The directions mentioned in the question are not given, so I have called the angle between their directions A. If Allyson and Adrian are moving in the same direction A is 0; if they are moving in opposite directions, A is 180; if they move at right angles A=90. Whatever the angle we can use the cosine rule to determine the length of the bungee cord after time t. The question suggests that the cord is slack to start with because they start at the same location. However, I have assumed that they are initially distance d apart, where d could be 30 feet, so that the cord is taut but not stretched. d could also be zero, indicating that Allyson and Adrian start from the same spot.
After time t, Adrian moves a distance 8t and Allyson 10t. Using the cosine rule we can write:
a^2=64t^2+(10t+d)^2-16t(10t+d)cosA
where a is the length of cord. You can see that at t=0, a=d. This equation applies up to t=5.5 seconds when Adrian stops. At this time, Adrian has moved 44 feet and Allyson 55 feet. After t=5.5 seconds, the equation changes to:
a^2=44^2+(10t+d)^2-88(10t+d)cosA, assuming Allyson continues to move in the same direction.
At t=7, this becomes a^2=1936+(70+d)^2-88(70+d)cosA. So we can find out what a is for d=0 and 30 and A=0, 180º, 90º, 60º if we put all these values into a table:
| d |
A |
a |
| 0 |
0 |
26 |
| 0 |
60 |
61.29 |
| 0 |
90 |
82.68 |
| 0 |
180 |
114 |
| 30 |
0 |
56 |
| 30 |
60 |
86.81 |
| 30 |
90 |
109.25 |
| 30 |
180 |
144 |
Allyson's position from her starting point is 7*10=70 feet. From Adrian's starting point it is either 70 or 100, depending on d.
To draw a picture of the situation at t=7 seconds, draw a triangle ABD angle A being as defined above. Let AB be Adrian's position, so AB=44. Allyson's position is 70+d so AD=70+d and C is positioned along AD d feet from A. So AC=0 if d=0 and 30 if d=30.
When a=90, the maximum stretch length of a, we have: 8100=1936+(10t+d)^2-88(10t+d)cosA. We first calculate (10t+d) as a unit (let x=10t+d and solve for x, Allyson's position, given A). We can then work out t having found x and substituting for d. When we substitute for A we get four values for x=134, 103.54, 78.51 and 46. These represent Allyson's position. So, depending on whether d=0 or 30, we get 13.4, 10.354, 7.851 and 4.6 for t seconds. If we subtract 3 from each of these we appear to get the answer for d=30. However, the two smallest values for t would be less than 5.5 seconds when Adrian is still moving, so we conclude that neither of them would be able to move once maximum stretch was reached; therefore we can eliminate certain values of d and A. We can eliminate 4.6 seconds (A=180 and d=0); and we can eliminate 4.851 and 1.6 seconds (A=90 and 180 and d=30). This means that Adrian and Allyson could not have been moving in opposite directions (A=180), and they could not have been moving at right angles if they were 30 feet apart at the start (d=30). So we're left with A=0, 60, 90 and d=0 or A=0 and 60 and d=30. A=0 means that they are moving in the same direction, but why would the question refer to the directions shown if they were both in the same direction? A=0 could then be eliminated. Also, why would the unstretched cord length be mentioned if it doesn't figure in the calculations? This leaves A=60 and d=30 as the likely conditions. A could be different from 60 (45 or 30 perhaps), but I picked 60 because cos60=1/2, a manageable fraction. The relevant answers would be a=86.81 feet after 7 seconds, and x=73.54 feet (Allyson's position from her starting point 30 feet from Adrian, when a=90).
