A child is in danger of drowning in the Merimac river. The Merimac river has a current of 3.1 km/hr to the east. The child is 0.6 km from the shore and 2.5 km upstream from the dock. A rescue boat with speed 24.8 km/hr (with respect to the water) sets off from the dock at the optimum angle to reach the child as fast as possible. How far from the dock does the boat reach the child?
d =....................km?
To find the distance from the dock to where the boat reaches the child, we need to use the concept of relative velocity.
Let's denote the velocity of the boat with respect to the water as v_boat and the current velocity as v_current. The resultant velocity of the boat in still water will be (v_boat^2 + v_current^2)^0.5.
Given:
v_boat = 24.8 km/hr (with respect to the water)
v_current = 3.1 km/hr to the east
Using Pythagoras' theorem, we can find the resultant velocity of the boat:
v_resultant = (v_boat^2 + v_current^2)^0.5
= (24.8^2 + 3.1^2)^0.5
= (615.04 + 9.61)^0.5
= 624.65^0.5
≈ 24.98 km/hr
Now, we need to find the time taken by the boat to reach the child.
Distance covered by the boat (d) = 2.5 km (upstream) + 0.6 km (child's initial distance from shore)
= 3.1 km
Time taken (t) = Distance covered / Velocity
= 3.1 km / 24.98 km/hr
≈ 0.1243 hr
Now, to find how far from the dock the boat reaches the child, we can use the formula:
Distance from dock (d) = Velocity * Time
= 24.98 km/hr * 0.1243 hr
≈ 3.10 km
So, the boat reaches the child at a distance of approximately 3.10 km from the dock.
To solve this problem, we need to break it down into three components: the horizontal motion of the boat, the vertical motion of the child, and the relative motion of the boat and the river.
Let's start by considering the horizontal motion. We know the boat's speed is 24.8 km/hr and the river's current is 3.1 km/hr to the east. These velocities will add up when the boat is moving in the same direction as the current and subtract when moving against the current.
So, the effective horizontal velocity of the boat when moving downstream (along with the current) is 24.8 km/hr + 3.1 km/hr = 27.9 km/hr.
Next, let's consider the vertical motion of the child. Since the child is in danger of drowning, it is important for the boat to reach the child as fast as possible. The fastest way to reach the child is to reach a point downstream directly across from the child's position.
To achieve this, the boat should aim to reach a point downstream that is parallel to the child's position. In this scenario, the vertical distance between the initial position of the child and the dock remains constant.
Now, let's consider the time it takes for the boat to reach the child. The time can be calculated by dividing the horizontal distance (2.5 km) between the child and the dock by the effective horizontal velocity of the boat (27.9 km/hr).
Time = Distance / Velocity
Time = 2.5 km / 27.9 km/hr
Now that we have the time, we can calculate the vertical distance d using the child's rate of movement. Since the river's current is 3.1 km/hr and the child is 0.6 km from the shore, the child's rate of movement (relative to the dock) will be 3.1 km/hr + 0.6 km/hr = 3.7 km/hr.
d = t * rate of movement
d = (2.5 km / 27.9 km/hr) * 3.7 km/hr
d ≈ 0.3328 km
Therefore, the boat reaches the child approximately 0.3328 km from the dock.
To summarize, the boat reaches the child approximately 0.3328 km from the dock by accounting for the horizontal motion of the boat, the vertical motion of the child, and the relative motion of the boat and the river.