Questions: A ranger in tower A spots a fire at a direction of 347°. A ranger in tower B, located 60 mi at a direction of 59° from tower A, spots the fire at a direction of 317°. How far from tower A is the fire? How far from tower B? The fire is from tower A. (Round to the nearest whole number.) The fire is from tower B.

A ranger in tower A spots a fire at a direction of 347°. A ranger in tower B, located 60 mi at a direction of 59° from tower A, spots the fire at a direction of 317°. How far from tower A is the fire? How far from tower B?

The fire is from tower A. (Round to the nearest whole number.) The fire is from tower B.
Transcript text: A ranger in tower $A$ spots a fire at a direction of $347^{\circ}$. A ranger in tower B , located 60 mi at a direction of $59^{\circ}$ from tower A, spots the fire at a direction of $317^{\circ}$. How far from tower $A$ is the fire? How far from tower $B$ ? The fire is $\square$ $\square$ from tower $A$. (Round to the nearest whole number.) The fire is $\square$ $\checkmark$ from tower B.
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Solution

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Solution Steps

Step 1: Understand the Problem

We need to find the distances from two towers (A and B) to a fire. The directions from the towers to the fire are given, and the distance between the towers is known.

Step 2: Identify Given Information
  • Tower A spots the fire at a direction of 347°.
  • Tower B spots the fire at a direction of 59°.
  • The distance between Tower A and Tower B is 40 miles.
  • The angle between the directions from the towers to the fire is 317° - 59° = 258°.
Step 3: Use the Law of Sines

We can use the Law of Sines to find the distances from the towers to the fire. The Law of Sines states: asin(A)=bsin(B)=csin(C) \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} where aa, bb, and cc are the sides of the triangle, and AA, BB, and CC are the opposite angles.

Step 4: Calculate the Angles
  • Angle at Tower A (A) = 360° - 347° = 13°
  • Angle at Tower B (B) = 59°
  • Angle at the fire (C) = 180° - (13° + 59°) = 108°
Step 5: Apply the Law of Sines

Using the Law of Sines: ABsin(C)=asin(A)=bsin(B) \frac{AB}{\sin(C)} = \frac{a}{\sin(A)} = \frac{b}{\sin(B)} where AB=40AB = 40 miles, C=108°C = 108°, A=13°A = 13°, and B=59°B = 59°.

Step 6: Solve for Distances

40sin(108°)=asin(13°)=bsin(59°) \frac{40}{\sin(108°)} = \frac{a}{\sin(13°)} = \frac{b}{\sin(59°)}

Solving for aa (distance from Tower A to the fire): a=40sin(13°)sin(108°) a = \frac{40 \cdot \sin(13°)}{\sin(108°)}

Solving for bb (distance from Tower B to the fire): b=40sin(59°)sin(108°) b = \frac{40 \cdot \sin(59°)}{\sin(108°)}

Final Answer

  • Distance from Tower A to the fire: a9a \approx 9 miles (rounded to the nearest whole number)
  • Distance from Tower B to the fire: b35b \approx 35 miles (rounded to the nearest whole number)
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