Find the radius, volume, and hieght of the right- circlar cylindar that can be inscribed in a right- circular cone with a radius of 6 inches and a hieght of 10 inches.

There are an infinite number of solutions to this. Did you mean the right cylinder with the max volume? Or surface area?

If it is the right cylinder of max volume. Consider this.

The height and radius of the cylinder are related:
h= 10 - 10r/6

Volume cylinder = PI * r^2 * h
= PI*r^3 *(10 - 10r/6 )

Take the derivatative, set to zero . and solve.

To find the radius, volume, and height of the right-circular cylinder that can be inscribed in a right-circular cone, we need to consider the relationship between the height and radius of the cylinder.

The formula for the height of the cylinder in terms of its radius is given by:
h = 10 - (10r/6)

Now, to find the volume of the cylinder, we use the formula:
Volume = π * r^2 * h
Substituting the formula for h, we get:
Volume = π * r^2 * (10 - (10r/6))

To find the radius of the cylinder that maximizes the volume, we take the derivative of the volume equation with respect to r and set it to zero. This will give us the critical points, which correspond to the maximum volume.

So let's differentiate the volume equation:
dV/dr = π * (2r * (10 - (10r/6))) + π * r^2 * (-10/6)

Setting this derivative equal to zero, we can solve for r:
0 = π * (2r * (10 - (10r/6))) + π * r^2 * (-10/6)

Simplifying the equation and solving for r will give us the radius of the cylinder with maximum volume.

Note: If you intended to find the cylinder with maximum surface area, then you would need to differentiate the surface area equation with respect to r instead of the volume equation.