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Eric Connaly, Functions Modelling Change: A Preparation for Calculus, Fourth Edition, Wiley 2011
21. You have been asked to build a ramp for Dan’s Daredevil Motorcycle Jump. The dimensions you are given are indicated in Figure 6.114. Find all the other dimensions.
22. A kite flier wondered how high her kite was flying. She used a protractor to measure an angle of 38◦ from level ground to the kite string. If she used a full 100 yard spool of string, how high, in feet, was the kite?
23. The top of a 200-foot vertical tower is to be anchored by cables that make an angle of 30◦ with the ground. How long must the cables be? How far from the base of the tower should anchors be placed?
24. Find approximately the acute angle formed by the line y = − 2x + 5 and the x-axis.
25. The front door to the student union is 20 feet above the ground, and it is reached by a flight of steps. The school wants to build a wheel-chair ramp, with an incline of 15 degrees, from the ground to the door. How much horizontal distance is needed for the ramp?
26. A ladder 3 meters long leans against a house, making an angle α with the ground. How far is the base of the ladder from the base of the wall, in terms of α? Include a sketch.
27. A plane is flying at an elevation of 35,000 feet when the Gateway Arch in St. Louis, Missouri comes into view. The pilot wants to estimate her horizontal distance from the arch, so she notes the angle of depression, θ, between the horizontal and a line joining her eye to a point on the ground directly below the arch. Make a sketch. Express her horizontal distance to that point as a function of θ.
28. You see a friend, whose height you know is 5 feet 10 inches, some distance away. Using a surveying device called a transit, you determine the angle between the top of your friend’s head and ground level to be 8°. You want to find the distance d between you and your friend.
(a) First assume you cannot find your calculator to evaluate trigonometric functions. Find d by approximating the arc length of an 8° angle with your friend’s height. (See Figure 6.115.)
(b) Now assume that you have a calculator. Use it to find the distance d. (See Figure 6.116.)
(c) Would the difference between these two values for d increase or decrease if the angle were smaller?
60. You have two money machines, both of which increase any money inserted into them. The first machine doubles your money. The second adds five dollars. The money that comes out is described by d(x) = 2x, in the first case, and a(x) = x + 5, in the second, where x is the number of dollars inserted. The machines can be hooked up so that the money coming out of one machine goes into the other. Find formulas for each of the two possible composition machines. Is one machine more profitable than the other?
34. An average of 50,000 people visit Riverside Park each day in the summer. The park charges $15.00 for admission. Consultants predict that for each $1.00 increase in the entrance price, the park would lose an average of 2500 customers per day. Express the daily revenue from ticket sales as a function of the number of $1.00 price increases. What ticket price maximizes the revenue from ticket sales?
33. Three ounces of broiled ground beef contains 245 calories. Is the number of calories directly or inversely proportional to the number of ounces? Explain your reasoning and write a formula for the proportion. How many calories are there in 4 ounces of broiled hamburger?
34. A 30-second commercial during Super Bowl XL in 2006 cost advertisers $2.5 million. For the first Super Bowl in 1967, an advertiser could have purchased approximately 28.699 minutes of advertising time for the same amount of money.
(a) Assuming that cost is proportional to time, find the cost of advertising, in dollars/second, during the 1967 and 2006 Super Bowls.
(b) How many times more expensive was Super Bowl advertising in 2006 than in 1967?
35. A group of friends rent a house at the beach for spring break. If nine of them share the house, it costs $150 each. Is the cost to each person directly or inversely proportional to the number of people sharing the house? Explain your reasoning and write a formula for the proportion. How many people are needed to share the house if each student wants to pay a maximum of $100 each?
36. Driving at 55 mph, it takes approximately 3.5 hours to drive from Long Island to Albany, NY. Is the time the drive takes directly or inversely proportional to the speed? Explain your reasoning and write a formula for the proportion. To get to Albany in 3 hours, how fast would you have to drive?
37. On a map, 1/2 inch represents 5 miles. Is the map distance between two locations directly or inversely proportional to the actual distance which separates the two locations? Explain your reasoning and write a formula for the proportion. How far apart are two towns if the distance between these two towns on the map is 3.25 inches?
38. A volcano erupts in a powerful explosion. The sound from the explosion is heard in all directions for many hundreds of kilometers. The speed of sound is about 340 meters per second.
(a) Fill in Table 9.5 showing the distance, d, that the sound of the explosion has traveled at time t. Write a formula for d as a function t.
(b) How long after the explosion will a person living 200 km away hear the explosion?
