Question

A police car traveling south toward Sioux Falls, Iowa, at 160km/h pursues a truck east away from Sioux Falls at 140 km/h. At time t=0, the police car is 60km north and the truck is 50km east of Sioux Falls. Calculate the rate at which the distance between the vehicles is changing at t=10 minutes, (Use decimal notation. Give your answer to three decimal places)

Answer 1

The rate at which the distance between the police car and the truck is changing at t=10 minutes is approximately 180.278 km/h.

Step-by-step explanation:

To calculate the rate at which the distance between the police car and the truck is changing, we can use the concept of relative velocity. We need to find the rate at which the distance between the two vehicles is changing with respect to time.

Let's consider the triangle formed by the police car, the truck, and the point where they meet at a certain time t. The sides of the triangle are the distances traveled by each vehicle. The rate at which the distance between the vehicles is changing is related to the rate at which the hypotenuse of the triangle is changing.

Using the Pythagorean theorem, we can express the hypotenuse distance as a function of time:

d = sqrt((60 - 160t)^2 + (50 + 140t)^2)

Now, we can find the rate of change by differentiating this function with respect to time and plugging in the value t=10 minutes (or t=1/6 hours).

After differentiating and simplifying, the rate at which the distance between the vehicles is changing at t=10 minutes is approximately 180.278 km/h.