A ship is 15 miles east of O and moving west at 20mph; ship B is moving 60miles south of O and moving north at 15mph. Are they approaching or separating after 1 hour and at what rate?
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A ship is 15 miles east of O and moving west at 20mph; ship B is moving 60miles south of O and moving north at 15mph. Are they approaching or separating after 1 hour and at what rate?
Hello, ^_^Engineer_Adam^_^!
Quote:
Ship A is 15 miles east of O and moving west at 20 mph;
ship B is 60 miles south of O and moving north at 15 mph.
Are they approaching or separating after 1 hour and at what rate?
Code:: 1520t A 20t P
 O *         *      *  
 *
 *
6015t  * x
 *
 *
B *

15t 

Q *
Ship A starts at point P, 15 miles east of O.
. . In t hours, it has traveled 20t miles west to point A.
. . Hence: OA = 15  20t
Ship B starts at point Q, 60 miles south of O.
. . In t hours, it has traveled 15t miles north to point B.
. . Hence: OB  60  15t
Let x = AB.
From Pythagorus, we have: .x² .= .(15  20t)² + (60  15t)²
. . which simplifies to: .x² .= .625t²  2400t + 3825 .[1]
Differentiate with respect to time: . 2x(dx/dt) .= .1250t  2400
. . and we have: .dx/dt .= .25(25t  48)/x .[2]
When t = 1, [1] gives us: .x² .= .625  2400 + 3825 .= .2050 . → . x .= .5√82
Substitute into [2]: .dx/dt . = . 25(25  48)/(5√82) . = . 12.6996255
. . The distance is decreasing.
Therefore, they are approaching at about 12.7 mph.