(c) Fill Table 9.5 showing the land area, A, over which the explosion can be heard as a function of time. Write a formula for A as a function of t.
(d) The average population density around the volcano is 31 people per square kilometer. Write a formula for P as function of t, where P is the number of people who have heard the explosion at time t.
(e) Graph the function P = f(t). How long will it take until 1 million people have heard the explosion?
39. The thrust, T , delivered by a ship’s propeller is proportional to the square of the propeller rotation speed, R, times the fourth power of the propeller diameter, D.
(a) Write a formula for T in terms of R and D.
(b) What happens to the thrust if the propeller speed is doubled?
(c) What happens to the thrust if the propeller diameter is doubled?
(d) If the propeller diameter is increased by 50%, by how much can the propeller speed be reduced to deliver the same thrust?
40. Two oil tankers crash in the Pacific ocean. The spreading oil slick has a circular shape, and the radius of the circle is increasing at 200 meters per hour.
(a) Express the radius of the spill, r, as a power function of time, t, in hours since the crash.
(b) Express the area of the spill, A, as a power function of time, t.
(c) Clean-up efforts begin 7 hours after the spill. How large an area is covered by oil at that time?
44. You wish to pack a cardboard box inside a wooden crate. In order to have room for the packing materials, you need to leave a 0.5-ft space around the front, back, and sides of the box, and a 1-ft space around the top and bottom of the box. If the cardboard box is x feet long, (x + 2) feet wide, and (x − 1) feet deep, find a formula in terms of x for the amount of packing material needed.
45. Take an 8.5 by 11-inch piece of paper and cut out four equal squares from the corners. Fold up the sides to create an open box. Find the dimensions of the box that has maximum volume.
12. A person leaves home and walks 2 miles due west. She then walks 3 miles southwest. How far away from home is she? In what direction must she walk to head directly home?
13. The person from Problem 12 next walks 4 miles southeast. How far away from home is she? In what direction must she walk to head directly home?
14. A hockey puck starts on the edge of the rink and slides with a constant velocity v, at a speed of 7 ft/sec and an angle of 35° with the edge. After 2 seconds, how far has the puck traveled? How far is it from the edge?
15. Oracle Road heads due north from its intersection with Route 10, which heads 20° west of north.
(a) If you travel 5 miles up Route 10 from the intersection, how far are you from Oracle Road?
(b) How far do you have to travel up Route 10 from the intersection to be 2 miles from Oracle Road?
18. (a) A kite on a 50 foot string is flying at an angle of 20° with flat ground. What is the magnitude and the direction of the vector from the kite to the ground?
(b) A wind blows the kite upward so that its angle with the ground is 40°. How far is the new position from the original position?
19. A ball is thrown horizontally at 5 feet per second relative to still air. At the same time a wind blows at 3 feet per second at an angle of 45◦ to the ball’s path. What is the velocity of the ball relative to the ground? (There are two answers)
17. Shortly after takeoff, a plane is climbing northwest through still air at an airspeed of 200 km/hr and rising at a rate of 300 m/min. Resolve into components its velocity vector in a coordinate system in which the x-axis points east, the y-axis points north, and the z-axis points up.
19. Which is traveling faster, a car whose velocity vector is 21i +35j , or a car whose velocity vector is 40i , assuming that the units are the same for both directions?
20. A truck is traveling due north at 30 km/hr toward a crossroad. On a perpendicular road a police car is traveling west toward the intersection at 40 km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car.
22. (a) A man swims northeast at 5 mph. Give a vector representing his velocity.
(b) He now swims in a river at the same velocity relative to still water; the river’s current flows north at 1.2 mph. What is his velocity relative to the riverbed?
23. The coastline is along the x-axis and the sea is in the region y > 0. A sailing boat has velocity v = 2.5i +3.1j mph. What is the speed of the boat? What angle does the boat’s path make with the coastline?
11. There are five students in a class. Their scores on the midterm (out of 100) are given by the vector v = (73, 80, 91, 65, 84). Their scores on the final (out of 100) are given by w = (82, 79, 88, 70, 92). The final counts twice as much as the midterm. Find a vector giving the total scores (out of 100) of the students.
12. An airplane is heading northeast at an airspeed of 700 km/hr, but there is a wind blowing from the west at 60 km/hr. In what direction does the plane end up flying? What is its speed relative to the ground?
13. An airplane is flying at an airspeed of 600 km/hr in a cross-wind that is blowing from the northeast at a speed of 50 km/hr. In what direction should the plane head to end up going due east?
14. Two children are throwing a ball back-and-forth straight across the back seat of a car. Suppose the ball is being thrown at 10 mph relative to the car and the car is going 25 mph down the road.
