At 8:00am, ship A is 120km due east of ship B. Ship A is moving north at 20km/hr, and ship B is moving east at 25km/hr. How fast is the difference between the two ships changing at 8:30am?
After t hours,
ship A has traveled 20 t km
ship B has traveled 25t
I sketched a right-angled triangle with
legs 120-25t and 20t
let D be the distance between them , making D the hypotenuse
D^2 = ((120- 25t)^2 + (20t)^2
at 8:30 , t = .5
D^2 = 11656.25
D = 107.964..
2D dD/dt = 2(120-25t)(-25) + 2(20t)(20)
so when t= .5 and D = 107.964..
dD/dt = ( (120-12.5)(-25) + 10(20) )/107.964
= -23.04..
so at 8:30 , the distance between them is decreasing at appr 23 km/h
check my calculations
To find the rate at which the difference between the two ships is changing at 8:30am, we can use the concept of relative motion.
Let's break down the problem into different components and find the rate of change for each component separately.
First, we need to find the distance between the two ships at 8:30am.
Ship A is moving north at 20 km/hr for 30 minutes (from 8:00am to 8:30am). Therefore, the northward distance covered by Ship A is:
DistanceA = (20 km/hr) * (30 min) = 10 km
Ship B is moving east at 25 km/hr for 30 minutes. Therefore, the eastward distance covered by Ship B is:
DistanceB = (25 km/hr) * (30 min) = 12.5 km
Now, we can find the total distance between the two ships at 8:30am by using the Pythagorean theorem:
Distance between the two ships = √(DistanceA^2 + DistanceB^2)
= √(10^2 + 12.5^2)
= √(100 + 156.25)
= √256.25
≈ 16.01 km
Next, we need to find the rates at which the distances are changing for both Ship A and Ship B.
The rate at which Ship A is moving north does not change with time, so it remains constant at 20 km/hr.
The rate at which Ship B is moving east also does not change with time, so it remains constant at 25 km/hr.
To find the rate at which the difference between the two ships is changing at 8:30am, we need to find the rate of change for each distance component separately.
The rate of change for the northward distance is 20 km/hr.
The rate of change for the eastward distance is 25 km/hr.
Now, we can apply the Pythagorean theorem to find the rate at which the difference between the two ships is changing:
Rate of change = √[(Rate of change for DistanceA)^2 + (Rate of change for DistanceB)^2]
= √[(20 km/hr)^2 + (25 km/hr)^2]
= √[400 + 625]
= √1025
≈ 32.02 km/hr
So, the difference between the two ships is changing at a rate of approximately 32.02 km/hr at 8:30am.