300. 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 sphere, this article will help.
https://math.stackexchange.com/questions/3079286/sphere-inscribed-in-a-cone
The proportion shown gives the formula
R = radius of cone
h = height of cone
r = radius of sphere
√(R^2+h^2) / R = (h-r)/r
Knowing r, you can easily get the volume.
5.35
Well, that's quite a geometric puzzle you have there! Let's break it down step by step, but don't worry, I won't be using any fancy mathematical jargon.
First, we need to find the radius of the cone. Since the diameter is given as 24 cm, the radius would be half of that, which is 12 cm. Easy-peasy!
Next, we want to find the radius of the largest sphere that can fit inside the cone. If the sphere is tangent to the base of the cone, it means that the radius of the sphere is equal to the radius of the cone. So, the radius of the sphere is also 12 cm.
Now, let's calculate the volume of the cone and the volume of the sphere. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the cone and h is the height. However, the height is not given, so we'll have to get a bit creative.
Since the slant height of the cone is also 24 cm, we can use the Pythagorean theorem to find the height. Imagine the slant height, the height, and the radius forming a right-angled triangle. Using good old Pythagoras, we have h^2 + 12^2 = 24^2. Solving this equation, we get h ≈ 20.7846 cm (rounded to 4 decimal places).
Now we can calculate the volume of the cone using V = (1/3)πr^2h. Substituting the values, we get V_cone ≈ (1/3)π(12^2)(20.7846) ≈ 9978.4097 cm³.
The volume of a sphere is given by the formula V = (4/3)πr^3. Substituting the radius we found earlier, we get V_sphere ≈ (4/3)π(12^3) ≈ 7238.2295 cm³.
To estimate the percentage of the cone's volume occupied by the sphere, we can calculate V_sphere / V_cone. Substituting the values, we get (7238.2295 / 9978.4097) ≈ 0.7257.
So, the estimate is around 72.57%.
Now, if you want the exact percentage, we need to divide V_sphere by V_cone and multiply by 100. So, (7238.2295 / 9978.4097) * 100 ≈ 72.57%.
Therefore, the sphere occupies approximately 72.57% of the cone's volume. Quite a bit of space, isn't it? But hey, that's one spherical cone!
To find the radius of the largest sphere that can be placed inside the cone, we can use the Pythagorean theorem to relate the diameter, radius, and slant height of the cone.
Let's denote the radius of the sphere as r and the slant height of the cone as l.
Using the Pythagorean theorem, we have:
r^2 + (d/2)^2 = l^2
Substituting the given values, we have:
r^2 + (24/2)^2 = 24^2
Simplifying the equation, we get:
r^2 + 12^2 = 24^2
r^2 + 144 = 576
r^2 = 432
r ≈ √432
r ≈ 20.79 cm (rounded to two decimal places)
Now, to estimate the percentage of the cone’s volume occupied by the sphere, we can use a formula that relates the volume of a cone to the volume of a sphere.
The volume of a cone is given by:
V_cone = (1/3)πr^2h
The volume of a sphere is given by:
V_sphere = (4/3)πr^3
Now, consider the ratio of the volume of the sphere to the volume of the cone:
(V_sphere / V_cone) = [(4/3)πr^3] / [(1/3)πr^2h]
Simplifying the equation, we get:
(V_sphere / V_cone) = 4r / h
To estimate the percentage, we can substitute the approximate values:
(V_sphere / V_cone) ≈ 4(20.79) / 24
Calculating the approximate percentage:
(V_sphere / V_cone) ≈ 3.48 / 24 ≈ 0.145
Therefore, the estimated percentage of the cone's volume occupied by the sphere is approximately 0.145, or 14.5%.
To find the radius of the largest sphere that can be placed inside the cone, we can start by finding the radius of the cone's base.
The diameter of the cone is given as 24 cm, which means the radius of the cone's base is half of that, or 12 cm. Since the sphere is tangent to the base of the cone, the radius of the sphere must be equal to the radius of the cone's base.
Therefore, the radius of the largest sphere is 12 cm.
To find the percentage of the cone's volume occupied by the sphere, we need to compare the volume of the sphere with the volume of the cone.
The volume of a sphere is given by the formula V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
The volume of a cone is given by the formula V_cone = (1/3) * π * r^2 * h, where r is the radius of the cone's base and h is the height of the cone.
Since the radius of the sphere and the radius of the cone's base are the same, we can substitute r = 12 cm into the formulas.
V_sphere = (4/3) * π * 12^3
V_cone = (1/3) * π * 12^2 * h
To estimate the percentage of the cone's volume, we can compare the volumes of the sphere and the cone using rough values for π and h.
Assuming π ≈ 3.14 and h ≈ 24 cm, we can calculate the estimated volume percentage:
Estimated Percentage = (V_sphere / V_cone) * 100
Now we can substitute the calculated values and solve for the estimated percentage:
Estimated Percentage = ((4/3) * 3.14 * 12^3) / ((1/3) * 3.14 * 12^2 * 24) * 100
Calculating this expression will give you the estimated percentage of the cone's volume occupied by the sphere.
To obtain a more accurate calculation, you can use the actual value of π (approximately 3.14159) and the given height of the cone.
Actual Percentage = ((4/3) * π * 12^3) / ((1/3) * π * 12^2 * 24) * 100
Calculating this expression will give you the actual percentage of the cone's volume occupied by the sphere.