(a) Make a sketch showing the relevant velocity vectors.
(b) If one child does not catch the ball and it goes out an open window, what angle does the ball’s horizontal motion make with the road?
15. A man walks 5 miles in a direction 30° north of east. He then walks a distance x miles due east. He turns around to look back at his starting point, which is at an angle of 10° south of west.
(a) Make a sketch. Give vectors in i and j components, for each part of the man’s walk.
(b) What is x?
(c) How far is the man from his starting point?
16. Three different electric charges q1, q2, and q3 exert forces on a test charge Q. The forces are, respectively, F1 = (3, 6), F2 = (−2, 5), and F3 = (7, −4). The net force, Fnet is given by
Fnet = F1 + F2 + F3
(a) Calculate Fnet
(b) If a fourth charge q4 is added, what force F4 must it exert on Q so that Q feels no net force at all, that is, so that Fnet = 0?
11. The force of gravity acting on a ball is 2 lb downward. How much work is done by gravity if the ball
(a) Falls 3 feet?
(b) Moves upward 5 feet?
12. How much work is done in pushing a 350 lb refrigerator up a 12 ft ramp which makes a 30° angle with the floor?
22. (a) Bread, eggs, and milk cost $3.00 per loaf, $2.00 per dozen, and $4.00 per gallon, respectively, at Acme Store. Use a price vector a and a consumption vector c to write a vector equation that describes what may be bought for $40.
(b) At Beta Mart, where the food is fresher, the price vector is b = (3.20, 1.80, 4.50). Explain the meaning of (b − a) ·c in practical terms. Is b − a ever perpendicular to c?
(c) Some people think Beta Mart’s freshness makes each grocery item at Beta equivalent to 110% of the corresponding Acme item. What does it mean for a consumption vector to satisfy the inequality
(1/1.1) b ·c < a ·c?
24. A basketball gymnasium is 25 meters high, 80 meters wide and 200 meters long. For a half time stunt, the cheerleaders want to run two strings, one from each of the two corners of the gym above one basket to the diagonally opposite corners of the gym floor. What is the angle made by the strings as they cross?
Review Exercises and Problems for Chapter 10
25. Find a formula for F = f(θ), the sliding force (in lbs) exerted on a block if the plank makes an angle of θ with the ground.
26. One end of the plank is lifted at a constant rate of 2 ft per second, while the other end rests on the ground.
(a) Find a formula for F = h(t), the sliding force exerted on the block as a function of time.
(b) Suppose the block begins to slide once the sliding force equals 3 lbs. At what time will the block begin to slide?
27. The 5-lb force exerted on the block by gravity can be resolved into two components, the sliding force FS parallel to the ramp and the normal force FN perpendicular to the ramp. Use this information to show that your formula in Problem 25 is correct.
28. A plane is heading due east and climbing at the rate of 80 km/hr. If its airspeed is 480 km/hr and there is a wind blowing 100 km/hr to the northeast, what is the ground speed of the plane?
29. A particle moving with speed v hits a barrier at an angle of 60° and bounces off at an angle of 60°
in the opposite direction with speed reduced by 20 percent, as shown in Figure 10.33. Find the velocity vector of the object after impact.
30. Figure 10.34 shows a molecule with four atoms at O, A, B and C. Show that every atom in the molecule is 2 units away from every other atom.
31. Two cylindrical cans of radius 2 and height 7 are shown in Figure 10.35. The cans touch down the side. Let A be the point at the top rims where they touch. Let B be the front point on the bottom rim of the left can, and C be the back point on the bottom rim of the right can. The origin is at A and the z-axis points upward; the x-axis points forward (out of the paper) and the y-axis points to the right.
(a) Write vectors in component form for AB and AC.
(b) What is the angle between AB and AC?
26. During 2004, about 81 million barrels of oil a day were consumed worldwide. Over the previous decade, consumption had been rising at 1.2% a year; assume that it continues to increase at this rate.
(a) Write the first four terms of the sequence an giving daily oil consumption n years after 2003; give a formula for the general term an.
(b) In what year is consumption expected to exceed 100 million barrels a day?
37. Calculate the distance, S7, the object falls in 7 seconds.
39. Find a formula for f(n), the distance fallen by the object in n seconds.
40. If the object falls from 1000 feet, how long does it take to hit the ground?
41. A boy is dividing M&Ms between himself and his sister. He gives one to his sister and takes one for himself. He gives another to his sister and takes two for himself. He gives a third one to his sister and takes three for himself, and so on.
