This problem can be solved using trigonometry and the concept of right triangles. Let's break it down step by step:

Step 1: Define variables and geometry

• Let the distance from the first observation point to the base of the mountain be $x$.
• Let the height of the mountain be $h$.

From the problem:

1. At the first observation point, the angle of elevation is $27^\circ$.
2. From a point 1000 feet closer, the angle of elevation is $32^\circ$.

We have two right triangles:

1. Larger triangle with angle $27^\circ$, base $x$, and height $h$.
2. Smaller triangle with angle $32^\circ$, base $x - 1000$, and height $h$.

Using the tangent function for each triangle: $\tan(27^\circ) = \frac{h}{x}$ $\tan(32^\circ) = \frac{h}{x - 1000}$

Step 2: Solve for $h$ and $x$

From the equations above, we can express $h$ in terms of $x$: $h = x \cdot \tan(27^\circ)$ $h = (x - 1000) \cdot \tan(32^\circ)$

Equating the two expressions for $h$: $x \cdot \tan(27^\circ) = (x - 1000) \cdot \tan(32^\circ)$

Step 3: Simplify and solve for $x$

Expand the equation: $x \cdot \tan(27^\circ) = x \cdot \tan(32^\circ) - 1000 \cdot \tan(32^\circ)$ Rearrange terms: $x \cdot (\tan(27^\circ) - \tan(32^\circ)) = -1000 \cdot \tan(32^\circ)$ Solve for $x$: $x = \frac{1000 \cdot \tan(32^\circ)}{\tan(32^\circ) - \tan(27^\circ)}$

Step 4: Calculate $x$ and $h$

Using approximations for the tangent values: $\tan(27^\circ) \approx 0.5095, \quad \tan(32^\circ) \approx 0.6249$

Substitute these values into the formula for $x$: $x = \frac{1000 \cdot 0.6249}{0.6249 - 0.5095} = \frac{624.9}{0.1154} \approx 5414.6 \, \text{feet}$

Now substitute $x$ into the equation for $h$: $h = x \cdot \tan(27^\circ)$ $h = 5414.6 \cdot 0.5095 \approx 2757.6 \, \text{feet}$

Final Answer:

The height of the mountain is approximately 2758 feet.

Let me know if you’d like further clarifications! Here are some related questions to explore:

1. How would the result change if the angles were measured in radians instead of degrees?
2. What if the second observation point were 2000 feet closer instead of 1000 feet?
3. How can this method be applied to real-world surveying scenarios?
4. Can this problem be solved graphically using software tools?
5. What assumptions are implicit in using tangent for this calculation?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems.