Find the volume of a right circular cone with height h and radius of base r.
Introduce a coordinate system to measure the height of the cone.
A cross-section of the cone is a circle.
Using similar triangles, it can be shown that the radius of the cross-section is r(h-x)/h.
Hence, the area of the circle A(x) = π*(radius)2 = π*[ r(h-x)/h]2.
The volume of the cone is 0∫h A(x)dx = 0∫h π*[ r(h-x)/h]2dx.
You may also remember that the formula for the volume of a cone is 1/3*(area of base)*height = 1/3*πr2h. Let's see if these two formulas give the same value for a cone.
Using the TI-83/84
Measure the height h and the radius r of a cone. Store these values in H and R.
An example.
Suppose my cone has a radius of 3 cm and a height of 5 cm.
3→R
5→H
\Y1=π(R(H-X)/H)2
\Y2=1/3*πR2H
You find fnInt under MATH. From the home screen, enter
fnInt(Y1,X,0,H)
ENTER
47.1238898
Y2
ENTER
47.1238898
The volume of my cone is about 47 cubic centimeters.
Task.
You have a Dixie cup. Find its volume using rice, using the regular formula, and using calculus!