(a) On the nth round, how many M&Ms does the boy give his sister? How many does he take himself?
(b) After n rounds, how many M&Ms does his sister have? How many does the boy have?
42. An auditorium has 30 seats in the first row, 34 seats in the second row, 38 seats in the third row, and so on. If there are twenty rows in the auditorium, how many seats are there in the last row? How many seats are there in the auditorium?
16. Figure 11.3 shows the quantity of the drug atenolol in the blood as a function of time, with the first dose at time t = 0. Atenolol is taken in 50 mg doses once a day to lower blood pressure.
(a) If the half-life of atenolol in the blood is 6.3 hours, what percentage of the atenolol present at the start of a 24-hour period is still there at the end?
(b) Find expressions for the quantities Q0, Q1, Q2, Q3, . . ., and Qn shown in Figure 11.3. Write the expression for Qn in closed-form.
(c) Find expressions for the quantities P1 , P2 , P3 , . . ., and Pn shown in Figure 11.3. Write the expression for Pn in closed-form.
17. Annual deposits of $3000 are made into a bank account earning 5% interest per year. What is the balance in the account right after the 15th deposit if interest is calculated
(a) Annually (b) Continuously
18. A bank account with a $75,000 initial deposit is used to make annual payments of $1000, starting one year after the initial $75,000 deposit. Interest is earned at 2% a year, compounded annually, and paid into the account right before the payment is made.
(a) What is the balance in the account right after the 24th payment?
(b) Answer the same question for yearly payments of $3000.
19. A deposit of $1000 is made once a year, starting today, into a bank account earning 3% interest per year, compounded annually. If 20 deposits are made, what is the balance in the account on the day of the last deposit?
20. What effect does doubling each of the following quantities (leaving other quantities the same) have on the answer to Problem 19? Is the answer doubled, more than doubled, or less than doubled?
(a) The deposit
(b) The interest rate
(c) The number of deposits made
21. To save for a new car, you put $500 a month into an account earning interest at 3% per year, compounded continuously.
(a) How much money do you have 2 years after the first deposit, right before you make a deposit?
(b) When does the balance first go over $10,000?
22. A bank account in which interest is earned at 4% per year, compounded annually, starts with a balance of $50,000. Payments of $1000 are made out of the account once a year for ten years, starting today. Interest is earned right before each payment is made.
(a) What is the balance in the account right after the tenth payment is made?
(b) Assume that the tenth payment exhausts the account. What is the largest yearly payment that can be made from this account?
13. A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction.
Consider the decimal 0.232323 . . . .
(a) Use the fact that 0.232323 . . . = 0.23 + 0.0023 + 0.000023 + · · · to write 0.232323. . . as a geometric series.
(b) Use the formula for the sum of a geometric series to show that 0.232323 . . . = 23/99.
19. You have an ear infection and are told to take a 250 mg tablet of ampicillin (a common antibiotic) four times a day (every six hours). It is known that at the end of six hours, about 4% of the drug is still in the body. What quantity of the drug is in the body right after the third tablet? The fortieth? Assuming you continue taking tablets, what happens to the drug level in the long run?
20. In Problem 19 we found the quantity Qn , the amount (in mg) of ampicillin left in the body right after the nth tablet is taken.
(a) Make a similar calculation for Pn , the quantity of ampicillin (in mg) in the body right before the nth
tablet is taken.
(b) Find a simplified formula for Pn .
(c) What happens to Pn in the long run? Is this the same as what happens to Qn ? Explain in practical terms why your answer makes sense.
28. What is the present value of a $1000 bond that pays $50 a year for 10 years, starting one year from now? Assume interest rate is 6% per year, compounded annually.
29. What is the present value of a $1000 bond that pays $50 a year for 10 years, starting one year from now? Assume the interest rate is 4% per year, compounded annually.
30. (a) What is the present value of a $1000 bond that pays $50 a year for 10 years, starting one year from now? Assume the interest rate is 5% per year, compounded annually.
(b) Since $50 is 5% of $1000, this bond is often called a 5% bond. What does your answer to part (a) tell you about the relationship between the principal and the present value of this bond when the interest rate is 5%?
(c) If the interest rate is more than 5 % per year, compounded annually, which one is larger: the principal or the value of the bond? Why do you think the bond is then described as trading at discount?
(d) If the interest rate is less than 5 % per year, compounded annually, why is the bond described as trading at a premium?
Review Exercises and Problems for Chapter 13
10. Figure 13.4 shows the first four members in a sequence of square-shaped grids. In each successive grid, new dots are shown in black. For instance, the second grid has 3 more dots than the first grid, the third has 5 more dots than the second, and the fourth has 7 more dots than the third.
