6. Applications of Linear Functions
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Chapter 3
6.
Applications of Linear Functions
Linear functions are central to the realm of algebra. The lesson focuses on the application and interpretation of these functions within various contexts. These functions help in depicting relationships, forecasting trends, and making informed decisions in numerous fields, from finance to physics. A solid grasp on the concept of a function in algebra provides a foundational understanding that is beneficial for both academic pursuits and real-world applications. Embracing the nuances of linear functions enables individuals to navigate complex situations with greater precision and confidence.
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Lesson Settings & Tools
| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Applications of Linear Functions
Slide of 9
Every day people have problems that the application of linear functions can solve. In this lesson, some common situations will be presented and solved seamlessly using them.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
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Analyzing Two Linear Functions
Dominika and her friend Zosia live relatively close to each other. Each morning they leave their homes to go to school, and at some point, they meet and continue the trip together. Usually, when walking separately, they walk at different speeds. 
The following two linear functions describe the way the girls walk to school separately.
f(x)=5x+a and g(x)=3x+b
Here, x denotes the time the girls are walking, while f and g represent each girls' distance from Dominika's home. The graphs of these functions and their point of intersection are demonstrated in the applet.
External credits: @pikisuperstar
a If Dominika walks faster than Zosia, which function corresponds to her? Label the graphs of f and g on the coordinate plane.
b Locate a and b on the coordinate plane. Then, interpret those values in terms of the given situation.
c Consider the points with the x-coordinates of m+2 on the functions f and g, where (m,n) is their point of intersection. What is the difference between the y-coordinates of these points?
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Comparing Rates of Changes
Dominika, a budding citizen scientist, is curious about the changes in temperature where she lives in Phoenix, Arizona. Since she walks to school everyday, the temperature affects her walk. Dominika decides to measure the temperature at three different times on the same day. Here are the measurements she recorded.
| Time | Temperature |
|---|---|
| 16:00 | 94^(∘)F |
| 19:00 | 88^(∘)F |
| 21:00 | 78^(∘)F |
When was the temperature decreasing the fastest: between 16:00 and 19:00 or between 19:00 and 21:00?
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Hint
Calculate the change in time and temperature. Use the formula for the rate of change.
Solution
In order to determine when the temperature was decreasing the fastest, the rate of change should be calculated. First, recall its definition. Rate of change=Δ y/Δ x In this formula, Δ y represents the change in the output variable and Δ x the change in the input variable. Because the change of temperature T over the change of time t needs to be found, the formula for the rate of change can be rewritten in terms of T and t. Rate of change=Δ T/Δ t To apply this formula, the changes in time and temperature will be calculated.
| Time Interval | Δ t | Δ T |
|---|---|---|
| 16:00- 19:00 | 3 hours | 94-88= - 6^(∘)F |
| 19:00- 21:00 | 2 hours | 88-78= - 10^(∘)F |
Now, substitute the obtained values of Δ T and Δ t into the formula to find the rate of change of each time interval.
| Time Interval | Δ t | Δ T | Rate of Change |
|---|---|---|---|
| 16:00- 19:00 | 3 hours | - 6^(∘)F | - 6/3=- 2^(∘)F per hour |
| 19:00- 21:00 | 2 hours | - 10^(∘)F | - 10/2=- 5^(∘)F per hour |
As can be seen, the temperature decreased at an average rate of 2^(∘)F in the first time interval and 5^(∘)F in the second time interval. Therefore, the temperature decreased the fastest between 19:00 and 21:00.
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Using Function Notation to Model a Problem
Along the walk to school, Dominika passes her father's 24-hour cafe. Running the cafe requires close attention to customer behavior. Her father notices that it is typical to serve 1 customer each night from 2:00AM till 4:00AM. Then, starting from 6:00AM, the cafe slowly begins to fill with regulars.
