Need A Tutor?    |   Need Homework Help?                                                                             Help and Support     | Join or Cancel

Assume that the situation can be expressed as a linear cost function. Find the cost function.
Fixed cost is ​$200; 20 items cost ​$600 to produce.

C(x) = mx + b
600 = m (20) + 200
Subtract 200 for both sides
400 = m(20)
Divide both sides by 20
M = 20
C(x) = 20x + 200
   
Assume that the situation can be expressed as a linear cost function. Find the cost function in this case.
Marginal​ cost: ​$40​; 170 items cost ​$9000 to produce.

 
The linear cost function is ​C(x)= 40x + 2200
 
C(x) – 9000 = 40(x – 170)
C(x) – 9000 = 40x – 6800 (add 9000 to each side)
C(x) =
C(x) = 40x + 2200
   
To produce x units of a religious medal costs C(x) = 12x + 33. The revenue R(x) = 23x. Both cost and revenue are in dollars.
a. Find the​ break-even quantity.
b. Find the profit from 370 units.
c. Find the number of units that must be produced for a profit of ​$110
 

 
3 units is the​ break-even quantity.
23x = 12x + 33 (subtract 12x from both sides)
11x = 33 divide both sides by 11
x = 3
 
b. Find the profit from 370 units.
P (x) = 23x – (12x + 33) remove the parentheses and perform the subtraction in the right side
P (x) = 11x – 33 (substitute 370 for x in the equation above)
P (370) = 11(370) – 33 = P (370) = $4,037                  Profits units for 370 units is $4,037.
 
c.
Find the number of units that must be produced for a profit of ​$110
110 = 11x – 33 (add 33 to both sides)
143 = 11x (divide both sides by 11)
13 units make a profit of $110.
   
Joanne sells​ silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one​ T-shirt is 2.50.
Her total cost to produce 60 T-shirts is $240 and she sells them for ​$6 each.
a. Find the linear cost function for​ Joanne's T-shirt production.
b. How many​ T-shirts must she produce and sell in order to break​ even?
c. How many​ T-shirts must she produce and sell to make a profit of ​$800​?
 

Find the linear cost function for​ Joanne's T-shirt production.
 
Explanation
C(X) = mx + b
240 = 2.50(60) + b =
240 = 150 + b (subtract 150 from both sides)
90 = b
linear cost function is c(x) =
2.50x + 90
 
How many​ T-shirts must she produce and sell in order to break​ even
 
Explanation
P(x) = r(x) – c(x)
6x = 2.50x + 90 (subtract 2.50 from both sides)
3.50x = 90 (divide both sides by 3.50)
x = 25.7
 
Joanne must produce and sell 26 t-shirts to break even.
How many​ T-shirts must she produce and sell to make a profit of ​$800​?
 
Explanation
800 = 6x – (2.50x + 90) (subtract 2.50x from 6x)
800 = 3.50x – 90 (add 90 to both sides)
890 = 3.50x (divide 3.50 by both sides)
254.28
254.28 must be raised to 255 for Joanne to make a profit of $800
Joanne must produce 255 t-shirts to make a profit of $800.
  The manager of a restaurant found that the cost to produce 200 cups of coffee is ​$20.12​,
while the cost to produce 400 cups is ​$38.92.
Assume the cost​ C(x) is a linear function of​ x, the number of cups produced. Answer parts a through f.
Find a formula for​ C(x). Choose the correct answer below.
 
 
m = y₂ – y₁ / x₂ – x₁
(x₁, y₁) = (200, 20.12)
(x2, y2) = (400, 38.92)
38.92 – 20.12 / 400 – 200
18.8 / 200 = .094
20.12 = .094(200) + b
20.12 = 18.8x + b (subtract 18.8 from both sides)
1.32 = b
 
C(x) = 0.094x + 1.32
 
What is the fixed​ cost?
b = fixed cost so
$1.32 is the fixed cost
 
The total cost of producing 1000 cups is $
C (1000) = 0.094(1000) + 1.32
C (1000) = 95.32
 
The total cost of producing 1000 cups is $95.32.
 
