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Algebra / Linear functions Difficulty: Hard

An economist modeled the demand Q for a certain product as a linear function of the selling price P. The demand was 20,000 units when the selling price was $40 per unit, and the demand was 15,000 units when the selling price was $60 per unit. Based on the model, what is the demand, in units, when the selling price is $55 per unit?

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Explanation

Choice A is correct. Let the economist’s model be the linear function $Q equals, m P plus b$ , where Q is the demand, P is the selling price, m is the slope of the line, and b is the y-coordinate of the y-intercept of the line in the xy-plane, where $y equals Q$ . Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs $P comma Q$ , two points that satisfy the function are $40 comma 20,000$ and $60 comma 15,000$ . The slope m of the function can be found using the formula $m equals, the fraction with numerator Q sub 2, minus Q sub 1, and denominator P sub 2, minus P sub 1, end fraction$ . Substituting the given values into this formula yields $m equals, the fraction with numerator 15,000 minus 20,000, and denominator 60 minus 40, end fraction$ , or $m equals, negative 250$ . Therefore, $Q equals, negative 250 P plus b$ . The value of b can be found by substituting one of the points into the function. Substituting the values of P and Q from the point $with coordinates 40 comma 20,000$ yields $20,000 equals, negative 250 times 40, plus b$ , or $20,000 equals, negative 10,000 plus b$ . Adding 10,000 to both sides of this equation yields $b equals 30,000$ . Therefore, the linear function the economist used as the model is $Q equals, negative 250 P plus 30,000$ . Substituting 55 for P yields $Q equals, negative 250 times 55, plus 30,000, equals 16,250$ . It follows that when the selling price is $55 per unit, the demand is 16,250 units.

Choices B, C, and D are incorrect and may result from calculation or conceptual errors.