Calculus and Applications

Question

A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 6 ft/s from a point 500 ft due east of P. At what rate (in ft/s) are the people moving apart 15 minutes after the woman starts walking? (Round your answer to two decimal places.)

Answer 1

4ft/sec= 240 ft/min north

in 5 minutes the man walked 5x240= 1200 feet north

in 15 minutes he's 3600 feet north

6 ft/sec= 360 ft/min south

in 5 minutes she's 1200 feet south and 500 feet east of the man

in 15 minutes she's 15x360 = 5400 feet south of the man's starting point and 9000 feet south of the man

construct a right triangle

use the Pythagorean Theorem

d^2 = a^2+b^2

take derivatives with respect to time

divide by 2

dd' = aa' +bb' where a'=600,b'=0, a=9000,b=500, d = sqr(9000^2+500^2) =sqr(81000000+250000)=9013.88

plug in and solve for d'= how fast distance between them is increasing

= (600)(9000)/9013.88

= 599.08 ft/minute

= 9.98 feet per second

(no guarantees this calculation is error free. but it looks reasonable, in the ball park

major thing is keep the units the same, all minutes or all seconds during the calculations

and use the Pythagorean Theorem)