A car is traveling west at 67 mi./hr and a truck is traveling south at 59 mi./hr. Both are headed for the intersection of the two roads. At what rate are the car and truck approaching each other when the car is 0.2 mi. and the truck is 0.5 mi. from the intersection?
i have a similar problem like that but dont quite get whether dx/dt= 67mi./hr or 59mi./hr
well, usually x is east-west and y is north-south, dontcha think?
and both are negative, since they are approaching (0,0).
dx/dt = -67
dy/dt = -59
The distance z is
z^2 = x^2+y^2
2z dz/dt = 2x dx/dt + 2y dy/dt
now just plug in z at the desired moment, and then the rest of your numbers.
To find the rate at which the car and truck are approaching each other, we can use the concept of relative velocity.
Let's first visualize the scenario. We have a car traveling west and a truck traveling south. The intersection point is the point where the car and truck will meet.
To solve the problem, we need to break down the respective velocities of the car and truck into their horizontal and vertical components.
Given:
Car's velocity (west) = 67 mi./hr
Truck's velocity (south) = 59 mi./hr
We can represent the car's velocity as (-67, 0) mi./hr, indicating that it is moving in the negative x-direction (west) with no vertical component.
Similarly, the truck's velocity can be represented as (0, -59) mi./hr, indicating no horizontal component and moving in the negative y-direction (south).
Now, let's find the distance between the car and the truck at any given time t using the Pythagorean theorem:
Distance between the car and the truck = √[(0.2 - (-0.5))^2 + (0)^2]
= √[(0.7)^2]
= 0.7 mi.
Next, let's differentiate the distance with respect to time t to find the rate at which the distance is changing:
d(distance)/dt = d(√[(0.2 - (-0.5))^2 + (0)^2])/dt
Using the chain rule, we get:
d(distance)/dt = (1/2)(0.7)^(-1/2) × d(0.7 - t)/dt
d(distance)/dt = (1/2)(0.7)^(-1/2) × (-1)
Simplifying further:
d(distance)/dt = -0.707 mi./hr
Therefore, the car and truck are approaching each other at a rate of 0.707 mi./hr when the car is 0.2 mi. and the truck is 0.5 mi. from the intersection.