Answer :
We can see here that the distance between the truck and Uniontown is changing at a rate of approximately 69.61 miles per hour.
How we arrived at the solution?
To find the rate at which the distance between the truck and Uniontown is changing, we can use the concept of related rates.
Let's break it down step by step:
1. The truck starts off 242 miles directly west from Uniontown. This means that the initial distance between the truck and Uniontown is 242 miles.
2. The truck travels due north at a speed of 51 miles per hour. This tells us that the truck is moving away from Uniontown. We can consider this as the rate at which the truck is moving in the y-direction.
3. After traveling 193 miles, we want to find how fast the distance between the truck and Uniontown is changing. To do this, we can use the Pythagorean theorem.
The distance between the truck and Uniontown is given by the formula:
Distance = √((distance in x-direction)² + (distance in y-direction)²)
We know the distance in the x-direction is still 242 miles, and the distance in the y-direction is 193 miles (as the truck traveled that distance north).
4. To find how fast the distance is changing, we can take the derivative of the distance formula with respect to time. This gives us:
d(Distance)/dt = (1/2)×(1/√((distance in x-direction)² + (distance in y-direction)²)) × (2×(distance in x-direction)×(dx/dt) + 2×(distance in y-direction)×(dy/dt))
Here, dx/dt represents the rate of change of the x-direction (which is 0 as the truck is not moving east or west) and dy/dt represents the rate of change of the y-direction (which is given as 51 miles per hour).
5. Substituting the known values, we have:
d(Distance)/dt = (1/2)×(1/√((242)² + (193)²)) × (2×(242)×(0) + 2×(193)×(51))
= (1/2)×(1/√(58513)) × 0 + (2×193×51)/(2×√(58513))
= 9837/(2×√(58513))
6. Evaluating this expression, we get:
d(Distance)/dt ≈ 69.61 miles per hour
Therefore, the distance between the truck and Uniontown is changing at a rate of approximately 69.61 miles per hour.
Learn more about distance on https://brainly.com/question/26550516
#SPJ4
