At what rate is the boat approachng the dock when it is 8m from the dock?
Illustration:
Required: $\displaystyle \frac{dx}{dt}$ when $x = 8m$
Solution:
By using Pythagorean Theorem we have,
$z^2 = 1^2 + x^2 \qquad$ Equation 1
taking the derivative with respect to time
$
\begin{equation}
\begin{aligned}
\cancel{2} z \frac{dz}{dt} =& \cancel{2}x \frac{dx}{dt}
\\
\\
\frac{dx}{dt} =& \frac{z}{x} \frac{dz}{dt} \qquad \text{Equation 2}
\end{aligned}
\end{equation}
$
We can get the value of $z$ by substituting $x = 8$ in equation 1
$
\begin{equation}
\begin{aligned}
z^2 =& 1^2 +x^2
\\
\\
z^2 =& 1^2 + 8^2
\\
\\
z =& \sqrt{65} m
\end{aligned}
\end{equation}
$
Now, using equation 2 to solve for the unknown
$
\begin{equation}
\begin{aligned}
\frac{dx}{dt} =& \frac{\sqrt{65}}{8} (1)
\\
\\
\frac{dx}{dt} =& \frac{\sqrt{65}}{8} m/s
\end{aligned}
\end{equation}
$
This means that the boat is approaching the dock at a rate of $\displaystyle \frac{\sqrt{65}}{8} m/s$ when the boat is $8 m$ from the dock.
Saturday, January 31, 2015
Single Variable Calculus, Chapter 3, 3.8, Section 3.8, Problem 20
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