A____________B 9 miles is between A and B?
at 3pm Andy began travelling South at a rate of 4mi/hr from point A
at the same time Bill began travelling West at a rate of 5mi/hr from point B to A
both stop wlking when Bill reaches point A
What time will the distance between Bill and Andy be minimized?
Let A be (0,0) and B = (9,0)
The distance d between Bill and Andy after h hours can be found using
d^2 = (4h)^2 + (9-5h)^2
= 16h^2 + 81 - 90h + 25h^2
= 41h^2 - 90h + 81
d = √(41h^2 - 90h + 81)
dd/dh = (82h - 90)/√(41h^2 - 90h + 81)
dd/dh = 0 when h = 90/82 = 1.1 hr = 1:06
3:00 + 1:06 = 4:06
To find the time when the distance between Bill and Andy is minimized, we need to determine when they will be closest to each other.
Let's assume that t hours have passed since 3pm.
Since Andy is traveling south at a rate of 4 mi/hr, after t hours, he would have traveled 4t miles.
Similarly, since Bill is traveling west at a rate of 5 mi/hr, after t hours, he would have traveled 5t miles.
The distance between Bill and Andy at any given time can be found using the Pythagorean theorem:
Distance = √((4t)^2 + (9 - 5t)^2)
To find the time when the distance is minimized, we can differentiate the above expression with respect to t and set it equal to 0. Then we can solve for t.
Differentiating the distance expression and setting it equal to 0:
d(Distance)/dt = 0
Simplifying the expression and solving for t:
2(4t)(4) + 2((9 - 5t)(-5)) = 0
32t - 50(9 - 5t) = 0
32t - 450 + 250t = 0
282t = 450
t ≈ 1.60 hours
To convert this to minutes, multiply by 60:
t ≈ 1.60 * 60 ≈ 96 minutes
Therefore, the time when the distance between Bill and Andy is minimized is approximately 96 minutes after 3pm, which is around 4:36 pm.