Atrain is heading due west from st. louis. at noon, a plane flying horizontally due north at a fixed altitude of 4 miles passes directly over the train. 1 hour later the train had traveled 1 mile and is going 80 miles per hour, while the plane traveled 5 miles and is going 500 miles per hour. at 1pm how fast is the distance between the train and plane increasing?

505mph

Explanation:

Let Dh = Distance between Train & Airplane at horizontal level (ground)

At initial Stage

Dh² = (1)² + (5)²

Dh² = 26

Dh = 5.09 miles

Let Dhv = vertical distance between Plane & Train

Dhv² = 5.09² + 4²

Dhv² = 26 + 16

Dhv = 6.48 miles

After an hour:

Dh1 = Distance between Train & Airplane at horizontal level (ground) after 1 hour

Dh1² = [80 m/h (1 hr) + 1miles]² + [500 m/h(1 hr) + 5miles]²

Dh1² = (81)²+ (505)²

Dh1 = √(6561) + (255025)

Dh1 = √261586

Dh1 = 511.45

Left Dh1v = vertical distance between Plane & Train after 1 hour

Dh1v² = 511.45² + 4²

Dh1v = √261586 + 16

Dh1v = √261602

Dh1v = 511.47 mile

In 1 hours time they will be Dh1v - Dhv apart

Dh1v - Dhv

511.47 mile - 6.48 miles = 504.99 mph = 505mph

vapour or gas assuming atmospheriv pressure