3.9 Related Rates

Question

At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 30 km/h. How fast (in km/hr) is the distance between the ships changing at 4:00 p.m.? (Round your answer to three decimal places.)

 

 km/h

Answer 1

A is 150 km west of B at noon

A is moving east 35 km/hr = A'=-35 4 hours later A=150-4(35) = 10

B is moving north 30 km/hr = B' 4 hours later B=4(30) = 120

current distance between them = 150 km

in 4 hours distance between them = d = sqr(10^2+120^2) = sqr14500= 10sqr145 = about 120+ km

how fast is the distance between them changing at 4:00 pm?

Use the Pythagorean theorem, take the derivative and plug in the known values, solve for d'

d^2 = A^2 + B^2

2dd' = 2AA' +2BB'

divide by 2

dd' = AA' +BB'

d' = (AA' + BB')/d

= (10(-35) +120(30))/10sqr145

= (-350+3600)/10sqr145

= 3250/10sqr145

=325sqr145

= 325sqr145/145

= (325/145)sqr145

= about 26.990 km/hr = d' = rate of change of distance changing between ship A and ship B

no guarantee this answer is error free

tedious calculations are prone to errors

but the answer should be close to 30 km/hr and a little less than 30 km/hr