Two buses are driving along parallel freeways that are 5mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55mph, find the rate at which the distance between the buses is changing when they are 13mi apart, heading toward each other.
I assume highways are east / west and the13 miles is the hypotenuse
13^2 = 5^2 + x^2
55* 2 = 110 =difference in speed east
t in hours
x distance apart = Xi - 110 t
y distance apart = 5
x difference at t = 0 , Xi = sqrt(13^2 - 5^2) = sqrt(169-25) = 12 miles apart east/west at t = 0
s = distance apart = sqrt (x^2 + 5^2) = (x^2+25)^.5
ds/dt = 0.5 (x^2+25)^-.5 * 2 x dx/dt
at t = 0, x = 12 and dx/dt is always 110 mph
ds/dt = 0.5 * 24 *110 mph /(144+25)^.5
= 1320 mph /(13) = 101.53 mph
Well, if the buses are driving towards each other, we can consider their combined speed. So, the total speed would be 55mph + 55mph, which is 110mph. Now, we know that the rate of change of distance is the same as the rate at which they are getting closer to each other. Therefore, the rate at which the distance between the buses is changing would be 110mph. So, they are getting closer at a rate of 110mph. Just make sure you don't get caught in between them!
To find the rate at which the distance between the buses is changing, we can use the concept of related rates. Let's denote the distance traveled by the eastbound bus as x (in miles) and the distance traveled by the westbound bus as y (in miles).
Given that each bus is driving at a constant speed of 55 mph, we can express the rates of change as dx/dt = 55 mph and dy/dt = 55 mph.
The distance between the two buses at any time is given by the equation x + y = 5 miles.
To find the rate at which the distance between the buses is changing (d(x + y)/dt), we need to differentiate this equation with respect to time (t).
Differentiating both sides of the equation with respect to t, we get:
dx/dt + dy/dt = 0
Since dx/dt = 55 mph and dy/dt = 55 mph, we can substitute these values into the equation:
55 + 55 = 0
110 = 0
This equation is not true, which means there is a contradiction. It implies that the assumption of the buses being 5 miles apart and traveling at 55 mph is not possible.
Therefore, there is an error in the given information, and we cannot determine the rate at which the distance between the buses is changing when they are 13 miles apart heading toward each other.
To solve this problem, we need to use the concept of related rates. We are given that the buses are driving along parallel freeways that are 5 miles apart and each bus is driving at a constant speed of 55 mph. We want to find the rate at which the distance between the two buses is changing when they are 13 miles apart, heading towards each other.
Let's start by assigning some variables. Let's say that the distance traveled by the east-bound bus is x(t) and the distance traveled by the west-bound bus is y(t), where t represents time. Since the buses are driving towards each other, the total distance between them can be represented as x(t) + y(t), and we want to find the rate at which this distance is changing, d/dt (x(t) + y(t)), when they are 13 miles apart.
We know that the east-bound bus is driving at a speed of 55 mph, so dx/dt = 55. Similarly, the west-bound bus is also driving at a speed of 55 mph, so dy/dt = 55.
Now, we need to find an equation that relates x(t) and y(t). Since the buses are driving along parallel freeways that are 5 miles apart, we can write the equation x(t) - y(t) = 5.
To find the rate at which the distance between the buses is changing, we take the derivative of this equation with respect to time:
d/dt (x(t) - y(t)) = d/dt (5).
Using the chain rule, we get:
dx/dt - dy/dt = 0.
Substituting the known values, we have:
55 - 55 = 0.
Therefore, dx/dt - dy/dt = 0.
Now, we can rewrite the equation x(t) + y(t) = 13 as x(t) = 13 - y(t).
Taking the derivative of this equation with respect to time, we get:
dx/dt = d/dt (13 - y(t)).
Using the chain rule again, we have:
dx/dt = -dy/dt.
Substituting the known values, we have:
dx/dt = -55 mph.
Finally, we can substitute the values of dx/dt and dy/dt into the equation dx/dt - dy/dt = 0 to find the rate at which the distance between the buses is changing when they are 13 miles apart:
-55 - 55 = -2dy/dt.
Simplifying, we get:
-110 = -2dy/dt.
Dividing by -2, we have:
dy/dt = 55 mph.
Therefore, the rate at which the distance between the buses is changing when they are 13 miles apart, heading toward each other, is 55 mph.