Let P represent the number of people in the cafe and t represent the time in hours since midnight.
a Is t a function of P? Explain your answer.
b Is P a function of t? If yes, consider a function P=f(t) and interpret the meaning of f(3) in the context of the given situation.
c Mr. Beckett noticed that the number of customers usually increases by an average of 20 people during the lunch hours from 12:00PM to 2:00PM. Which of the following equations best represents that observation?
Answer
a No, because the input value of P=1 is associated with three output values t=1, t=2, t=3.
b Yes, f(3) represents the number of customers in the cafe at 3:00AM.
c (ii) f(14)-f(12)=20
Hint
a Recall the definition of a function. Investigate whether each input is assigned to exactly one output.
b To interpret the meaning of f(3), think about what variables t and P denote.
c Analyze the meaning of each given equation. How can the number of customers at 12:00PM and 2:00PM be written using a function notation?
Solution
a To determine whether t is a function of P, recall the definition of a function.
b Contrary to Part A, if P is a function of t, each input t is assigned to exactly one output P.
c In order to identify the equation which best describes Mr. Beckett's observation, each given equation will be analyzed separately.
| Equation | Meaning |
|---|---|
| f(12)=20 | At 12:00PM, there are 20 customers in the cafe. |
| f(14)-f(12)=20 | The difference between the number of customers at 12:00PM and the number of customers at 2:00PM is 20 people. |
| f(14)-f(12)/2=20 | The average change of the number of customers per hour between 12:00PM and 2:00PM is 20 people. |
| P=f(12)+20 | The number of people at the cafe equals the number of customers at 12:00PM plus 20 more customers. |
The second equation can be rephrased as the number of customers at 2:00PM increased by 20 since 12:00PM, which coincides with what Mr. Beckett observed. Therefore, this equation best represents the described situation.
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Identifying the Domain and Range of a Function
While walking to school, Zosia was thinking about a story her auntie just told her. In the windy city crazy winds, flurries of snow, and a max capacity of 61 500 cheering fans have been a part of the legacy of Soldier Field, where the Chicago Bears have played on the gridiron each Sunday for decades. The average cost of ticket to a game was $45 at some point in time.
Zosia's auntie went to a game against there rivals, only to hear that the owner was thinking of selling the team. If only they could make more money from ticket sales! The amount of money earned from the game her auntie attended is a function of the number of people n who attended it.
a What is the domain of this function? Write the answer as a compound inequality.
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b What is the range of this function if M is the amount of money earned from the game? Write the answer as a compound inequality.
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c Write the function for the average profit P from the game if the cost of organizing the game is $100 000.
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Hint
a Analyze what minimum and maximum values n can have.
c First, find the expression for the earnings from the game. Then subtract the cost of holding the game.
Solution
a It is given that the amount of money earned from the game is a function of the number of people n who will attend the game. Let M be the amount of money earned from the game. Now, the described relation can be written as follows.
M=f(n) Recall that a domain is the set of all inputs for which the function is defined. In this case, the inputs of the function are the values of n. Since n represents the number of people attending the game, it can only have non-negative values. n≥ 0 Also, Soldier Field is said to have a maximum capacity of 61 500. Therefore, n cannot have values greater than 61 500. n≤ 61 500 Additionally, since n is the number of people, it can only have integer values. Gathering all the found information, the domain of the function can now be determined. D={n∈ Z: 0≤ n≤ 61 500}
b The range of a function is the set of all the outputs of the function. In this case, the outputs consist of all M values, representing the amount of money earned from selling tickets. If nobody attended the game — meaning no tickets were sold — the earnings would be 0. This is the minimum value that M could have.
M≥ 0 In the opposite situation, the stadium would be completely full. This would mean that all tickets were sold, and the earnings would be the maximum possible. In average, a ticket costs $45, and the stadium's total capacity is 61 500, so the product of these values will give the maximum possible earnings. $45* 61 500=$2 767 500 ⇓ M≤ 2 767 500 Finally, because both the price of a ticket and the number of people are integers, M will also be an integer number. With this information, the range of the function can now be summarized. R={M∈ Z:0 ≤ M≤ 2 767 500}
c Because the cost of organizing the game is $100 000, the profit P
can be determined by subtracting this quantity from the money earned by selling the tickets. Note that the money M earned by selling the tickets is equal to the product of the price of a ticket and the number of people n that will attend the game.
