Given:
A plane flying a straight course observes a mountain at a bearing of 35° to the right of its course.
The distance between plan and mountain is 10 km.
A short time later, the bearing to the mountain becomes 45°.
Here NM is the distance between the plane from the mountain when the second bearing is taken.
We need to find the measure of NM.
[tex]We\text{ know that }\angle INM\text{ and }\angle\text{XNM are supplementary angles.}[/tex]The sum of the supplementary angles is 180 degrees.
[tex]\angle INM+\angle XNM=180^o[/tex][tex]\text{Substitute }\angle XNM=45^o\text{ in the equation.}[/tex][tex]\angle INM+45^o=180^o[/tex][tex]\angle INM=180^o-45^o[/tex][tex]\angle INM=135^o[/tex]We know that the sum of all three angles of the triangle is 180 degrees.
[tex]\angle INM+\angle NIM+\angle INM=180^o[/tex][tex]\text{Substitute }\angle INM=135^o\text{ and }\angle NIM=35^o\text{ in the equation.}[/tex][tex]135^o+35^o+\angle INM=180^o[/tex][tex]170^o+\angle INM=180^o[/tex][tex]\angle INM=180^o-170^o[/tex][tex]\angle INM=10^o[/tex]Consider the sine law.
[tex]\frac{\sin N}{IM}=\frac{\sin M}{IN}=\frac{\sin I}{NM}[/tex]Take the equation to find the measure of NM.
[tex]\frac{\sin N}{IM}=\frac{\sin I}{NM}[/tex][tex]\text{Substitute }\angle N=135^o,\angle I=35^o,\text{ and IM=10 in the equation.}[/tex][tex]\frac{\sin 135^o}{10}=\frac{\sin35^o}{NM}[/tex][tex]NM=\frac{\sin 35^o}{\sin 135^o}\times10[/tex][tex]NM=8.11[/tex]Hence the measure of NM is 8.1 km.
The plane is 8.1 km far from the mountain when the second bearing is taken.
