Austin is driving to Denver. Suppose that the remaining distance to drive (in miles) is a linear function of his driving time (in minutes). When graphed, the function gives a line with a slope of -0.85 . See the figure below.
Austin has 52 miles remaining after 41 minutes of driving. How many miles will be remaining after minutes of driving?
We are given that the slope of the linear function is -0.85. This means that for every one minute of driving time, the distance remaining decreases by 0.85 miles.
We can set up a proportion to find the number of miles remaining after a certain number of minutes of driving:
(41 minutes) / (52 miles remaining) = (x minutes) / (miles remaining)
Cross-multiplying gives us:
41 * (miles remaining) = 52 * (x minutes)
Simplifying further,
(miles remaining) = (52 * x minutes) / 41
Substituting in the given value of 41 minutes for x,
(miles remaining) = (52 * 41 minutes) / 41
(miles remaining) = 52
Therefore, after 41 minutes of driving, there are 52 miles remaining.
To find the number of miles remaining after a different number of minutes of driving, we can substitute that value into the equation. For example, if we want to find the number of miles remaining after 60 minutes of driving:
(miles remaining) = (52 * 60 minutes) / 41
(miles remaining) = 75.60975609756098
Therefore, after 60 minutes of driving, there will be approximately 75.6 miles remaining.
To find the number of miles remaining after a certain number of minutes of driving, we can use the equation of a line in slope-intercept form: y = mx + b, where y is the number of miles remaining, x is the number of minutes of driving, m is the slope of the line, and b is the y-intercept.
From the given information, we know that the slope of the line is -0.85. The y-intercept can be found by substituting the values of x and y from one of the given points, in this case, (41, 52):
52 = (-0.85)(41) + b
52 = -34.85 + b
b = 52 + 34.85
b = 86.85
So the equation of the line is:
y = -0.85x + 86.85
To find the number of miles remaining after a certain number of minutes, we substitute the value of x into the equation and solve for y. In this case, the number of minutes is not given, so we'll call it "m":
y = -0.85m + 86.85