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A forest ranger in the west observation tower spots a fire 41° north of east. Fifteen miles directly east, the forest ranger in the east tower spots the same fire at 56° north of west. How far away is the ranger who is closest to the fire? Approximate the distance by rounding to the nearest hundredth of a mile. 9. 91 mi 11. 87 mi 12. 53 mi 18. 95 mi.

Sagot :

Using Pythagoras theorem and trigonometric rule, the ranger who is closest to the fire is 10.18 miles.

What is trigonometry?

Trigonometry deals with the relationship between the sides and angles of a right-angle triangle.

Given

A forest ranger in the west observation tower spots a fire 41° north of the east.

Fifteen miles directly east, the forest ranger in the east tower spots the same fire at 56° north of west.

Let AC be the x and BC will be 15 -x.

And FC is the h.

In ΔACF.

[tex]\rm h = x tan 41\\\\h = 0.8693 \ x[/tex]...1

In ΔBCF.

[tex]\rm h = (15-x) tan 34\\\\h = 10.1176 - 0.6745x[/tex]...2

From equation 1 and 2

[tex]\rm 0.8693 \ x = 10.1176 -0.6745x[/tex]

On simplifying, we have

x = 6.554

Then 15-x will be 8.446

And h will be 5.697

Then by Pythagoras theorem, the ranger who is closest to the fire will be

FB² = 8.446² + 5.697²

FB  = 10.18 miles

The diagram is shown below.

Thus, the ranger who is closest to the fire is 10.18 miles.

More about the trigonometric link is given below.

https://brainly.com/question/22698523

View image jainveenamrata
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