Question
Use the image to answer the question.
A cone is marked with a height of 12 centimeters and a diameter of 11 centimeters, both shown as perpendicular dotted lines. The slant height is 9 centimeters.
What is the height of the right circular cone?
11 cm
12 cm
5.5 cm
9 cm
Since the slant height of the cone is 9 centimeters, the height of the right circular cone can be found by using the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the height is one of the other sides, the slant height is the hypotenuse, and the radius (half of the diameter) is the other side.
Using the Pythagorean theorem: height^2 + radius^2 = slant height^2
Substituting the given values: height^2 + (11/2)^2 = 9^2
height^2 + 121/4 = 81
height^2 = 81 - 121/4
height^2 = 324/4 - 121/4
height^2 = (203 - 121) / 4
height^2 = 82 / 4
height^2 = 41 / 2
Taking the square root of both sides: height = √(41/2)
So, the height of the right circular cone is approximately 5.7 cm. However, none of the given answer choices match this value. The closest answer choice is 5.5 cm, so that is the best option available.