4. A gas station stands at the intersection of a north-south road and an east-west road. A police car is traveling towards the gas station from the east, chasing a stolen truck which is traveling north away from the gas station. The speed of the police car is 100 mph when it is 3 miles from the gas station. At the same time the truck is 4 miles from the gas station going 80 mph. At this moment:
a. Is the distance between the car and the truck increasing or decreasing? How fast?
b. Repeat part a) if the truck is going 70 mph instead of 80 mph
To answer these questions, we need to determine the rates at which the distance between the car and the truck is changing.
Let's start by defining some variables:
- Let t be the time in hours since the car and truck began moving.
- Let d(t) be the distance between the car and the gas station at time t.
- Let x(t) be the distance between the truck and the gas station at time t.
We know the following information:
- The speed of the car is 100 mph.
- The car is 3 miles from the gas station when it started chasing the truck.
- The speed of the truck is initially 80 mph.
- The truck is 4 miles from the gas station when it started being chased by the car.
Using this information, we can set up equations to describe the motion of the car and the truck.
For the car:
- d(t) = 3 + 100t
- The distance between the car and the truck is x(t) - d(t).
For the truck:
- x(t) = 4 - 80t
Now, let's calculate the rate at which the distance between the car and the truck is changing.
a) To determine whether the distance between the car and the truck is increasing or decreasing, we need to find the derivative of the difference between the two distances with respect to time (t).
Differentiating the equation x(t) - d(t) gives us:
- (d/dt)(x(t) - d(t)) = (d/dt)(4 - 80t - (3 + 100t))
Simplifying, we have:
- (d/dt)(x(t) - d(t)) = -80 - 100 = -180
Since the derivative is negative (-180), the distance between the car and the truck is decreasing.
To find the rate at which the distance is changing, we take the absolute value of the derivative:
- |(d/dt)(x(t) - d(t))| = |-180| = 180 mph.
Therefore, the distance between the car and the truck is decreasing at a rate of 180 mph.
b) If the truck is going 70 mph instead of 80 mph, we need to repeat the calculation.
For the truck:
- x(t) = 4 - 70t
Differentiating the equation x(t) - d(t) gives us:
- (d/dt)(x(t) - d(t)) = (d/dt)(4 - 70t - (3 + 100t))
Simplifying, we have:
- (d/dt)(x(t) - d(t)) = -70 - 100 = -170
Again, the derivative is negative (-170), indicating that the distance between the car and the truck is decreasing.
Taking the absolute value of the derivative:
- |(d/dt)(x(t) - d(t))| = |-170| = 170 mph.
Therefore, if the truck is going 70 mph instead of 80 mph, the distance between the car and the truck is decreasing at a rate of 170 mph.