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Here are a few recommended readings before getting started with this lesson.
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?
Calculate the change in time and temperature. Use the formula for the rate of change.
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.
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.
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.
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.
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}
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}
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
Help Dominika find the needed functions so that she can finish her app.
x= 0, f(x)= -459.67
Zero Property of Multiplication
Identity Property of Addition
Rearrange equation
x= 310.15, f(x)= 98.6
LHS+459.67=RHS+459.67
Rearrange equation
.LHS /310.15.=.RHS /310.15.
LHS+459.67=RHS+459.67
Rewrite 1.8 as 18/10
a/b=.a /2./.b /2.
LHS * 5/9=RHS* 5/9
Rearrange equation
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.
Number of Quesadillas Sold | 10 | 25 | 50 | 75 | 100 | 200 |
---|---|---|---|---|---|---|
Total Cost | ||||||
Cost per Quesadilla | ||||||
Sales Price per Quesadilla |
Price of One Quesadilla: $4
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 |
x= 25
Multiply
Add terms
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 |
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.
(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.
Graph:
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
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