A car is heading east toward an intersection at the rate of 40 mph. A truck is heading south, away from the same intersection at the rate of 60 mph. At what rate is the distance between the car and the truck changing when the car is 8 miles from the intersection and the truck is 15 from the intersection?
let's look at t hrs after the above condition
let the distance between them be D miles
then
D^2 = (40t+8)^2 + (15+60t)^2
2D dD/dt = 2(40t+8)(40) + 2(15+60t)(60)
dD/dt = [ 40(40t+8) + 60(15+60t) ]/D
when t = 0 , the above condition,
D = √8^2+15^2) = 17
and
dD/dt = [ 40(8) + 60(15) ]/17 = 71.76
At that moment they are separating at 71.8 mph
To solve this problem, we can use the concept of related rates. Let's assume that the car is moving along the x-axis and the truck is moving along the y-axis.
Let's define:
- x = distance covered by the car from the intersection (measured along the x-axis)
- y = distance covered by the truck from the intersection (measured along the y-axis)
- r = distance between the car and the truck
We are given that the car is heading east at a speed of 40 mph, so dx/dt = 40 mph. The truck is heading south at a speed of 60 mph, so dy/dt = -60 mph (negative because the truck is moving away).
We need to find dr/dt, the rate at which the distance between the car and the truck is changing.
From the Pythagorean theorem, we have:
r^2 = x^2 + y^2
Differentiating both sides of this equation implicitly with respect to time t, we get:
2r dr/dt = 2x dx/dt + 2y dy/dt
Simplifying, we have:
dr/dt = (x dx/dt + y dy/dt) / r
Now let's substitute the given values:
- x = 8 miles (car is 8 miles from the intersection)
- y = 15 miles (truck is 15 miles from the intersection)
- dx/dt = 40 mph (car's speed)
- dy/dt = -60 mph (truck's speed)
Plugging in these values into the equation for dr/dt, we have:
dr/dt = (8 * 40 + 15 * -60) / r
Now we need to find the value of r, which is the distance between the car and the truck. Using the Pythagorean theorem:
r^2 = x^2 + y^2
r^2 = 8^2 + 15^2
r^2 = 64 + 225
r^2 = 289
r = 17
Substituting r = 17 into the equation for dr/dt:
dr/dt = (8 * 40 + 15 * -60) / 17
dr/dt = (320 - 900) / 17
dr/dt = -580 / 17
dr/dt ≈ -34.12 mph
Therefore, the distance between the car and the truck is changing at a rate of approximately -34.12 mph when the car is 8 miles from the intersection and the truck is 15 miles from the intersection. Note that the negative sign indicates that the distance is decreasing.
To find the rate at which the distance between the car and the truck is changing, we need to use the concept of related rates. We can consider the car and the truck as two points moving towards or away from the same intersection.
Let's define the following variables:
- x: represents the horizontal distance of the car from the intersection (measured eastwards).
- y: represents the vertical distance of the truck from the intersection (measured southwards).
- d: represents the distance between the car and the truck.
We are given the following information:
- dx/dt = 40 mph (rate at which the car is moving eastwards)
- dy/dt = -60 mph (rate at which the truck is moving southwards)
We want to find the rate at which the distance, d, between the car and the truck is changing when x = 8 miles and y = 15 miles.
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (d) is equal to the sum of the squares of the other two sides (x and y):
d^2 = x^2 + y^2
Differentiating both sides of this equation with respect to time (t) gives:
2d * dd/dt = 2x * dx/dt + 2y * dy/dt
Since we're interested in finding dd/dt (the rate at which d is changing), we can rearrange the equation to solve for it:
dd/dt = (x * dx/dt + y * dy/dt) / d
Substituting the given values:
x = 8 miles, dx/dt = 40 mph (eastwards)
y = 15 miles, dy/dt = -60 mph (southwards)
d = sqrt(x^2 + y^2) = sqrt(8^2 + 15^2) = sqrt(289) = 17 miles
Now we can substitute the values into the equation to find dd/dt:
dd/dt = (8 * 40 + 15 * -60) / 17
Simplifying further:
dd/dt = (320 - 900) / 17
dd/dt = -580 / 17
Therefore, the rate at which the distance between the car and the truck is changing when the car is 8 miles from the intersection and the truck is 15 miles from the intersection is approximately -34.12 mph.