A fixed slant height on a right cone is a cue to use the Pythagorean theorem. The radius, perpendicular height, and slant height form a right triangle. Find the height in terms of the radius, then substitute it into the cone-volume formula. The tempting slip is to use the slant height as the height in that formula.
Hints
- Hint 1
In a right cone, the radius runs across the base and the perpendicular height runs straight up. They meet at a right angle, so the slant height is the side across from that angle. Which side goes alone in the Pythagorean equation?
- Hint 2
You need the height, not its square. After getting alone, take the positive square root because a height is a length.
- Hint 3
A cone’s volume is one-third of its base area times its perpendicular height. The circular base has area , so keep the radius squared when you substitute for the height.
Step-by-step
Find the height, then use cone volume
Step 1Connect the three lengths
No Desmos needed. The right triangle and cone formula give an exact expression in terms of . The radius and perpendicular height are the legs of a right triangle. The slant height is its hypotenuse, the side across from the right angle, so the Pythagorean theorem gives:
- Step 2
Get the height squared alone
The volume formula needs , so first subtract from both sides:
- Step 3
Find the perpendicular height
Take the positive square root, since a height cannot be negative:
- Step 4
Choose the cone-volume formula
A cone has one-third the volume of a cylinder with the same circular base and perpendicular height, so:
Volume uses the perpendicular height , not the slant height.
- Step 5
Write volume in terms of the radius
Substitute the height you found for :
This gives the cone’s volume in cubic centimeters for radius . Choice C.