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Analyze the unit rates of change of translated rational functions by modeling
per-person cost, Practice Set C
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Date:
1. You and your friends have decided to rent a limousine for prom. The cost of the
limousine for the evening is a fixed cost of $500, to be split evenly among all attendees,
plus an extra gas cost of $5 per person.
Create a chart using x-values of 1, 2, 3, 4, 5, ... ,
99, 100, … , 999, and 1,000 and use it to find the cost per person, as well as the unit change
in per person cost from 1-2 attendees, 2-3 attendees, 3-4 attendees, 4-5 attendees, 99-100
attendees, and 999-1,000 attendees. Then use your chart to answer the questions below.
a. Describe how the unit change in per person cost is changing as the
number of people increases.
b. Will the per person cost of the limousine ever be $0? Explain.
c. Will the unit change in per person cost ever be $0? Explain.
d. If we were to plot the cost per person relative to the number of people, would
the graph be linear? Explain.
2. You and your friends are looking to put a big celebration on for the end of the school
year. You found the perfect venue to rent for your party, and to rent the space for a 3-hour
long party, there is a fixed cost of $750, to be split evenly among all attendees, plus a
cleaning fee of $10 per person. Create a chart using x-values of 1, 2, 3, 4, 5, ... , 99, 100, … ,
999, and 1,000 and use it to find the cost per person, as well as the unit change in per
person cost from 1-2 attendees, 2-3 attendees, 3-4 attendees, 4-5 attendees, 99-100
attendees, and 999-1,000 attendees. Then use your chart to answer the questions below.
a. Describe how the unit change in per person cost is changing as the
number of people increases.
b. Will the per person cost of the limousine ever be $0? Explain.
c. Will the unit change in per person cost ever be $0? Explain.
d. If we were to plot the cost per person relative to the number of people, would
the graph be linear? Explain.
Analyze the unit rates of change of translated rational functions by modeling
per-person cost, Practice Set C
Answer Key
1. You and your friends have decided to rent a limousine for prom. The cost of the
limousine for the evening is a fixed cost of $500, to be split evenly among all attendees,
plus an extra gas cost of $5 per person.
Create a chart using x-values of 1, 2, 3, 4, 5, ... ,
99, 100, … , 999, and 1,000 and use it to find the cost per person, as well as the unit change
in per person cost from 1-2 attendees, 2-3 attendees, 3-4 attendees, 4-5 attendees, 99-100
attendees, and 999-1,000 attendees. Then use your chart to answer the questions below.
Number of
people in
limousine
Cost per person
Unit change in per person cost
(Show your work below)
(Show your work below)
1
$500 + $5(1 𝑝𝑒𝑟𝑠𝑜𝑛)
1 𝑝𝑒𝑟𝑠𝑜𝑛
= $505
2
$500 + $5(2 𝑝𝑒𝑜𝑝𝑙𝑒)
2 𝑝𝑒𝑜𝑝𝑙𝑒
= $255
$255 − $505
= −$250
2−1
3
$500 + $5(3 𝑝𝑒𝑜𝑝𝑙𝑒)
3 𝑝𝑒𝑜𝑝𝑙𝑒
= $171.67
$171.67 − $255
3−2
= −$83.33
4
$500 + $5(4 𝑝𝑒𝑜𝑝𝑙𝑒)
4 𝑝𝑒𝑜𝑝𝑙𝑒
= $130
$130 − $171.67
4−3
= −$41.67
5
$500 + $5(5 𝑝𝑒𝑜𝑝𝑙𝑒)
5 𝑝𝑒𝑜𝑝𝑙𝑒
= $105
$105 − $130
= −$25
5−4
99
$500 + $5(99 𝑝𝑒𝑜𝑝𝑙𝑒)
99 𝑝𝑒𝑜𝑝𝑙𝑒
= $10.05
100
$500 + $5(100 𝑝𝑒𝑜𝑝𝑙𝑒)
100 𝑝𝑒𝑜𝑝𝑙𝑒
= $10
$10 − $10.05
100 − 99
= −$0.05
999
$500 + $5(999 𝑝𝑒𝑜𝑝𝑙𝑒)
999 𝑝𝑒𝑜𝑝𝑙𝑒
= $5.5005
1,000
$500 + $5(1000 𝑝𝑒𝑜𝑝𝑙𝑒)
1000 𝑝𝑒𝑜𝑝𝑙𝑒
= $5.50
$5.50 − $5.005
1000 − 999
= −$0.005
a. Describe how the unit change in per person cost is changing as the
number of people increases.
