Final answer:
To find the rate at which the car and truck are separating at 1:00 p.m., we need to determine their velocities at that time and calculate the distance between them. The car has traveled 90 miles and the truck has traveled 80 miles. Using the Pythagorean theorem, the distance between them is found to be 380 miles.
Step-by-step explanation:
To find the rate at which the car and truck are separating at 1:00 p.m., we need to determine their velocities at that time. From the given information, we know that the car is traveling east at a constant speed of 30 miles per hour and the truck is traveling north at a constant speed of 40 miles per hour. Since they started at different times, we need to calculate the distances they have traveled when it is 1:00 p.m.
Since the car has been traveling for 3 hours (from 10:00 a.m. to 1:00 p.m.), it has traveled a distance of 30 miles per hour * 3 hours = 90 miles. On the other hand, the truck has been traveling for 2 hours (from 11:00 a.m. to 1:00 p.m.), so it has traveled a distance of 40 miles per hour * 2 hours = 80 miles.
To find the separation rate, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the distance between the car and truck forms the hypotenuse, and the distances traveled by each vehicle form the other two sides.
Distance between car and truck = sqrt((90 miles)^2 + (80 miles)^2) = sqrt(8100 miles^2 + 6400 miles^2) = sqrt(144500 miles^2) = 380 miles.
Therefore, the rate at which the car and truck are separating at 1:00 p.m. is 380 miles.