Suppose that the distance (in miles) is a linear function of his total driving time (in minutes). Eric has 51 miles to his destination after 24 minutes of driving, and he has 39 miles to his destination after 40 minutes of driving. How many miles will he have to his destination after 44 minutes of driving
x = m t + b
d - 51 = m * 24 + b
d - 39 = m * 40 + b
---------------------------- subtract
0 -12 = - 16 m
m = 0.75
d - x = .75 * 44 + b
d - 39 = .75*40 + b
------------------------------- subtract again
39 - x = .75*4
x = 39 - 3 = 36 miles
or just say 4 more minuites at 0.75 miles/ minute after passing 39 :)
To find the linear function that relates the distance to the driving time, we can use the formula for a line given two points.
Let's call the distance (in miles) "d" and the driving time (in minutes) "t".
We are given two points on the line:
(24, 51) - representing 51 miles after 24 minutes of driving
(40, 39) - representing 39 miles after 40 minutes of driving
Using the formula for a line, we can find the equation of the line:
slope (m) = (change in y) / (change in x)
m = (39 - 51) / (40 - 24) = -12 / 16 = -3/4
Now that we have the slope, we can use one of the points to find the y-intercept (b) using the formula:
y = mx + b
Using point (24, 51):
51 = (-3/4)(24) + b
51 = -18 + b
b = 51 + 18 = 69
So the equation of the line relating distance (d) to driving time (t) is:
d = (-3/4)t + 69
Now, let's find out how many miles Eric will have to his destination after 44 minutes of driving:
d = (-3/4)(44) + 69
d = -33 + 69
d = 36
Therefore, after 44 minutes of driving, Eric will have 36 miles to his destination.
To find the answer to this question, we can first set up a linear equation using the given information.
Let's assume that the distance Eric has to his destination after x minutes of driving is represented by the variable D.
We can set up two equations based on the given information:
1) After 24 minutes, Eric has 51 miles to his destination:
D = mx + b, where m is the rate of change (slope) and b is the y-intercept.
51 = m(24) + b ---- Equation 1
2) After 40 minutes, Eric has 39 miles to his destination:
D = mx + b
39 = m(40) + b ---- Equation 2
Now, we have a system of linear equations:
51 = 24m + b
39 = 40m + b
To solve this system of equations, we can use the method of substitution or elimination. In this case, let's use the method of substitution.
1) Solve equation 1 for b:
b = 51 - 24m
2) Substitute the value of b in equation 2:
39 = 40m + (51 - 24m)
Simplify the equation:
39 = 40m + 51 - 24m
Combine like terms:
39 - 51 = 40m - 24m
-12 = 16m
Divide both sides by 16:
-12/16 = m
-3/4 = m
Now, substitute the value of m back into equation 1 to find the value of b:
51 = 24(-3/4) + b
51 = -18 + b
b = 51 + 18
b = 69
Therefore, the equation representing the relationship between distance and time is:
D = (-3/4)x + 69
To find the distance Eric will have to his destination after 44 minutes, we can substitute x = 44 into the equation for D:
D = (-3/4)(44) + 69
Simplify the equation:
D = -33 + 69
D = 36
Therefore, Eric will have 36 miles to his destination after 44 minutes of driving.