sailboat related rate problem

**cruxkitty** · January 7th 2009, 06:04 PM

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

**skeeter** · January 7th 2009, 06:21 PM

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 ...

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

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

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

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

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

**sfgiants13** · January 8th 2009, 07:10 PM

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

**skeeter** · January 9th 2009, 04:36 PM

b) how fsst is the boat's azimuth angle changing for an observer in the lighthouse

$\theta = \arctan\left(\frac{x}{5}\right)$

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

$\frac{d\theta}{dt} = \frac{5}{25+12^2} \cdot (-6)<br />$ rad/hr

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