Cyclists A and B rode along a straight one way road starting from the same location and they both rode for an hour....
Cyclists A and B rode along a straight one way road starting from the same location and they both rode for an hour. Cyclist A rode half the time with speed 10 miles/ hour and the other half time with speed 30 miles/ hour. Cyclist B rode half the total distance with speed 10 miles/ hour and half the distance with speed 30/ hours. Sketch a graph of the distance function for both cyclists. Calculate the distance covered by each cyclist.
To solve this problem, we need to use the speed formula `v=d/t` which can be rearranged as `d=vt`.
For the first 1/2 hour A has gone 5 miles. In the time up to the 1/2 hour, the distance is the straight line function `y=10t`. In the second 1/2 hour, A is moving at speed 30 mph, so the distance is an additional 15 miles, so the distance function is `y=30t+b`. To find b, we may sub in the point `(1/2,5)` which gives the equation `5=15+b` so the second half is the straight line function `y=30t-10`.
This is slightly more complicated, since we don't know how far B has gone, but the total distance was spent at 10 mph and 30 mph. Let d be the distance. Then let `t` be the first time and let `1-t_` be the second time. We know that `d/2=10t=30(1-t)`, which we can solve to find t.` `
This means that `t=3-3t` so `4t=3`
which means that the first speed is for `3/4=0.75` hours and the second speed is for `1/4=0.25` hours.
After the first 45 minutes, the total distance was 7.5 miles and an additional 7.5 miles came from the next 15 minutes. This means the straight line function for the first part is `y=10t`
and the straight line function for the second part is `y=30t+b` where we solve for b using the point `(3/4,7.5)` which gives the equation `7.5=22.5+b`. This means the function for the second point is `y=30t-15`.
The graphs are then given (A in green, B in blue). Notice that A and B are the same function until t=.5