6. Applications of Linear Functions
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Chapter 3
6.
Applications of Linear Functions
| Student Learning Objectives: |
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Why
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.
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.
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Challenge
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.
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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Example
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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Example
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?
I.& f(12)=20 [0.3em] II.& f(14)-f(12)=20 [0.3em] III.& f(14)-f(12)/2=20 [0.3em] IV.& P=f(12)+20
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.
From 2:00AM till 4:00AM there is usually 1 customer in the cafe. If t is a function of P, written as t=f(P), then the values of P are inputs and the values of t are outputs. Here, the input is 1, the customer, and the outputs are the hours 2:00AM, 3:00AM, and 4:00AM, labeled 2, 3, and 4, respectively. This relation can be visualized with a mapping diagram.
In the diagram, it can be observed that there would be more than one output t associated with the single input value of P=1. This fact contradicts the definition of a function. Therefore, t cannot be a function of P. t=f(P) *
b Contrary to Part A, if P is a function of t, each input t is assigned to exactly one output P.
Note that it does not matter that different input values are assigned to the same output value. The important thing is that they are not assigned to more than one output. Therefore, P is a function of t. P=f(t) ✓ To interpret the meaning of f(3), recall that the input t represents the time in hours since midnight, while the output P represents the number of people in the cafe.
The expression f(3) represents the corresponding value of P when t=3. In this situation, t=3 represents 3:00AM and f(3) is the number of people in the cafe at that time.
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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Example
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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Example
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.
- 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 For this situation, the inputs x will represent a temperature in Kelvins and the outputs f(x) will represent the corresponding temperature in degrees Fahrenheit. It is known that 0 Kelvin equals -459.67^(∘) F. Therefore, the value of the constant b can be calculated by substituting 0 for x and -459.67 for f(x).
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
Now that the value of b is found, substitute it into the above formula of a linear function. f(x)=ax+ b ⇓ f(x)=ax- 459.67 Using a similar reasoning, the given body temperatures in Kelvin and degrees Fahrenheit can be used to find the value of a. This time, 310.15 will be substituted for x and 98.6 for f(x) into the partial function found previously.
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
Finally, the function that converts Kelvins into degrees Fahrenheit can be completed. f(x)=1.8x-459.67
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.
y=1.8 x-459.67 Next, because the inverse of a function reverses x and y, switch x and y places in the function rule. x=1.8 y-459.67 Now, the obtained equation should be solved for y.
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)
Therefore, the inverse function of f has been found. In the context of this situation, this inverse function converts degrees Fahrenheit into Kelvins. f^(- 1)(x)=5/9(x+459.67) Dominika can now use these two functions to implement the desired feature on her amazing 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.
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Example
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 $75. Then, think about how that cost is distributed between quesadillas depending on the number of quesadillas sold.
Solution
a First, by substituting 25 for x, the total cost of making 25 quesadillas can be found.
C(x)=0.7x+75
Substitute
x= 25
C( 25)=0.7( 25)+75
Multiply
Multiply
C(25)=17.5+75
AddTerms
Add terms
C(25)=92.5
Next, divide the total cost by 25 to calculate the cost of making one quesadilla. Cost per Quesadilla: 92.5/25=$ 3.70 It is given that the chef wants to earn $0.30 from each quesadilla. By adding that value to the cost of making one quesadilla, the price of one quesadilla can be determined. Price of a Quesadilla: 3.70+0.30=$4.00
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.
Average Cost per Quesadilla: 0.7x+75/x Next, add $0.30. This is the profit the chef wants to earn from each quesadilla sold. In doing this, the ideal sales price P of making one quesadilla can be found. P(x)=0.7x+75/x+0.3 Finally, to find the number of quesadillas the chef should sell, substitute P with $1.5 and solve the equation for x.
Therefore, the chef should sell 150 quesadillas each week for the price of one quesadilla to be $1.50.
c The total cost of making 10 quesadillas can be found by substituting 10 for x into the given function C(x).
The cost per quesadilla can be calculated either by dividing the found total cost by 10 or by substituting x with 10 into the expression for the average cost of a quesadilla found in Part B. Cost per Quesadilla: 82/10=$ 8.2 To find the sale price of one quesadilla, add $0.30 to the cost of one quesadilla. Also, the function P(x) from Part B can be used. Price of a Quesadilla: 8.2+0.30=$8.50 Similarly, the total cost, cost per quesadilla, and sales price per quesadilla can be calculated for the rest of the given numbers of sold quesadillas.
| 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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Example
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.
