Question

a boat is traveling east across a river that is 112 meters wide at 8 meters per second. if the river has a northward current of 5 meters per second, what is the resultant speed of the motorboat rounded to the nearest tenth?
3.0 m/s
8.6 m/s
9.4 m/s
13.0 m/s

Explanation:

Step1: Identify the velocities as perpendicular vectors

The boat's velocity east is \( v_{east} = 8\space m/s \) and the river's current velocity north is \( v_{north} = 5\space m/s \). These two velocities are perpendicular to each other (east and north are perpendicular directions).

Step2: Use the Pythagorean theorem to find the resultant speed

The resultant speed \( v \) of two perpendicular vectors is given by the Pythagorean theorem: \( v=\sqrt{v_{east}^{2}+v_{north}^{2}} \).

Substitute \( v_{east} = 8 \) and \( v_{north}=5 \) into the formula:
\[

$$\begin{align*} v&=\sqrt{8^{2}+5^{2}}\\ &=\sqrt{64 + 25}\\ &=\sqrt{89} \end{align*}$$

\]

Step3: Calculate the numerical value and round to the nearest tenth

Calculate \( \sqrt{89} \approx 9.43398 \). Rounding to the nearest tenth, we look at the hundredth place. The digit in the hundredth place is 3, which is less than 5, so we round down. Thus, \( \sqrt{89}\approx9.4 \).

Answer:

9.4 m/s