Stewart Precalc 6e Section 1.6: Modeling with Equations

1 Concept - Modeling · Level 1

Explain in your own words what it means for an equation to model a real-world situation, and give an example.

2 Concept - Simple Interest · Level 1

In the formula $I = P r t$ for simple interest, $P$ stands for ____, $r$ for ____, and $t$ for ____.

3 Concept - Geometry Formulas · Level 1

Give a formula for the area of the geometric figure.

(a) A square of side $x$: $A =$ ____

Answer for 3(a)

(b) A rectangle of length $l$ and width $w$: $A =$ ____

Answer for 3(b)

(c) A circle of radius $r$: $A =$ ____

Answer for 3(c)

Enter your answer directly below each part above.

4 Concept - Percent · Level 1

Balsamic vinegar contains 5% acetic acid, so a 32-oz bottle of balsamic vinegar contains ____ ounces of acetic acid.

5 Concept - Rate of Work · Level 1

A painter paints a wall in $x$ hours, so the fraction of the wall that she paints in 1 hour is ____.

6 Concept - Distance Formula · Level 1

The formula $d = r t$ models the distance $d$ traveled by an object moving at the constant rate $r$ in time $t$. Find formulas for the following quantities. $r =$ ____ $t =$ ____

7 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The sum of three consecutive integers; $n =$ first integer of the three.

8 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The sum of three consecutive integers; $n =$ middle integer of the three.

9 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The average of three test scores if the first two scores are 78 and 82; $s =$ third test score.

10 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The average of four quiz scores if each of the first three scores is 8; $q =$ fourth quiz score.

11 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The interest obtained after one year on an investment at $2 \dfrac{1}{2}$% simple interest per year; $x =$ number of dollars invested.

12 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The total rent paid for an apartment if the rent is \$795 a month; $n =$ number of months.

13 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The area (in ft²) of a rectangle that is three times as long as it is wide; $w =$ width of the rectangle (in ft).

14 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The perimeter (in cm) of a rectangle that is 5 cm longer than it is wide; $w =$ width of the rectangle (in cm).

15 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The distance (in mi) that a car travels in 45 min; $s =$ speed of the car (in mi/h).

16 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The time (in hours) it takes to travel a given distance at 55 mi/h; $d =$ given distance (in mi).

17 Skill - Express Quantity · Level 2

Express the given quantity in terms of the indicated variable: The concentration (in oz/gal) of salt in a mixture of 3 gal of brine containing 25 oz of salt to which some pure water has been added; $x =$ volume of pure water added (in gal).

18 Skill - Express Quantity · Level 3

Express the given quantity in terms of the indicated variable: The value (in cents) of the change in a purse that contains twice as many nickels as pennies, four more dimes than nickels, and as many quarters as dimes and nickels combined; $p =$ number of pennies.

19 Application - Renting a Truck · Level 2

Renting a Truck. A rental company charges \$65 a day and 20 cents a mile for renting a truck. Michael rented a truck for 3 days, and his bill came to \$275. How many miles did he drive?

20 Application - Cell Phone Costs · Level 2

Cell Phone Costs. A cell phone company charges a monthly fee of \$10 for the first 1000 text messages and 10 cents for each additional text message. Miriam's bill for text messages for the month of June is \$38.50. How many text messages did she send that month?

21 Application - Investments · Level 3

Investments. Phyllis invested \$12,000, a portion earning a simple interest rate of $4 \dfrac{1}{2}$% per year and the rest earning a rate of 4% per year. After 1 year the total interest earned on these investments was \$525. How much money did she invest at each rate?

22 Application - Investments · Level 3

Investments. If Ben invests \$4000 at 4% interest per year, how much additional money must he invest at $5 \dfrac{1}{2}$% annual interest to ensure that the interest he receives each year is $4 \dfrac{1}{2}$% of the total amount invested?

23 Application - Investments · Level 2

Investments. What annual rate of interest would you have to earn on an investment of \$3500 to ensure receiving \$262.50 interest after 1 year?

