# Thread: sailboat related rate problem

1. ## sailboat related rate problem

a sailboat is moving at 6 knots (nautical miles) due east on a course which will take it 5 knots south of a lighthouse. when the boat is 13 miles from the lighthouse....
a) how fast is the boat approaching the lighthousee
b) how fsst is the boat's azimuth angle changing for an observer in the lighthouse

2. 1 knot = 1 nautical mile per hour

let the horizontal displacement = $x$

$\frac{dx}{dt} = -6$ nm/hr

distance from the lighthouse = $z$

using Pythagoras ...

$x^2 + 5^2 = z^2$

$\frac{d}{dt}(x^2 + 5^2 = z^2)$

$2x \cdot \frac{dx}{dt} = 2z \cdot \frac{dz}{dt}$

$\frac{x}{z} \cdot \frac{dx}{dt} = \frac{dz}{dt}$

$\frac{12}{13} \cdot (-6) = \frac{dz}{dt}$

3. Can anyone do part 2? I have a similar problem and it would probably help if this was solved and I could use it as a guide for mine

4. b) how fsst is the boat's azimuth angle changing for an observer in the lighthouse
$\theta = \arctan\left(\frac{x}{5}\right)$

$\frac{d\theta}{dt} = \frac{5}{25+x^2} \cdot \frac{dx}{xt}$

$\frac{d\theta}{dt} = \frac{5}{25+12^2} \cdot (-6)
$