A child in danger of drowning in a river is being carried downstream by a current that has a speed of 2.10 km/h. The child is 0.600 km from shore and 0.800 km upstream of a boat landing when a rescue boat sets out.

(a) If the boat proceeds at its maximum speed of 19.0 km/h relative to the water, what heading relative to the current should the pilot take?
-- ° relative to the direction of the current

(b) What angle does the boat velocity make with the current?
--- ° relative to the direction of the current

(c) How long does it take the boat to reach the child?
----s

To solve this problem, we need to consider the velocities and distances involved. Let's break it down step by step.

(a) To determine the heading the pilot should take relative to the current, we need to find the resultant velocity that will cancel out the effects of the current and move the boat towards the child.
The boat's maximum speed relative to the water is given as 19.0 km/h, and the current speed is 2.10 km/h downstream. To counteract the current, the boat's velocity relative to the ground will have to be equal to the sum of the child's velocity relative to the ground (which is equal to the current's velocity, 2.10 km/h) and the desired velocity of the boat relative to the ground towards the child.

Let's denote the desired velocity of the boat relative to the ground towards the child as Vb. Thus, we have:
Vb = Vc + Vd
Vc = current velocity = 2.10 km/h

Since velocity is a vector quantity, we need to consider their magnitudes and directions in this question.

Now, let's break down the child's velocity relative to the ground:
The child is being carried downstream by the current at a speed of 2.10 km/h. This means that the child's velocity downstream is 2.10 km/h.

Now, let's break down the boat's velocity relative to the ground:
The boat has a maximum speed of 19.0 km/h relative to the water. To simplify the calculation, we'll assume the boat is traveling directly towards the child. If the boat is heading directly towards the child, its velocity will be equal to the child's velocity downstream plus the boat's velocity relative to the water:
Vb = 2.10 km/h + 19.0 km/h.

Now we can solve for Vb:
Vb = 21.10 km/h.

So, the boat should have a resultant velocity of 21.10 km/h relative to the ground towards the child.

To find the heading relative to the current, we need to consider that the resultant velocity (Vb) forms a right triangle with the boat's velocity relative to the water and the current velocity. The heading we need to find is the angle between Vb and Vc.

To find this angle using trigonometry, we can use the inverse tangent function (tan^(-1)):

Angle = tan^(-1)(Vb / Vc) = tan^(-1)(21.10 / 2.10) = tan^(-1)(10) = 84.29°

Therefore, the pilot should take a heading relative to the current of 84.29°.

(b) To find the angle the boat's velocity makes with the current, we need to find the difference between the heading (84.29°) and the direction of the current (downstream). Since the current is downstream, its direction can be taken as 0°.

Angle = Heading - Direction of Current = 84.29° - 0° = 84.29°

Therefore, the boat's velocity makes an angle of 84.29° relative to the direction of the current.

(c) To find the time it takes for the boat to reach the child, we need to consider the distance between them and the velocity of the boat. The child is 0.600 km from shore and 0.800 km upstream of a boat landing.

Given that the boat's velocity relative to the ground towards the child is 21.10 km/h, we can calculate the time it takes for the boat to reach the child using the formula:
Time = Distance / Velocity

Time = (0.600 km + 0.800 km) / 21.10 km/h = 1.400 km / 21.10 km/h

Now, let's convert km/h to km/s:
1 km = 1000 m
1 hour = 60 minutes = 60 * 60 seconds = 3600 seconds

21.10 km/h = 21.10 * (1000 m / 1 km) * (1 hour / 3600 s) = 0.00586 km/s

Time = 1.400 km / 0.00586 km/s = 238.90 s

Therefore, it takes the boat approximately 238.90 seconds to reach the child.

To summarize:
(a) The pilot should take a heading relative to the current of 84.29°.
(b) The boat's velocity makes an angle of 84.29° relative to the direction of the current.
(c) It takes the boat approximately 238.90 seconds to reach the child.