A man starts walking north at 4 ft/sec from point P. Five minutes later a woman starts walking south at 5 ft/sec from a point 500 ft due east of P. At what rate ae the people moving apart 15 minutes after the woman starts walking? Draw a diagram.
1) What rate is given? State answer in terms of variables from diagram.
2) What rate are you asked to find? State answer in terms of variables from diagram.
3) Find the required rate
1) The rate given in the problem is the rate at which the man is walking, which is 4 ft/sec.
2) The rate we are asked to find is the rate at which the people are moving apart 15 minutes after the woman starts walking.
3) To find the required rate, we need to determine how far each person has traveled in 15 minutes and then find the rate at which they are moving apart.
Let's start by drawing a diagram to visualize the situation:
```
|
|
| 500 ft
|
P |
-------|-------
|
|
```
In the diagram, point P represents the starting point. The man starts walking north from point P, while the woman starts 500 ft due east of point P.
Since the man walks at a rate of 4 ft/sec and 15 minutes is equivalent to 15 * 60 = 900 seconds, we can calculate the distance the man has traveled in 15 minutes:
Distance traveled by the man = rate * time = 4 ft/sec * 900 sec = 3600 ft.
The woman starts walking 5 minutes after the man, which means she starts 5 * 60 = 300 seconds after the man. She walks at a rate of 5 ft/sec, so the distance she covers in 15 minutes is:
Distance traveled by the woman = rate * time = 5 ft/sec * 900 sec = 4500 ft.
Now, let's calculate the rate at which the people are moving apart:
Distance between them = Distance traveled by the woman - Distance traveled by the man = 4500 ft - 3600 ft = 900 ft.
The people are moving apart by 900 ft in 15 minutes. To find the rate at which they are moving apart, we divide this distance by the time:
Rate at which the people are moving apart = Distance / Time = 900 ft / 900 sec = 1 ft/sec.
Therefore, the rate at which the people are moving apart 15 minutes after the woman starts walking is 1 ft/sec.
1) The rate given is that the man starts walking north at 4 ft/sec.
2) The rate we are asked to find is the rate at which the people are moving apart 15 minutes after the woman starts walking.
3) To find the required rate, we need to first draw a diagram to visualize the situation.
Let's label the starting point P. The man starts walking north, so we can label his direction as "up."
After 5 minutes, the woman starts walking south from a point 500 ft due east of P. We can label this point as Q and draw a line connecting P and Q.
Now, let's label the rate at which the woman is walking south as -5 ft/sec, since it is in the opposite direction of the man's movement. Additionally, let's label the rate at which the man is walking north as +4 ft/sec.
To find the rate at which they are moving apart 15 minutes after the woman starts walking, we can use the Pythagorean theorem to calculate the distance between the man and the woman at that time.
The distance between the man and the woman can be represented as the hypotenuse of a right-angled triangle, with the horizontal distance being 500 ft (from P to Q) and the vertical distance being the distance covered by the man in 15 minutes.
Since the man is walking at a rate of 4 ft/sec, the distance covered by him in 15 minutes is 4 ft/sec * 15 min = 60 ft.
Using the Pythagorean theorem, the distance between the man and the woman after 15 minutes can be calculated as follows:
Distance^2 = (500 ft)^2 + (60 ft)^2
Distance = sqrt((500 ft)^2 + (60 ft)^2)
Finally, to find the rate at which they are moving apart, we need to find the derivative of the distance equation with respect to time (t):
Rate of moving apart = d(Distance)/dt
By differentiating the distance equation, we can find the rate at which they are moving apart 15 minutes after the woman starts walking.
at time t>=300 (5 min = 300 sec) the man has gone 4t ft north, and the woman has gone 5(t-300) ft south.
The distance d between them is thus
d^2 = 500^2 + (4t + 5(t-300))^2
= 81t^2 - 27000t + 2,500,000
at t=15 min = 900 seconds,
d = 6619 ft
2d dd/dt = 162t - 27000
so, at t=900
2(6619) dd/dt = 162(900)-27000
dd/dt = 8.97 ft/s
This makes sense, since the farther apart they are, the less the 500 ft between their paths matters, and their relative speed will approach 5+4=9 ft/s.