Look into the fact that if you sum up integers, the running tally will grow quad

Question

Look into the fact that if you sum up integers, the running tally will grow quadratically. Now, armed with this knowledge, consider the following question. Each unit of brand awareness of a product generated by an ad campaign costs $.10 more than the previous one (i.e., the first costs .1, the second costs .1 more than that (.2), the third costs .1 more than that, (.3), etc.), and the overhead cost of $50 is needed to begin the ad campaign. Each unit of awareness generates exactly $5 in revenue. Find a function that models this in terms of awareness and profit. Profit will be a function of the accumulated awareness. Graph it. How much awareness gets us to break even? How much maximizes the profit?

Explanation / Answer

each unit generates $ 5 in revenue

so if x units are generated revenue is 5x

cost of producing x units

50 + ( .1 + .2 +.3 +....0.1x )

50 + 0.05x ( x+1)

profit = revenue -cost

p(x) = 5x - 50 - 0.05x(x+1)

p(x) = -0.05x^2 + 4.95x - 50

for breakeven point set p(x) = 0

-0.05x^2 + 4.95x - 50 = 0

two breakeven points are

x = 11 , 88

maximum profit occurs at

x = - 4.95 /2 (-.05)

x = 49.5 units

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