The total cost of producing 1001 cups is $
 
C(1001) = 0.094(1001) + 1.32
C(1000) = 95.41
 
Find the marginal cost of the 1001st cup.
 
0.094 x 1000 = 9.4
9.4 cents
 
What does the marginal cost of a cup of coffee mean to the​ manager?
A. The marginal cost of a cup of coffee is the cost of producing one additional cup.
B. The marginal cost of a cup of coffee is the cost of producing the first cup.
C. The marginal cost of a cup of coffee is the cost of producing a given number of cups.
  A firm produces a product that has the production cost function ​C(x) = 240x + 3720 and the revenue function ​
R(x) = 300x.  No more than 58 units can be sold. Find and analyze the​ break-even quantity, then find the profit function.
R(x) = C(x)
 

 
300x = 240x + 3720 (subtract 240x from both sides)
60x = 3720 (divide by 60)
x = 62
 
The​ break-even quantity is 62 units.
 
If the company can produce and sell no more than 58 units, should it do​ so?
D. No. Since 58 is less than the​ break-even quantity, production of the product cannot produce a profit.
 
Write the profit function.
P(x) = R(X) – C(X)
300x – (240x + 3720) subtract 240x
 
P(x) = 60x – 3720
   
A person is the manager of a firm. He is considering manufacturing a new​ product, so he asks the accounting
department for cost estimates and the sales department for sales estimates. After he receives the​ data, he must
decide whether to go ahead with production of the new product. Analyze the following data​ (find a​ break-even quantity)
and then decide what the manager would do in this case.​ Also, write the profit function.
​  C(x) = 160x + 1400 and ​R(x) = 140x
 

140x = 160x + 1,400 (subtract 160x)
-20x = 1,400 (divide by -20)
x = break even -70
 
What would the manager decide to do in this​ case?
A. The manager would decide to not go ahead with the production.
B. The manager would decide to go ahead with the production
 
Write the profit function
​P(x) = ​R(x) - ​C(x)
140x – (160x + 1400)
-20x – 1400
 
Suppose that the fixed cost for a product is ​$260 and the​ break-even quantity is 65.
Find the marginal profit​ (the slope of the linear profit​ function).
The marginal profit is
 
Margin profits = fixed costs / units sold
Margin profits = 260 / 65 = 4
   
Suppose that the supply function for honey is p = S(q) = 0.3q + 2.7, where p is the price in dollars for an 8​-oz container
and q is the quantity in barrels.
Suppose also that the equilibrium price is ​$4.50 and the demand is 2 barrels when the price is ​$6.70.
Find an equation for the demand​ function, assuming it is linear.
 

4.50 = .3q + 2.7 (subtract 2.7)
1.8 = .3q
Q = 6
the demand is 2 barrels when the price is ​$6.70 and the demand is 6 barrels when the price is ​$4.50
6.70 – 4.50 / 2 – 6
-.55
P - 6.70 = -.55(q – 2)
P – 6.70 = -.55q + 1.1 (add 6.70)
P or D(q) = -0.55q + 7.80
   
Let one​ week's supply and demand functions for gasoline be given by p=D(q)=294−5/3q and p=S(q)=2/3​,
where p is the price in dollars and q is the number of​ 42-gallon barrels.
 
(a)   Graph these equations on the same axes.
(b) Find the equilibrium quantity.
(c) Find the equilibrium price.
 

