Math Problem Statement

Jess is driving to Pittsburgh. Suppose that the distance to her destination (in miles) is a linear function of her total driving time (in minutes). Jess is 49 miles away from her destination after 45 minutes of driving, and she is 37 miles away from her destination after 61 minutes of driving. Find the function and determine how many miles away from her destination she will be after 73 minutes of driving?

Solution

To solve this problem, we need to find the linear function that represents Jess's distance from her destination as a function of time, and then use that function to determine how far she will be from her destination after 73 minutes.

Step 1: Define the linear function.

Let d(t)d(t) be the distance (in miles) from her destination as a function of driving time tt (in minutes). Since it's a linear function, we can express it as:

d(t)=mt+bd(t) = mt + b

where mm is the slope (rate of change in distance per minute) and bb is the y-intercept (initial distance from her destination).

Step 2: Use the given points to create equations.

We know the following:

  • After 45 minutes of driving, Jess is 49 miles away. So, d(45)=49d(45) = 49.
  • After 61 minutes of driving, Jess is 37 miles away. So, d(61)=37d(61) = 37.

These two points allow us to set up the following system of equations:

  1. 49=45m+b49 = 45m + b
  2. 37=61m+b37 = 61m + b

Step 3: Solve for mm and bb.

We start by subtracting the first equation from the second to eliminate bb:

(3749)=(61m+b)(45m+b)(37 - 49) = (61m + b) - (45m + b)

Simplifying:

12=16m-12 = 16m

Solving for mm:

m=1216=34m = \frac{-12}{16} = -\frac{3}{4}

Now, substitute m=34m = -\frac{3}{4} into one of the original equations to find bb. Using the first equation:

49=45(34)+b49 = 45\left(-\frac{3}{4}\right) + b

Simplifying:

49=1354+b49 = -\frac{135}{4} + b

Convert 49 to a fraction with denominator 4:

49=196449 = \frac{196}{4}

Now solve for bb:

1964=1354+b\frac{196}{4} = \frac{-135}{4} + b

b=1964+1354=3314b = \frac{196}{4} + \frac{135}{4} = \frac{331}{4}

So, b=3314=82.75b = \frac{331}{4} = 82.75.

Step 4: Write the linear function.

The linear function for the distance is:

d(t)=34t+82.75d(t) = -\frac{3}{4}t + 82.75

Step 5: Determine the distance after 73 minutes.

To find how many miles Jess is away from her destination after 73 minutes, substitute t=73t = 73 into the function:

d(73)=34(73)+82.75d(73) = -\frac{3}{4}(73) + 82.75

Simplifying:

d(73)=2194+82.75=54.75+82.75=28d(73) = -\frac{219}{4} + 82.75 = -54.75 + 82.75 = 28

So, after 73 minutes of driving, Jess will be 28 miles away from her destination.

Final Answer:

Jess will be 28 miles away from her destination after 73 minutes of driving.


Would you like more details on any step or have any questions?

Here are 5 related questions to explore:

  1. How do we calculate the slope of a linear function from two points?
  2. What does the slope represent in this driving problem?
  3. How would the equation change if Jess started driving closer to her destination?
  4. Can you find out how long it would take for Jess to reach her destination (i.e., when d(t)=0d(t) = 0)?
  5. How would you describe the relationship between distance and time if Jess were driving faster?

Tip: When working with linear functions, remember that the slope mm represents the rate of change in the dependent variable per unit of the independent variable.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Algebra

Formulas

Linear function formula: d(t) = mt + b
Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Slope-intercept form of a linear equation

Suitable Grade Level

Grades 8-10