1) Two people are 3.21 miles apart and both spot a hot air balloon. One person observes the balloon at an angle of elevation of 25 degrees and the other observes the balloon at an angle of elevation of 38 degrees. Determine the altitude of the ballon and round your answer to the nearest hundrendth and be sure to label your answer.

2) A runner starts at point A and runs 860 feet due East to point B. The runner then turns and runs at a bearing of S12°E for 175 feet to point C. How far is point C from point A?

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1) To determine the altitude of the balloon, we can use trigonometry and the concept of similar triangles.

Let's label some points to make it easier to visualize:
- Point A is where the observer with the 25-degree angle of elevation is standing.
- Point B is where the observer with the 38-degree angle of elevation is standing.
- Point C is the location of the balloon.

We can create a right triangle ABC, where AB represents the horizontal distance between the observers and BC represents the altitude of the balloon.

Given that the distance between the observers is 3.21 miles, we know that AB = 3.21 miles.

Now, let's focus on triangle ABC. The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

For observer A, the tangent of the angle of elevation (25 degrees) is equal to BC / AB:
tan(25 degrees) = BC / 3.21 miles

Similarly, for observer B, the tangent of the angle of elevation (38 degrees) is equal to BC / AB:
tan(38 degrees) = BC / 3.21 miles

To solve for BC, we can rearrange these equations:
BC = tan(25 degrees) * 3.21 miles
BC = tan(38 degrees) * 3.21 miles

Using a calculator, we can evaluate the tangent of 25 degrees and the tangent of 38 degrees. Let's assume the tangent of 25 degrees is approximately 0.466 and the tangent of 38 degrees is approximately 0.781.

Now we can calculate BC:
BC = 0.466 * 3.21 miles = 1.49786 miles (rounded to the nearest hundredth)

Therefore, the altitude of the balloon is approximately 1.50 miles.

2) To find how far point C is from point A, we can construct a right triangle based on the given information.

Let's label some additional points:
- Point A is the starting point.
- Point B is the endpoint after running 860 feet due East.
- Point C is the location after running at a bearing of S12°E for 175 feet.

From point A to point B, we only move horizontally along the East direction, so the distance is 860 feet.

Now, from point B to point C, we move in a specific bearing, which is S12°E. This means we move 12 degrees south of due east. Note that a bearing is always taken clockwise from the north direction.

Since the bearing says "S12°E," we can break this movement into two components:
- Southward movement: sin(12 degrees) * 175 feet
- Eastward movement: cos(12 degrees) * 175 feet

Using a calculator, we can evaluate sin(12 degrees) and cos(12 degrees). Let's assume sin(12 degrees) is approximately 0.208 and cos(12 degrees) is approximately 0.978.

Now we can calculate the distance from point B to point C:
Distance BC = Southward movement + Eastward movement
Distance BC = 0.208 * 175 feet + 0.978 * 175 feet

Simplifying this expression:
Distance BC = (0.208 + 0.978) * 175 feet

Calculating the value in parentheses:
Distance BC = 1.186 * 175 feet
Distance BC ≈ 207.55 feet (rounded to the nearest hundredth)

Therefore, the distance from point C to point A is approximately 207.55 feet.