Final answer:
To determine how fast the distance between two trucks is changing, we can use the Pythagorean Theorem and calculus. By setting up the appropriate equations and applying differentiation, we can calculate the rate at which the distance between the trucks is changing at a specific time.
Step-by-step explanation:
The student's question requires the application of the Pythagorean Theorem and calculus to find the rate at which the distance between two vehicles is changing. At 12:00 PM, the FedEx truck is 100 miles east of the UPS truck. The FedEx truck is traveling west at 55 mph, and the UPS truck is traveling north at 75 mph. To solve this problem, we can let x represent the distance of the FedEx truck from the starting point, and y represent the distance of the UPS truck from its starting point. Then, the distance between the trucks, z, can be represented by the equation z² = x² + y².
By 2:00 PM, each truck had been traveling for 2 hours. Thus, the FedEx truck has traveled 110 miles (55 mph * 2 hours), and the UPS truck has traveled 150 miles (75 mph * 2 hours). The rate of change of x with respect to time is dx/dt = -55 mph, and the rate of change of y with respect to time is dy/dt = 75 mph.
Using implicit differentiation on the equation for z, we get 2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt), which simplifies to dz/dt = (x(dx/dt) + y(dy/dt)) / z. Substituting the values for x, y, dx/dt, and dy/dt, we can solve for dz/dt to find the rate at which the distance between the two trucks is changing at 2:00 PM.