Answer:
The piecewise function d(t) models Jamie’s sightseeing trip, where d(t) is the distance in miles from the center of town at time t in hours. The function is defined as follows:
d(t) = 30t for 0 <= t < 2
d(t) = 60 for 2 <= t < 3
d(t) = 45t - 75 for 3 <= t < 4
d(t) = 105 for 4 <= t < 5
d(t) = -25t + 230 for 5 <= t < 6
d(t) = 80 for 6 <= t < 7
d(t) = -40t + 360 for 7 <= t < 9
To find the time when the distance is 40 miles, we need to solve the equation d(t) = 40 for each piece of the function and check if the solution falls within the valid range of t for that piece. Let’s do that:
For the first piece, we have 30t = 40, which gives t = 40/30 = 1.33. This is within the range of t for this piece (0 <= t < 2), so 1.33 hours is one possible answer.
For the second piece, the distance is constant at 60 miles, which is not equal to 40, so there are no solutions in this range.
For the third piece, we have 45t -75 =40, which gives t=115/45=2.56. This is within the range of t for this piece (3 <= t <4), so 2.56 hours is another possible answer.
For the fourth piece, the distance is constant at 105 miles, which is not equal to 40, so there are no solutions in this range.
For the fifth piece, we have -25t +230=40, which gives t=190/25=7.6. This is not within the range of t for this piece (5 <= t <6), so there are no solutions in this range.
For the sixth piece, the distance is constant at 80 miles, which is not equal to 40, so there are no solutions in this range.
For the seventh piece, we have -40t +360=40, which gives t=320/40=8. This is within the range of t for this piece (7 <= t <9), so 8 hours is another possible answer.
So, Jamie would be 40 miles away from town center at times: approximately at 1.33 hours (or around one hour and twenty minutes), approximately at 2.56 hours (or around two hours and thirty-four minutes), and at exactly eight hours into her trip.