24 Application - Investments · Level 3

Investments. Jack invests \$1000 at a certain annual interest rate, and he invests another \$2000 at an annual rate that is one-half percent higher. If he receives a total of \$190 interest in 1 year, at what rate is the \$1000 invested?

25 Application - Salaries · Level 2

Salaries. An executive in an engineering firm earns a monthly salary plus a Christmas bonus of \$8500. If she earns a total of \$97,300 per year, what is her monthly salary?

26 Application - Salaries · Level 2

Salaries. A woman earns 15% more than her husband. Together they make \$69,875 per year. What is the husband's annual salary?

27 Application - Inheritance · Level 2

Inheritance. Craig is saving to buy a vacation home. He inherits some money from a wealthy uncle, then combines this with the \$22,000 he has already saved and doubles the total in a lucky investment. He ends up with \$134,000 — just enough to buy a cabin on the lake. How much did he inherit?

28 Application - Overtime Pay · Level 3

Overtime Pay. Helen earns \$7.50 an hour at her job, but if she works more than 35 hours in a week, she is paid $1 \dfrac{1}{2}$ times her regular salary for the overtime hours worked. One week her gross pay was \$352.50. How many overtime hours did she work that week?

29 Application - Labor Costs · Level 3

Labor Costs. A plumber and his assistant work together to replace the pipes in an old house. The plumber charges \$45 an hour for his own labor and \$25 an hour for his assistant's labor. The plumber works twice as long as his assistant on this job, and the labor charge on the final bill is \$4025. How long did the plumber and his assistant work on this job?

30 Application - A Riddle · Level 2

A Riddle. A father is four times as old as his daughter. In 6 years, he will be three times as old as she is. How old is the daughter now?

31 Application - A Riddle · Level 3

A Riddle. A movie star, unwilling to give his age, posed the following riddle to a gossip columnist: "Seven years ago, I was eleven times as old as my daughter. Now I am four times as old as she is." How old is the movie star?

32 Application - Career Home Runs · Level 2

Career Home Runs. During his major league career, Hank Aaron hit 41 more home runs than Babe Ruth hit during his career. Together they hit 1469 home runs. How many home runs did Babe Ruth hit?

33 Application - Value of Coins · Level 2

Value of Coins. A change purse contains an equal number of pennies, nickels, and dimes. The total value of the coins is \$1.44. How many coins of each type does the purse contain?

34 Application - Value of Coins · Level 3

Value of Coins. Mary has \$3.00 in nickels, dimes, and quarters. If she has twice as many dimes as quarters and five more nickels than dimes, how many coins of each type does she have?

35 Application - Geometry · Level 2

Length of a Garden. A rectangular garden is 25 ft wide. If its area is 1125 ft², what is the length of the garden?

36 Application - Geometry · Level 2

Width of a Pasture. A pasture is twice as long as it is wide. Its area is 115,200 ft². How wide is the pasture?

37 Application - Geometry · Level 3

Dimensions of a Lot. A square plot of land has a building 60 ft long and 40 ft wide at one corner. The rest of the land outside the building forms a parking lot. If the parking lot has area 12,000 ft², what are the dimensions of the entire plot of land?

38 Application - Geometry · Level 3

Dimensions of a Lot. A half-acre building lot is five times as long as it is wide. What are its dimensions? [Note: 1 acre $=$ 43,560 ft².]

39 Application - Geometry · Level 2

Dimensions of a Garden. A rectangular garden is 10 ft longer than it is wide. Its area is 875 ft². What are its dimensions?

40 Application - Geometry · Level 2

Dimensions of a Room. A rectangular bedroom is 7 ft longer than it is wide. Its area is 228 ft². What is the width of the room?

41 Application - Geometry · Level 3

Dimensions of a Garden. A farmer has a rectangular garden plot surrounded by 200 ft of fence. Find the length and width of the garden if its area is 2400 ft².

42 Application - Geometry · Level 3

Dimensions of a Lot. A parcel of land is 6 ft longer than it is wide. Each diagonal from one corner to the opposite corner is 174 ft long. What are the dimensions of the parcel?

