Question

a diner serves lunch for $6. an average of 90 lunches are sold per week. a market survey indicates that for each 25-cent increase in the price, the number of customers per week will decrease by 3.
a) determine a function R(x) that models the weekly revenue after x 25-cent increases in the price.
b) what price should the diner charge to maximize the weekly revenue? what is the maximum revenue?

Answer 1

please help me!

Answer 2

So R(0) = # of lunches sold after NO prices increases. What is that equal to?

Answer 3

6

Answer 4

No, how many lunches are sold today.

Answer 5

wait i mean 90

Answer 6

Right. R(1) = # lunches sold after 1 25 cent increase. What's that equal to?

Answer 7

not sure sorry

Answer 8

The question says: "a market survey indicates that for each 25-cent increase in the price, the number of customers per week will decrease by 3."

Hence if the price increases by 25 cents, how many customers will there be?

Answer 9

There is 90. If you take 90 and decrease it by 3, what do you get?

Answer 10

87

Answer 11

Yes, so R(1) = 90 - 3

Now what if there are two 25 cent increases? What is
R(2)
equal to?

Answer 12

84

Answer 13

Right, R(2) = 90 - 2*3

Hence in general, if there are x 25 cent increases, how many customers will there be
R(x) ?

Answer 14

R(0) = 90
R(1) = 90 - 1*3
R(2) = 90 - 2*3
...

Answer 15

R(x) = 90-3x ??

Answer 16

correct.

Answer 17

so how about part b?

Answer 18

Now you want to maximize revenue. Revenue = (Price) x (# customers)

Write Q(x) for revenue, P(x) for price and R(x) for customers. You just found an expression for R(x).

Find now an expression for P(x). Then

Q(x) = P(x).R(x)

Answer 19

Then you want to use calculus to maximize the function Q(x).

Answer 20

can you tell me what the answers are and then i can ask you if i dont understand it?

Answer 21

Nope. Use the same procedure we just used to find R(x) to find P(x).

P(0) = $6.00
if we have one price increase the price is what? That's P(1). Keep on going until you figure out what P(x) is.

Answer 22

i dont get it

Answer 23

Where are you lost? Do you understand where R(x) came from?

Answer 24

R(x) = 90 - 3x

Answer 25

right i get that

Answer 26

Do you agree that revenue = price * units sold ?

Answer 27

yes

Answer 28

Hence if we want to maximize Revenue, we need to find an expression for Revenue.

We only have one piece of the equation: units sold. Now we need the other piece of the equation price.

Write P(x) as the function for price.
What is P(0) = price with no price increases?

Answer 29

6

Answer 30

And what is P(1) ?

Answer 31

6.25

Answer 32

Hence what is P(x) in general?

Answer 33

6+.25x ?

Answer 34

correct, P(x) = 6 + 0.25x.

hence what is the equation for revenue as a function of x. Write that as Q(x).

Q(x) = what?

Answer 35

not sure sorry!

Answer 36

Revenue = Price * Units sold.

Answer 37

90 (6+.25x)

Answer 38

Price = P(x)
units sold = R(x)

Hence, revenue = Q(x) = ... what?

Answer 39

90-3x (6+.25x) ?

Answer 40

Q(x) = R(x).P(x)
= (90 - 3x)(6 + 0.25x)

Answer 41

Now use calculus to find the maximum for Q(x). I.e., you're solving the problem: what is the number of prices increases x such that revenue Q(x) is maximized.

Answer 42

how do i find that....?

Answer 43

Do you know calculus?

Answer 44

im in precalc and my teacher didnt teach us this. can you just please tell me?

Answer 45

In which case, your teacher is expecting you to use other things you know about quadratic equations.

Q(x) = R(x).P(x)
= (90 - 3x)(6 + 0.25x)

Expand that expression out and put it in standard form for a quadratic:
ax^2 + bx + c

Then use what you know about quadratic equations to find its maximum. In this case, the maximum will be at the vertex of the graph of the quadratic.

Answer 46

the question wants price and the maxium revenue...when i find the vertex on the graph is that for price or maximum revenue

Answer 47

for maximum revenue because the function you are graphing is Q(x), the expression for revenue.

Answer 48

ok and what is maximum price then?

Answer 49

That will be the value of x corresponding to where Q(x) is maximized.

Answer 50

I mean it will be P(x) for the x where Q(x) is maximized.

Answer 51

what does that mean

Answer 52

x is the number of price increases. The price for x price increases is P(x) = 6 + 0.25x.

Answer 53

You are maximizing revenue, Q(x), as a function of x, that is, as a function of the number of 25 cent price increases. That's the way the problem has been set up for you.

Answer 54

i got .03 for maximum revenue? that doesnt make sense

Answer 55

No. What is your expression for Q(x) written as a quadratic in standard form?

Answer 56

Q(x) = R(x).P(x)
= (90 - 3x)(6 + 0.25x)

Answer 57

-.75x^2 + 4.5x +540

Answer 58

exactly, good. Now, for what value of x is that maximized?

Answer 59

now i got 2.9 ??

Answer 60

for what value of x is
ax^2 + bx + c
have a vertex?

Answer 61

*does

Answer 62

i dont know!!!

Answer 63

x = -b/2a

Answer 64

which is 3 which is what i got (2.9)

Answer 65

Hence Q(x) = -.75x^2 + 4.5x +540
has a vertex when
\[ x = \frac{-4.5}{2(-1.5)} = 3 \]

Answer 66

This is the value of x for which Q(x) is maximized.

Answer 67

That corresponds to 3 prices increases of 25 cents.

Answer 68

thats the rmaximum revenue?????

Answer 69

The maximum revenue is the value of Q(x) when x = 3.

Answer 70

546. 75

Answer 71

and what is the price then

Answer 72

Yes, Q(3) = $546.75.

You tell me what the price is. What does x = 3 mean? We've written down what it means for price about four times already.

Answer 73

6.75 ?

Answer 74

yes. P(x) = 6 + 0.25x. Hence P(3) = $6.75

Answer 75

thanks for working through this with me.

Answer 76

Candy, Whenever I have a new hard problem like this one. After I've figured it out, I take a blank piece of paper and rewrite the solution. Then I do it again

When I can do that from a blank piece of paper without looking at the earlier solutions, I know I understand it. This is a great way to reinforce what you've learnt and I recommend you try it. I know learning new things is painful. But it's very satisfying when you get it.

Answer 77

thanks. is there any way to simple this down:\[x^3-5x^2-2x+24+5IX^2-5IX-30I\]

Answer 78

i got that from multiplying (x^2 -x -6) (x-4+5i)