A fire is spotted 29 degrees west of South. The fire is 58 km from the look out station. A fire fighting base is 62 km due west of the first station. How far is the fire from the fighting base?

Question

Elijah Alexander · Accepted Answer

If the lookout is at the origin, the base is at #(-62,0)# and the fire is at #(-58 sin 29^circ, -58 cos 29^circ)# so the distance is

# d=\sqrt{(62-58 \ sin 29^circ)^2+58 \ cos^2 29^circ} approx 34.5# km

Samantha Abate · Answer

undefined Using trigonometry, we can find the distance from the fire to the fighting base. Since the fire is 29 degrees west of South, it forms a right triangle with the lookout station and the fighting base.

We can use the tangent function to find the distance from the fire to the fighting base.

Tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Let's denote the distance from the fire to the fighting base as $ x $.

Thus, $ \tan(29^\circ) = \frac{58}{x} $.

Solving for $ x $, we get $ x = \frac{58}{\tan(29^\circ)} $.

Using a calculator, we find that $ \tan(29^\circ) \approx 0.5543 $.

So, $ x \approx \frac{58}{0.5543} $.

$ x \approx 104.61 $ kilometers.

Therefore, the fire is approximately 104.61 kilometers from the fighting base.