43 Application - Geometry · Level 3

Dimensions of a Lot. A rectangular parcel of land is 50 ft wide. The length of a diagonal between opposite corners is 10 ft more than the length of the parcel. What is the length of the parcel?

44 Application - Geometry · Level 3

Dimensions of a Track. A running track has the shape shown in the figure, with straight sides and semicircular ends. If the length of the track is 440 yd and the two straight parts are each 110 yd long, what is the radius of the semicircular parts (to the nearest yard)?

45 Application - Geometry · Level 3

Length and Area. Find the length $x$ in the figure. The area of the shaded region is given.

46 Application - Geometry · Level 3

Length and Area. Find the length $y$ in the figure. The area of the shaded region is given.

47 Application - Geometry · Level 3

Framing a Painting. Ali paints with watercolors on a sheet of paper 20 in. wide by 15 in. high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture?

48 Application - Geometry · Level 3

Dimensions of a Poster. A poster has a rectangular printed area 100 cm by 140 cm and a blank strip of uniform width around the edges. The perimeter of the poster is $1 \dfrac{1}{2}$ times the perimeter of the printed area. What is the width of the blank strip?

49 Application - Pythagorean Theorem · Level 2

Reach of a Ladder. A $19 \dfrac{1}{2}$-foot ladder leans against a building. The base of the ladder is $7 \dfrac{1}{2}$ ft from the building. How high up the building does the ladder reach?

50 Application - Pythagorean Theorem · Level 4

Height of a Flagpole. A flagpole is secured on opposite sides by two guy wires, each of which is 5 ft longer than the pole. The distance between the points where the wires are fixed to the ground is equal to the length of one guy wire. How tall is the flagpole (to the nearest inch)?

51 Application - Similar Triangles · Level 3

Length of a Shadow. A man is walking away from a lamppost with a light source 6 m above the ground. The man is 2 m tall. How long is the man's shadow when he is 10 m from the lamppost? [Hint: Use similar triangles.]

52 Application - Similar Triangles · Level 3

Height of a Tree. A woodcutter determines the height of a tall tree by first measuring a smaller one 125 ft away, then moving so that his eyes are in the line of sight along the tops of the trees and measuring how far he is standing from the small tree (see the figure). Suppose the small tree is 20 ft tall, the man is 25 ft from the small tree, and his eye level is 5 ft above the ground. How tall is the taller tree?

53 Application - Mixture Problem · Level 2

Mixture Problem. What quantity of a 60% acid solution must be mixed with a 30% solution to produce 300 mL of a 50% solution?

54 Application - Mixture Problem · Level 2

Mixture Problem. What quantity of pure acid must be added to 300 mL of a 50% acid solution to produce a 60% acid solution?

55 Application - Mixture Problem · Level 3

Mixture Problem. A jeweler has five rings, each weighing 18 g, made of an alloy of 10% silver and 90% gold. She decides to melt down the rings and add enough silver to reduce the gold content to 75%. How much silver should she add?

56 Application - Mixture Problem · Level 3

Mixture Problem. A pot contains 6 L of brine at a concentration of 120 g/L. How much of the water should be boiled off to increase the concentration to 200 g/L?

57 Application - Mixture Problem · Level 3

Mixture Problem. The radiator in a car is filled with a solution of 60% antifreeze and 40% water. The manufacturer of the antifreeze suggests that for summer driving, optimal cooling of the engine is obtained with only 50% antifreeze. If the capacity of the radiator is 3.6 L, how much coolant should be drained and replaced with water to reduce the antifreeze concentration to the recommended level?

58 Application - Mixture Problem · Level 3

Mixture Problem. A health clinic uses a solution of bleach to sterilize petri dishes in which cultures are grown. The sterilization tank contains 100 gal of a solution of 2% ordinary household bleach mixed with pure distilled water. New research indicates that the concentration of bleach should be 5% for complete sterilization. How much of the solution should be drained and replaced with bleach to increase the bleach content to the recommended level?

