The solution to this problem involves determining when the distance between the car and boat is minimized. This occurs when the derivative of the distance equation equals zero.
1. Represent the variables:
The speed of the river is 7.5 m/s, the speed of the car is 10.2 m/s, and the distance to the bridge is 530 m.
2. Define the unknown factor:
Let z represent the time difference when the boat and the car are at the closest distance.
3. Apply the Pythagorean Theorem to derive the distance between the car and the boat as they are moving at right angles to each other:
The formula is D = sqrt((river_speed * z)^2 + (distance - car_speed * z)^2).
4. Differentiate the distance formula to find the minimum:
The derivative of the distance formula with respect to time 'z' gives an equation that helps to find the minimal distance.
5. Solve the equation for the derived function:
Set the derivative of the distance formula equal to zero and solve the resulting equation for z.
By solving the equation, we find that the time difference 'z' when the boat and the car are at the closest distance is roughly 33.73 seconds. This means that the boat will be closest to the car approximately 33.73 seconds after the boat approaches the bridge and the car starts crossing it.