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Over the ravine!

You are helping to design a road for a high mountain pass. There are two routes over the pass, but both have to cross steep ravines. Use what you know about solving radical functions to design a bridge that will safely cross the ravine.

Your bridge

1. What is the horizontal distance that your bridge must span to clear the ravine? (1 point)

The figure below represents your bridge, where the horizontal distance from one edge of the ravine to the other is x, the rise of the bridge from one side of the ravine to the other is y, and the length of the bridge deck is r.

2. Write a function that models the relationships among x, y, and r. (Hint: Use the Pythagorean theorem.) Then solve the equation for r in terms of x and y.
(3 points: 1 point for the correct Pythagorean setup, 1 point for work, 1 point for the solution)

3. There are several possible heights at which the higher end of the bridge can be attached to the higher mountain. Fill in the table below to use 5 possible values for y, and calculate the resulting values for r. (5 points:1 point for each line of the chart)

Maximum slope of your bridge

For automobile safety, the maximum slope that a highway can have is 6%, or 0.06. A surveyor measures the angle from one edge of the ravine to the other and determines that your bridge should have a slope of 0.06.

You need to find the length of your bridge if it has a slope of 0.06. This will tell you how much pavement is needed for the bridge deck

4. Use the definition of slope to calculate the exact rise of the bridge, y. (3 points: 2 points for work, 1 point for the solution)

5. How long is the deck of the bridge? Show how you calculated r. (3 points: 2 points for work, 1 point for the solution)

Hold on tight!

One of the construction workers likes to listen to music on the job. Unfortunately, he also has a bad habit of accidentally dropping his music player into the ravine! Luckily, his daughter is good with physics and has built a safety balloon for his latest music player that will release after the music player has been falling for 1 second. 
The time it takes for the music player to fall from the bridge to the bottom of the ravine can be modeled by the following equation: 

where t is the amount of time since the construction worker dropped his music player. 
Will the safety balloon release in time?

6. First, how long will it take for the music player to fall to the bottom of the ravine? Solve the equation and find the values of t. (2 points: 1 point for each value)

7. Your solution produces two roots, but you should make sure that both times are valid. Check your work by plugging each root back into the original equation and simplifying each expression. Identify the answer that is actually a solution. (2 points: 1 point for each time)

8. Will the music player land safely? How do you know? (1 point)