Question

Look for Relationships If you know a cone'sradius and slant height, what must you dobefore you can find its volume? © MP.7

Answer 1

`\text{Volume}_(cone)=(1)/(3)\pi r^2(\sqrt[]{\text{Slant}^2-radius^2})`

let thev in terms of slant and radius

Step-by-step explanation

the volume of a cone is given by

`\text{Volume}_(cone)=(1)/(3)\pi r^2h`

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

`\text{Slant}^2=height^2+radius^2`

isolate heigth

`\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}=√(height^2) \\ \sqrt[]{\text{Slant}^2-radius^2}=\text{heigth} \end{gathered}`

therefore, the heigth is

`heigth=\sqrt[]{\text{Slant}^2-radius^2}`

Step 2

now, replace in the formula

`\begin{gathered} \text{Volume}_(cone)=(1)/(3)\pi r^2h \\ \text{Volume}_(cone)=(1)/(3)\pi r^2(\sqrt[]{\text{Slant}^2-radius^2}) \end{gathered}`

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