Activity 36.
A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow as in Figure 166. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when the length of rope from bow to pulley is 13 feet?
(a)
In the right triangle in the Figure 166, indicate which sides are constant over time and which sides vary. Then, label each side of the triangle with a variable or a constant accordingly.
Hint.
The boat will always be 5 feet higher than the bow of the boat. But the other distances in this problem vary.
Answer.
Label the height of the triangle as 5 (since it is constant). Label the base of the right triangle as \(x\) and the hypotenuse as \(z\text{.}\)
(b)
Using derivative notation, state which rate is given and which rate is to be calculated.
Hint.
The derivatives are all functions of time \(t\text{.}\)
Answer.
It is known that \(\frac{dz}{dt} = -2\) feet per second. We look to calculate the rate \(\frac{dx}{dt}\) when \(z=13\) feet.
(c)
Determine an equation that relates the two variables identified in (a).
Hint.
Use the Pythagorean Theorem.
Answer.
\(x(t)^2 + 5^2 = z(t)^2\)
(d)
Use implicit differentiation on the equation found in (c) to relate the two rates identified in (b). Use this to determine how fast the boat is approaching the dock when the length of the rope from bow to pulley is 13 feet.
Hint.
Will the rate in question be positive or negative? Be sure you get a sign on the answer that you expect.
Answer.
\(-\frac{13}{6}\) feet per second
Solution.
Since \(\displaystyle 2x \cdot \frac{dx}{dt} + 0 = 2z \cdot \frac{dz}{dt}\text{,}\) we have that \(\displaystyle \frac{dx}{dt} = \frac{2z \cdot \frac{dz}{dt}}{2x}\text{.}\) When \(z = 13\) feet, \(x(t)^2 + 5^2 = 13^2\) so that \(x=12\) feet. Thus,
\begin{align*}
\frac{dx}{dt}\Big|_{z=13} = \amp \mathstrut \frac{2(13)(-2)}{2(12)}\\
= \amp \mathstrut -\frac{13}{6} ft/sec
\end{align*}
