A man who can row in still water at 3 mph heads directly across a river flowing at 7 mph.

At what angle to the line on which he is heading does he drift downstream and at what speed does he drift downstream?

He drifts downstream at the river's speed, 7 mph. He has a 3 mph component across the river.

His direction with respect to the heading angle (straight across) is
tan^-1 7/3 = 66.8 degrees (towards downstream)

To find the angle at which the man drifts downstream, we need to use trigonometry. Here's how you can calculate it:

Step 1: Determine the speed at which the man is moving with respect to the ground.
The man's overall speed across the river will be the vector sum of his own rowing speed and the speed of the river current. Given that the man can row at 3 mph and the river is flowing at 7 mph, his total speed can be calculated using the Pythagorean theorem:

Speed = √(man's rowing speed^2 + river current speed^2)
= √(3^2 + 7^2)
= √(9 + 49)
= √58
≈ 7.62 mph

So, the man's total speed across the river is approximately 7.62 mph.

Step 2: Determine the speed at which the man drifts downstream.
Since the man is rowing directly across the river, the angle between his heading and the current direction is 90 degrees. In this case, the speed at which he drifts downstream can be calculated using trigonometry:

Speed downstream = man's total speed × sin(angle between heading and current direction)

In this scenario, the angle is 90 degrees, so sin(90) = 1. Therefore,

Speed downstream = 7.62 mph × sin(90)
= 7.62 mph × 1
= 7.62 mph

So, the man drifts downstream at a speed of 7.62 mph.

Step 3: Determine the angle at which the man drifts downstream.
Using trigonometry, we can calculate the angle at which the man drifts downstream using:

Angle downstream = sin^(-1)(speed downstream / man's total speed)

Substituting in the values we found earlier:

Angle downstream = sin^(-1)(7.62 mph / 7.62 mph)
= sin^(-1)(1)
= 90 degrees

Therefore, the man drifts downstream at an angle of 90 degrees to the line on which he is heading.

In summary, the man drifts downstream at a speed of 7.62 mph and at an angle of 90 degrees relative to the line on which he is heading.