M=45n Now, the cost of organizing the game can be subtracted from the money earned to get the profit. P=45n-100 000 It can be seen that P is a function of n. This can be explicitly indicated by writing the equation for the average profit in function notation. P(n)=45n-100 000
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Using a Function and Inverse Function to Convert Temperatures
As Dominika was studying temperatures previously, she came across the temperature scales of both Fahrenheit and Celsius. She saw the Kelvin temperature scale — often used in scientific research — for the first time. She finds an online program to make a simple app that converts Kelvins to Fahrenheit and Fahrenheit to Kelvins.
At the moment, she only knows two facts that she has already added to the app.
f(x)=x
Therefore, substitute f(x) with x and solve the equation using inverse operations.
It can be concluded that 574.6 Kelvins equals 574.6^(∘)F.
- The zero point on the Kelvin scale is the lowest possible temperature in the universe and is equal to - 459.67^(∘)F.
- The body temperature in Kelvins is 310.15, while in degrees Fahrenheit, it is 98.6.
Help Dominika find the needed functions so that she can finish her app.
a Write a linear function f that assigns to a temperature in Kelvins its equivalent in degrees Fahrenheit.
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b Find the inverse of the function f and explain its meaning in terms of temperature conversions.
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Hint
a Since f is a linear function, it has the form of y=ax+b. Use the given information about the zero point and body temperature to find the values of a and b.
b To find the inverse of a function, switch the x and y places, and then solve the equation for y.
c If there is a temperature which is the same in Kelvins and degrees Fahrenheit, then f(x) would be equal to x for that value.
Solution
a This part asks for a linear function that converts a given temperature in Kelvins into degrees Fahrenheit. The following expression represents the general form of a linear function.
f(x)=ax+b
SubstituteII
x= 0, f(x)= -459.67
-459.67=a( 0)+b
▼
Solve for b
ZeroPropMult
Zero Property of Multiplication
-459.67=0+b
IdPropAdd
Identity Property of Addition
-459.67=b
RearrangeEqn
Rearrange equation
b=-459.67
f(x)=ax-459.67
SubstituteII
x= 310.15, f(x)= 98.6
98.6=a( 310.15)-459.67
▼
Solve for a
AddEqn
LHS+459.67=RHS+459.67
558.27=310.15a
RearrangeEqn
Rearrange equation
310.15a=558.27
DivEqn
.LHS /310.15.=.RHS /310.15.
a=1.8
b In order to find the inverse of the function f, first replace f(x) with y, as f(x)=y describes the input-output relationship of the function.
x=1.8y-459.67
▼
Solve for y
AddEqn
LHS+459.67=RHS+459.67
x+459.67=1.8y
Rewrite
Rewrite 1.8 as 18/10
x+459.67=18/10y
ReduceFrac
a/b=.a /2./.b /2.
x+459.67=9/5y
MultEqn
LHS * 5/9=RHS* 5/9
5/9(x+459.67)=y
RearrangeEqn
Rearrange equation
y=5/9(x+459.67)
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Applying a Linear Function to a Real-Life Problem
Dominika's father, the chef, knows that his daughter and her friends love to stop by his cafe along their walk home from school on Fridays. He is eager to expand his menu and add flor de calabaza quesadillas because it is his daughter's favorite!
