Answer:
The boat is approaching the dock when it is 7 m from the dock at a rate of 1.0102 m
Explanation:
The situation is drawn in the image shown below.
From the image,
Applying Pythagorean theorem as:
a² + 1² = b² .....1
Differentiation both side w.r.t. time as:
[tex]2a\times \frac{\partial a}{\partial t}+0=2b\times \frac{\partial b}{\partial t}[/tex]
or,
[tex]a\times \frac{\partial a}{\partial t}=b\times \frac{\partial b}{\partial t}[/tex]...2
Given:
The rate of the pulling of the rope in = 1 m/s
Thus, [tex]\frac{\partial b}{\partial t}=1[/tex]
We have to find [tex]\frac{\partial a}{\partial t}[/tex] when a = 7 m
Using equation 1 to calculate b as:
7² + 1² = b²
b = √50 m
Using equation 2 as:
[tex]7\times \frac{\partial a}{\partial t}=\sqrt {50}\times 1[/tex]
Thus,
[tex]\frac{\partial a}{\partial t}=\frac {\sqrt {50}}{7}\ m/s=1.0102\ m/s[/tex]
Thus, the boat is approaching the dock when it is 7 m from the dock at a rate of 1.0102 m
