Question

How do I answer this navigation problem?

Find the width of a river that flows due east, if a surveyor finds that a tree on the opposite bank has a bearing of N2230'E from a certain point and a bearing of N15W from a point 400 feet downstream.

Answer 1

`color(blue)(188.887)` feet (approximately)

Explanation:

Here is what the situation looks like:

We have
`color(white)("XXXXXX")a+b=400color(white)("xx")rarrcolor(white)("xx")b=400-a`
and
`color(white)("XXX")w=a * sin(67^circ30')=b * sin(75^circ)`
`color(white)("XXXXXXXXXXXXXXX")=(400-a) * sin(75^circ)`

`rArrcolor(white)("XXX")a=(400 * sin(75^circ))/(sin(67^circ30')+sin(75^circ))`

with the aid of a calculator (and remembering that `67^circ30' = 67.5^circ`)

we can determine that
`color(white)("XXX")a=204.4498015187` feet (approx.)
and
`color(white)("XXX")w=a * sin(67^circ30')=188.8869870491` feet (approx.)