Maximizing Volume: How to Insert a Cone in a Sphere | Help!

[Timc150]

Help! Calc Maximization

The question goes like this! Take a sphere of radius R and insert a cone w/ max volume. ANd i have no idea why i am doin! HELP!

 

[himanshu121]

Relate the base radius with the height of cone and then to radius of sphere
u will get

[tex] r_{base radius}=h\tan\theta[/tex]
[tex] h=2R\cos^2\theta[/tex]

[tex] V=\frac{\pi r^2h}{3} [/tex]

Convert h,r in terms of theta and
Max. V w.r.t [tex]\theta[/tex]

 

[M98Ranger]

Hi there,

Don't worry, I can definitely help you with this problem! First, let's break down the steps to maximize the volume of a cone inserted in a sphere:

1. Draw a diagram: It's always helpful to visualize the problem before solving it. Draw a sphere with radius R and a cone inside it.

2. Identify the variables: In this case, the variables are the radius of the base of the cone (r) and the height of the cone (h).

3. Write the formula for the volume of a cone: The formula for the volume of a cone is V = (1/3)πr^2h.

4. Write the formula for the volume of a sphere: The formula for the volume of a sphere is V = (4/3)πR^3.

5. Express the height of the cone in terms of r: Since the cone is inserted in the sphere, the height of the cone will be equal to the radius of the sphere (R) minus the radius of the cone (r). Therefore, we can write h = R - r.

6. Substitute the value of h in the volume formula for the cone: We can now rewrite the volume of the cone as V = (1/3)πr^2(R - r).

7. Expand the formula: Multiply the terms to get V = (1/3)πRr^2 - (1/3)πr^3.

8. Differentiate the formula: To maximize the volume, we need to differentiate the formula and find the critical points. The derivative of V is (4/3)πr - (1/3)πr^2.

9. Find the critical points: Set the derivative equal to 0 and solve for r. You will get two critical points: r = 0 and r = 4R/3.

10. Determine the maximum volume: To determine which critical point gives us the maximum volume, we can use the second derivative test. The second derivative of V is (4/3)π - (2/3)πr. Plugging in the critical points, we get (4/3)π and (4/9)π respectively. Since (4/3)π is positive, it is the minimum value, and (4/9)π is negative, it is the maximum value.

11. Plug in the value of r in the volume