Part 3 Module 7 Real-world problems involving distance

Two useful facts From elementary geometry we have two familiar facts that can be useful if we are trying to calculate a distance. One of these is the Pythagorean Theorem, and the other is the formula for the circumference of a circle.

Exercise #1 The GEOMETRIZER The figure below shows an aerial view of The Hurl-O-Matic, a carnival ride in which the passengers are seated in a car, attached to the end of an arm which rotates rapidly around a central hub. Suppose that the length ( r ) of the arm is 39 feet, and that, at full speed, it takes 4 seconds for the car to complete one revolution. Find the speed of the car. A. 819 miles per hour B. 54 miles per hour C. 90 miles per hour D. 21 miles per hour E. 42 miles per hour

Solution #1 Suppose that the length ( r ) of the arm is 39 feet, and that, at full speed, it takes 4 seconds to for the car to complete one revolution. Find the speed of the car. In four seconds, the distance the cars travels is the circumference of a circle whose radius is 39 feet. 2π39 = feet Since the car travels feet in 4 seconds, its speed is /4 = feet per second. We must convert feet per second to miles per hour. First, multiply by 60 since there are 60 seconds in a minute x 60 = The car travels feet in one minute. Multiply again by 60, since there are 60 minutes in an hour x 60 = 220,536 The car travels 220,536 feet in one hour. Finally, divide by 5280, since there are 5280 feet in one mile 220,536/5280 = The car travels miles in one hour, so its speed is miles per hour. The best choice is E.

Exercise #2 The GEOMETRIZER Homer and Aristotle are loitering on a street corner when Plato, to whom they owe money, suddenly approaches. Homer begins running northward at 12 miles per hour, and Aristotle begins running eastward at 15 miles per hour. How far apart are Homer and Aristotle after 10 minutes? A. 1.6 miles B. 4.5 miles C. 3.2 miles D miles

Solution #2 Homer begins running northward at 12 miles per hour, and Aristotle begins running eastward at 15 miles per hour. How far apart are Homer and Aristotle after 10 minutes? We are trying to find the hypotenuse of a right triangle. We will first find the distance (D) between them after one hour, then scale that back to 10 minutes. After one hour, Homer has traveled 12 miles and Aristotle has traveled 15 miles. Since they are miles apart after one hour, and 10 minutes is 10/60 of an hour, after ten minutes we have D = x 10/60 = 3.2 miles

Exercise #3 The GEOMETRIZER Archimedes is a traveling salesperson who needs to drive from Podunck to Boonies (see map below). However, Archimedes is too cheap to take the Boonies Expressway (which is a toll road), so instead he takes the Podunck Parkway to Sticks and then takes Sticks Highway to Boonies. The direct distance from Podunck to Boonies is 64 miles, and the distance from Sticks to Boonies is 45 miles. How much gas will Archimedes use, assuming his Yugo gets 5 miles per gallon? A. 9.1 gallons B gallons C gallons D gallons

Solution #3 To find the number of gallons of gas used, we need to find the number of miles traveled, then divide by 5. The number of miles traveled is x We use the Pythagorean Theorem to find x. The number of miles traveled is = 90.5 The number of gallons of gas used is 90.5/5 = 18.1

Exercise from Part 3 Module 6 From The Big UNIT-izer How many square yards are in square feet? A B C D. 3888

Solution How many square yards are in square feet? A B C. 1296D To find the correct conversion factor between square yards and square feet, we need to use the fact that 1 yard = 3 feet, along with the meaning of the word “square.” “1 square yard” means “1 yard x 1 yard” 1 sq. yard = 1 yd. x 1 yd. = 3 feet x 3 feet = 9 square feet We have figured out that 1 square yard = 9 square feet. To convert 11,664 square feet to square yards, we divide by 9. 11,664/9 = 1296The correct choice is C.

Exercise #4 The GEOMETRIZER Study the race track shown below. If Gomer runs 32 laps around the track, how many miles will he have run? A miles B miles C miles D miles

Solution #4 We need to find the perimeter of the figure, the multiply by 32 (since he runs 32 laps), then divide by 5280 (to convert from feet by miles. To find the perimeter ( P ) of the figure, we will assume that the figure consists of a rectangle with exactly half a circle joined to each end. P = C1 + C2 where C1 + C2 is the circumefrence of a circle whose diameter is 374 feet. 374π = 1175 feet So P = = 2739 feet In one lap he runs 2739 feet, so in 32 laps he runs 32 x 2739 = 87,648 feet 87648/5280 = 16.6 miles

Exercise #5 The GEOMETRIZER According to classical mythology, Sisyphus was condemned to spend eternity rolling a large rock up a hill. Upon reaching the top of the hill, the rock would roll down the other side, whereupon Sisyphus would repeat the process. Suppose the figure below (definitely not drawn to scale -- note the cosmic size of the hill) shows his situation. Suppose also that the rock is roughly circular with a diameter of 3.6 feet, and that on average it takes him 10 months to roll the rock one complete revolution. How long will it take him to roll the rock to the top of the hill? A years B years C years D years E years

Solution #5 Suppose the figure below (definitely not drawn to scale -- note the cosmic size of the hill) shows his situation. Suppose also that the rock is roughly circular with a diameter of 3.6 feet, and that on average it takes him 10 months to roll the rock one complete revolution. How long will it take him to roll the rock to the top of the hill? The amount of time required to get to the top of the hill is determined by the number of revolutions of the rock needed to get to the top. The number of revolutions of the rock depends upon the length of the hillside and the distance the rock travels in one revolution. The distance the rock travels in one revolution is the circumference ( C ) of the rock. C = 3.6π = feet Each time the rock turns one complete revolution, it has traveled feet. The length ( L ) of the hillside is the hypotenuse of a right triangle whose legs measure 500 miles and 2000 miles, respectively. The length of the hillside is miles. Convert this to feet, so that it is comparable to the circumference of the rock x 5280 = 10,885,248 feet

Solution #5, page 2 Each time the rock turns one complete revolution, it has traveled feet ( C ), and the length of the hillside ( L ) is 10,885,248 feet. Divide L by C to get the number of revolutions of the rock needed to get from the bottom to the top of the hill. Number of revolutions = L/C = /11.31 = The rock will turn 962, revolutions, and each revolution takes 10 months, so number of months = x 10 = Finally, convert from months to years by dividing by 12: /12 = years. The best choice is E ( years). Of course, our answer isn’t an exact match, due to the fact that we rounded our results during several of the preliminary steps.