Question
What is the radius of a cone with a height of 99ft and a slant of 101ft?
Answers
GPT-4o mini
To find the radius of the cone, we can use the relationship between the height, radius, and slant height (also termed the lateral height) of a cone. This relationship can be expressed using the Pythagorean theorem.
In a right triangle formed by the radius (r), height (h), and slant height (s), the relationship is given by:
\[
s^2 = r^2 + h^2
\]
Where:
- \( s \) is the slant height (101 ft)
- \( h \) is the height (99 ft)
- \( r \) is the radius we want to find.
We can rearrange the formula to solve for the radius \( r \):
\[
r^2 = s^2 - h^2
\]
Now, substituting the values for \( s \) and \( h \):
\[
r^2 = 101^2 - 99^2
\]
Calculating \( 101^2 \) and \( 99^2 \):
\[
101^2 = 10201
\]
\[
99^2 = 9801
\]
Now, subtract:
\[
r^2 = 10201 - 9801 = 400
\]
Taking the square root of both sides to find \( r \):
\[
r = \sqrt{400} = 20
\]
So, the radius of the cone is \( \boxed{20} \) feet.
In a right triangle formed by the radius (r), height (h), and slant height (s), the relationship is given by:
\[
s^2 = r^2 + h^2
\]
Where:
- \( s \) is the slant height (101 ft)
- \( h \) is the height (99 ft)
- \( r \) is the radius we want to find.
We can rearrange the formula to solve for the radius \( r \):
\[
r^2 = s^2 - h^2
\]
Now, substituting the values for \( s \) and \( h \):
\[
r^2 = 101^2 - 99^2
\]
Calculating \( 101^2 \) and \( 99^2 \):
\[
101^2 = 10201
\]
\[
99^2 = 9801
\]
Now, subtract:
\[
r^2 = 10201 - 9801 = 400
\]
Taking the square root of both sides to find \( r \):
\[
r = \sqrt{400} = 20
\]
So, the radius of the cone is \( \boxed{20} \) feet.