Final answer:
To meet the businesswoman driving at an average speed of 70 mph over a distance of 420 miles, the plane must fly at a minimum of 50 mph over the straight-line distance of 300 miles. Given the answer choices, the plane must at least fly at 70 mph to meet at La Crosse at the same time.
Step-by-step explanation:
The question asks us to determine how fast a plane needs to fly in order to meet a businesswoman traveling by car at the same time in La Crosse. The businesswoman drives 180 miles east and then 240 miles north, totaling a distance of 420 miles (since 180 + 240 = 420). She travels this distance at an average speed of 70 mph.
To calculate the time it takes for her to reach La Crosse, we use the formula:
Time (in hours) = Distance (in miles) / Speed (in mph).
Time = 420 miles / 70 mph = 6 hours.
In order for the partner to meet the businesswoman at the same time, the partner's plane must cover the distance between Des Moines and La Crosse directly. Assuming the plane takes a straight flight path, we can calculate the straight-line distance using the Pythagorean theorem:
Distance = √(1802 + 2402) miles.
Distance = √(32400 + 57600) miles.
Distance = √90000 miles.
Distance = 300 miles.
Now that we have the distance, we need to calculate the speed required for the plane to travel 300 miles in 6 hours:
Speed (in mph) = Distance (in miles) / Time (in hours).
Speed = 300 miles / 6 hours = 50 mph.
This calculation shows that for the plane to meet at La Crosse at the same time as the businesswoman, it must fly at a minimum speed of 50 mph. However, this speed is not listed in the answer options, hence the actual flight speed must be the lowest available option from the choices provided, which is 70 mph (option a).