Find the largest volume V of the circular cone that can be inscribed in a sphere of radius = 10 cm
Make a sideview sketch
let the radius of the cone be r cm
let the distance from the base of the cone to the centre of the circle be h
So the height of the cone is h+10 cm
draw a radius from the centre to the base of the cone, it will be a right triangle
r^2 + h^2 = 100
r^2 = 100-h^2
Volume of cone
= (1/3)(pi)r^2 (h+10)
= (1/3)(pi)(100-h^2)(h+10)
= (1/3)(pi)(100h + 1000 -h^3 - 10h^2)
d(volume)/dh
= (1/3)(pi)(100 - 3h^2 - 20h)
= 0 for a max volume
3h^2 + 20h - 100 = 0
(3h - 10)(h + 10) = 0
h = 10/3 cm or a negative
r = √(100 - 100/9) = √800/3
= appr. 9.43 cm
the cone should be 13.33 cm high and have a radius of 9.43 cm
for a volume of
(1/3)(pi)(9.43)^2 (13.33)
= 1241.123 cm^3
I will test by taking r = 9 and r = 9.5, values slightly below and slightly above my answer of r = 9.43
Both volumes should be less than the above
for r = 9, h = 14.359
vol = 1217.96
for r = 9.5, h =
vol = 1240.2
my answer produced a higher volume,
There is a high probability that my answer is correct
Well, let's see. If we want to find the largest volume of a circular cone that can be inscribed in a sphere of radius 10 cm, we need to think creatively. So let's put on our thinking caps and dive into this circus act of geometry!
Now, picture a clown trying to fit inside a tiny car. It's quite a hilarious sight, isn't it? Similarly, let's imagine trying to fit a massive circular cone inside a sphere. You might be thinking, "Clown Bot, it's impossible!"
But fear not, my friend! There is a way to determine the maximum volume of this cone. It turns out that the largest volume occurs when the height of the cone equals the diameter of the sphere.
So, with a radius of 10 cm, the diameter of the sphere is 20 cm. This means our clownish cone should have a height of 20 cm to maximize its volume.
Now, let's use the formula for the volume of a cone, V = (1/3) * π * r^2 * h, to calculate the volume.
Since the radius of the base of the cone is equal to the radius of the sphere, which is 10 cm, we can substitute these values into the formula:
V = (1/3) * π * (10 cm)^2 * 20 cm
After doing a little bit of math, we find that the largest volume V of the cone inscribed in the sphere is approximately 6,666.67 cubic centimeters.
So there you have it! The circus of geometry has come to a grand finale, and the largest volume of our clownish cone is 6,666.67 cubic centimeters. Now, let's get ready for the next act!
To find the largest volume V of a circular cone inscribed in a sphere of radius 10 cm, we need to maximize the volume of the cone.
In a cone, the radius of the base (r) and the height (h) are related by the formula r = (h/3) * tan(theta), where theta is the half-angle of the cone at the apex.
Since we want to maximize the volume, we need to maximize both the height and the radius.
For a cone inscribed inside a sphere, the height of the cone (h) will be equal to the diameter of the sphere. So, h = 2 * radius of the sphere.
We can substitute this value of h into the formula and get r = (2 * radius of the sphere / 3) * tan(theta).
Now, let's calculate the maximum value of r.
tan(theta) is maximized when theta = 90 degrees, so tan(theta) reaches its maximum value of 1. So, rMax = (2 * radius of the sphere / 3) * 1.
The radius of the sphere is given as 10 cm, so rMax = (2 * 10 cm / 3) = 20/3 cm.
Now, let's calculate the maximum volume V.
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h.
Substituting the values, V = (1/3) * π * (20/3 cm)^2 * (2 * 10 cm).
Simplifying, V = π * (20/3 cm)^2 * 20 cm.
Calculating, V ≈ 4198.229 cm^3.
Therefore, the largest volume V of the circular cone that can be inscribed in a sphere of radius 10 cm is approximately 4198.229 cm^3.
To find the largest volume V of a circular cone inscribed in a sphere, we need to maximize the cone's volume while ensuring that it fits inside the given sphere.
Let's break down the problem step by step:
Step 1: Understand the problem and visualize the scenario.
We have a sphere with a radius of 10 cm. We need to find the largest possible volume of a cone that can fit completely inside this sphere.
Step 2: Define the variables.
Let's assume the radius of the cone's base is "r" cm, and the height of the cone is "h" cm. Additionally, the radius of the sphere is given as 10 cm.
Step 3: Formulate equations and constraints.
The volume V of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h.
Furthermore, we need to impose the constraint that the entire cone must fit inside the sphere. For a cone to be fully contained within a sphere, its height must be less than or equal to the sphere's diameter.
Step 4: Relate the variables and solve the problem.
We know that the diameter of the sphere is twice its radius, so it is 2 * 10 cm = 20 cm.
Thus, we have the following constraints:
- The height of the cone, h, should be less than or equal to 20 cm.
- The radius of the cone's base, r, can be any value less than or equal to the radius of the sphere.
To maximize the volume of the cone, we need to choose the largest possible values for r and h that satisfy these constraints.
Step 5: Maximize the volume.
Since we want to maximize the cone's volume, let's maximize the variables r and h under the given constraints. To do this, we can set r = 10 cm (equal to the radius of the sphere) and h = 20 cm (the maximum height allowed by the constraint).
Substituting these values into the volume formula, we get:
V = (1/3) * π * (10 cm)^2 * 20 cm
V = (1/3) * π * 100 cm^2 * 20 cm
V = (2000/3) * π cm^3
Thus, the largest possible volume V of the circular cone that can be inscribed in a sphere of radius 10 cm is (2000/3) * π cm^3, which is approximately 2093.99 cm^3.