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)
cm
To find the length of the height of the cone, we can use the Pythagorean Theorem.
The diameter of the base is given as 10 centimeters, so the radius (r) is half of that, which is 5 centimeters.
The slant height of the cone is given as 13 centimeters.
Let h be the height of the cone.
Using the Pythagorean Theorem, we can set up the equation:
r^2 + h^2 = slant height^2
5^2 + h^2 = 13^2
25 + h^2 = 169
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.
In the given image, we know that the slant height of the cone is 13 centimeters and the diameter is 10 centimeters.
Using the Pythagorean theorem, we can solve for the height. 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, the slant height is the hypotenuse, the height is one of the other sides, and the radius (half the diameter) is the other side.
Let's call the height "h" and the radius "r". The equation based on the Pythagorean theorem can be written as:
h^2 + r^2 = slant height^2
Plugging in the known values, we have:
h^2 + (5 cm)^2 = (13 cm)^2
Simplifying this equation, we have:
h^2 + 25 cm^2 = 169 cm^2
Subtracting 25 cm^2 from both sides, we have:
h^2 = 144 cm^2
Taking the square root of both sides, we have:
h = √144 cm
Therefore, the length of the height of the cone is:
h = 12 cm
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, the height of the cone represents the vertical side of the right triangle, and the slant height represents the hypotenuse. The base of the cone, which is the diameter, represents the horizontal side of the right triangle.
We are given that the slant height is 13 cm and the diameter is 10 cm. Let's use the Pythagorean theorem to calculate the height.
According to the Pythagorean theorem:
(h)^2 + (r)^2 = (s)^2
Where:
h = height of the cone
r = radius (half of the diameter), which is equal to 10 cm / 2 = 5 cm
s = slant height of the cone, which is 13 cm
Substituting the given values into the formula:
(h)^2 + (5 cm)^2 = (13 cm)^2
Simplifying the equation:
(h)^2 + 25 cm^2 = 169 cm^2
Subtracting 25 cm^2 from both sides:
(h)^2 = 169 cm^2 - 25 cm^2
(h)^2 = 144 cm^2
Taking the square root of both sides:
h = √144 cm^2
h = 12 cm
Therefore, the length of the height of the cone is 12 cm.