A man starts walking north at 2ft/s from a point P. Twenty-five seconds later a woman starts walking south at 3ft/s from a point 400ft due east of P. At what rate are the people moving apart 50 seconds after the woman starts walking? Do not include units in your answer.
Man starts north at t = 0 from origin
Woman starts south at t = 25 from (400,0)
where are they at t = 75?
man 2*75 = 150 so at (0,150)
woman 3*50 = 150 so at (400,-150)
distance apart is hypotenuse of triangle with legs 300 and 400 so 500 ft
How fast is this changing?
The north south difference is changing at 2+3 = 5 ft/s
The East west distance is not changing at all
start new t now at 0
N-S distance = 300 + 5 t
E-W distance = 400
h is hypotenuse, the distance
h^2 = 400^2 + (300+5t)^2
2 h dh = 0 + 2 (300+5t)(5)dt
h dh = 5(300+5t) dt
at h = 500 and t = 0 this is
500 dh = 5(300) dt
dh/dt = .01(300) = 3
let the times passed since the woman started walking be t seconds
make a diagram
distance covered by man = 50 + 2t
distance covered by woman = 3t
let the distance between them be D
I see a right-angled triangle with hypotenuse D,
the vertical line as (50+2t+3t)
and the horizontal as 400
D^2 = (5t+50)^2 + 400^2
2D dD/dt = 2(5t+50)(5) + 0
dD/dt = 5(5t+50)/D
when t = 50,
D^2 = 300^2+400^2
D = 500
dD/dt = 5(300)500 = 3
thank you very much
To find the rate at which the two people are moving apart, we need to find the rate at which their distance is increasing. Let's first calculate the distance traveled by both the man and the woman during the given time intervals.
The man starts walking north at 2 ft/s from point P. In 25 seconds, he will have traveled a distance of:
Distance by man = rate × time = 2 ft/s × 25 s = 50 ft
The woman starts walking south at 3 ft/s from a point 400 ft east of point P. In 50 seconds (25 seconds after the woman starts walking), she will have traveled a distance of:
Distance by woman = rate × time = 3 ft/s × 50 s = 150 ft
Now, we need to calculate the distance between the man and the woman after 50 seconds. To do so, we can calculate the vertical (north-south) and horizontal (east-west) components separately and then use the Pythagorean theorem to find the distance.
The vertical component between the man and the woman is simply the distance covered by the man, i.e., 50 ft.
The horizontal component between the man and the woman is the difference between their positions. The woman starts 400 ft east of point P, and after 50 seconds, she will be 150 ft south of her starting point. The horizontal distance between the two is then:
Horizontal component = distance east - distance west
= 400 ft - 150 ft
= 250 ft
Now we can find the distance between the man and the woman after 50 seconds using the Pythagorean theorem:
Distance = sqrt(vertical component² + horizontal component²)
= sqrt(50 ft² + 250 ft²)
= sqrt(62500 ft²)
= 250 ft
Therefore, the rate at which the people are moving apart 50 seconds after the woman starts walking is 250 ft.