The unit change in per person cost is decreasingly decreasing.
b. Will the per person cost of the limousine ever be $0? Explain.
No, the per person cost will get infinitely close to $0, but never actually be
$0. The reason for this is if you take $500 and divide by a larger and larger
amount, you will always get a number greater than zero.
c. Will the unit change in per person cost ever be $0? Explain.
No, the unit change in per person cost will get infinitely close to $0, but
never actually be $0. The per person cost will never be $0, therefore the
unit change in per person cost can also never be $0.
d. If we were to plot the cost per person relative to the number of people, would
the graph be linear? Explain.
The graph would not be linear. A linear graph means that the unit rate of
change is constant, and in our case, the unit rate of change is not constant;
it is decreasingly decreasing.
2. You and your friends are looking to put a big celebration on for the end of the school
year. You found the perfect venue to rent for your party, and to rent the space for a 3-hour
long party, there is a fixed cost of $750, to be split evenly among all attendees, plus a
cleaning fee of $10 per person. Create a chart using x-values of 1, 2, 3, 4, 5, ... , 99, 100, … ,
999, and 1,000 and use it to find the cost per person, as well as the unit change in per
person cost from 1-2 attendees, 2-3 attendees, 3-4 attendees, 4-5 attendees, 99-100
attendees, and 999-1,000 attendees. Then use your chart to answer the questions below.
Number
of
people
Cost per person
Unit change in per person cost
(Show your work below)
(Show your work below)
1
$750 + $10(1 𝑝𝑒𝑟𝑠𝑜𝑛)
1 𝑝𝑒𝑟𝑠𝑜𝑛
= $760
2
$750 + $10(2 𝑝𝑒𝑜𝑝𝑙𝑒)
2 𝑝𝑒𝑜𝑝𝑙𝑒
= $385
$385 − $760
= −$375
2−1
3
$750 + $10(3 𝑝𝑒𝑜𝑝𝑙𝑒)
3 𝑝𝑒𝑜𝑝𝑙𝑒
= $260
$260 − $385
= −$125
3−2
4
$750 + $10(4 𝑝𝑒𝑜𝑝𝑙𝑒)
4 𝑝𝑒𝑜𝑝𝑙𝑒
= $197.50
$197.50 − $260
4−3
= −$62.50
5
$750 + $10(5 𝑝𝑒𝑜𝑝𝑙𝑒)
5 𝑝𝑒𝑜𝑝𝑙𝑒
= $160
99
$750 + $10(99 𝑝𝑒𝑜𝑝𝑙𝑒)
99 𝑝𝑒𝑜𝑝𝑙𝑒
= $17.57
100
$750 + $10(100 𝑝𝑒𝑜𝑝𝑙𝑒)
100 𝑝𝑒𝑜𝑝𝑙𝑒
= $17.50
999
$750 + $10(999 𝑝𝑒𝑜𝑝𝑙𝑒)
999 𝑝𝑒𝑜𝑝𝑙𝑒
= $10.7507
1,000
$750 + $10(1000 𝑝𝑒𝑜𝑝𝑙𝑒)
1000 𝑝𝑒𝑜𝑝𝑙𝑒
= $10.75
$160 − $197.50
5−4
= −$37.50
$17.50 − $17.57
100 − 99
= −$0.07
$10.75 − $10.7507
1000 − 999
= −$0.0007
a. Describe how the unit change in per person cost is changing as the
number of people increases.
The unit change in per person cost is decreasingly decreasing.
b. Will the per person cost of the limousine ever be $0? Explain.
No, the per person cost will get infinitely close to $0, but never actually be
$0. The reason for this is if you take $500 and divide by a larger and larger
amount, you will always get a number greater than zero.
c. Will the unit change in per person cost ever be $0? Explain.
No, the unit change in per person cost will get infinitely close to $0, but
never actually be $0. The per person cost will never be $0, therefore the
unit change in per person cost can also never be $0.
d. If we were to plot the cost per person relative to the number of people, would
the graph be linear? Explain.
The graph would not be linear. A linear graph means that the unit rate of
change is constant, and in our case, the unit rate of change is not constant;
it is decreasingly decreasing.