MAT 126 Fall Final Review Packet #1

Transcription

1 MAT 126 Fall Final Review Packet #1 Numeric Response 1. The given numbers are coordinates of points A and B, respectively, on a coordinate line. Find the distance,. 2. Approximate the real-number expression to four decimal places. 3. Simplify the expression. 4. Archeologists can determine the height of a human without having a complete skeleton. If an archeologist finds only a humerus, then the height of the individual can be determined by using a simple linear relationship. (The humerus is the bone between the shoulder and the elbow.) For a female, if x is the length of the humerus (in centimeters), then her height h (in centimeters) can be determined using the formula. For a male, should be used. A female skeleton having a 23-centimeter humerus is found. Find the woman's height at death. cm 5. A construction firm is trying to decide which of two models of a crane to purchase. Model A costs $50,000 and requires $3,500 per year to maintain. Model B has an initial cost of $48,000 and a maintenance cost of $4,000 per year. For how many years must model A be used before it becomes more economical than B? years 6. A consumer is trying to decide whether to purchase car A or car B. Car A costs $10,000 and has an mpg rating of 30, and insurance is $450 per year. Car B costs $12,000 and has an mpg rating of 50, and insurance is $500 per year. Assume that the consumer drives 15,000 miles per year and that the price of gas remains constant at $1.25 per gallon. Based only on these facts, determine how long it will take for the total cost of car B to become less than that of car A. years 7. Find the distance d (A, B) between the points A (- 3, 10) and B (0, - 9). Please round the answer to the nearest hundredth.

2 8. Vertical wind shear occurs when wind speed varies at different heights above the ground. Wind shear is of great importance to pilots during takeoffs and landings. If the wind speed v 1 is at height h 1 and v 2 at height h 2, then the average wind shear s is given by the slope formula If the wind speed at ground level is 17 mi/hr and s has been determined to be 0.06, find the wind speed 150 feet above the ground. speed = mi/hr 9. A cable television firm presently serves 2,700 households and charges $24 per month. A marketing survey indicates that each decrease of $1 in the monthly charge will result in 270 new customers. Let R(x) denote the total monthly revenue when the monthly charge is x dollars. Find the value of x that results in maximum monthly revenue. 10. Several values of two functions T and S are listed in the following tables: t T(t) x???????? S(x)???????? Find: 11. A herd of 45 deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. Suppose that the number of deer after years is given by the following formula. where How many years does it take for the population to become extinct? years 12. If, find a number such that the graph of contains the point (2,48). 13. Use the remainder theorem to find

3 , 14. A storage tank for propane gas is to be constructed in the shape of a right circular cylinder with a hemisphere attached to the top. If the total height of the structure is 40 feet, determine the radius of the cylinder that results in a total volume of. r = feet 15. Poiseuille's law states that the blood flow rate F ( in L/min ) through a major artery is directly proportional to the product of the fourth power of the radius r and the blood pressure P. During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by 9%, approximately how much harder must the heart pump? Round your answer to two decimal places. About times as hard 16. Examine the expression y / x for the set of data points of the form ( x, y ) : Find the constant of variation. 17. Examine the expression x 2 y for the set of data points of the form ( x, y ) : Find the constant of variation. 18. If a savings fund pays interest at a rate of 11% per year compounded semi-annually, how much money invested now will amount to $9,000 after 1 year? 19. Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. Suppose that a language originally had basic words and that at time, measured in millennia (1 millennium = 1,000 years), the number of basic words that remain in common use is given by. Approximate the percentage of basic words lost every 100 years. Round the answer to two decimal places, if necessary. % 20. Solve the equation: x =

4 21. Solve the equation: x = 22. If a certain make of automobile is purchased for C dollars, its trade-in value at the end of t years is given by: If the original cost is $8,378, calculate the value after 6 years. 23. A certain department store requires its credit-card customers to pay interest on unpaid bills at the rate of 18% per year, compounded monthly. If a customer buys a television set for $500 on credit and makes no payments for one year, how much is owed at the end of the year? 24. Some lending institutions calculate the monthly payment M on a loan of L dollars at an interest rate r (expressed as a decimal) by using the formula: where k = [1 + ( r / 12 )] 12t and t is the number of years that the loan is in effect. An automobile dealer offers customers no-down-payment 9-year loans at an interest rate of 15%. If a customer can afford to pay $124 per month, find the price of the most expensive car that can be purchased. 25. Graph: Use the graph to estimate x if y = 2. Give the answer to two decimal places, if necessary. x = 26. If 100 dollars is deposited in a savings account that pays interest at a rate of per year compounded continuously, find the balance after 13 years.

5 Round your answer to the nearest cent. 27. The effective yield (or effective annual interest rate) for an investment is the simple interest rate that would yield at the end of one year the same amount as is yielded by the compounded rate that is actually applied. Approximate, to the nearest 0.01% if necessary, the effective yield corresponding to an interest rate of 10% per year compounded quarterly. % 28. How much money, invested at an interest rate of 11% per year compounded continuously, will amount to $91,000 after 17 years? 29. The 1980 population of the United States was approximately 227 million, and the population has been growing continuously at a rate of 0.7% per year. Predict the population in the year 2030 if this growth trend continues. Give your answer in millions, rounded to the nearest integer. million 30. Solve the equation. 31. Solve the equation. 32. An urban density model is a formula that relates the population density D (in thousands/ mi 2 ) to the distance x (in miles) from the center of the city. The formula for the central density a and coefficient of decay b has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970, a = 5.5 and b = At approximately what distance was the population density 2,800 per square mile? Round the answer to the nearest tenth, if necessary. miles from the center 33. Solve the equation. x =

6 34. Solve the equation. 35. One method of estimating the thickness of the ozone layer is to use the formula, where I o is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, and k is the absorption constant of ozone for that wavelength. Suppose a wavelength of cm with k = 0.35, is measured as Approximate the thickness of the ozone layer to the nearest 0.01 centimeter, if necessary. cm 36. The growth in height of trees is frequently described by a logistic equation. Suppose the height h (in feet) of a tree at age t (in years) is (as illustrated by the graph in the figure). What is the height of the tree at age 10? Round the answer to the nearest hundredth, if necessary. feet 37. Use the compound interest formula to determine how long it will take for a sum of money to double if it is invested at a rate of 6% per year compounded monthly. Round the answer to the nearest hundredth, if necessary. years 38. Estimate using the change of base formula. Give the answer to four decimal places, if necessary.

7 39. Evaluate the expression using the change of base formula (without a calculator). 40. Use the change of base formula to approximate the y-intercept. Give the answer to four decimal places, if necessary. x =