Math Problem Statement

A motorboat moves across a lake at a constant speed. When it begins, it is 64 km from the shore. After 17 minutes, it is 30 km from the shore. Which function describes the motorboat's distance from the shore?

Solution

The question asks to find a function that describes the motorboat's distance from the shore based on the given information:

  • Initially, the motorboat is 64 km from the shore.
  • After 17 minutes, the motorboat is 30 km from the shore.

This setup implies a linear relationship where the motorboat is moving toward the shore, decreasing its distance from the shore over time.

Steps to solve:

  1. Slope (rate of change): The motorboat’s speed is constant, meaning the rate of change in distance is linear. We can calculate the slope using the points:

    • Initial distance: (0, 64) at time 0 minutes.
    • After 17 minutes: (17, 30).

    The formula for the slope mm is: m=y2y1x2x1=3064170=3417=2m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 64}{17 - 0} = \frac{-34}{17} = -2

  2. Equation of the line: The general form of a linear equation is: y=mx+by = mx + b Where:

    • m=2m = -2 (the slope),
    • b=64b = 64 (the initial distance from the shore, i.e., the y-intercept when x=0x = 0).

    So, the equation becomes: y=2x+64y = -2x + 64

Conclusion:

The correct function is Option C: y=2x+64y = -2x + 64.

Would you like a more detailed explanation or have any other questions?

Here are five related questions you might find useful:

  1. How do you determine the slope of a line when given two points?
  2. What is the physical meaning of the slope in this context?
  3. How do you interpret the y-intercept in real-world linear functions?
  4. Can a motorboat ever have a positive slope in this scenario?
  5. How can you verify a linear function using additional data points?

Tip: Always check the units when interpreting real-world linear equations, especially time and distance!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Linear equation: y = mx + b

Theorems

Linear function properties

Suitable Grade Level

Grade 8