{cone}=\frac{1}{3}\pi r^2(\sqrt[]{\text{Slant}^2-radius^2}) \end{gathered}[/tex]Now, we finally get a expression to find the volume when the slant and radius are given.
let the height in terms of slant and radius
I hope this helps you
