Math Problem Statement
You have 4320043200 cm2 of cardboard to make a box with a square base and an open top. What is the largest possible volume of such a box?
Answer: cm3
Solution
To solve this problem, we need to maximize the volume of the box given the constraints. Let's go step by step:
Variables and Formulae
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Let the side length of the square base be (in cm), and the height of the box be (in cm).
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The surface area of the box, which is given, consists of:
- The square base ().
- Four vertical sides ().
Hence, the total surface area constraint is:
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The volume of the box is given by:
Express in Terms of
From the surface area constraint:
Substitute into the Volume Formula
Simplify:
Maximize the Volume
To find the maximum volume, differentiate with respect to , and set the derivative equal to zero:
Solve for
Substitute back into the formula for :
Calculate the Maximum Volume
Substitute and into the volume formula:
Final Answer:
Would you like detailed derivations or clarifications on specific steps?
Here are five related questions to expand the concept:
- How does adding a top to the box change the optimization problem?
- What happens to the volume if the surface area is doubled?
- Can you derive the maximum volume of a box with a rectangular base instead of a square base?
- How would the solution change if the box must have a height equal to its base side length?
- How does calculus help in solving real-world optimization problems like this one?
Tip: Always check the critical points in optimization problems for feasibility under given constraints.
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Math Problem Analysis
Mathematical Concepts
Calculus (Optimization)
Surface Area
Volume of a Box
Differentiation
Formulas
Surface Area: x² + 4xh = 4320043200
Volume: V = x²h
Height: h = (4320043200 - x²) / (4x)
Derivative of Volume: dV/dx = (4320043200 - 3x²) / 4
Theorems
Critical Points in Optimization
First Derivative Test for Maxima and Minima
Suitable Grade Level
Grades 11-12, College
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