An economist modeled the demand Q for a certain product as a linear function of the selling price P .

Question

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?

A.

16,250

B.

16,500

C.

16,750

D.

17,500

Accepted Answer

Choice A is correct. Let the economist&rsquo;s model be the linear function , 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 . Two pairs of the selling price P and the demand Q are given. Using the coordinate pairs , two points that satisfy the function are and . The slope m of the function can be found using the formula . Substituting the given values into this formula yields , or . Therefore, . 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 yields , or . Adding 10,000 to both sides of this equation yields . Therefore, the linear function the economist used as the model is . Substituting 55 for P yields . 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.

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