# A right cylinder is inscribed in a sphere of radius r. How do you find the largest possible volume of such a cylinder?

There are several steps to this optimization problem.

1.) Find the equation for the volume of a cylinder inscribed in a sphere.

2.) Find the derivative of the volume equation.

3.) Set the derivative equal to zero and solve to identify the critical points.

4.) Plug the critical points into the volume equation to find the maximum volume.

The best place to start is by drawing a diagram. The picture below shows the cylinder inscribed in the sphere. Given the height,

Note that

To find the volume of our cylinder, we need to multiply the area of the top by the total height of the cylinder. In other words;

This is our volume function. Next we take the derivative of the volume function and set it equal to zero. If we move the

The

After some rearranging;

Take the square root of both sides.

This is our optimized height. To find the optimized volume, we need to plug this into the volume function.

Simplify.

This is the optimized volume for the cylinder. Its a good check to notice that

By signing up, you agree to our Terms of Service and Privacy Policy

To find the largest possible volume of a right cylinder inscribed in a sphere of radius ( r ), you can use the formula for the volume of a cylinder, which is ( V = \pi r^2 h ), where ( r ) is the radius of the base and ( h ) is the height. In this case, since the cylinder is inscribed in the sphere, the diameter of the base of the cylinder is equal to the diameter of the sphere, which is ( 2r ).

To maximize the volume of the cylinder, we need to maximize the height ( h ). The height of the cylinder can be found using the Pythagorean theorem, as it forms the hypotenuse of a right triangle with the radius ( r ) and the height of the sphere. The height of the sphere can be calculated as ( h_s = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} ).

Now, using the Pythagorean theorem again for the cylinder, we have ( h^2 = (r\sqrt{2})^2 - r^2 = 2r^2 - r^2 = r^2 ), which gives ( h = r ).

Substitute ( h = r ) into the volume formula for the cylinder: ( V = \pi r^2 h = \pi r^2 r = \pi r^3 ).

Therefore, the largest possible volume of the cylinder inscribed in the sphere is ( \pi r^3 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the local linearization of #y = sin^-1x # at a=1/4?
- How do you find the rate at which a batter's distance from second base decreases when he is halfway to first base if the baseball diamond is a square with side 90 ft and a batter hits the ball and runs toward first base with a speed of 24 ft/s?
- How do you find the linear approximation L to f at the designated point P. compare the error in approximating f by L at the specified point Q with the distance between P and Q given #f(x,y) = 1/sqrt(x^2+y^2)#, P(4,3) and Q(3.92, 3.01)?
- Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10 m?
- A girl of height 120 cm is walking towards a light on the ground at a speed of 0.6 m/s. Her shadow is being cast on a wall behind her that is 5 m from the light. How is the size of her shadow changing when she is 1.5 m from the light?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7