59 Application - Mixture Problem · Level 3

Mixture Problem. A bottle contains 750 mL of fruit punch with a concentration of 50% pure fruit juice. Jill drinks 100 mL of the punch and then refills the bottle with an equal amount of a cheaper brand of punch. If the concentration of juice in the bottle is now reduced to 48%, what was the concentration in the punch that Jill added?

60 Application - Mixture Problem · Level 3

Mixture Problem. A merchant blends tea that sells for \$3.00 a pound with tea that sells for \$2.75 a pound to produce 80 lb of a mixture that sells for \$2.90 a pound. How many pounds of each type of tea does the merchant use in the blend?

61 Application - Sharing a Job · Level 2

Sharing a Job. Candy and Tim share a paper route. It takes Candy 70 min to deliver all the papers, and it takes Tim 80 min. How long does it take the two when they work together?

62 Application - Sharing a Job · Level 3

Sharing a Job. Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

63 Application - Sharing a Job · Level 3

Sharing a Job. Betty and Karen have been hired to paint the houses in a new development. Working together, the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6 h to paint a house alone. How long does it take Karen to paint a house working alone?

64 Application - Sharing a Job · Level 3

Sharing a Job. Next-door neighbors Bob and Jim use hoses from both houses to fill Bob's swimming pool. They know that it takes 18 h using both hoses. They also know that Bob's hose, used alone, takes 20% less time than Jim's hose alone. How much time is required to fill the pool by each hose alone?

65 Application - Sharing a Job · Level 4

Sharing a Job. Henry and Irene working together can wash all the windows of their house in 1 h 48 min. Working alone, it takes Henry $1 \dfrac{1}{2}$ h more than Irene to do the job. How long does it take each person working alone to wash all the windows?

66 Application - Sharing a Job · Level 4

Sharing a Job. Jack, Kay, and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 h to deliver all the flyers, and it takes Lynn 1 h longer than it takes Kay. Working together, they can deliver all the flyers in 40% of the time it takes Kay working alone. How long does it take Kay to deliver all the flyers alone?

67 Application - Distance/Speed/Time · Level 3

Distance, Speed, and Time. Wendy took a trip from Davenport to Omaha, a distance of 300 mi. She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged 40 mi/h, and the train averaged 60 mi/h. The entire trip took $5 \dfrac{1}{2}$ h. How long did Wendy spend on the train?

68 Application - Distance/Speed/Time · Level 2

Distance, Speed, and Time. Two cyclists, 90 mi apart, start riding toward each other at the same time. One cycles twice as fast as the other. If they meet 2 h later, at what average speed is each cyclist traveling?

69 Application - Distance/Speed/Time · Level 3

Distance, Speed, and Time. A pilot flew a jet from Montreal to Los Angeles, a distance of 2500 mi. On the return trip, the average speed was 20% faster than the outbound speed. The round-trip took 9 h 10 min. What was the speed from Montreal to Los Angeles?

70 Application - Distance/Speed/Time · Level 4

Distance, Speed, and Time. A woman driving a car 14 ft long is passing a truck 30 ft long. The truck is traveling at 50 mi/h. How fast must the woman drive her car so that she can pass the truck completely in 6 s, from the position shown in figure (a) to the position shown in figure (b)? [Hint: Use feet and seconds instead of miles and hours.]

71 Application - Distance/Speed/Time · Level 3

Distance, Speed, and Time. A salesman drives from Ajax to Barrington, a distance of 120 mi, at a steady speed. He then increases his speed by 10 mi/h to drive the 150 mi from Barrington to Collins. If the second leg of his trip took 6 min more time than the first leg, how fast was he driving between Ajax and Barrington?

72 Application - Distance/Speed/Time · Level 3

Distance, Speed, and Time. Kiran drove from Tortula to Cactus, a distance of 250 mi. She increased her speed by 10 mi/h for the 360-mi trip from Cactus to Dry Junction. If the total trip took 11 h, what was her speed from Tortula to Cactus?

