Use the image to answer the question.
An illustration shows a cone that rests flat on the circular base. The visible circumference of the base and sides are in solid lines. The circumference of the base at the back that is not visible is indicated by a dashed line. Parts are labeled. A large point is marked in the center of the base. A solid line passes through this point and extends from one edge of the base to the other to indicate the diameter of the base. The line is labeled d equals 10 centimeters. The right side of the cone extending from where the diameter meets the edge of the base up to the point of the cone shows a slant height of 13 centimeters. A dotted line extends from the interior of the vertex of the cone down to the point marked in the center of the base. A right angle is indicated where the dotted height line meets the diameter line in the center of the base.
Find the length of the height of the cone.
(1 point)
The length of the height of the cone is not given in the description.
To find the length of the height of the cone, we can use the Pythagorean theorem.
Let's label the height of the cone as "h".
We already know the diameter of the base (d = 10 cm) and the slant height of the side (s = 13 cm).
Using the Pythagorean theorem, the equation we can use is:
s^2 = r^2 + h^2
Since the diameter is twice the radius (r), we can substitute 10 cm for d:
13^2 = (10/2)^2 + h^2
169 = 25 + h^2
Subtracting 25 from both sides:
144 = h^2
Taking the square root of both sides:
h = √144
h = 12 cm
Therefore, the length of the height of the cone is 12 centimeters.
To find the length of the height of the cone, we can use the Pythagorean theorem. The slant height of the cone (13 centimeters) represents the hypotenuse of a right triangle formed by the height, the slant height, and the radius of the base.
Since the diameter of the base is given as 10 centimeters, the radius is half of that, which is 5 centimeters. Now, we can set up the equation using the Pythagorean theorem:
(height)^2 + (radius)^2 = (slant height)^2
Let's substitute the values we have:
(height)^2 + (5 cm)^2 = (13 cm)^2
Simplifying the equation:
(height)^2 + 25 cm^2 = 169 cm^2
Subtracting 25 cm^2 from both sides:
(height)^2 = 144 cm^2
Taking the square root of both sides:
height = √(144 cm^2)
Calculating the square root:
height = 12 cm
Therefore, the length of the height of the cone is 12 centimeters.