Boris is driving to Memphis. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Boris has 58 miles to his destination after
21 minutes of driving, and he has
42.6 miles to his destination after
43 minutes of driving. How many miles will he have to his destination after 59 minutes of driving?
y = mx+b
21m+b = 58
43m+b = 42.6
Solve for m and b, and then figure y when x=59.
Or, note that 22 minutes of driving covered 15.4 miles.
So, after 16 more minutes, he covers (16/22)(15.4) more miles. Subtract that from 42.6
To find the distance Boris will have to his destination after 59 minutes of driving, we need to determine the equation of the linear function.
Let's denote the total driving time in minutes with "t" and the distance to the destination in miles with "d".
First, let's find the slope (m) of the linear function. The slope represents the change in distance per change in time.
m = (d2 - d1) / (t2 - t1)
Given data points:
(d1, t1) = (58 miles, 21 minutes)
(d2, t2) = (42.6 miles, 43 minutes)
m = (42.6 - 58) / (43 - 21)
m = -15.4 / 22
m ≈ -0.7 (rounded to one decimal place)
Next, we can use the slope-intercept form of a linear equation to find the y-intercept (b). The equation is given by:
d = mt + b
Using the point (58, 21):
58 = -0.7 * 21 + b
58 = -14.7 + b
b = 58 + 14.7
b ≈ 72.7 (rounded to one decimal place)
Thus, the equation representing the distance Boris will have to his destination (d) after t minutes of driving is:
d = -0.7t + 72.7
Now, we can substitute t = 59 into the equation to find the distance after 59 minutes:
d = -0.7 * 59 + 72.7
d ≈ -41.3 + 72.7
d ≈ 31.4
Therefore, Boris will have approximately 31.4 miles to his destination after 59 minutes of driving.
To solve this problem, we can use the concept of a linear function. Let's start by finding the rate at which Boris is driving.
We are given that after 21 minutes of driving, Boris has 58 miles left to his destination. This gives us one point on the line.
We can express this information as the coordinates (21, 58), where 21 represents the time in minutes and 58 represents the distance in miles.
Similarly, after 43 minutes of driving, Boris has 42.6 miles left to his destination. This gives us another point on the line, (43, 42.6).
Now, using the two points, we can calculate the slope of the line. The formula for slope is:
Slope = (change in y)/(change in x)
In this case, the change in y is the change in distance and the change in x is the change in time:
Slope = (42.6 - 58) / (43 - 21)
Slope = (-15.4) / (22)
Slope ≈ -0.7
Now that we have the slope, we can use it to find the equation of the line. The equation of a line in slope-intercept form is:
y = mx + b
Where m is the slope and b is the y-intercept.
Using the slope (-0.7) and one of the points, let's substitute the values into the equation:
58 = (-0.7)(21) + b
Simplifying:
58 = - 14.7 + b
Adding 14.7 to both sides:
72.7 = b
So, the equation of the line representing Boris's distance to his destination is:
y = -0.7x + 72.7
Now, we can find the distance after 59 minutes:
y = -0.7(59) + 72.7
Simplifying:
y = -41.3 + 72.7
y ≈ 31.4
Therefore, Boris will have approximately 31.4 miles left to his destination after 59 minutes of driving.