A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters α = 3, θ = 0.5, where θ is in units of days per 1000 units of rolls. It costs $ 2 to make a unit that sells for $ 5 on the first day when the rolls are fresh. Any leftover units are sold on the second day for $ 1. How many units should be made to maximize the expected value of the profit?

The value  is  

Step-by-step explanation:

From the question we are told that

   The  parameters  are  α = 3, θ = 0.5

    The cost of making a unit on the first day  is  c = $2

    The selling price of a  unit on the first day is  s = $5

    The selling price of a leftover unit on the second day is  v  = $ 1

Generally the profit of a unit on the first day is

The profit of a unit on the second day is

=>    

Generally the probability of making profit greater than $ 1 is mathematically represented as

=>  

Now from the gamma distribution table  we have that

Generally the probability of making profit less than or  equal to  $ 1 is mathematically represented as

=>    

=>        

So  the probability of making  $3  is    

and  the probability of making  -$1  is    

Generally the value of profit per day is mathematically represented as

=>    

=>    

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.

2

step-by-step explanation: