Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.?
A) mph
B) mph
C) mph
D) 32 mph
E) 24 mph

Question

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.? ​
A) mph 
B) mph 
C) mph 
D) 32 mph 
E) 24 mph

Quizplus · Accepted Answer

We can use the Pythagorean theorem to find the distance between the two ships at any given time: 
distance^2 = (northward distance)^2 + (eastward distance)^2

At noon (when they both depart), the distance between them is: 
distance = sqrt((0)^2 + (0)^2) = 0

One hour later, the northward distance traveled by Ship A is: 
northward distance = 1 hour * 21 mph = 21 miles

The eastward distance traveled by Ship B is: 
eastward distance = 1 hour * 12 mph = 12 miles

So the distance between the two ships after one hour is: 
distance = sqrt((21)^2 + (12)^2) ≈ 24.186

To find how fast the distance is changing at 1 p.m., we need to find the derivative of distance with respect to time:

distance^2 = (northward distance)^2 + (eastward distance)^2

2(distance)(rate of change of distance) = 2(northward distance)(rate of change of northward distance) + 2(eastward distance)(rate of change of eastward distance)

Substituting the values we know:

2(√(21^2 + 12^2))(rate of change of distance) = 2(21)(0) + 2(12)(-12)

(rate of change of distance) = (-12*12)/(√(21^2 + 12^2)) ≈ -1.977

So the distance between the two ships is decreasing at a rate of approximately 1.977 mph. Answer choice C is the closest option.

Two Ships Leave the Same Port at Noon