To determine the function that describes the motorboat's distance from the shore, let's go through the problem step by step:
1. Initial Position:
At the start, the motorboat is 64 km from the shore. This is our [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]).
2. Distance After Time:
After 17 minutes, the boat is 30 km from the shore.
3. Time Elapsed:
The time taken to reach this new position is 17 minutes.
4. Speed Calculation:
We can determine the motorboat's speed using the formula:
[tex]\[
\text{Speed} = \frac{\text{Distance Travelled}}{\text{Time Taken}}
\][/tex]
The distance travelled can be calculated as the difference between the initial distance and the distance after 17 minutes:
[tex]\[
\text{Distance Travelled} = 64 \, \text{km} - 30 \, \text{km} = 34 \, \text{km}
\][/tex]
Therefore, the speed is:
[tex]\[
\text{Speed} = \frac{34 \, \text{km}}{17 \, \text{minutes}} = 2 \, \text{km/min}
\][/tex]
5. Direction of Movement:
Since the boat is moving towards the shore, this speed is considered in the negative direction when developing our linear equation. Hence, the slope ([tex]\( m \)[/tex]) is:
[tex]\[
m = -2 \, \text{km/min}
\][/tex]
6. Linear Function Formulation:
The general form of the linear function describing the motorboat's distance from the shore, [tex]\( y \)[/tex], as a function of time in minutes, [tex]\( x \)[/tex], is:
[tex]\[
y = mx + b
\][/tex]
Substituting the slope [tex]\( m \)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
[tex]\[
y = -2x + 64
\][/tex]
So, the function that describes the motorboat's distance from the shore is:
[tex]\[
\boxed{y = -2x + 64}
\][/tex]
Therefore, the correct answer is:
B. [tex]\( y = -2x + 64 \)[/tex]
