1. CS123 Quiz 3

2. Question 5
a) How many units of ammonia must be manufactured and sold daily to maximize the profit? How much money will be made each day for that amount of production?
So we need a function to represent daily profit.
Assume the daily production is x units
The daily production cost is
C:=x-> *x+1/10*x^2

3. Given the selling price is 590 dollars per unit
We have the daily income is 590*x dollars
So the daily profit is the income minus the cost
profit := x -> 590 * x - C(x);
Considering that the daily capacity is at most units, we can plot this function as
plot(profit(x), x= );

5. Then, get the maximum profit
maxProfit := Optimization[Maximize](profit(x),x= );
The output should be like
[ *10^5, [x = 2850.]]

6. Get the daily units that maximize the daily profit
Take the 2nd element of the list
maxProfit[2]; we will get [x=2850.]
Remove the square bracket
op(maxProfit[2]); we will get x=2850.
Take the right hand side of the equation
rhs(op(maxProfit[2])); we will get 2850
Round it to the nearest integer
round(rhs(op(maxProfit[2]))); we will get the final result 2850
Get the aassociated daily profit (use “round” function to get the nearest integer)
round(maxProfit[1])

7. b)
prices := [600, 610, 620, 630, 640, 650, 660, 670, 680, 690, 700, 710, 720, 730, 740, 750, 760, 770, 780, 790, 800, 810, 820, 830, 840, 850, 860, 870, 880, 890]:
Use the same C function for the cost
C:=x-> *x+(1/10)*x^2:
Create a table for new profits based on new prices
profitTab := table();

8. Create a list of daily profits
for i from 1 to nops(prices) do
profitTab[i]:=prices[i]*x-C(x);
end do;
Conver the table to a list
profitList:=convert(profitTab,list):
Use the same technique in a) to define a function to for the unit that make maximize the profit and also output the nearest integer
maxProf:=x->round(rhs(op(Optimization[Maximize](x)[2])));

9. Use “map” function to apply this function to the list of daily profits expressions
map(maxProf,profitList);
Then you will get a list of units that maximize the daily profits based on different daily prices.
[2900, 2950, 3000, 3050, 3100, 3150, 3200, 3250, 3300, 3350, 3400, 3450, 3500, 3550, 3600, 3650, 3700, 3750, 3800, 3850, 3900, 3950, 4000, 4050, 4100, 4150, 4200, 4250, 4300, 4350];