Applied Optimization (Calculus I)
Free printable Calculus I worksheet on real-world optimization: minimizing the surface area of a box or can with fixed volume, maximizing a fenced area along a river, and number problems. Each solved in code by setting the derivative to zero.
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Calculus I · Math worksheet
Name
Date
Math
Applied Optimization
For each problem: name a variable, write the quantity to optimize as a function of that one variable (use the constraint), differentiate, solve the derivative equal to zero, and confirm max or min with the second derivative. State the answer with units.
- 1.An open-top box with a square base must hold exactly 108 cm³. Find the base side length and the height that minimize the amount of material (the surface area: base plus four sides), and state that minimum surface area.
- 2.A farmer has 152 m of fencing for a rectangular paddock along a straight river. The river side needs no fence. Find the dimensions that maximize the enclosed area, and state that maximum area.
- 3.Two positive numbers add to 46. Find the two numbers whose product is as large as possible, and state that maximum product.
- 4.A closed cylindrical can must hold exactly 16π cm³. Find the radius and height that minimize the total surface area (top, bottom and side).
- 5.Two positive numbers have product 144. Find the two numbers whose sum is as small as possible, and state that minimum sum.
- 6.An open-top box with a square base must hold exactly 500 cm³. Find the base side length and the height that minimize the amount of material (the surface area: base plus four sides), and state that minimum surface area.
- 7.A farmer has 116 m of fencing for a rectangular paddock along a straight river. The river side needs no fence. Find the dimensions that maximize the enclosed area, and state that maximum area.
- 8.A closed cylindrical can must hold exactly 250π cm³. Find the radius and height that minimize the total surface area (top, bottom and side).
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