A hemisphere of radius 5 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.

To find the radius and height of the cylinder that maximizes its volume, we need to set up an equation to express the volume and then differentiate it with respect to the variables (radius and height) and find where the derivative is zero. Let's go step by step:

Let's call the radius of the cylinder r and its height h.

Since the cylinder's base is centered at the center of the hemisphere, the height of the cylinder (h) is the sum of the radius of the hemisphere (5) and the radius of the cylinder (r). Therefore, h = 5 + r.

The volume of the cylinder is given by the formula V = πr²h.

Substituting the value of h from above, we get V = πr²(5 + r).

To maximize the volume, we need to find the critical points by differentiating with respect to r and setting the derivative equal to zero.

Differentiating V with respect to r, we get dV/dr = 2πr(5 + r) + πr².

Setting dV/dr equal to zero, we have 2πr(5 + r) + πr² = 0.

Simplifying the equation, we get 2πr² + 10πr + πr² = 0.

Combining like terms, we have 3πr² + 10πr = 0.

Factorizing out πr, we get πr(3r + 10) = 0.

This equation is satisfied when r = 0 (which is not physically meaningful in this context) or when 3r + 10 = 0.

Solving for r, we find r = -10/3.

Since the radius cannot be negative, we discard r = -10/3 and conclude that the only critical point is when r = 0.

Now we need to check the endpoints of the feasible interval. In this case, since the radius of the hemisphere is 5, the maximum value of r can only be 5.

So, we evaluate the volume at the critical point r = 0 and at the endpoint r = 5.

When r = 0, the height h = 5 + r = 5 + 0 = 5. Therefore, the volume V is π(0)²(5) = 0.

When r = 5, the height h = 5 + r = 5 + 5 = 10. Therefore, the volume V is π(5)²(10) = 250π.

Comparing the volumes, we find that V = 0 at r = 0 and V = 250π at r = 5.

Hence, the maximum volume of the cylinder is 250π, which occurs when r = 5.

Therefore, the radius of the cylinder of maximum volume is 5 and the height is 5 + r = 5 + 5 = 10.

In conclusion, the radius of the cylinder of maximum volume is 5 units, and the height is 10 units.

To find the radius and height of the cylinder with maximum volume, let's start by considering the given information and breaking down the problem into steps:

Step 1: Identify the variables.
Let:
- R be the radius of the hemisphere.
- r be the radius of the cylinder.
- h be the height of the cylinder.

Step 2: Determine the limitations/constraints.
Since the cylinder stands with its axis vertical, the top circular rim of the cylinder touches the hemisphere. This means that the height of the cylinder (h) plus the radius of the cylinder (r) should be equal to the radius of the hemisphere (R). Mathematically, this can be expressed as:
h + r = R ...(equation 1)

Step 3: Express the volume of the cylinder in terms of the given variables.
The volume of a cylinder is given by the formula V = π * r² * h.

Step 4: Express the constraints in terms of a single variable.
From equation 1, we can rewrite it as:
r = R - h ...(equation 2)

Step 5: Express the volume in terms of a single variable using the constraints.
Using equation 2, the volume of the cylinder can be expressed as:
V = π * (R - h)² * h ...(equation 3)

Step 6: Find the maximum volume using calculus.
To find the maximum volume, we need to find the critical points of equation 3 by differentiating it with respect to h and setting the derivative equal to zero. Then solve for h.

Step 7: Differentiate equation 3 with respect to h and set it equal to zero.
dV/dh = π * (2R - 3h) * (R - h)² = 0

Step 8: Solve for h.
If (2R - 3h) = 0 or (R - h) = 0, we have the points where the derivative is equal to zero. Solving these equations, we get:
2R - 3h = 0 ...(equation 4)
R - h = 0 ...(equation 5)

From equation 5, we can solve for h:
h = R

Step 9: Determine the maximum volume.
Now that we have the value of h, we can substitute it back into equation 3 to find the maximum volume:
V = π * (R - h)² * h
= π * (R - R)² * R
= 0

This means that the maximum volume is 0, which implies that the cylinder has zero height and zero volume.

Step 10: Interpretation of the result.
From the steps above, it can be concluded that the maximum volume of the cylinder is zero. This implies that the cylinder cannot exist in the given conditions, as it touches the hemisphere exactly at the rim.

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