Question

A manufacturer of a product has fixed costs totaling $8295. He finds he can never sell any of the product unless the price is below $97, and for each decrease of $5 in the price, he is able to sell 100 more units. Find the cost function, demand function, revenue function, and profit function

Answer 1

Each unit costs an additional $47 to produce

Answer 2

C(x) = 8925 + 47 x , thats all i got, cant do the demand equation

Answer 3

what do you mean by "each unit costs and additional" ? is that just 45 per unit?

Answer 4

47 if i read it right lol

Answer 5

, yes, per item it costs 47 to produce

Answer 6

, i need help with the demand equation

Answer 7

fixed costs means like overhead, building and stuff

Answer 8

rent

Answer 9

fixed costs = 8295.

price < 97

-5 in the price, sell +100

Find the cost function, demand function, revenue function, and profit function

Cost(x) = 47x + 8295

Revenue(x) = x*Quantity(x)

Quantity(x) = has a slope of 100/-5 = -20
= -20x +20(Px) +Py, if its linear

Answer 10

Quantity would mean Demand

Answer 11

how much can they sell at $97 tho

Answer 12

where is your demand function

Answer 13

still working on it, but itd be nice to know how much can be sold at a given price

Answer 14

97,0 maybe

Answer 15

i guess we start with 97 as the price

Answer 16

(97,0) and (92, 100)

Answer 17

Demand(x) = has a slope of 100/-5 = -20
= -20x +20(97)
= -20x +194

Answer 18

let quantity demanded be a function of price

Answer 19

i dropped a zero :)

Demand(x) = -20x+1940

Answer 20

, wait, where did you get x?

Answer 21

x is just what im using for price

Answer 22

i have D = -20 p + 97

Answer 23

x should stand for units sold

Answer 24

your wrong on that

Answer 25

we used x earlier for units sold

Answer 26

change x to p then :)

Answer 27

D(p) = -20p +1940

Answer 28

actually , the units sold are the quantity demanded

Answer 29

so we have x = -20 p + 97

Answer 30

not 97

Answer 31

at p=97 you sell 0

Answer 32

one sec

Answer 33

0 = -20(97) + C

Answer 34

C = 1940

Answer 35

hmmm, i thought (97,0) is the y intercept

Answer 36

x intercept :)

Answer 37

or rather, price intercept

Answer 38

graphic wise it can be whatever axis you want to put it on

Answer 39

we have (97,0) and (92,100)

Answer 40

we are drawing the line through those two points

Answer 41

slope was given as 100/-5 to begin with = -20

Answer 42

(price, units sold/quantity demanded)

Answer 43

the price intercept (97,0)

Answer 44

so it should be x = -20 p + 97

Answer 45

oh no

Answer 46

D(p)= -20p + C , at a price intercept would to the independant variable

Answer 47

youre right

Answer 48

thats why i used "x" to show price

Answer 49

ok so x = -20 p + 1940

Answer 50

but p is fine, just as long as we know where its going :)

Answer 51

why did you call it C for price intercept

Answer 52

Demand(price) = -20(price) + 1940

Revenue(p) = price*Demand(price)

Answer 53

..C just means some constant that has to be determined

Answer 54

C is the demand intercept if anything

Answer 55

ok, thought it was cost

Answer 56

C is the quantity demanded intercept

Answer 57

yes

Answer 58

or 1940

Answer 59

ok now what do we do

Answer 60

Now we can determine Revenue(with regards to price)

Answer 61

R(p) = p*D(p)

R(p) = -20p^2 +1940p

Answer 62

Revenue tells us how much we bring in for a given price

Answer 63

Profit = Revenue - Costs

Answer 64

but original problem was in terms of x , not p

Answer 65

i assume that units sold (x) is equal to quantity demanded

Answer 66

thats fine

Answer 67

but x will be a function of p

Answer 68

otherwise we cant go further

Answer 69

since quantity depends on price

Answer 70

right, ill be back in 3 minutes

Answer 71

can you hold a sec

Answer 72

unless you have the answer

Answer 73

Q(p) = -20p +1940

R(p) = -20p^2 +1940p

C(Q(p)) = 47(Q(p)) + 8295
= 47(-20p +1940) + 8295

P(p) = -20p^2 +1940p -47(-20p +1940) -8295

Answer 74

simplify is wanted

Answer 75

max profits are when p=$72 if its right :)

Answer 76

hmm, so you did everything in terms of price ?

Answer 77

interesting, brb

Answer 78

yes

Answer 79

since the number of items depends on the number sold; x is a function of p; or rather x = Q(p)

Answer 80

but is that a fair assumption, the quantity demanded is the same as the number of units sold?

Answer 81

you dont want to mass produce; that just wastes money if noone will buy

Answer 82

let the amount that you can sell determine the number you will make

Answer 83

so the number of units demanded or sold will be the number of units you will produce or supply?

Answer 84

yes, ideally

Answer 85

you dont want a shortage or a surplus

Answer 86

but in wikipedia there is a supply curve and a demand curve

Answer 87

and they are not the same

Answer 88

but here we will just assume they are the same ?

Answer 89

and where the two shall meet is the ideal condition