Allyson and Adrian have decided to connect their ankles with a bungee cord; one end is tied to each person's ankle. The cord is 30 feet long, but can stretch up to 90 feet. They both start from the same location. Allyson moves 10 ft/sec and Adrian moves 8 ft/sec in the directions indicated. Adrian stops moving at time t =5.5 sec, but Allyson keeps on moving 10 ft/sec in the indicated direction.
(a) Sketch an accurate picture of the situa-tion at timet =7 seconds. Make sure to label the locations of Allyson and Adrian; also, compute the length of the bungee cord att =7 seconds.
(b) Where is Allyson when the bungee reaches its maximum length?
We have no indication of the directions.
(a) To sketch the situation at t = 7 seconds, we need to determine the distances traveled by Allyson and Adrian in that time period.
Allyson's distance traveled = speed × time = 10 ft/sec × 7 sec = 70 ft
Adrian's distance traveled = speed × time = 8 ft/sec × 5.5 sec = 44 ft
Since they started from the same location, at t = 7 seconds, Allyson would be 70 ft ahead in the indicated direction and Adrian would be 44 ft behind in the opposite direction.
We can represent this situation on a number line:
Adrian ------------------------- Allyson
At t = 7 seconds, the length of the bungee cord can be calculated by subtracting the distance traveled by Adrian from the total length of the cord.
Length of bungee cord = Total length - Distance traveled by Adrian
Length of bungee cord = 30 ft - 44 ft (since Adrian moved in the opposite direction)
Length of bungee cord = -14 ft
Note that the negative value indicates that the bungee cord is stretched beyond its original length.
(b) The bungee reaches its maximum length when the total distance traveled by Allyson equals the length of the bungee cord when it is fully stretched. In this case, the length of the bungee cord when fully stretched is 90 ft.
Distance traveled by Allyson = 10 ft/sec × time (t) = 10t ft
To find when the bungee reaches its maximum length, we need to solve for t:
10t ft = 90 ft
t = 90 ft / 10 ft/sec
t = 9 sec
Therefore, Allyson is at the maximum length of the bungee cord at t = 9 seconds.
To solve this problem, let's break it down into two parts.
(a) To sketch the situation at time t=7 seconds, we need to find the locations of Allyson and Adrian at that time. We can do this by multiplying their speeds by the time elapsed.
Allyson's distance moved = 10 ft/sec * 7 sec = 70 ft
Adrian's distance moved = 8 ft/sec * 5.5 sec = 44 ft
Since Adrian stopped moving at t=5.5 seconds, his distance remains constant at 44 ft.
Now, let's sketch the situation. Draw a horizontal line to represent the ground. Mark a point for Allyson 70 ft to the right of the starting point, and another point for Adrian 44 ft to the right of the starting point.
Now let's calculate the length of the bungee cord at t=7 seconds. This can be found by taking the difference between their distances from the starting point.
Length of the bungee cord = Allyson's distance - Adrian's distance
Length of the bungee cord = 70 ft - 44 ft = 26 ft
So, the length of the bungee cord at t=7 seconds is 26 feet.
(b) To find where Allyson is when the bungee reaches its maximum length, we need to consider that Allyson keeps moving at a constant speed while Adrian has already stopped. Since Allyson continues moving at 10 ft/sec, she will keep moving away from Adrian.
The maximum length of the bungee cord is 90 feet, so when Allyson reaches this point, the bungee cord will be at its maximum length.
To calculate the time it takes for Allyson to reach the maximum length, we can set up an equation:
Allyson's distance = Maximum length of bungee cord
10 ft/sec * t = 90 ft
Solving for t, we get:
t = 90 ft / 10 ft/sec
t = 9 seconds
Therefore, Allyson will be at the maximum length of the bungee cord at t = 9 seconds.