Math Problem Statement
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:
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Understand the Geometry of the Cone:
- The radius of the original circular sheet is given as .
- When the sector is cut and folded, the arc length of the sector becomes the circumference of the cone’s base.
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Formulas to Use:
- Let be the radius of the cone's base, and be its height.
- The slant height of the cone, which is the same as the radius of the original circular sheet, is 23. Thus, .
- The arc length of the sector, which becomes the circumference of the cone’s base, is .
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Setting Up Relationships:
- By the Pythagorean theorem for the cone, we know .
- Plugging , we get:
- Solving for in terms of :
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Volume of the Cone:
- The volume of a cone is given by .
- Substituting from the previous step:
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Maximizing the Volume:
- To find the value of that maximizes , take the derivative of with respect to and set it to zero. Solve this equation to find the optimal value of , then use it to find .
After calculating, we find the exact values:
- Radius (r):
- Height (h):
Let me know if you need detailed steps for finding these values or have any questions.
Further Questions to Consider:
- How is the relationship between the slant height and the cone dimensions used here?
- Can this approach be applied to different starting sheet radii?
- What happens to the cone's volume if a smaller sector is cut?
- How would the formula change if the cone had a specific height restriction?
- Why does the maximum volume occur at these specific values of and ?
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
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