Her father has calculated that there would be a weekly fixed cost of $75 for adding a new dish plus an additional $0.7 for the ingredients per quesadilla. The total cost C for preparing x quesadillas can be represented by a linear function. C(x)=0.7x+75 Now, the chef needs to figure out how much to charge for one quesadilla. Ideally, he wants to earn $0.30 per each quesadilla sold.
a If the chef sells 25 quesadilla per week, what will be the cost of making one quesadilla? At this cost, what should the price of one quesadilla be?
b If the chef wants to be sure that the price of one quesadilla is not greater than $1.50. How many quesadillas should he sell every day to set such a price and still make the desired profit?
c Complete the table showing the cost of making one quesadilla depending on how many quesadillas the chef sells.
| Number of Quesadillas Sold | 10 | 25 | 50 | 75 | 100 | 200 |
|---|---|---|---|---|---|---|
| Total Cost | ||||||
| Cost per Quesadilla | ||||||
| Sales Price per Quesadilla |
d Explain why the price of a quesadilla becomes smaller as the number of quesadillas sold increases.
Answer
a Cost per Quesadilla: $3.70
Price of One Quesadilla: $4
b $1.50
c
| Number of Quesadillas Sold | 10 | 25 | 50 | 75 | 100 | 200 |
|---|---|---|---|---|---|---|
| Total Cost | $82 | $92.50 | $110 | $127.50 | $145 | $215 |
| Cost per Quesadilla | $8.20 | $3.70 | $2.20 | $1.70 | $1.45 | $1.08 |
| Sales Price per Quesadilla | $8.50 | $4.00 | $2.50 | $2.00 | $1.75 | $1.38 |
d As more quesadillas are sold, the fixed weekly cost is shared by more quesadillas, which lowers the average cost of making one quesadillas.
Hint
a Calculate the total cost of making 25 quesadillas and then add $0.30 to find the price of one quesadilla.
b Find the expression for the average cost per quesadilla and then use it to form an equation for the price P of one quesadillas.
c Follow the steps used in Part A or use the found expressions for the average cost per quesadilla and the price of one quesadilla from Part B.
d Remember that the chef has a fixed weekly cost of $75. Then, think about how that cost is distributed between quesadillas depending on the number of quesadillas sold.
Solution
a First, by substituting 25 for x, the total cost of making 25 quesadillas can be found.
C(x)=0.7x+75
Substitute
x= 25
C( 25)=0.7( 25)+75
Multiply
Multiply
C(25)=17.5+75
AddTerms
Add terms
C(25)=92.5
b To determine how many quesadillas the chef should sell for the price of one quesadilla to be $1.50, the function for the price of one quesadilla should be found. First, by dividing the total cost of making quesadillas by the number of quesadillas x, the average cost of one quesadilla can be calculated.
c The total cost of making 10 quesadillas can be found by substituting 10 for x into the given function C(x).
| Number of Quesadillas Sold | 10 | 25 | 50 | 75 | 100 | 200 |
|---|---|---|---|---|---|---|
| Total Cost | $82 | $92.50 | $110 | $127.50 | $145 | $215 |
| Cost per Quesadilla | $8.20 | $3.70 | $2.20 | $1.70 | $1.45 | $1.08 |
| Sales Price per Quesadilla | $8.50 | $4.00 | $2.50 | $2.00 | $1.75 | $1.38 |
d The price of one quesadilla gets smaller as the number of sold quesadillas increases because there is a fixed weekly cost of adding a new dish to the menu. As more quesadillas are sold, this fixed cost is shared by more quesadillas. As a result, the average cost and, consequently, the price per quesadilla gets smaller.
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Finding a Linear Function Given Two Points
Let M(x,y) and N(z,w) be two different points. 
a Is there always a linear function whose graphs goes through M and N? Please explain.
b Find a linear equation in the standard form such that (2,5) and (- 3,1) are the solutions to the equation.
c Find a linear equation in the standard form such that (4,- 2) and (4,4) are the solutions to the equation.
Answer
a No, if the x-coordinates of the points are the same.
b Example Solution: x-2.5y=- 8
c Example Solution: x=4
Hint
a Consider the case where the x-coordinates of M and N are the same. Use the definition of a function.
c Examine the coordinates of the given points. What can be said about the line on which they lie?
Solution
a Recall that a function is a relation in which each input is assigned to exactly one output. Now, consider the situation where M and N have the same x-coordinates.