73 Application - Distance/Speed/Time · Level 3

Distance, Speed, and Time. It took a crew 2 h 40 min to row 6 km upstream and back again. If the rate of flow of the stream was 3 km/h, what was the rowing speed of the crew in still water?

74 Application - Speed of a Boat · Level 3

Speed of a Boat. Two fishing boats depart a harbor at the same time, one traveling east, the other south. The eastbound boat travels at a speed 3 mi/h faster than the southbound boat. After two hours the boats are 30 mi apart. Find the speed of the southbound boat.

75 Modeling - Law of the Lever · Level 2

**Law of the Lever.** For a seesaw to balance, the product of the weight and its distance from the fulcrum must be the same on each side; that is, $w_1 x_1 = w_2 x_2$. A woman and her son are playing on a seesaw. The boy is at one end, 8 ft from the fulcrum. If the son weighs 100 lb and the mother weighs 125 lb, where should the woman sit so that the seesaw is balanced?

76 Modeling - Law of the Lever · Level 2

**Law of the Lever.** A plank 30 ft long rests on top of a flat-roofed building, with 5 ft of the plank projecting over the edge. A worker weighing 240 lb sits on one end of the plank. What is the largest weight that can be hung on the projecting end of the plank if it is to remain in balance? (Use the law of the lever stated in Exercise 75.)

77 Modeling - Box Volume · Level 3

**Dimensions of a Box.** A large plywood box has a volume of 180 ft^3. Its length is 9 ft greater than its height, and its width is 4 ft less than its height. What are the dimensions of the box?

78 Modeling - Sphere Volume · Level 3

**Radius of a Sphere.** A jeweler has three small solid spheres made of gold, of radius 2 mm, 3 mm, and 4 mm. He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be?

79 Modeling - Box Volume · Level 3

**Dimensions of a Box.** A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides. The box is to hold 100 in^3. How big a piece of cardboard is needed?

80 Modeling - Cylinder Volume · Level 2

**Dimensions of a Can.** A cylindrical can has a volume of $40 \pi$ cm^3 and is 10 cm tall. What is its diameter?

81 Modeling - Sphere Volume · Level 3

**Radius of a Tank.** A spherical tank has a capacity of 750 gallons. Using the fact that one gallon is about 0.1337 ft^3, find the radius of the tank (to the nearest hundredth of a foot).

82 Modeling - Right Triangle Geometry · Level 4

**Dimensions of a Lot.** A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot?

83 Modeling - Construction Cost Optimization · Level 4

**Construction Costs.** The town of Foxton lies 10 mi north of an abandoned east-west road that runs through Grimley. The point on the abandoned road closest to Foxton is 40 mi from Grimley. County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost \$100,000 per mile, whereas building a new road would cost \$200,000 per mile. How much of the abandoned road should be used if the officials intend to spend exactly \$6.8 million? Would it cost less than this amount to build a new road connecting the towns directly?

84 Modeling - Distance and Time · Level 4

**Distance, Speed, and Time.** A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 min 45 s?

85 Modeling - Cone Volume · Level 3

**Volume of Grain.** Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1000 ft^3 of grain?

86 Modeling - Screen Geometry · Level 4

**TV Monitors.** Two television monitors sitting side by side on a shelf in an appliance store have the same screen height. One has a conventional screen, which is 5 in. wider than it is high. The other has a wider, high-definition screen, which is 1.8 times as wide as it is high. The diagonal measure of the wider screen is 14 in. more than the diagonal measure of the smaller screen. What is the height of the screens, correct to the nearest 0.1 in.?

87 Modeling - Composite Solid Volume · Level 4

**Dimensions of a Structure.** A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof. The height of the roof is one-third the height of the entire structure. If the total volume of the structure is $1400 \pi$ ft^3 and its radius is 10 ft, what is its height?

