A man starts walking north at 3ft/s from a point P. Five minutes later, a woman starts walking south at 6ft/s from a point 500ft due east of P. At what rate are the people moving apart 15min after the woman starts walking?
To find the rate at which the people are moving apart, we can use the concept of relative velocity.
Step 1: Determine the positions of the man and woman after 15 minutes.
The man starts walking north from point P, so after 15 minutes (0.25 hours), he has traveled:
Distance_man = Rate_man * Time = 3 ft/s * 0.25 hours = 0.75 ft
The woman starts walking south from a point 500 ft due east of P, so after 15 minutes (0.25 hours), she has traveled:
Distance_woman = Rate_woman * Time = 6 ft/s * 0.25 hours = 1.5 ft
Step 2: Calculate the position of each person relative to the other.
Since the woman is 500 ft due east of point P, her position relative to the man is:
Relative_position_woman = Distance_woman - Distance_man = 1.5 ft - 0.75 ft = 0.75 ft (east)
The man's position relative to the woman is simply the negative of the woman's position:
Relative_position_man = -Relative_position_woman = -0.75 ft (west)
Step 3: Determine the rate at which the people are moving apart.
The rate at which the people are moving apart is the sum of their individual rates:
Rate_apart = Rate_woman + Rate_man
= 6 ft/s + 3 ft/s
= 9 ft/s
Therefore, the people are moving apart at a rate of 9 ft/s, 15 minutes after the woman starts walking.