Apex

2.1.4 practice: modeling: multistep linear equations

you are traveling to an island that is accessible only by boat. you plan to take one boat there and a different boat back. determine how fast each boat travels.

in this assignment, you may work alone, with a partner, or in a small group. discuss the results of your work and/or any lingering questions with your teacher.

1. circle the two boats you picked for your trip.

steamboat and tall ship

"the steamboat makes the trip in 5 hours. the tall ship makes the trip in 10 hours. the tall ship is 10 knots slower than the steamboat. for each boat, the trip time multiplied by the speed equals the distance traveled. you know both boats will travel the same distance. how fast is each boat traveling, in knots (nautical miles per hour)?

2. make sense of the problem

what do you know? (list four things you know about the time, the speed, and the distance the boats travel.)

what do you want to find out?

what kind of answer do you expect?

writing some expressions

3. write an expression for the distance going to the island. let x be the speed of the boat you are taking to the island. distance is equal to speed times time. (2 points)

4. write an expression for the speed of the boat coming back from the island in terms of the other boat’s speed. let x be the speed of the boat you are taking to the island. (2 points)

5. write an expression for the distance returning from the island. use the expression for speed you found in question 4. remember that distance is equal to speed times time. (2 points)

6. each ship travels the same distance. set the distances equal to each other to get the equation for the speed of the ship you are taking to the island. (1 point)

7. solve the equation from question 6 for x. what does this tell you? (5 points: 1 point for showing your work, 3 points for solving for x, 1 point for explaining the answer)

8. how fast does the second boat travel? show your work. (2 points: 1 point for showing your work, 1 point for the answer)

1. The two boats picked for the trip are the steamboat and the tall ship. Let us assume that we will take the steamboat going to the island, and then we will take the tall ship for the return trip. We will then relate the distances travelled by both ships to each other.

2. We know that the steamboat takes five hours to complete the trip. The tall ship takes more time, at ten hours to complete the trip. We do not have the exact speeds of the steamboat or of the tall ship, but we do know that the tall ship is 10 knots slower than the steamboat. We likewise do not know the exact distance travelled by either ship, but we do know that both travel the same distance. We want to find out how fast each boat travels. We expect the answers to be in knots, with a difference of 10.

3. We know that distance is equivalent to the product of speed of a boat multiplied by the time of travel. For the trip going to the island, we will use the steamboat. Let its speed be x knots (equivalent to x nautical miles per hour), and let the distance going to the island be d nautical miles. Given that the time takes is 5 hours, this means that d = 5x.

4. If we let x be the speed of the boat you are taking to the island (the steamboat), then we know that the speed of the other boat (the tall ship) is 10 knots less than the steamboat's. So the speed of the tall ship (for the return trip) is (x - 10) knots.

5. Similar to part 3: we will multiply speed by time to determine the distance from the island. From part 4, we have determined that the speed of the tall ship to be used in returning is (x - 10) knots. Meanwhile, the given in the problem says that the tall ship will take 10 hours to make the trip. Therefore the distance will be equal to d = 10(x - 10) = 10x - 100 nautical miles.

6. We can assume that the distance travelled going to the island is the same distance travelled coming back. Therefore, we can equate the formula for distance from part 3 for the steamboat, to the distance from part 5 for the tall ship.
5x = 10x - 100

7. Solving for x: 5x = 10x - 100
-5x = -100
x = 20
Since x is the speed of the steamboat, x = 20 means that the steamboat's speed is 20 knots.

8. We determined in part 4 that the speed of the second boat (in our case, the tall ship) is (x - 10) knots. Since we have calculated in part 7 that the steamboat travels at x = 20 knots, then the speed of the tall ship is (x - 10) = 20 - 10 = 10 knots.

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