Tony is driving to Denver. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes), as shown in the figure below. Tony has 67 miles to his destination after 35 minutes of driving, and he has 55.6 miles to his destination after 47 minutes of driving. How many miles will he have to his destination after 61 minutes of driving
To find the number of miles Tony will have to his destination after 61 minutes of driving, we need to determine the linear function that represents the relationship between the distance and the driving time.
Let's use the given information to form an equation for this linear function.
Step 1: Identify the variables:
Let "d" be the distance to the destination (in miles), and "t" be the driving time (in minutes).
Step 2: Write down the information given:
We know that Tony has 67 miles to his destination after 35 minutes of driving, and he has 55.6 miles to his destination after 47 minutes of driving.
When t = 35 minutes, d = 67 miles.
When t = 47 minutes, d = 55.6 miles.
Step 3: Write the equation for the linear function:
To find the equation of a line, we can use the formula: y = mx + b.
Let's rewrite the equation in terms of our variables:
d = mt + b, where d represents the distance, t represents the time, m represents the slope, and b represents the y-intercept.
Step 4: Find the slope:
We can find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1).
Using the given information:
x1 = 35 minutes
y1 = 67 miles
x2 = 47 minutes
y2 = 55.6 miles
m = (55.6 - 67) / (47 - 35) = -11.4 / 12 = -0.95
So, the slope of the line is -0.95.
Step 5: Find the y-intercept:
To find the y-intercept (b), we can substitute one of the given points into the equation.
Using the point (35, 67):
d = mt + b
67 = -0.95(35) + b
67 = -33.25 + b
b = 67 + 33.25 = 100.25
So, the y-intercept of the line is 100.25.
Step 6: Write the final equation:
Using the slope and y-intercept values we found, we can write the equation for the linear function:
d = -0.95t + 100.25
Step 7: Find the distance after 61 minutes:
We can now use this equation to find the distance (d) after 61 minutes (t = 61).
d = -0.95(61) + 100.25
d = -57.95 + 100.25
d = 42.3
Therefore, Tony will have approximately 42.3 miles to his destination after 61 minutes of driving.