Answer:
22 minutes after 7:00 P.M. they will be closest.
Step-by-step explanation:
A boat heading south travelling for t hours at the rate of 20 km/h, so the distance x = 20t
The another boat will reach the dock after travelling another 1-t hours at the rate of 15 km/h, so the distance =
y = 15 - 15t
D = d² = x² + y²
D = (20t)² + (15 - 15t)²
dD/dt = -2(15² )( 1-t ) +2 × 20² × t
dD/dt = 2 (15² + 20²) × t -2 ( 15 )² = 0
t = [tex]\frac{2(15)^{2}}{(2\times15^{2}+2\times20^{2})}[/tex]
t = 0.36 hours = 0.36 × 60 = 21.6 minutes ≈ 22 minutes
Therefore, the distance is minimized 22 minutes after 7 pm.
