Required information
Sheena can row a boat at $3.20 \mathrm{mih}$ in still water. She needs to cross a river that is 1.20 mi wide with a current flowing at $1.70 \mathrm{mi} / \mathrm{h}$. Not having her calculator ready, she guesses that to go straight across, she should head upstream at an angle of $25.0^{\circ}$ from the direction straight across the river.

How far upstream or downstream from her starting point will she reach the opposite bank? If upstream, enter a positive value and if downstream, enter a negative value.
mi

Step 1 ：First, calculate the resultant velocity of the boat using the Pythagorean theorem to add the velocity of the boat in still water and the velocity of the current. The angle between these two velocities is 90 degrees, so the Pythagorean theorem can be used directly. The formula is \(v_{resultant} = \sqrt{v_{boat}^2 + v_{current}^2}\).

Step 2 ：Next, calculate the time it takes for Sheena to cross the river. This can be done by dividing the width of the river by the component of the resultant velocity that is perpendicular to the river. The formula is \(time = \frac{width}{v_{resultant} \cdot \cos(angle)}\).

Step 3 ：Finally, calculate the distance upstream or downstream from her starting point. This can be done by multiplying the time it takes for her to cross the river by the component of the resultant velocity that is parallel to the river. The formula is \(distance = time \cdot v_{resultant} \cdot \sin(angle)\).

Step 4 ：However, the question asks for the distance downstream if she heads upstream at an angle of 25 degrees. This means that the direction of the current is opposite to the direction she is heading. Therefore, the distance should be negative.

Step 5 ：Final Answer: The distance downstream from her starting point that Sheena will reach the opposite bank is \(\boxed{-2.57}\) miles.