Related Rates: Two cars approach the intersection of two highways. The one coming from the west is traveling at 60mph and the one traveling from north is traveling at 50 mph. At what rate is the distance between the cars changing when the car from the west is 3 miles from the intersection and the car from the north is 4 miles from the intersection?
no ideas at all? Maybe the Pythagorean Theorem? The distance z is
z^2 = x^2+y^2
2z dz/dt = 2x dx/dt + 2y dy/dt
or
z dz/dt = x dx/dt + y dy/dt
Remembering your good old 3-4-5 right triangle, you can see that z=5 when x=3 and y=4.
Now, you have all the numbers you need to find dz/dt.
No I don't have any idea. And I had started out with the 3-4-5 triangle but didn't know what to do besides that
To find the rate at which the distance between the cars is changing, we need to use the concept of related rates and apply the Pythagorean theorem. The distance between the cars can be represented as the hypotenuse of a right triangle, with the car from the west representing one leg and the car from the north representing the other leg.
Let's denote the distance between the car from the west and the intersection as x (in miles), and the distance between the car from the north and the intersection as y (in miles). The distance between the two cars is given by the equation:
d^2 = x^2 + y^2
Taking the derivative of both sides with respect to time (t), we have:
2d * dd/dt = 2x * dx/dt + 2y * dy/dt
Since we are interested in finding the rate at which the distance between the cars is changing (dd/dt), we need to solve for that. We are given that the car from the west is 3 miles from the intersection (x = 3) and the car from the north is 4 miles from the intersection (y = 4).
Let's differentiate the equation:
dd/dt = (2x * dx/dt + 2y * dy/dt) / (2d)
Substituting the given values:
dd/dt = (2(3)(dx/dt) + 2(4)(dy/dt)) / (2d)
Simplifying further:
dd/dt = (3(dx/dt) + 4(dy/dt)) / d
To find the values of dx/dt and dy/dt, we need to differentiate the x and y equations with respect to time.
Given that the car from the west is traveling at 60mph, dx/dt = 60 mph.
Given that the car from the north is traveling at 50mph, dy/dt = 50 mph.
Substituting these values into the equation for dd/dt:
dd/dt = (3(60) + 4(50)) / d
Now we just need to find the value of d when x = 3 and y = 4. Using the Pythagorean theorem, we have:
d^2 = 3^2 + 4^2
Simplifying,
d^2 = 9 + 16 = 25
Taking the square root of both sides,
d = 5
Substituting this value into the equation for dd/dt:
dd/dt = (3(60) + 4(50)) / 5
Simplifying,
dd/dt = (180 + 200) / 5
dd/dt = 380 / 5
dd/dt = 76 mph
Therefore, the distance between the cars is changing at a rate of 76 mph when the car from the west is 3 miles from the intersection and the car from the north is 4 miles from the intersection.