linear function

Question

You and your company partner are researching how many coffee mugs must be produced and sold for maximum revenue. After extensive research, you concluded that you could give away 50 coffee mugs without charging anything. If you charge $10.00, nobody will buy the coffee mugs. Your job is to find the maximum possible revenue from selling the coffee mugs.

2. Find the equation of the linear function S(p) that describes the situation.

3. The revenue is determined by R = S p. Express the revenue as a function of the price p.

4. Graph the revenue function. Make sure to label the x - and the y - axis with the appropriate units.

5. At which price per coffee mug do you not make any money?

6. Find the price per coffee mug that makes the most revenue.

7. How many coffee mugs must be sold to maximize the revenue?

8. What is the maximum revenue?

9. How many coffee mugs must be sold to make at least $50 in revenue?

Answer 1

1. The demand line as a function of price has points (0,50) and (10,0);

[50 given away and none sold when price is $10]

2. The equation of the demand line...

slope is m=50/-10 = -5

The intercept, using (10,0) is B = y - mx = 0 - (-5)(10) = 50

S(p) = -5p + 50

3. Revenue function R(p) = p (-5p+50) = -5p^2 + 50p

4. graph functions at desmos dot com

5. 0 = p(-5p + 50)

p=0 or -5p + 50 = 0

p=0 or p=10

You make no money if given away for free, or if price is $10, as giv en

6. Maximizing the revenue function:

7. A=-5 B=50

8. max occurs at -B/(2a) = -50/-10 = 5

R(5) = 5( -5*5+50) = 5 ( 25)= 125

max revenue of $125 when price is $5

9. -5p^2 + 50p>50

-5p^2 + 50p - 50 > 0

p^2 - 10p + 10 < 0

From the graph,

$50 revenue between p=1.18 and 8.88