Question

Suppose that two boats leave a dock at different times. One heads due north, the other due east. Find the rate at which the distance between the boats is changing when the first boat is 70 miles from the dock traveling at a speed of 45 miles per hour and the second boat is 81 miles from the dock traveling at a speed of 40 miles per hour.

Answer 1

looks like a pythagoras problem to me

Answer 2

i have no idea what that means. i cant even enter things into the calculator without problems

Answer 3

let x be the the distance of the boat heading north and y be the distance of the boat heading east. then you know the distance between them is
\[\sqrt{x^2+y^2}\]

Answer 4

the derivative is the rate of change of the distance so you know that
\[x'=45\] and
\[y'=40\]

Answer 5

btw it is always easier to work with the square of the distance than the distance, so we can get rid of the annoying square root sign. put
\[d^2=x^2+y^2\] and take the derivative of both side wrt time to get
\[2d d' = 2x x' + 2y y'\]

Answer 6

or if you prefer (i certainly do)
\[dd'=xx'+yy'\]

Answer 7

now plug in the numbers and solve for
\[d'\]

Answer 8

\[x=70,x'=45,y=81,y'=40\] and
\[d=\sqrt{70^2+81^2}\]

Answer 9

107.0560601?

Answer 10

59.35569251?