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Linear functions SAT question
Math · Algebra · Hard · Answer it here, explanation included.
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?
ExplanationShow
Choice A is correct. Let the economist’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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