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.
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 know that the slant height (labeled as 13 cm) is the hypotenuse of a right triangle formed by the height, the base radius (labeled as 5 cm), and the slant height.
Using the Pythagorean Theorem, we have:
h^2 + 5^2 = 13^2
Simplifying, we get:
h^2 + 25 = 169
Subtracting 25 from both sides, we have:
h^2 = 144
Taking the square root of both sides, we get:
h = 12
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 Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle formed by the height of the cone (h), the radius of the base (r), and the slant height of the cone (l).
We know that the diameter of the base (d) is equal to 10 centimeters, so the radius (r) is half of that, or 5 centimeters.
The slant height of the cone (l) is given as 13 centimeters.
Let's label the right triangle:
- The hypotenuse is l (the slant height)
- One leg is r (the radius)
- The other leg is h (the height)
According to the Pythagorean theorem, we have the equation: l^2 = r^2 + h^2
Plugging in the known values, we get: 13^2 = 5^2 + h^2
Simplifying, we have: 169 = 25 + h^2
Subtracting 25 from both sides, we get: 144 = h^2
Taking the square root of both sides, we find: h = √144
Evaluating the square root, h ≈ 12
Therefore, the length of the height of the cone is approximately 12 centimeters.
To find the length of the height of the cone, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the hypotenuse is the slant height of the cone, which is given as 13 centimeters. The other two sides are the height of the cone (which we are trying to find) and half of the diameter of the base (which is half of 10 centimeters, or 5 centimeters).
So we have the equation:
height^2 + (1/2 * diameter)^2 = slant height^2
Let's substitute the known values into the equation:
height^2 + (1/2 * 10 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
Therefore, the length of the height of the cone is 12 centimeters.