1. 24 – Piecewise Function Modeling – Day 1 Calculator Required
Piecewise Investigations
24 – Piecewise Function Modeling – Day 1 Calculator Required

2. Parking rates at Millennium Park in Chicago (as of October 2017) are as follows:
$14, for 0 to 3 hours
$18, more than 3 and up to 8 hours
$22, more than 8 and up to 24 hours
If t represents the number of hours parked and f(t) represents the total cost, create a piecewise function which represents the above rates.

3. Parking rates at Millennium Park in Chicago (as of October 2017) are as follows:
$14, for 0 to 3 hours
$18, more than 3 and up to 8 hours
$22, more than 8 and up to 24 hours
If t represents the number of hours parked and f(t) represents the total cost, create a piecewise function which represents the above rates.

4. Parking rates at Millennium Park in Chicago (as of October 2017) are as follows:
$14, for 0 to 3 hours
$18, more than 3 and up to 8 hours
$22, more than 8 and up to 24 hours
If t represents the number of hours parked and f(t) represents the total cost, create a piecewise function which represents the above rates.

5. Parking rates at Millennium Park in Chicago (as of October 2017) are as follows:
$14, for 0 to 3 hours
$18, more than 3 and up to 8 hours
$22, more than 8 and up to 24 hours
If t represents the number of hours parked and f(t) represents the total cost, create a piecewise function which represents the above rates.

6. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
A. Create a piecewise function which represents the total royalties.

7. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
A. Create a piecewise function which represents the total royalties.
The author receives royalties of 12 x 0.10 = $1.20 per book for the first 10,000 copies sold
If the author sells 10,000 books, the author receives royalties of
1.2 x 10,000 = $12,000.

8. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
A. Create a piecewise function which represents the cost for each plan.
If the author sells 10,000 books, the author receives royalties of
1.2 x 10,000 = $12,000.
If the author sells more than 10,000 books, the author receives $12,000 plus…
…0.15 x 12 = $1.80 per book beyond 10,000.
(x – 10,000)

9. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
B. If the author sells 12,000 books, how much will the author earn?

10. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
C. How many books have to be sold in order for the author to earn $9,000?
Maximum royalties for 10,000 or less books is $12,000.
Since $9,000 < $12,000, use first piece.
x = 7,500 books

11. 2. A paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold, and 15% on any additional copies. Let x represent the number of copies sold, and let R(x) represent the total royalties.
D. How many books have to be sold in order for the author to earn $30,000?
Maximum royalties for 10,000 or less books is $12,000.
Since $30,000 > $12,000, use second piece.
x = 20,000 books

12. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
A. Create a piecewise function which represents the scenario.

13. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
A. Create a piecewise function which represents the scenario.
After two hours, 2 inches of snow has fallen.

14. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
A. Create a piecewise function which represents the scenario.
After two hours, 2 inches of snow has fallen.
For the next six hours, the total snowfall: 2 plus…
2 per hour (after 2 hours).
(t – 2)
After eight hours, 2 + 2(8 – 2) = 14 inches of snow has fallen.

15. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
A. Create a piecewise function which represents the scenario.
After eight hours, 14 inches of snow has fallen.
After 8 hours, the total snowfall: 14 plus…
1 per hour (after 8 hours).
(t – 8)

16. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
B. What was the total snowfall for the storm?

17. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
C. If the storm began at 9 AM, how much snow had fallen as of 3 PM?

18. 3. During a nine hour snowstorm, it snows at a rate of 1 inch per hour for the first two hours, 2 inches per hour for the next six hours, and 1 inch per hour for the final hour. Let t represents the number of hours since the storm began, and let f(t) represent the total amount of snowfall during the storm t hours into the storm.
D. If the storm began at 9AM and Jamie was able to clear the driveway of snow at 12 PM, how much snow remains to be cleared after the storm?
After three hours, 2 + 2(3 – 2) = 4 inches of snow had fallen…cleared by Jamie.
The total amount of snow was 15 inches.
Jamie has 15 – 4 = 11 inches of snow remaining to be cleared.