M(3,y) and N(3,w) In such a case, for the input x=3, the function will have two different outputs, y and w. This contradicts the definition of a function. Therefore, there is not always a function whose graph passes M and N. However, if the points have different x-coordinates, there is always a function whose graph contains those points.
b Start by reviewing the standard form of an equation.
c Start by analyzing the coordinates of the given points.
(4,- 2) and (4,5) As can be noticed, the points have the same x-coordinates. Therefore, they both lie on the vertical line x=4.
Additionally, by multiplying both sides of this equation by some number, an equivalent equation that also contains these points can be found. 2x=8 or - 3x=- 12 Keep in mind that these equations do not describe a function, as for one input of 4, there are many different outputs.
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Making Conclusions About Two Linear Functions
Finally, the challenge presented at the beginning will be solved. It stated that Dominika and her friend Zosia live relatively close to each other. Each morning they leave their homes to walk to school, and at some point, they meet and continue the trip together. Usually, when walking separately, they walk at different speeds. 
The following two linear functions describe the way the girls go to school separately.
f(x)=5x+a and g(x)=3x+b
Here, x denotes the time the girls are walking, while f and g represent each girls' distance from Dominika's home. The graphs of these functions and their point of intersection are demonstrated in the applet.
External credits: @pikisuperstar
a If Dominika walks faster than Zosia, which function corresponds to her? Label the graphs of f and g on the coordinate plane.
b Locate a and b on the coordinate plane. Then, interpret those values in terms of the given situation.
c Consider the points with the x-coordinates of m+2 on the functions f and g, where (m,n) is their point of intersection. What is the difference between the y-coordinates of these points?
Answer
a Dominika: f(x), Zosia: g(x)
Graph:
b The value a represents Dominika's distance before she has begun her walk from her home to school. Likewise, b denotes Zosia's distance from Dominika's when she, too, has yet to begin her walk.
c 4
Hint
b Remember that the constant in the slope-intercept form represents the y-intercept of the function. Recall what x, f, and g denote.
c Identify the slopes of the functions from their equations. Use the slopes to find how much each function will rise when moving 2 units to the right.
Solution
a By examining the given functions, it can be noted that they are given in the slope-intercept form. The slopes of these functions represent the walking speeds of the girls. Since f has a greater slope, this function must correspond to Dominika. Therefore, g corresponds to Zosia.
b In the slope-intercept form, a constant represents the y-intercept of the function. Therefore, a and b are the y-coordinates of the points where the functions intersect the y-axis.
It is given that x represents the time spent walking, and f and g represent each girl's distance from Dominika's home. From the diagram, it can be concluded that a equals 0. This makes sense, since Dominika is going to school from her home, so when she has yet to start her walk, her distance from her house is 0. a= Dominika's distance from her house when she has not started walking to school yet In the case of Zosia, b denotes her distance from Dominika's house when she has not started her walk to school. If she is also starting from her own home, then b is the distance between the houses of the two girls. b= Zosia's distance from Dominika's house when she has not started walking to school yet
c To find the difference between the y-coordinates of the given two points, the slope of the functions will be used. First, identify the slopes of the functions using their equations.
Slope off(x): m_1= 5 Slope of g(x): m_2= 3 Recall that the slope gives the number of units a function rises when moving 1 unit to the right. By using this information, the number of units that each function rises when moving 2 units to the right can be determined. f(x):& 5*2=10 units up g(x):& 3*2=6 units up Now, by adding this values to n, the y-coordinates of the points with the x-coordinate of m+2 can be found. f(x):& (m+2,n+10) g(x):& (m+2,n+6) Finally, the difference between the found y-coordinates can be determined. (n+10)-(n+6)=4
Applications of Linear Functions
Exercise 2.1
Izabella and her family had a trip to the mountains. Yesterday they climbed the mountain Cloudy Peak, which is about 3750 meters above the ground. Izabella made the following graph to describe their trip from the top of the mountain to the ground.
