Answer:
Q4. Product types A , B , C ≥ = 0
Step-by-step explanation:
Defining variables
let type A product = a
type B = b
type C = c
Model for the information
For cleaning, the total time is
4a + 6b + 8c = 3.5
For cutting, the total time is
2a + 4b + 2c = 2.5
For packaging, the total time is
2a + 2b + 4c = 1.5
The restrictions are
x , y , z >= 0
Q5.
Price per unit is defined as p=50−0.1q and revenue R(q)= qp(q)
R(q)= q(50-0.1q) Equation 1
R(q)= 50q−0.1q² Equation 2
The cost function as provided is
C(q)=1500+10q Equation 3
The profit function is given by
P(q)= R(q)- C(q) Equation 4 (a)
={50q−0.1q²} - {1500+10q}
=50q−0.1q² - 1500-10q
=40q−0.1q² - 1500 Equation 4(b)
Since we already have a price function as p=50−0.1q then multiplying each side by 10 and re-arranging and making q the subject we have
q=500-10p
Equation 4(b) can now be written while substituting q with 500-10p hence we obtain equation 5
40(500-10p)−0.1(500-10p)² - 150
20000-400p-0.1(250000-10000p+100p²)-1500
20000-400p-25000+1000p-10p²-1500
-6500+600p-10p² Equation 5
Therefore, the weekly profit is expressed as -10p²+600p-6500
b) Equation 5 is a parabolic equation which when compared with typical ax²+bx+c=0 hence in our case a=-10, b=600 and c=-6500 from equation 5
The maximum point is at [tex] (\frac{-b}{2a} , \frac{-d}{4a} ) [/tex]
Therefore, our [tex] \frac{-b}{2a} =\frac{-600}{2*-10} =30 [/tex]
[tex] \frac{-d}{4a}= \frac{-b^2-4ac}{4a}= \frac{4ac- b^2}{4a}= \frac{4(-10)(-6500)- 600^2}{4(-10)}[/tex]
= [tex] \frac{260000-360000}{-40} [/tex]
Therefore, our vertex is (30, 2500)
The price of a bottle to realize maximum profit is therefore $30
c)
As already shown in equations for part b, the maximum profit is $2500
Q6.
Since P is given as 4(1 – (Exp)-0.037t) then in one week t is 7 days for 1 week
hence P will be 4(1 – (Exp)-0.037*7)= 0.912 million
b)
The target is 4 million hence to get 60% means 60/100 of 4 million =2.4 million people
Substituting 2.4 where there's P our equation is
2.4= 4(1 – (Exp)-0.037t)
Dividing both sides of the above equation by 4 we obtain
0.6= (1 – (Exp)-0.037t)
Rearranging the above equation we obtain
(Exp)-0.037t= 1-0.6
(Exp)-0.037t= 0.4
t=24.76 days
t= 25 days the the nearest day
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