The diameter and the slant height of a cone are both 24 cm. Find the radius of the largest sphere that can be placed inside the cone. (The sphere is therefore tangent to the base of the cone.) The sphere occupies a certain percentage of the cone’s volume. First estimate this percentage, then calculate it.
To find the radius of the largest sphere that can be placed inside the cone, we can use the concept of similar triangles.
Let's start by constructing a cross-section of the cone with the sphere inscribed in it. We can represent the cone as a right triangle, where the diameter of the base is 24 cm and the height is the slant height of the cone.
Now, let's consider the similarity between the smaller cone formed by the inscribed sphere and the original cone. The two cones are similar because the corresponding angles are equal.
The ratio of the corresponding sides of similar triangles is equal. We can express this as:
r/s = (r + h)/l,
where r is the radius of the inscribed sphere, s is the slant height of the cone, h is the height of the smaller cone, and l is the slant height of the smaller cone.
Since the diameter of the base of the cone is 24 cm, the radius of the cone is 12 cm. And since the diameter and slant height of the cone are both 24 cm, the height of the cone is 12 cm.
Now, let's find the value of h (the height of the smaller cone). Using the Pythagorean theorem, we have:
h^2 + r^2 = s^2.
Substituting the given values, we get:
12^2 + r^2 = 24^2,
144 + r^2 = 576,
r^2 = 432.
Taking the square root of both sides, we find that r ≈ 20.79 cm.
Now that we have the value of r, we can calculate the volume of the sphere using the formula V = (4/3)πr^3. Plugging in the value of r, we get:
V = (4/3)π(20.79)^3 ≈ 36215.24 cm^3.
Next, let's find the volume of the cone using the formula V = (1/3)πr^2h. Substituting the given values, we get:
V = (1/3)π(12^2)(12) = 576π cm^3.
To find the percentage of the cone's volume occupied by the sphere, we can calculate the ratio of the volumes:
Percentage = (V_sphere / V_cone) * 100.
= (36215.24 / 576π) * 100.
Calculating this expression numerically, we find that the percentage is approximately 62.74%.
Therefore, the radius of the largest sphere that can be placed inside the cone is approximately 20.79 cm, and the sphere occupies approximately 62.74% of the cone's volume.
To find the radius of the largest sphere that can fit in the cone, we need to consider the relationships between the cone, the diameter, the slant height, and the sphere.
Since the diameter of the cone is given as 24 cm, the radius can be calculated as half of the diameter, which is 12 cm.
To find the height of the cone, we can use the Pythagorean theorem. The slant height is the hypotenuse, and the radius is one of the legs. Let's call the height of the cone 'h'.
Using the Pythagorean theorem, we have:
\(12^2 + h^2 = 24^2\)
Simplifying, we get:
\(144 + h^2 = 576\)
Subtracting 144 from both sides, we get:
\(h^2 = 576 - 144\)
\(h^2 = 432\)
Taking the square root of both sides, we get:
\(h = \sqrt{432}\)
Simplifying, we find:
\(h = 12\sqrt{3}\)
Now, let's find the radius of the largest sphere that can fit inside the cone. The radius of the sphere is equal to the radius of the cone, which is 12 cm.
To calculate the volume of the cone and the volume of the sphere, we can use the formulas:
Volume of the cone = (1/3) * π * r^2 * h
Volume of the sphere = (4/3) * π * r^3
Substituting the values, we get:
Volume of the cone = (1/3) * π * (12^2) * (12√3)
Volume of the sphere = (4/3) * π * (12^3)
To find the percentage that the sphere occupies of the cone's volume, we can divide the volume of the sphere by the volume of the cone and multiply by 100.
Let's calculate the percentage:
\(Percentage = \frac{Volume of the sphere}{Volume of the cone} \times 100\)
\(Percentage = \frac{(4/3) * π * (12^3)}{(1/3) * π * (12^2) * (12√3)} \times 100\)
\(Percentage = \frac{(4/3) * (12^3)}{(1/3) * (12^2) * (12√3)} \times 100\)
Simplifying, we find:
\(Percentage = \frac{4 * (12^3)}{(12^2) * (12√3)} \times 100\)
Cancelling out common factors, we have:
\(Percentage = \frac{4 * 12}{12 * √3} \times 100\)
\(Percentage = \frac{4}{√3} \times 100\)
Approximating the value of √3 to 1.732, we find:
\(Percentage ≈ \frac{4}{1.732} \times 100\)
Simplifying, we get:
\(Percentage ≈ 230.94\)
Therefore, the sphere occupies approximately 230.94% of the cone's volume.