A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.35 km/hr. The child is 0.625 km from shore and 0.800 km upstream of a boat landing when a rescue boat sets out. If the boat proceeds at its maximum speed of 18.6 km/hr relative to the water, what heading relative to the shore should the pilot take?
What angle does the boat velocity make with the shore?
How long does it take the boat to reach the child?
tan(theta)=0.800/0.625
To solve this problem, we can break it down into three parts:
1. Determine the velocity of the boat relative to the shore
2. Find the angle the boat velocity makes with the shore
3. Calculate the time it takes for the boat to reach the child
1. Determining the velocity of the boat relative to the shore:
The velocity of the boat relative to the shore is the vector sum of the boat's velocity in still water and the velocity of the current. Since the boat's maximum speed is given as 18.6 km/hr and the speed of the current is 2.35 km/hr, we can find the relative velocity using the Pythagorean theorem.
Relative velocity = sqrt((boat velocity)^2 - (current velocity)^2)
Relative velocity = sqrt((18.6 km/hr)^2 - (2.35 km/hr)^2)
= sqrt(345.96 - 5.5225) km/hr
= sqrt(340.4375) km/hr
≈ 18.44 km/hr
2. Finding the angle the boat velocity makes with the shore:
To find the angle, we can use trigonometric functions. Let's consider the angle θ.
cos(θ) = (boat velocity along the shore) / (relative velocity)
cos(θ) = (boat velocity along the shore) / (18.44 km/hr)
The boat velocity along the shore is the component of the boat's velocity relative to the shore perpendicular to the current. This can be calculated using the Pythagorean theorem:
(boat velocity along the shore) = sqrt((boat velocity)^2 - (current velocity)^2)
= sqrt((18.6 km/hr)^2 - (2.35 km/hr)^2)
= sqrt(340.4375) km/hr
≈ 18.44 km/hr
cos(θ) = 18.44 km/hr / 18.44 km/hr
= 1
Since cos(90 degrees) = 0, we can conclude that θ ≈ 90 degrees.
Therefore, the boat velocity makes an angle of approximately 90 degrees with the shore.
3. Calculating the time it takes for the boat to reach the child:
The time it takes for the boat to reach the child can be calculated by dividing the distance between the boat landing and the child (0.800 km) by the relative velocity of the boat:
Time = (distance) / (relative velocity)
= 0.800 km / 18.44 km/hr
≈ 0.0434 hours
Convert this time to minutes (1 hour = 60 minutes):
Time ≈ 0.0434 hours × 60 minutes/hour ≈ 2.61 minutes
Therefore, it will take approximately 2.61 minutes for the boat to reach the child.
To determine the heading the pilot should take, we need to consider the relative velocities of the boat and the current.
1. Calculate the relative velocity of the boat with respect to the shore:
Boat velocity relative to the shore = Boat velocity - Current velocity
Boat velocity relative to the shore = 18.6 km/hr - 2.35 km/hr
Boat velocity relative to the shore = 16.25 km/hr
2. Calculate the angle the boat velocity makes with the shore:
Tan(θ) = (Current velocity) / (Boat velocity relative to the shore)
Tan(θ) = 2.35 km/hr / 16.25 km/hr
θ = arctan(0.1446)
θ ≈ 8.267 degrees
Therefore, the pilot should take a heading of approximately 8.267 degrees relative to the shore.
3. To find the time it takes for the boat to reach the child, we need to determine the effective speed of the boat.
Effective speed of the boat = Boat velocity relative to the shore / Cos(θ)
Effective speed of the boat = 16.25 km/hr / Cos(8.267 degrees)
Effective speed of the boat ≈ 16.37 km/hr
Time = Distance / Speed
Time = (0.625 km + 0.800 km) / 16.37 km/hr
Time ≈ 0.0871 hours
Therefore, it takes approximately 0.0871 hours or 5.22 minutes for the boat to reach the child.