You have been hired as a marketing consultant to Big Book Publishing, Inc., and you have been approached to determine the best s
elling price for the hit calculus text by Whiner and Istanbul entitled Fun with Derivatives. You decide to make life easy and assume that the demand equation for Fun with Derivatives has the linear form q = mp + b, where p is the price per book, q is the demand in annual sales, and m and b are certain constants you must determine.(a) Your market studies reveal the following sales figures: when the price is set at $54.00 per book, the sales amount to 10,000 per year; when the price is set at $79.00 per book, the sales drop to 1000 per year. Use these data to calculate the demand equation.(b) Now estimate the unit price that maximizes annual revenue. $ Predict what Big Book Publishing, Inc.'s annual revenue will be at that price.
2 answers:
Answer:
(a) q = -360p +29440
(b) maximum revenue of $601,884 at p = 40.89
Step-by-step explanation:
(a) <u>Demand Equation</u>
You are given to pairs of p and q:
(p, q) = (54, 10000) and (79, 1000)
These can be used in the 2-point form of the equation for a line:
q = (q2 -q1)/(p2 -p1)(p -p1) +q1
q = (1000 -10000)/(79 -54)(p -54) +10000 . . . . . substitute givens
q = (-9000/25)(p -54) +10000 . . . . . . . . . . . . . . . simplify a bit
q = -360p +29440 . . . . . . . . . . . . . . . . . . . . . . . . . write in slope-intercept form
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(b) <u>Maximum Revenue</u>
Revenue from book sales will be the product of the quantity sold (q) and the price at which they are sold (p). Here, that means ...
r(p) = p·q = p(-360p +29440) = 360p(81 7/9 -p)
This is the equation of a parabola that opens downward. Its axis of symmetry is at the price point halfway between the zeros of p=0 and p=81.78. That is, the revenue will be maximized at a unit price of 40.89. That maximum revenue will be about ...
r(40.89) ≈ $601,884.44
Big Book Publishing's annual revenue is predicted to be about $601,884 at a book price of $40.89.
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<em>Comment on the numbers</em>
Both the price and the revenue are repeating decimal fractions if these equations are taken at face value. If quantity is rounded to whole books and price is rounded to cents, then the maximum revenue is predicted to be $601,900.80 on sales of 14,720 books at a price of $40.89 each.
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