Math Problem Statement
Solution
To solve this problem, we need to determine the radius and height of a cylindrical can with a volume of 650 cm³, that minimizes the surface area of the can (since this will minimize the material needed). The can is open at the top, meaning the surface area consists only of the bottom circle and the cylindrical side.
Step 1: Define the Volume of the Cylinder
The volume of a cylinder is given by: We're told the volume is 650 cm³, so: This can be rearranged to express in terms of :
Step 2: Define the Surface Area to be Minimized
The surface area of an open-top cylinder is the sum of the area of the bottom and the lateral surface area: Substitute from the volume equation: Simplify the equation:
Step 3: Find the Minimum Surface Area
To find the value of that minimizes , take the derivative of with respect to and set it to zero: Set : Multiply both sides by : Solve for :
Step 4: Calculate
Substitute back into the equation for : Simplify:
Final Answer
So the values for and that minimize the surface area are:
Would you like further explanation on any part of this solution?
Follow-up Questions
- How would the formula change if the can were closed on top?
- What would the effect be on and if the volume were doubled?
- How can we verify if this solution gives a minimum or maximum surface area?
- What is the approximate decimal value of ?
- How does this approach apply to other shapes with fixed volumes?
Tip
When working with optimization problems, always define your constraints first and express one variable in terms of the others to reduce complexity.
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Math Problem Analysis
Mathematical Concepts
Optimization
Calculus
Geometry of Solids
Cylinders
Formulas
Volume of a cylinder: V = πr²h
Surface area of an open-top cylinder: A = πr² + 2πrh
Derivative for optimization: dA/dr = 0
Theorems
Optimization by differentiation
Suitable Grade Level
Grades 11-12
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