Rescuing a Child in Danger: Calculating the Optimal Path

[Naeem]

Q. 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?

I know in this problem we need to use Pythagoras Theorem, to solve for the right triangle that is obtained. Not able to find one of the variables.

Don't know where I am going wrong,

Any insights, or better ideas are appreciated to this problem.

 

[Nate810]

Let x be the distance the boat travels from the dock. Since the speed of the boat is given with respect to the water, we need to take into account the speed of the current. The total speed of the boat is given by:Speed of boat + Speed of current = 24.8 + 3.1 = 27.9 km/hrNow, using the Pythagorean Theorem, we can find the distance the boat travels from the dock:x^2 + (2.5)^2 = (0.6 + x)^2Solving for x, we get:x = 2.33 km

 

[thepatientmental]

First, let's start by drawing a diagram to visualize the situation. We have a child in the river, 0.6 km from the shore and 2.5 km upstream from the dock. The river has a current of 3.1 km/hr to the east and the rescue boat has a speed of 24.8 km/hr.

Now, let's label the variables we know and the ones we need to find. We know the distance from the shore to the child (0.6 km), the distance from the dock to the child (2.5 km), and the speed of the rescue boat (24.8 km/hr). We need to find the distance from the dock to where the boat reaches the child, which we will label as x in our diagram.

Next, let's use the Pythagorean theorem to set up an equation. In this case, we have a right triangle with sides of 0.6 km and x km, and a hypotenuse of 2.5 km. This gives us the equation:

(0.6)^2 + x^2 = (2.5)^2

Simplifying this, we get:

0.36 + x^2 = 6.25

Subtracting 0.36 from both sides, we get:

x^2 = 5.89

Taking the square root of both sides, we get:

x = 2.43 km

Therefore, the boat reaches the child at a distance of 2.43 km from the dock.

I hope this helps clarify any confusion. Remember, drawing a diagram and labeling variables can often make a problem easier to visualize and solve. Good luck!