Ax+By=C Next, substitute the coordinates of the two given points, ( 2, 4) and ( - 3, 2), to obtain two equations. 2A+ 4B=C (I) - 3A+ 2B=C (II) To eliminate one variable, for example B, multiply the second equation by 2 so that the coefficients before B match in both equations. - 3A&+2B=C | * 2 &⇕ - 6A&+4B=2C Now, subtract the obtained equation from Equation (I). 2A + 4B &= C ^- -6A + 4B &= 2C 8A &= - C There is not enough information to solve for both A and C. Therefore, choose a convenient value of C, and then use it to solve the equations for A and B. For example, let C be equal to - 8. Use this value to calculate A.
Now, the only unknown value is B. A=1 B= ? C=- 8 By substituting the known values of A and C into either Equation (I) or (II), the value of B can be calculated.
Finally, the equation can be completed. x-2.5y=- 8 Note that if another value of C was chosen, the coefficients of A and B would be different. For instance, if C=16, then A=- 2 and B=5. Then, a different equation is obtained. - 2x+5y=16 However, these are equivalent equations of the same line, as the first equation multiplied by - 2 is the same as the second equation. x-2.5y=- 8 | * (- 2) ⇕ - 2x+5y=16 If those equations were rewritten in slope-intercept form, they would be the same. This observation strongly emphasizes an advantage of the slope-intercept form: for every line, there is only one equation that describes it.
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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Closure
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.
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.
Dominika: f(x)=5x+a Zosia: g(x)=3x+b Note that a function with a greater slope has a steeper graph. On the given diagram, the blue line is steeper than the purple line. With this information, the functions can now be labeled.
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
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Applications of Linear Functions
Exercises
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To find the linear function corresponding to the line, we need to identify its y-intercept represented by b and the slope represented by m. y=mx+b From the graph, we see that f(x) intersects the y-axis at (0, 600). Let's substitute these values into the equation. f(x)&=mx+ 600 To determine the slope of the line, we need to look at how much the function rises along the y-axis when we move one step to the right along the x-axis.
As we can see, f(x) has a slope of 300. With this information we can complete our function rule. f(x)&= 300x+600
We will apply the same process as before in Part B to write a function rule for g(x). Since the graph of g(x) is passing through the origin the y-intercept is 0, which corresponds to the value of b. y=mx+ 0 ⇔ y=mx Next we will determine the slope. Let's have a look at the graph to see the change along the y-axis when we move to the right along the x-axis.
As we can see, g(x) has a slope of 400. Let's now complete our function rule. g(x)&= 400x
By equating the functions we can determine for how many hours the editors should be hired for the cost to be the same.
When the editors are hired for 6 hours, the costs of hiring them are the same.
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We know that 7 hours after the freezer broke the temperature was - 39^(∘) C, while after 12 hours it was - 24^(∘) C. Using this information, we can identify two points that lie on the line representing the situation. (7,- 39) and (12,- 24) We are asked to use a linear function to determine how the temperature increases. Therefore, we can write the equation in slope-intercept form. y=mx+b First, let's calculate the slope by substituting the two points into the Slope Formula.
The slope between the points is 3. We can substitute this value for m into the equation. y= mx+b ⇓ y= 3x+b Next, we can substitute either of the points into this equation and calculate the value of b. For example, let's substitute (7,- 39).
Therefore, the y-intercept is - 60^(∘) C. The x-coordinate of this point is 0, which means that this was the temperature at the moment the freezer broke. Finally, we can complete the equation. y=3x+( - 60) ⇓ y=3x-60
To determine the domain of the function we will first analyze the range, which is the set of all possible y-values.
Range
From Part A, we know that the freezer had an initial temperature of - 60^(∘) C. This is the minimum value of the temperature y. After breaking, the temperature of the freezer started to increase. Since the temperature of the freezer cannot exceed the room temperature, the maximum value of y is 21^(∘) C. - 60≤ y≤ 21
Domain
Since x represents the time since the freezer broke, negative values of x do not make sense in this situation. Therefore, x=0 is the minimum possible value. To find the maximum value we can substitute the maximum value of y, which is 21, into the equation and find the corresponding value of x.
