11.3.0: Exercises
- Page ID
- 171752
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, identify the states, the seats, and the state population (the basis for the apportionment) in the given scenarios.
A parent has 25 pieces of candy to split among their four children. They will earn the candy based on how many minutes of chores they children did this week.
The board of trustees of a college has recently approved the installation of 70 new emergency blue lights in five parking lots. The number of lights in each lot will be proportionate to the size of the parking lot, which is to be measured in acres.
The reading coach at an elementary school has 52 prizes to distribute to students as a reward for time spent reading.
(The Coronavirus Crisis, by Pien Huang, Shots Health News From NPR, npr.org, November 24, 2020)
For the following exercises, use the given information to find the standard divisor to the nearest hundredth. Include the units.
The total population is 2,235 automobiles, and the number of seats is 14 warehouses.
The total population is 135 hospitals, and the number of seats is 200 respirators.
For the following exercises, use the given information to find the standard quota. Include the units.
The state population is eight residents in a unit, and the standard divisor is 1.75 residents per parking space.
The state population is 52 ICU patients each week, and the number of seats is 6.5 patients per respirator.
The total population is 145 basketball players, the number of seats is 62 trophies, and the state population is 14 basketball players on Team Tigers.
The total population is 12 giraffes, the number of seats is nine water troughs, and the state population is three giraffes in Enclosure C.
For the following exercises, use the table below, which shows student head count, class section, and total faculty in each of four college departments.
| Department | (M) Math | (E) English | (H) History | (S) Science | (O) College Overall |
|---|---|---|---|---|---|
| (T) Student Head Count | 4800 | 2376 | 1536 | 2880 | 87118 |
| (C) Class Sections | 120 | 108 | 48 | 96 | 3712 |
| (F) Faculty Members | 30 | 27 | 12 | 24 | 928 |
Determine the F to T ratios for each department rounded to four decimal places as needed. What are the units?
Determine the C to F ratios for each department rounded to four decimal places as needed. What are the units?
What is the F to T ratio for the college overall? Include units. How does it compare to the F to T ratios for individual departments?
What is the overall C to F ratio? Include units. How does it compare to the C to F ratios for individual departments?
Does there appear to be a constant F to T ratio? If so, what is the ratio? If not, what implications does this have about the different departments?
Does there appear to be a constant C to F ratio? If so, what is the ratio? If not, what implications does this have about the different departments?
If the departments are the states, the students are the population, and the faculty members are the seats, use the College Overall column to determine the standard divisor for the apportionment of the faculty rounded to two decimal places as needed. Include the units.
If the departments are the states, the classes are the population, and the faculty members are the seats, use the Overall College column to determine the standard divisor rounded to two decimal places as needed. Include the units.
Use the standard divisor from question 18 to find the standard quota for each department rounded to two decimal places as needed. What are the units?
Use the standard divisor from question 19 to find the standard quota for each department rounded to two decimal places as needed.
For the following exercises, use this information: Wakanda, the domain of the Black Panther, King T’Challa has six fortress cities. In Wakandan, the word “birnin” means “fortress city.” King T’Challa has found 111 Vibranium artifacts that must be distributed among the fortress cities of Wakanda. He has decided to apportion the artifacts based on the number of residents of each birnin. The table below displays the populations of major Wakandan cities.
| Fortress Cities | Birnin Djata (D) | Birnin T’Chaka (T) | Birnin Zana (Z) | Birnin S’Yan (S) | Birnin Bashenga (B) | Birnin Azzaria (A) | Total Population |
|---|---|---|---|---|---|---|---|
| Residents | 26,000 | 57,000 | 27,000 | 18,000 | 64,000 | 45,000 | 237,000 |
Identify the states, the seats, and the state population (the basis for the apportionment) in this scenario.
Find the standard divisor for the apportionment of the Vibranium artifacts. Round to the nearest tenth as needed. Include the units.
Find each birnin’s standard quota for the apportionment of the Vibranium artifacts. Round to the nearest hundredth as needed. What are the units?
Find the sum of the standard quotas. Is it reasonably close to the number of artifacts available for distribution?
For the following exercises, suppose that 6.4 million doses of COVID-19 vaccine are to be distributed among U.S. states. The vaccines will either be distributed based on the total state population or based on the number of people over 65 years old, as shown in the table below.
| State | State Population | State Population Age 65+ | Percentage of State Population 65+ |
|---|---|---|---|
| (CA) California | 39,613,000 | 5,669,000 | 14.3% |
| (TX) Texas | 29,730,300 | 3,602,000 | 12.6% |
| (NY) New York | 19,300,000 | 3,214,000 | 16.4% |
| (FL) Florida | 21,944,600 | 4,358,000 | 20.5% |
| (PA) Pennsylvania | 12,804,100 | 2,336,000 | 18.2% |
| (US) United States | 330,151,000 | 52,345,000 | 15.8% |
Find the standard divisor for the apportionment of the vaccine doses by population using the estimate for the total U.S. population. Round to the nearest tenth as needed. Include the units.
Find each state's standard quota for the apportionment of the vaccine doses. Round to the nearest tenth as needed. What are the units?
Find the standard divisor for the apportionment of the vaccine doses by population age 65 and older using the estimate for the total U.S. population of people aged 65 and older. Round to the nearest tenth as needed. Include the units.
Find each state's standard quota for the apportionment of the vaccine doses by total state population. Round to the nearest tenth as needed. What are the units?
Compare the standard quota for each state based on the entire state population to the standard quota for each state based on the portion of the population age 65 and older. Which states would receive more doses of vaccine if the apportionment were based on the population of people age 65 and older?
Approximately 15.8 percent of U.S. residents are age 65 and older.
- Which of the five states listed have a percentage of residents age 65 and older greater than 15.8 percent?
- Which of the five states listed have a percentage of residents age 65 and older less than 15.8 percent?
- Explain the correlation.
For the following exercises, use this information: Children from five families—the Chorro family, the Eswaran family, the Javernick family, the Lahde family, and the Stolly family—joined a town-wide Easter egg hunt. When they returned with their baskets, they had 827 eggs! They decided to share their eggs among the families based on the number of children in each family, as shown in the table below.
| Family | Number of Children |
|---|---|
| (C) Chorro | 3 |
| (E) Eswaran | 2 |
| (J) Javernick | 4 |
| (L) Lahde | 1 |
| (S) Stolly | 5 |
Identify the states, the seats, and the state population (the basis for the apportionment) in this scenario.
Find the standard divisor for the apportionment of the Easter eggs. Round to five decimal places as needed. Include the units.
Find each family’s standard quota for the apportionment of the Easter eggs. Round to the nearest hundredth as needed. What are the units?
Find the sum of the standard quotas from exercise 34. Is the sum reasonably close to the number of Easter eggs available for distribution?