What is the largest volume of a cylinder you can inscribe into a cone with a height of 3 inches and a radius of 1.35 inches

Question

h = 3

r = 1.35

What is the largest volume of a cylinder you can inscribe into a cone with a height of 3 inches and a radius of 1.35 inches.

Answer 1

I would rather encourage you to think this through, rather than show the work and the answer.

The complexity of this problem can be reduced to 2 dimensions. Draw an isosceles triangle (representing the cone in 2d). Inscribe a rectangle (representing the cylinder). The dimensions of the triangle are given. Call the base of the rectangle x and the height y. Your goal is to get y in terms of x. Once you do that you can make an expression for the area of the rectangle with x as the only variable. Then differentiate as usual to find the max.

If you'd like to see an illustration of what I'm talking about, I'd be glad to make one.

Answer 2

isn't the general answer for any r,h

largest volume=(4/27)π(r²h)

reason, V=πx²z,,,,, and z(x)=h-(h/r)x

V(x)=πx²(h-(h/r)x)

V'(x)=2πhx-3π(h/r)x²

V'(x)=0,,,,x=2r/3,,,,,,that's not hard to envision.

plug x into V(x) equation