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Speed is calculated as the distance divided by time. Since the x-axis represents the time and the y-axis represents the distance, the speed is represented by the slope of the function. To calculate the slope, we will first identify two points on the graph.
Next, we can substitute these two points into the Slope Formula.
The slope is negative, because it shows that the height of Izabella's family was decreasing as they climbed down the mountain. However, a speed is always greater than or equal to 0. Therefore, their speed was 750 meters per hour.
From Part A, we know that the speed of Izabella and her family is 750 meters per hour. By dividing the height of the mountain above the ground of 3750 meters by the speed, we can determine how long it took them to reach the ground.
As we can see, it took 5 hours to reach the ground.
From the previous parts, we know that the function has a slope of - 750 and that it intersects the y-axis at 3750. With this information, we can write the complete function by substituting these values into the slope-intercept form. h= mt+ b ⇓ h= - 750t+ 3750 Note that here t represents the time of climbing and h represents the height above the ground.
Let's first determine the domain of the function, which is the set of possible t-values. It does not make sense for time to have negative values, so the minimum value is t=0. The family reached the bottom of the mountain after 5 hours of climbing down. They cannot go lower, so this is the maximum value. Domain: 0 ≤ t≤ 5 To determine the range, let's analyze all the possible h-values. The maximum height at which the family can be is the height of the mountain, which is 3750 meters. The lowest height is 0 meters above the ground. Range: 0 ≤ h≤ 3750
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Vincenzo has a part-time job alongside his studies. His salary depends on the number of hours he worked, and the salary is $14 per hour. However, due to certain regulations he is not allowed to work more than 25 hours a week.
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We know that Vincenzo earns $14 per hour. By multiplying this value by the number of hours t that he works, we can write the function for his total earnings. S(t)=14t
Let's first find the domain which is the set of all possible t-values. The minimum number of hours that Vincenzo can work is 0 and the maximum is 25. Domain: 0≤ t≤ 25 Now we can determine the range of the function by analyzing all possible values of S. If Vincenzo does not work, he earns nothing. Therefore, the minimum value of S is 0. He earns the most if he works 25 hours. Let's calculate how much that is by substituting 25 for t into the function.
Therefore, the maximum value of S is 320. Range: 0≤ S≤ 320
If Vincenzo was earning $16 per hour, then the following function would represent his total earnings. S(t)=16t Let's substitute 20 for t to calculate how much he would earn working 20 hours per week.
The maximum amount Vincenzo could have earned is $320 per week. This is less than the maximum of $350 that he can earn now.
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Kriz needs to compare the cost of printing a textbook that he needs for his studies. Digital Printer charges an initial cost of $1.20 and then 20 cents per copy. The printing service CopyFast charges no initial cost and 36 cents per copy.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"copies","answer":{"text":["219"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":" "},"formTextBefore":null,"formTextAfter":"copies","answer":{"text":["8"]}}
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Let's write a function describing all the expenses E for printing at Digital Printer in slope-intercept form. D= mx+ b Here, m is the slope and b is the y-intercept. We know that at Digital Printer the cost per copy is 20 cents, or $0.20. This means that for each copy Kriz prints, the total cost increases by $0.20. Therefore, the slope is m= 0.2. D= 0.2x+b We also know that Digital Printer's service has an initial cost of $1.20, so no matter how many copies Kriz prints they have to pay $1.20 in fixed fee. Therefore, the y-intercept b is 1.2. D=0.2x+ 1.2 By setting D=45 and solving for x, we determine how many copies Kriz can get for $45 at Digital Printer.
Kriz can print 219 copies for $45 at Digital Printer.
We know that CopyFast only charges per copy. Since this cost is 36 cents per copy, we get the following function. F=0.36x To find out how many copies Kriz should print for the expenses to be equal for both services, let's equate the functions D and F and solve for x.
The functions intersect when x=7.5. Let's draw them.
The function F is greater than D after the intersection, which means that the cost of copying at CopyFast is greater than at Digital Printer if Kriz needs to make 8 or more copies.
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