A circular cylinder is to fit inside a sphere of radius 10cm.Calculate the maximum posiible volume of the cylinder.
Take a look at this problem, it is a more general case than yours, but basically the same
You can replace r by 10 in her solution, and I would have assigned 2h as the height of the cylinder. That way some of the fractions could have been avoided.
https://www.youtube.com/watch?v=cw2wg6mWO2s
To find the maximum possible volume of the cylinder that can fit inside a sphere of radius 10 cm, we need to consider the dimensions and relationship between the cylinder and the sphere.
First, let's assume that the cylinder is placed inside the sphere such that the height of the cylinder is equal to the diameter of the sphere. This is the arrangement that will allow the cylinder to maximize its volume.
The diameter of the sphere is twice the radius, so it is 2 * 10 cm = 20 cm.
Therefore, the height of the cylinder can also be taken as 20 cm.
The maximum volume of a cylinder can be calculated using the formula:
Volume = π * r^2 * h
where r is the radius of the cylinder and h is its height.
Since the cylinder is placed inside the sphere, the radius of the cylinder is equal to the radius of the sphere, which is 10 cm.
Now, we can substitute the values into the formula:
Volume = π * (10 cm)^2 * 20 cm
= π * 100 cm^2 * 20 cm
= 2000 π cm^3
Hence, the maximum possible volume of the cylinder is 2000 π cm^3, which is approximately 6283.19 cm^3 (rounded to two decimal places).
To calculate the maximum possible volume of the cylinder, we need to consider its dimensions and the constraints provided by the sphere.
Let's assume that the cylinder has a radius of "r" and a height of "h." We are asked to find the maximum volume of the cylinder that can fit inside a sphere with a radius of 10 cm.
In a cylinder, the volume (V) is given by the formula: V = π(r^2)h.
To maximize the volume, we need to determine the values of "r" and "h" that satisfy the constraint of fitting inside the sphere.
In a sphere, the maximum diameter is equal to double the radius. Therefore, the diameter of the sphere is 2 * 10 cm = 20 cm.
For the cylinder to fit inside the sphere, the diameter of the cylinder (2 * r) must be smaller than or equal to the diameter of the sphere (20 cm). Hence, 2r ≤ 20 cm.
To maximize the volume, we need to choose the largest possible value for "r" that still satisfies the constraint.
From the inequality 2r ≤ 20 cm, we can solve for "r" by dividing both sides of the equation by 2:
r ≤ 20 cm / 2
r ≤ 10 cm
Therefore, the maximum value of "r" that satisfies the constraint is 10 cm.
Now that we have the value of "r," we can calculate the maximum height, "h," that still fits inside the sphere.
Using the Pythagorean theorem, we can find the relationship between the radius of the sphere (10 cm), the radius of the cylinder (r), and the height of the cylinder (h).
The diagonal of the cylinder (d) is equal to the diameter of the sphere, which is 20 cm.
Using the Pythagorean theorem, we have:
d^2 = r^2 + h^2
Substituting the known values:
20^2 = 10^2 + h^2
400 = 100 + h^2
300 = h^2
Taking the square root of both sides:
√300 = h
h ≈ 17.32 cm
Hence, the maximum possible volume of the cylinder is given by:
V = π(r^2)h
V = π(10^2)(17.32)
V ≈ 5,465.20 cm^3.
Therefore, the maximum volume of the cylinder that can fit inside the sphere is approximately 5,465.20 cm^3.