the pythagorean theorem in three dimensions practice
find the length of the height of a cone.
13 cm is the slant
10 cm is the base
help
To find the length of the height of a cone, you can use the Pythagorean theorem in three dimensions. The slant height acts as the hypotenuse, the base acts as one of the legs, and the height acts as the other leg.
Let's label the height of the cone as h, and the slant height as s. The base of the cone is not used in this calculation.
Using the Pythagorean theorem, we have:
s^2 = h^2 + r^2
Here, r represents the radius of the base of the cone. However, since we only have information about the slant height and base, we need to use the properties of a right triangle in a cone to find the value of r.
The radius is related to the height and base by the Pythagorean theorem in two dimensions:
r^2 = h^2 + (base/2)^2
In this case, the base of the cone is given as 10 cm, so we can substitute it in the equation above.
r^2 = h^2 + (10/2)^2
r^2 = h^2 + 25
r^2 - h^2 = 25
Replacing the value of r^2 - h^2 in the first equation with 25:
s^2 = h^2 + r^2
s^2 = h^2 + (r^2 - h^2)
s^2 = 25 + h^2
h^2 = s^2 - 25
Now, we can substitute the given values and calculate the height of the cone.
s = 13 cm
h^2 = (13 cm)^2 - 25
h^2 = 169 cm^2 - 25
h^2 = 144 cm^2
h = √144 cm
h = 12 cm
Therefore, the length of the height of the cone is 12 cm.