A cylinder is inscribed in a cone as shown in the diagram below. The cone has a height of 8cm and a fixed radius of R cm. The cylinder has a radius of x cm and a height of h cm. Show that the volume of the cylinder is maximised when x=kR, where k is a constant to be determined.

To find the maximum volume of the cylinder inscribed in the cone, we need to express the volume of the cylinder in terms of a single variable, either x or h. In this case, we will use x.

First, let's analyze the geometry of the problem. The cylinder is inscribed inside the cone, so the bases of the cylinder and the cone are congruent and lie on the same plane. This means that the radius of the cylinder (x) and the radius of the cone (R) are equal.

Let's denote the slant height of the cone as l. Using the Pythagorean theorem, we can relate the slant height, the height of the cone, and the radius of the cone:

l^2 = R^2 + (8^2)
l^2 = R^2 + 64
l = sqrt(R^2 + 64)

Now we can express the height of the cylinder (h) in terms of x, R, and l. Since the height of the cylinder is perpendicular to the slant height of the cone, we can use similar triangles to find h:

(R/l) = (x/h)
h = (x * l) / R
h = (x * sqrt(R^2 + 64)) / R

Now, we have expressions for both the radius (x) and the height (h) of the cylinder in terms of the radius of the cone (R). We can use these expressions to find the volume of the cylinder.

The volume of a cylinder is given by the formula:

V = π * r^2 * h

Plugging in the expressions for r (x) and h, we have:

V = π * (x^2) * ((x * sqrt(R^2 + 64)) / R)
V = (π * x^3 * sqrt(R^2 + 64)) / R

To maximize the volume, we need to find the value of x that maximizes this expression. We can do this by taking the derivative of V with respect to x, setting it equal to zero, and solving for x:

dV/dx = 3πx^2 * sqrt(R^2 + 64) / R = 0
3πx^2 * sqrt(R^2 + 64) = 0

Since the constant term 3π is non-zero, we can cancel it from both sides of the equation:

x^2 * sqrt(R^2 + 64) = 0

To maximize V, we need to find the value of x that makes this equation true. The only way this can happen is if x = 0. However, a cylinder with zero radius has zero volume, so this is not a valid solution.

Therefore, there is no maximum volume for the cylinder when x is a constant multiple of R. Instead, the volume of the cylinder keeps increasing as x approaches infinity.

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https://answers.yahoo.com/question/index?qid=20080427160136AACsLX6