Math 131. Implicit Derivatives/Related Rates
Name:
Hints and Answers
1. A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat.
Suppose the winch pulls in the rope at a rate of 4 feet per second.
(a) Determine the speed of the boat when there
is 15 feet of rope out.
(b) What happens to the speed of the boat as it gets
closer to the dock?
Answer. (a) Let x be the distance from the boat to the dock. From the Pythagorean theorem
dy
= −4 because the rope is being pulled in at a rate of 4 feet
x2 + 122 = y 2 , and we know
dt
per second. Differentiating implictly with respect to t yields
d 2
d
dx
dy
(x + 122 ) = (y 2 ) and so 2x
= 2y
dt
dt
dt
dt
and so
√
dx
y dy
=
. Given that y = 15, we find x = 152 − 122 = 9. Thus
dt
x dt
15
20
dx
=
· −4 = −
dt
9
3
The boat approaches the dock at a speed of 20/3 feet per second when there is 15 feet of rope
out.
(b) As the boat approaches the dock, y decreases to 12 and x decreases to 0. Thus the speed
at which the boat approaches the dock increases.
2. Find the tangent line to the Lemniscate 3(x2 + y 2 )2 = 100(x2 − y 2 ) at the point (4, 2).
Answer. Differenting the equation implicitly yields:
6(x2 + y 2 )(2x + 2yy 0 ) = 100(2x − 2yy 0 ) or 3(x2 + y 2 )(x + yy 0 ) = 50(x − yy 0 )
Then solving this for y 0 we obtain
[3(x2 + y 2 )y + 50y]y 0 = 50x − 3x(x2 + y 2 ) and so y 0 =
50x − 3x(x2 + y 2 )
3y(x2 + y 2 ) + 50y
When x = 4 and y = 2, this means y 0 = [200 − 12(20)]/(6(20) + 100) = −40/220 = −2/11.
2
2
Then the tangent line has equation y − 2 = − 11
(x − 4) or y = − 11
x + 30
.
11