# Related rates help again

• Mar 18th 2007, 07:29 AM
Related rates help again
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?
• Mar 18th 2007, 08:26 AM
Soroban

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:

```          :    15-20t      A    20t    P       - O * - - - - - - - - * - - - - - * - -           |              *           |          *   60-15t |        * 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: . .= .(15 - 20t)² + (60 - 15t)²

. . which simplifies to: . .= .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: . .= .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.