88 Modeling - Wire Cutting Geometry · Level 4

**Comparing Areas.** A wire 360 in. long is cut into two pieces. One piece is formed into a square, and the other is formed into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

89 Modeling - Pythagorean Theorem · Level 2

**An Ancient Chinese Problem.** This problem is taken from a Chinese mathematics textbook called *Nine Chapters on the Mathematical Art*, written about 250 B.C. A 10-ft-long stem of bamboo is broken in such a way that its tip touches the ground 3 ft from the base of the stem. What is the height of the break? (Use the Pythagorean Theorem.)

90 Discovery - Historical Research · Level 1

**Historical Research.** Read the biographical notes on Pythagoras (page 219), Euclid (page 497), and Archimedes (page 729). Choose one of these mathematicians and find out more about him from the library or on the Internet. Write a short essay on your findings. Include both biographical information and a description of the mathematics for which he is famous.

91 Discovery - Babylonian Quadratic · Level 4

**A Babylonian Quadratic Equation.** The ancient Babylonians knew how to solve quadratic equations. Here is a problem from a cuneiform tablet from about 2000 B.C.: "I have a reed, I know not its length. I broke from it one cubit, and it fit 60 times along the length of my field. I restored to the reed what I had broken off, and it fit 30 times along the width of my field. The area of my field is 375 square nindas. What was the original length of the reed?" Solve this problem. Use the fact that 1 ninda $= 12$ cubits.

92 Example - Renting a Car · Level 2

*Renting a Car* A car rental company charges \$30 a day and 15¢ a mile for renting a car. Helen rents a car for two days, and her bill comes to \$108. How many miles did she drive?

93 Example - Interest on an Investment · Level 3

*Interest on an Investment* Mary inherits \$100,000 and invests it in two certificates of deposit. One certificate pays 6% and the other pays $4 \dfrac{1}{2}$% simple interest annually. If Mary's total interest is \$5025 per year, how much money is invested at each rate?

94 Example - Dimensions of a Garden · Level 3

*Dimensions of a Garden* A square garden has a walkway 3 ft wide around its outer edge, as shown in Figure 1. If the area of the entire garden, including the walkway, is 18,000 $\text{ft}^2$, what are the dimensions of the planted area?

95 Example - Dimensions of a Building Lot · Level 3

*Dimensions of a Building Lot* A rectangular building lot is 8 ft longer than it is wide and has an area of 2900 $\text{ft}^2$. Find the dimensions of the lot.

96 Example - Similar Triangles (Modeling) · Level 2

A man who is 6 ft tall wishes to find the height of a certain four-story building. He measures its shadow and finds it to be 28 ft long, while his own shadow is 3.5 ft long. How tall is the building?

97 Example - Mixtures and Concentration · Level 3

A manufacturer of soft drinks advertises their orange soda as "naturally flavored," although it contains only 5% orange juice. A new federal regulation stipulates that to be called "natural," a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gal of orange soda to conform to the new regulation?

98 Example - Time to Do a Job · Level 3

Because of an anticipated heavy rainstorm, the water level in a reservoir must be lowered by 1 ft. Opening spillway A lowers the level by this amount in 4 hours, whereas opening the smaller spillway B does the job in 6 hours. How long will it take to lower the water level by 1 ft if both spillways are opened?

99 Example - Distance-Speed-Time · Level 3

A jet flew from New York to Los Angeles, a distance of $4200$ km. The speed for the return trip was $100$ km/h faster than the outbound speed. If the total trip took $13$ hours, what was the jet's speed from New York to Los Angeles?

100 Example - Energy Optimization (Bird Flight) · Level 4

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours, because air generally rises over land and falls over water in the daytime, so flying over water requires more energy. A bird is released from point A on an island, 5 mi from B, the nearest point on a straight shoreline. The bird flies to a point C on the shoreline and then flies along the shoreline to its nesting area D. The distance from B to D along the shoreline is 12 mi. Suppose the bird has 170 kcal of energy reserves. It uses 10 kcal/mi flying over land and 14 kcal/mi flying over water.

(a) Where should the point C be located so that the bird uses exactly 170 kcal of energy during its flight?

Answer for 100(a)

(b) Does the bird have enough energy reserves to fly directly from A to D?

Answer for 100(b)

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