A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.60km/h. The child is 0.565km from shore and 0.790km upstream of a boat landing when a rescue boat sets out. If the boat proceeds at its maximum speed of 21.9km/h relative to the water, what heading relative to the shore should the captain take? what angle (in degrees) does the boat velocity make with the shore? How long does it take the boat to reach the child?
To determine the heading the captain should take and the angle the boat velocity makes with the shore, we need to analyze the situation.
Let's first find the speed of the current relative to the shore:
Speed of the current = 2.60 km/h
Now, let's calculate the boat's effective speed relative to the shore by considering its maximum speed and the speed of the current:
Effective speed of the boat = Boat's maximum speed - Speed of the current
= 21.9 km/h - 2.60 km/h
= 19.30 km/h
We can now determine the angle the boat velocity makes with the shore. Since the boat velocity is the vector sum of the velocity of the boat relative to the water and the velocity of the water relative to the shore, we can use vector addition to find the angle.
Let θ be the angle between the boat velocity and the shore. Then:
tan(θ) = (Speed of the current) / (Effective speed of the boat)
= 2.60 km/h / 19.30 km/h
Using the inverse tangent function, we find:
θ = tan^(-1) (2.60 / 19.30)
≈ 7.7 degrees
Therefore, the captain should take a heading approximately 7.7 degrees relative to the shore.
To find the time it takes for the boat to reach the child, we need to calculate the distance the boat has to travel. The distance between the child and the boat landing can be found by adding their perpendicular distances from the shore:
Distance between child and boat landing = √[(Distance from child to shore)^2 + (Distance from boat landing to shore)^2]
= √[(0.565 km)^2 + (0.790 km)^2]
≈ 0.965 km
Now, we can use the following formula to calculate the time it takes for the boat to reach the child:
Time = Distance / Speed
= 0.965 km / 19.30 km/h
≈ 0.050 hours (or approximately 3 minutes)
Therefore, it takes the boat approximately 3 minutes to reach the child.
To solve this problem, we will break it down into three steps:
Step 1: Calculate the heading relative to the shore.
Step 2: Calculate the angle the boat velocity makes with the shore.
Step 3: Calculate the time it takes for the boat to reach the child.
Step 1: Calculate the heading relative to the shore.
We can use vector addition to determine the heading relative to the shore. The boat needs to travel directly to the location where the child is in order to rescue them.
Let's denote the direction of the shore as positive x-axis and the upstream direction as positive y-axis. The child is 0.790 km upstream, so the boat needs to head upstream to reach the child.
The heading relative to the shore can be represented by the angle θ, where:
tan(θ) = (0.790 km) / (0.565 km)
θ = tan^(-1)((0.790 km) / (0.565 km))
θ ≈ 54.22 degrees (rounded to two decimal places)
Therefore, the captain should take a heading of approximately 54.22 degrees relative to the shore.
Step 2: Calculate the angle the boat velocity makes with the shore.
The velocity of the boat can be found using vector addition. If we let Vb represent the velocity of the boat and Vc represent the velocity of the current, the equation becomes:
Vb + Vc = Vr (where Vr is the relative velocity of the boat with respect to the shore)
The magnitude of the relative velocity Vr is given by:
|Vr| = √((21.9 km/h)^2 + (2.60 km/h)^2)
|Vr| ≈ 22.04 km/h (rounded to two decimal places)
The angle the boat velocity makes with the shore can be represented by the angle φ, where:
sin(φ) = (2.60 km/h) / (22.04 km/h)
φ = sin^(-1)((2.60 km/h) / (22.04 km/h))
φ ≈ 6.78 degrees (rounded to two decimal places)
Therefore, the boat velocity makes an angle of approximately 6.78 degrees with the shore.
Step 3: Calculate the time it takes for the boat to reach the child.
To determine the time it takes for the boat to reach the child, we can use the formula:
Time = Distance / Velocity
The distance the boat needs to travel is the sum of the distances to the child and upstream of the landing:
Total distance = 0.565 km + 0.790 km
Total distance ≈ 1.355 km (rounded to three decimal places)
The velocity of the boat relative to the shore is 21.9 km/h, so:
Time = (1.355 km) / (21.9 km/h)
Time ≈ 0.062 h (rounded to three decimal places)
Therefore, it takes approximately 0.062 hours, or 3.72 minutes (rounded to two decimal places), for the boat to reach the child.