The maximum value of x is 27. This value tells us that the freezer will reach the room temperature of 21^(∘) C after 27 hours. Now we can write the domain. 0≤ x≤ 27
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If they use the coupon, the total cost of buying x tickets would be described by the following linear function. C(x)=24.5x-15
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We are given a function that describes the cost in dollars C of buying x tickets to a concert using a $15 coupon. C(x)=24.5x-15 If Ramsha goes with 5 of her friends, then 6 people need tickets. Calculating the cost of buying six tickets is the same as evaluating C(x) for x=6. Therefore, let's substitute 6 for every instance of x in the function.
With a $15 coupon, it costs $132 to buy 6 tickets to the concert.
In Part A we were given the number of tickets to buy and needed to calculate the cost. This time we are given the cost and need to calculate the number of tickets. C(x)=24.5x-15 To calculate the number of tickets that Ramsha can buy with $ 156.50, we need to substitute this amount for C(x) in the function and solve for x.
With $156.50 Ramsha can buy 7 tickets. This means that 6 friends will join her.
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The distance d that it travels in t hours is given by the following linear function. d(t)=17 500t
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We know the function that represents the distance d in miles that the Neowise Comet travels in t hours. d(t)=17 500t Calculating the distance that the comet travels in 12 hours is the same as evaluating d(t) for t=12. We can do this by substituting 12 for every instance of t in the function.
In 12 hours, the comet travels 210 000 miles.
In this case we are given the distance the comet travels, and we need to calculate the time it takes to go that distance. d(t)=17 500 t Calculating the time it would take for the comet to travel 1 000 000 miles means that we will need to substitute this distance for d(t) in the function and solve for t. Remember that we need to round the value we get to the nearest hour as well.
We found that the comet travels 1 million miles in approximately 57 hours.
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External credits: @rocketpixel
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We are told that the population of the United States increases linearly, so we can use an equation in slope-intercept form to describe this situation. y=mx+b Here x is the number of years since 1960 and y is the population of the United States in millions. By using the given information we can identify two points. (0,100) and (40,200) By substituting these points into the Slope Formula, we can determine the slope.
Note that we also know the value of b — which represents the y-intercept — because the line passes through the point (0,100). Therefore, b=100. With this information we can complete the equation. y = mx+ b ⇓ y = 2.5x+ 100
First, let's find how many years have passed from 1960 to 2022. 2022-1960=62 years To calculate the population in 2022, we can substitute x with 62 in the equation and solve for y.
According to the model, the population in year 2022 should be 255 million. The real value is around 330 million, so the model is slightly off.
We can determine in what year the population was 150 million by substituting 150 for y and solving for x.
Therefore, 20 years after 1960 (which was 1980) the population of the United States was about 150 million.
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Profit is calculated as the difference between revenue and expenses. Profit= Revenue- Expenses Let's calculate what revenue and expenses Emily's class has. The only revenue comes from the sale of the 100 tickets, and since each ticket is sold for $30, the total revenue is the product of these two numbers. Revenue=100* 30= $3000 The expenses are the venue's rental costs of $500 and the DJ who charges $1500. By adding these values, we can find the total amount of their expenses. Expenses=1500+500= $2000 Now that we know both the total amounts of revenue and expenses, we can determine the profit. Profit= 3000- 2000 ⇓ Profit=$1000
To write the function B that describes the profits or losses of Emily's class, we will use the formula for profit from Part A. Profit= Revenue- Expenses Let x be the number of tickets sold. The price of each ticket is $30. This implies that 30x represents the amount of revenue. As we calculated in Part A, the expenses equal $ 2000. With this information, we can write the function B. B(x)= 30x- 2000
In order to find the range of the function, let's first identify its domain. Since x represents the number of tickets sold it cannot have negative values, so the minimum value is 0. We also know that the maximum number of paying guests is 200. Domain: 0≤ x≤ 200 Using these values, we can now find the range of the function. Let's substitute x with 0 and find the minimum value of B.
The minimum of the function's value set is - 2000. The maximum possible profit is obtained when all 200 tickets have been sold. We can determine this value by substituting x=200 into the function.
Finally, we can write the range of the function B. Range: - 2000≤ B≤ 4000
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Applications of Linear Functions
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