Let x represent the number of units produced and sold, and let P represent the t

Question

Let x represent the number of units produced and sold, and let P represent the total profit. With a profit of $100 per unit.
a. What is the mathematical model for the total profit earned by producing and selling x units?
b. Furthermore, suppose the production capacity constraint is given by

5x ? 40
x?0

What is the optimal solution for this production model?

The average stock price for companies making up the Standard & Poor 500 was $30 per share and the standard deviation was $8.20 in 2003. Suppose the stock prices are normally distributed, how high does a stock price have to be to put a company in the top 1% of S&P500?

Explanation / Answer

Q1) a) P(x)= 100x

         b) If the constraint is 5x<=40, then x<=8. So, the optimal solution is x=8

Q2) Let X be the stock price. Then X follows a normal distribution with mu=$30 and sigma=$8.2

Now we look for a value "A" such that P(X>=A)=0.01

i.e.,   P(Z> (A-30)/8.2) =0.01

So, P(Z<(A-30)/8.2)=0.99

So, (A-30)/8.2=2.326

So,   A=30+2.326*8.2= 40.08 (dollars)

So, a stock price needs to be $40.08 to be in the top 1%

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