Question
13 cm
d = 10 cm
Find the length of the height of the cone.
(1 point)
d = 10 cm
Find the length of the height of the cone.
(1 point)
Answers
GPT-4o mini
To find the height of a cone when you know the radius and the slant height, you can use the Pythagorean theorem.
Let:
- \( r \) be the radius of the cone's base (in this case, \( r = 13 \, \text{cm} \))
- \( d \) be the slant height (in this case, \( d = 10 \, \text{cm} \))
- \( h \) be the height of the cone
According to the Pythagorean theorem, the relationship between the radius, height, and slant height of the cone can be given by the formula:
\[
d^2 = r^2 + h^2
\]
Substituting the given values:
\[
10^2 = 13^2 + h^2
\]
Calculating the squares:
\[
100 = 169 + h^2
\]
Now, we isolate \( h^2 \):
\[
h^2 = 100 - 169
\]
\[
h^2 = -69
\]
Since the result is negative, this implies that the given values \( r = 13 \, \text{cm} \) and \( d = 10 \, \text{cm} \) are not valid for a cone, as the slant height cannot be less than the radius of the cone's base. In a real cone, the slant height must be greater than or equal to the radius.
Thus, there is no valid height given these dimensions.
Let:
- \( r \) be the radius of the cone's base (in this case, \( r = 13 \, \text{cm} \))
- \( d \) be the slant height (in this case, \( d = 10 \, \text{cm} \))
- \( h \) be the height of the cone
According to the Pythagorean theorem, the relationship between the radius, height, and slant height of the cone can be given by the formula:
\[
d^2 = r^2 + h^2
\]
Substituting the given values:
\[
10^2 = 13^2 + h^2
\]
Calculating the squares:
\[
100 = 169 + h^2
\]
Now, we isolate \( h^2 \):
\[
h^2 = 100 - 169
\]
\[
h^2 = -69
\]
Since the result is negative, this implies that the given values \( r = 13 \, \text{cm} \) and \( d = 10 \, \text{cm} \) are not valid for a cone, as the slant height cannot be less than the radius of the cone's base. In a real cone, the slant height must be greater than or equal to the radius.
Thus, there is no valid height given these dimensions.