Scenario: The Zip Line
View the video found on page 1 of this Journal activity.
[This video begins with an image of a tree in a forest. Attached to the tree is diamond-shaped yellow sine that shows a picture of a person going down a zip line in a harness. The sign says “Caution Zip Line Crossing.” Click the button titled “Start” to begin the video.[Click.]Audio: My friend and I thought it would be fun to set up a zip line in the forest near our house. [A picture is shown of a harness, helmet, and shoes.]We’ve got the perfect tree, 200 feet of steel cable, and a tall adjustable ladder. [An image is shown of a tree, a coil of steel cable, and a tall metal ladder.]Before jumping into building, we did some research on the Internet to see what sorts of safety precautions we should consider. [The image of the ladder moves to the left side of the screen to make room for an image of a bright yellow cross.]We found out that setting up our ladder at a 75 degree angle is the safest. And, to get a moderately quick speed on the zip line, it should make about an 80 degree angle from the tree. [Two different pictures are shown of a metal ladder leaning up against a tree. A picture is shown of a boy wearing a helmet and harness attached to a zip line and standing on a platform about to go down the zip line.]My friend wants to use the entire length of the ladder, which is 40 feet. [A picture is shown of a girl wearing a helmet and hanging upside down in the midair secured by a harness.]That’s a little scary for me — I’d be happier with the ladder being only 25 feet long. [A picture is shown of a very tall ladder pointing into the sky. A picture is shown of a short step ladder.]We do agree we should connect the cable to the tree at the top of the ladder. [A diagram is shown of a vertical line labeled “tree” standing perpendicular to a line labeled “ground.” Another line is drawn from “ground” at an angle to “tree.” This line is labeled “ladder.” A line is drawn from the top of “ladder” down at an angle until it reaches “ground.”]But before we start building, we need to decide how long the ladder should be. [A picture is shown of a kitten hanging on to a cord.]We could just drag the cable up the tree using brute force and see how far we get. But we think a little math — maybe the law of sines — can save us a lot of time and trouble finding the right ladder length. Work smarter, not harder, right? On-screen text: .] video transcript
Using the information provided in the video, answer the questions below.
Show your work, including all calculations and diagrams.
Student Conjectures: Two friends are trying to decide how long their ladder should be for the zip line they are building.
1. What is each person suggesting? (1 point)
StudentConjectureWill (friend)Thomas (narrator)What do you think?
2. Which ladder length would you pick? Why? (1 point)
3. What problems would you encounter if the ladder were too short or too long? (1 point)
Draw a Diagram:
4. What key details are given in the scenario? (1 point)
5. Complete the diagram by identifying the missing angles. Use the length of the ladder you chose for the missing length. (2 points total: 1 for each angle)
6. In the space below, use the law of sines or trigonometric ratios to find each length. Round your answer to 1 decimal place. (4 points total: 1 point each)
a. The length of the zip line, z
b. The height of the point on the tree where the top of the ladder rests against it, h
c. The distance between the base of the ladder and the base of the tree, b
d. The distance between the base of the tree and the spot where the zip line is anchored to the ground, B
Make a Comparison:
7. Now let’s complete a diagram for the other ladder length. (1 point)
8. Use the law of sines or trigonometric ratios to find each length. Round your answer to 1 decimal place. (4 points total: 1 point each)
a. The length of the zip line
b. The height of the point on the tree where the top of the ladder rests against it
c. The distance between the base of the ladder and the base of the tree
d. The distance between the base of the tree and the spot where the zip line is anchored in the ground
Make a Decision:
9. Based on the calculations you did using the law of sines, did you make a good choice for which ladder to use? Why or why not? (2 points)
10. How long should the ladder be if they want to use all the cable they have? Use the law of sines to find the length, L.(2 points
11. The friends had so much fun that they are now thinking about going into business building zip lines. Analyze the concerns of a company that builds real zip lines. What issues would it have to deal with in real life that are different from those presented by the representations of triangles we have been looking at? (1 point)
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