(a) Write down the sequence of the total number of dots in each grid (black and white). Using the fact that each grid is a square, find a formula in terms of n for the number of dots in the nth grid.
(b) Write down the sequence of the number of black dots in each grid. Find a formula in terms of n for
the number of black dots in the nth grid.
(c) State the relationship between the two sequences in parts (a) and (b). Use the formulas you found in parts (a) and (b) to confirm this relationship.
11. The numbers in the sequence 1, 4, 9, 16, . . . from Problem 10 are known as square numbers because they describe the number of dots in successive square grids. Analogously, the numbers in the sequence 1, 3, 6, 10, . . . are known as triangular numbers because they describe the number of dots in successive triangular patterns, as shown in Figure 13.5: Find a formula for the nth triangular number.
12. In a workshop, it costs $300 to make one piece of furniture. The second piece costs a bit less, $280. The third costs even less, $263, and the fourth costs only $249. The cost for each additional piece of furniture is called the marginal cost of production. Table 13.3 gives the marginal cost, c, and the change in marginal cost, Δc, in terms of the number of pierces of furniture, n. As the quantity produced increases, the marginal cost generally decreases and then increases again.
(a) Assume that the arithmetic sequence −20 , −17 , −14 , . . . , continues. Complete the table for n = 5 , 6, . . . , 12.
(b) Find a formula for cn, the marginal cost for producing the nth piece of furniture. Use the fact that cn is found by adding the terms in an arithmetic sequence. Using your formula, find the cost for producing the 12th piece and the 50th piece of furniture.
(c) A piece of furniture can be sold at a profit if it costs less than $400 to make. How many pieces of furniture should the workshop make each day? Discuss.
13. A store clerk has 108 cans to stack. He can fit 24 cans on the bottom row and can stack the cans 8 rows high. Use arithmetic series to determine how he can stack the cans so that each row contains fewer cans than the row beneath it and that the number of cans in each row decreases at a constant rate.
14. A university with an enrollment of 8000 students in 2007 is projected to grow by 2% in each of the next three years and by 3.5% in each of the following seven years. Find the sequence of the university’s student enrollment for the next 10 years.
15. Each person in a group of 30 shakes hands with each other person exactly once. How many total handshakes take place?
16. A bank account with a $75,000 initial deposit is used to make annual payments of $1000, starting one year after the initial $75,000 deposit. Interest is earned at 2% a year, compounded annually, and paid into the account right before the payment is made.
(a) What is the balance in the account right after the 24th payment?
(b) Answer the same question for yearly payments of $3000.
17. One way of valuing a company is to calculate the present value of all its future earnings. A farm expects to sell $1000 worth of Christmas trees once a year forever, with the first sale in the immediate future. What is the present value of this Christmas tree business? Assume that the interest rate is 4 % per year, compounded continuously.
18. You inherit $100,000 and put the money in a bank account earning 3% per year, compounded annually. You withdraw $2000 from the account each year, right after the interest is earned. Your first $2000 is withdrawn before any interest is earned.
(a) Compare the balance in the account right after the first withdrawal and right after the second withdrawal. Which do you expect to be higher?
(b) Calculate the balance in the account right after the 20th withdrawal is made.
(c) What is the largest yearly withdrawal you can take from this account without the balance decreasing over time?
19. After breaking his leg, a patient retrains his muscles by going for walks. The first day, he manages to walk 300 yards. Each day after that he walks 50 yards farther than the day before.
(a) Write a sequence that represents the distances walked each day during the first week.
(b) How far is he walking after two weeks?
(c) How long until he is walking at least one mile?
20. Before email made it easy to contact many people quickly, groups used telephone trees to pass news to their members. In one group, each person is in charge of calling 4 people. One person starts the tree by calling 4 people. At the second stage, each of these 4 people call another 4 people. In the third stage, each of the people in stage two calls 4 people, and so on.
(a) How many people have the news by the end of the 5th stage?
(b) Write a formula for the total number of members in a tree of 10 stages.
(c) How many stages are required to cover a group with 5000 members?
21. A ball is dropped from a height of 10 feet and bounces. Each bounce is 3/4 of the height of the bounce before. Thus after the ball hits the floor for the first time, the ball rises to a height of 10(3/4) = 7.5 feet, and after it hits the floor for the second time, it rises to a height of 7.5(3/4) = 5.625 ft
(a) Find an expression for the height to which the ball rises after it hits the floor for the nth time.
(b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times.
(c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the nth time. Express your answer in closed form.
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