ACT Preparation / Unit 11 ACT 1018 worked example, 60 to 90 seconds
Finding a Maximum in a Word Problem
The idea
An optimization word problem with a quadratic is solved in four steps. First, name the variable and write the quantity to maximize or minimize as a function of it. Second, put it in the form ax² + bx + c and confirm that the parabola opens the right way. Third, find the vertex: x = −b/(2a). Fourth, answer the question asked, the input or the extreme value, with units.
In practice
Worked example: a farmer has 120 meters of fencing for a rectangular pen against a barn wall, fencing three sides. With width x, the length is 120 − 2x, and the area is A = x(120 − 2x) = −2x² + 120x. It opens down, so the vertex is a maximum: x = −120/(2 × −2) = 30 meters. The length is 60 meters, and the greatest area is 30 × 60 = 1,800 square meters.
Where people slip
Check that the answer fits the situation: the width must leave a positive length, so x lies between 0 and 60.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Builds on
Unlocks
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