A bear is directly opposite a tent on the other side of a river. A campfire is on the same side of the river as the tent. A right triangle is formed from lines connecting the bear, the tent, and the campfire with the right angle at the tent. The lines from the bear to the tent and from the bear to the campfire form a 33° angle. If the bear is 60 feet from the tent, how far is the campfire from the tent? Round your answer to the nearest foot.

Question

A bear is on one side of a river. A tent and a campfire are on the other side of a river.

(1 point)
The campfire is
feet from the tent.
Remaining Attempts : 3

Answer 1

To solve this problem, we can use trigonometry and the properties of right triangles.

Let's denote:

• \( T \) as the tent
• \( B \) as the bear
• \( C \) as the campfire

Given that:

• \( BT = 60 \) feet (the distance from the bear to the tent),
• the angle \( \angle TBC = 33^\circ \),
• \( \angle TCB = 90^\circ \) (because the right angle is at the tent).

We need to find the distance \( TC \) (the distance from the tent to the campfire).

In triangle \( BTC \):

1. We can use the tangent function: \[ \tan(\angle TBC) = \frac{TC}{BT} \] Here, \( BT = 60 \) feet and \( \angle TBC = 33^\circ \).

2. Rearranging this gives us: \[ TC = BT \cdot \tan(\angle TBC) \] Substituting the known values: \[ TC = 60 \cdot \tan(33^\circ) \]

3. Now we need to calculate \( \tan(33^\circ) \). Using a calculator: \[ \tan(33^\circ) \approx 0.6494 \]

4. Therefore: \[ TC \approx 60 \cdot 0.6494 \approx 38.964 \]

5. Rounding this to the nearest foot: \[ TC \approx 39 \text{ feet} \]

Thus, the campfire is approximately 39 feet from the tent.