Math Problem Statement

A cone is constructed by cutting a sector from a circular sheet of metal with radius 23. The cut sheet is then folded up and welded. Find the radius and height of the cone with maximum volume that can be formed in this way.

Solution

To solve this problem, we need to optimize the dimensions of a cone that’s formed by cutting a sector from a circular sheet with radius 23 and then welding the sheet's edges to create the cone.

Steps to Solve:

  1. Understand the Geometry of the Cone:

    • The radius of the original circular sheet is given as R=23R = 23.
    • When the sector is cut and folded, the arc length of the sector becomes the circumference of the cone’s base.
  2. Formulas to Use:

    • Let rr be the radius of the cone's base, and hh be its height.
    • The slant height of the cone, which is the same as the radius of the original circular sheet, is 23. Thus, s=23s = 23.
    • The arc length of the sector, which becomes the circumference of the cone’s base, is 2πr2\pi r.
  3. Setting Up Relationships:

    • By the Pythagorean theorem for the cone, we know s2=r2+h2s^2 = r^2 + h^2.
    • Plugging s=23s = 23, we get: 232=r2+h2529=r2+h2.23^2 = r^2 + h^2 \Rightarrow 529 = r^2 + h^2.
    • Solving for hh in terms of rr: h=529r2.h = \sqrt{529 - r^2}.
  4. Volume of the Cone:

    • The volume VV of a cone is given by V=13πr2hV = \frac{1}{3} \pi r^2 h.
    • Substituting hh from the previous step: V=13πr2529r2.V = \frac{1}{3} \pi r^2 \sqrt{529 - r^2}.
  5. Maximizing the Volume:

    • To find the value of rr that maximizes VV, take the derivative of VV with respect to rr and set it to zero. Solve this equation to find the optimal value of rr, then use it to find hh.

After calculating, we find the exact values:

  • Radius (r): 233\frac{23}{\sqrt{3}}
  • Height (h): 2323\frac{23\sqrt{2}}{\sqrt{3}}

Let me know if you need detailed steps for finding these values or have any questions.

Further Questions to Consider:

  1. How is the relationship between the slant height and the cone dimensions used here?
  2. Can this approach be applied to different starting sheet radii?
  3. What happens to the cone's volume if a smaller sector is cut?
  4. How would the formula change if the cone had a specific height restriction?
  5. Why does the maximum volume occur at these specific values of rr and hh?

Tip: Always check if derivatives are set up correctly when optimizing geometric shapes!

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Math Problem Analysis

Mathematical Concepts

Geometry
Optimization
Cone Volume Maximization

Formulas

Volume of a cone: V = (1/3) * π * r^2 * h
Relationship between slant height, radius, and height: s^2 = r^2 + h^2
Circumference of cone’s base: C = 2πr

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 11-12