110 CHAPTER 2 Functions and Their Graphs 55. Federal Income Tax Two 2024 tax rate schedules are given in the accompanying table. If x equals the taxable income and y equals the tax due, construct a function y f x( ) = for Schedule X. 2024 Tax Rate Schedules Schedule X—Single ScheduleY-1—Married Filing Jointly or Qualified Widow(er) If Taxable Income Is Over But Not Over TheTax Is This Amount Plus This % Of the Excess Over If Taxable Income Is Over But Not Over TheTax Is This Amount Plus This % Of the Excess Over $0 $11,600 $0 + 10% $0 $0 $23,200 $0 + 10% $0 11,600 47,150 $1,160.00 + 12% 11,600 23,200 94,300 $2,320.00 + 12% 23,200 47,150 100,525 5,426.00 + 22% 47,150 94,300 201,050 10,852.00 + 22% 94,300 100,525 191,950 17,168.50 + 24% 100,525 201,050 383,900 34,337.00 + 24% 201,050 191,950 243,725 39,110.50 + 32% 191,950 383,900 487,450 78,221.00 + 32% 383,900 243,725 609,350 55,678.50 + 35% 243,725 487,450 731,200 111,357.00 + 35% 487,450 609,350 183,647.25 + 37% 609,350 731,200 196,669.50 + 37% 731,200 56. Federal Income Tax Refer to the 2024 tax rate schedules in Problem 55. If x equals the taxable income and y equals the tax due, construct a function y f x( ) = for Schedule Y-1. 57. Cost of Transporting Goods A trucking company transports goods between Chicago and New York, a distance of 960 miles.The company’s policy is to charge, for each pound, $0.50 per mile for the first 100 miles, $0.40 per mile for the next 300 miles, $0.25 per mile for the next 400 miles, and no charge for the remaining 160 miles. (a) Graph the relationship between the per-pound cost of transportation in dollars and mileage over the entire 960-mile route. (b) Find the cost as a function of mileage for hauls between 100 and 400 miles from Chicago. (c) Find the cost as a function of mileage for hauls between 400 and 800 miles from Chicago. 58. Car Rental Costs An economy car rented in Florida from Enterprise® on a weekly basis costs $185 per week. Extra days cost $37 per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number of days used x, where x 7 14. ≤ ≤ Graph this function. 59. Mortgage Fees Fannie Mae charges a loan-level price adjustment (LLPA) on all mortgages, which represents a fee homebuyers seeking a loan must pay. The rate paid depends on the credit score of the borrower, the amount borrowed, and the loan-to-value (LTV) ratio. The LTV ratio is the ratio of amount borrowed to appraised value of the home. For example, a homebuyer who wishes to borrow $250,000 with a credit score of 730 and an LTV ratio of 80% will pay 0.75% (0.0075) of $250,000, or $1875. The table shows the LLPA for various credit scores and an LTV ratio of 80%. Credit Score Loan-level Price Adjustment Rate 659 ≤ 3.00% 660–679 2.75% 680–699 1.75% 700–719 1.25% 720–739 0.75% 740 ≥ 0.50% Source: Fannie Mae. (a) Construct a function C C s , ( ) = where C is the loanlevel price adjustment (LLPA) and s is the credit score of an individual who wishes to borrow $300,000 with an 80% LTV ratio. (b) What is the LLPA on a $300,000 loan with an 80% LTV ratio for a borrower whose credit score is 725? (c) What is the LLPA on a $300,000 loan with an 80% LTV ratio for a borrower whose credit score is 670? 60. Minimum Payments for Credit Cards Holders of credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than $10, the entire amount is due. For a bill of at least $10 but less than $500, the minimum due is $10. A minimum of $30 is due on a bill of at least $500 but less than $1000, a minimum of $50 is due on a bill of at least $1000 but less than $1500, and a minimum of $70 is due on bills of $1500 or more. Find the function f that describes the minimum payment due on a bill of x dollars. Graph f. 61. Wind Chill The wind chill factor represents the air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is ( )( ) ( ) = ≤ < − + − − ≤ ≤ − − > ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ W t v v v t v t v 0 1.79 33 10.45 10 33 22.04 1.79 20 33 1.5958 33 20 where v represents the wind speed (in meters per second) and t represents the air temperature C . ( ) ° Compute the wind chill for the following: (a) An air temperature of 10 C° and a wind speed of 1 meter per second (m/sec) (b) An air temperature of 10C° and a wind speed of 5 m sec (c) An air temperature of 10C° and a wind speed of 15 m sec (d) An air temperature of 10 C° and a wind speed of 25 m sec (e) Explain the physical meaning of the equation corresponding to v 0 1.79. ≤ < (f) Explain the physical meaning of the equation corresponding to v 20. >
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