A swimmer can swim 3 mph in still water. She decides to swim perpendicular to the river current which is 3 mph. If the river is 19.89 miles wide, it will take the swimmer ___ hours to cross the river.
To determine the time it will take for the swimmer to cross the river, we can use the concept of relative velocity.
Relative velocity is the vector difference between the velocity of an object and the velocity of the medium it is moving through. In this case, we need to consider the swimmer's velocity with respect to the still water and the river current.
Given that the swimmer can swim at a speed of 3 mph in still water, and the river current has a velocity of 3 mph, we can find the effective velocity of the swimmer using vector addition.
Since the swimmer is swimming perpendicular to the river current, the effective velocity will be the magnitude of the swimmer's velocity with respect to the still water, which is 3 mph, and the river current velocity, which is also 3 mph.
Using the Pythagorean theorem, we can calculate the effective velocity:
Effective velocity = √((3 mph)^2 + (3 mph)^2)
= √(9 mph^2 + 9 mph^2)
= √(18 mph^2)
= 3√2 mph
Now, we can calculate the time it will take for the swimmer to cross the river by dividing the width of the river by the effective velocity:
Time = Distance / Velocity
= 19.89 miles / (3√2 mph)
To simplify, you can approximate √2 as 1.41:
Time ≈ 19.89 miles / (3 × 1.41 mph)
= 19.89 miles / 4.23 mph
≈ 4.704 hours
Therefore, it will take the swimmer approximately 4.704 hours to cross the river.