 
 

294 – 5/3q = 2/3q (multiply by 3)
882 – 5q = 2q (add 5q)
882 = 7q (divide by 7)
Q = 126
 
P = 2/3(126) = $84
   
Use the information listed below to solve parts a through h.
Suppose that the demand and price for a certain model of a youth wristwatch are related by the following equation
p = ​D(q) = 16 = 1.25q where p is the price​ (in dollars) and q is the quantity demanded​ (in hundreds).
Find the price at each level of demand. Answer parts a through d.
 
  
p = ​D(q) = 16 = 1.25q
 
Find the price when the demand is 0 watches.
16 – 1.25q(0)
16 – 1.25
The price when the demand is 0 watches is 16
 
16 – 1.25(4)
16 – 5 = 11
 
The price when the demand is 400 watches is $11
 
Find the quantity demanded for the watch when the price is ​$6.
 
At a price of ​$6​, the demand is for ___ watches.
 
6 = 16 – 1.25q (subtract 16)
-10 = -1.25q (divide -1.25)
Q = 8 x 100 = 800
At a price of ​$6, the demand is for 800 watches.
 
Suppose the price and supply of the watch are related by the following equation p = S(q) = 0.75q
where p is the price​ (in dollars) and q is the quantity supplied​ (in hundreds) of watches.
Answer parts e through g.
p  = .75q
0 = .75q
 
P = .75q
 
(10) = .75 (divide by .75)
13.333 Multiply by 100 = 1333
 
16 – 1.25q = .75q (add 1.25)
16 = 2q (divide by 2)
8 multiply by 100 = 800 watches
 
.75(8) = $6
  Use the information listed below to solve parts a through h.
Suppose that the demand and price for a certain model of a youth wristwatch are related by the following equation.
p = ​D(q) = 16 = 1.25q where p is the price​ (in dollars) and q is the quantity demanded​ (in hundreds).
Find the price at each level of demand. Answer parts a through d.
 
  
p = ​D(q) = 16 = 1.25q
 
a.     Find the price when the demand is 0 watches.
 
16 – 1.25(0)
16 – 0
The price when the demand is 0 watches is 16.
 
b.     Find the price when the demand is 400 watches.
 
16 – 1.25(4)
16 – 5 = 11
The price when the demand is 400 watches is $11.
 
c.      Find the quantity demanded for the watch when the price is ​$6.
 
Explanation
At a price of ​$6​, the demand is for ___ watches.
 
6 = 16 – 1.25q (subtract 16)
-10 = -1.25q (divide -1.25)
Q = 8 x 100 = 800
 
At a price of ​$6, the demand is for 800 watches.
 
Suppose the price and supply of the watch are related by the following equation p = S(q) = 0.75q
where p is the price​ (in dollars) and q is the quantity supplied​ (in hundreds) of watches.
Answer parts e through g.
p = .75q
0 = .75q
 
Explanation
P = .75q
(10) = .75 (divide by .75)
10 / .75 = 1.3333333333
13.333 x 100 = 1333
 
16 – 1.25q = .75q
16 = 1.25q + .75q = 16 = 2q
q = 16 / 2 = q = 8
8 x 100 = 800
 
.75(8) = $6  
The table gives the​ revenue, R of a fast food​ chain, where R is in millions of dollars and t is in years since January​ 1, 2002. Complete parts ​a-d below
 

a) Find the average rate of change in revenue over the​ one-year intervals from 2002 to 2003, from 2003 to 2004 and from 2004 to 2005
f(b) – f(a) / b – a
 

b) Assuming the pattern​ repeats, find the average rate of change over the​ one-year intervals from 2005 to 2006 and from 2006 to 2007.

​​c) Study the table of values. What can be inferred about the function​ R(t) from t = 0 to t = 5​?

d) Report and interpret the slope if the revenue is linear.
Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice.

Report and interpret the​ y-coordinate of the vertical intercept if the revenue is linear.
Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice.

 
Revenue R increases steadily at 5 points
   
Choose the best answer for the limit.

 
Does not exist.
      lim f(x) = 8   and   lim f(x) = 8, but f(6) = -8
   x-->6^-                      x-->6⁺
 
What can you say about lim f(x)?
                                          x-->6

lim f(x)? is 8
 x --> 6

---

---