19. 4. When you have a job, a portion of the ‘deductions’ in your paycheck is Social Security. As of October 2017, 6.2% of your paycheck goes to Social Security; however, no additional social security is deducted after $127,200 in earnings during any particular year. Let w represents your total wages in a year, and S(w) represents the total deductions for Social Security.
A. Create a piecewise function which represents the total deductions for Social Security in any particular year.
6.2% (0.062) of your wages up to $127,200 is deducted for Social Security.
Wages of $127,200 will total $ in Social Security deductions.

20. 4. When you have a job, a portion of the ‘deductions’ in your paycheck is Social Security. As of October 2017, 6.2% of your paycheck goes to Social Security; however, no additional social security is deducted after $127,200 in earnings during any particular year. Let w represents your total wages in a year, and S(w) represents the total deductions for Social Security.
A. Create a piecewise function which represents the total deductions for Social Security in any particular year.
Wages of $127,200 will total $ in Social Security deductions.
After $127,200, no additional Social Security deductions occur

21. 4. When you have a job, a portion of the ‘deductions’ in your paycheck is Social Security. As of October 2017, 6.2% of your paycheck goes to Social Security; however, no additional social security is deducted after $127,200 in earnings during any particular year. Let w represents your total wages in a year, and S(w) represents the total deductions for Social Security.
B. Marie has a part time job that pays $200 per week. Assuming 52 weeks in a year, how much will Marie pay in Social Security deductions in a year?
Marie’s total wages in a year: 200 x 52 = $10,400.

22. 4. When you have a job, a portion of the ‘deductions’ in your paycheck is Social Security. As of October 2017, 6.2% of your paycheck goes to Social Security; however, no additional social security is deducted after $127,200 in earnings during any particular year. Let w represents your total wages in a year, and S(w) represents the total deductions for Social Security.
C. Ken has a full time job with an annual salary of $85,000. How much will Ken pay in Social Security deductions?

23. 4. When you have a job, a portion of the ‘deductions’ in your paycheck is Social Security. As of October 2017, 6.2% of your paycheck goes to Social Security; however, no additional social security is deducted after $127,200 in earnings during any particular year. Let w represents your total wages in a year, and S(w) represents the total deductions for Social Security.
D. Michelle is CEO of a large company, earning a salary of $1,200,000 per year. How much will Michelle pay in Social Security deductions?

24. 5. The height of the front of a camping tent can be modeled by the function
where x and f(x) are measured in feet and the x-axis
represents the ground.
A. Rewrite f(x) as a simplified piecewise function.

25. 5. The height of the front of a camping tent can be modeled by the function
where x and f(x) are measured in feet and the x-axis
represents the ground.
B. Find the height of the front of the tent at 2 feet.

26. 5. The height of the front of a camping tent can be modeled by the function
where x and f(x) are measured in feet and the x-axis
represents the ground.
C. Find the maximum height of the front of the tent.
The maximum height will occur at the vertex of the function, where x = 2.5.

27. 5. The height of the front of a camping tent can be modeled by the function
where x and f(x) are measured in feet and the x-axis
represents the ground.
D. Explain why it doesn’t make sense to find the height of the front of the tent at 10 feet.
This would be below the ground.

28. 5. The height of the front of a camping tent can be modeled by the function
where x and f(x) are measured in feet and the x-axis
represents the ground.
E. Find the width of the tent at ground level. Explain your answer.
The width of the tent is 5 – 0 = 5 feet.

29. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
A. Create a piecewise function which represents the scenario.

30. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
A. Create a piecewise function which represents the scenario.
f(x) = commission

31. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
A. Create a piecewise function which represents the scenario.
f(x) = commission
When sales are $25,000, the total salary is (25000) = $3,350

32. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
A. Create a piecewise function which represents the scenario.
When sales are $25,000, the total salary is (25000) = $3,350

33. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
B. Find the total salary for the salesperson if sales are $30,000 this month.

34. 6. An air conditioning salesperson receives a base salary of $2,850 per month plus a commission. The commission is 2% of the sales up to and including $25,000 for the month and 5% of the sales over $25,000 for the month. Let x represent the amount of sales, and f(x) represent the total salary.
C. If the maximum an employee can earn is $6,000 per month, find the sales necessary to attain the maximum total salary.