The situation is drawn in the diagram below:
We are interested in knowing the speed of the boat (dx/dt) when h = 10 ft and dh/dt = 2 ft/s.
Applying the Pythagorean Theorem:
[tex]h^2=x^2+6^2[/tex]
Differentiating with respect to the time:
[tex]2h\cdot\frac{dh}{dt}=2x\cdot\frac{dx}{dt}[/tex]
Simplifying and solving:
[tex]\frac{dx}{dt}=\frac{h}{x}\cdot\frac{dh}{dt}[/tex]
When h = 10, we can solve for x:
[tex]\begin{gathered} x^2=h^2-6^2=100-36=64 \\ \\ x=\sqrt{64}=8 \end{gathered}[/tex]
Substitute this value of x:
[tex]\frac{dx}{dt}=\frac{10}{8}\cdot2\text{ ft/s}[/tex]
Calculating:
[tex]\frac{dx}{dt}=\frac{5}{2}\text{ ft/s}[/tex]
b. To calculate the rate of change of the angle, we use the tangent ratio as follows:
[tex]\tan\theta=\frac{x}{6}[/tex]
