Diane rows a boat at 8.3 m/s directly across a river that flows at 5.6 m/s.

(a) What is the resultant speed of the boat?
m/s
(b) If the stream is 250 m wide, how long will it take Diane to row across?
s
(c) How far downstream will Diane be

To find the answers to these questions, we can break down the problem into components and use vector addition. Let's go step by step:

(a) Resultant Speed of the Boat:
Since the boat is rowing across the river, the resultant speed of the boat is the vector sum of its speed in still water (its rowing speed) and the speed of the river current. This can be calculated using vector addition.

To determine the resultant speed, we can use the Pythagorean theorem:
resultant speed^2 = (rowing speed)^2 + (current speed)^2

In this case, the rowing speed is given as 8.3 m/s, and the current speed is given as 5.6 m/s.

Plugging in the values:
resultant speed^2 = (8.3 m/s)^2 + (5.6 m/s)^2
resultant speed^2 = 68.89 m^2/s^2 + 31.36 m^2/s^2
resultant speed^2 = 100.25 m^2/s^2

Taking the square root of both sides:
resultant speed = √(100.25 m^2/s^2)
resultant speed ≈ 10.01 m/s

Therefore, the resultant speed of the boat is approximately 10.01 m/s.

(b) Time to Row Across the Stream:
To calculate the time it takes for Diane to row across the stream, we'll use the formula:
time = distance / speed

The distance she needs to cross is given as 250 m. The speed we need to use here is the resultant speed of the boat, which we found in part (a) to be approximately 10.01 m/s.

Plugging in the values:
time = 250 m / 10.01 m/s
time ≈ 24.98 s

Therefore, it will take Diane approximately 24.98 seconds to row across the stream.

(c) Distance Downstream:
To find out how far downstream Diane will be, we can use the formula:
distance downstream = current speed * time

The current speed is given as 5.6 m/s, and the time taken to row across the stream is approximately 24.98 seconds.

Plugging in the values:
distance downstream = 5.6 m/s * 24.98 s
distance downstream ≈ 139.89 m

Therefore, Diane will be approximately 139.89 meters downstream.