To solve this problem, we can use trigonometry. We have two points (A and B), with angles of elevation to the top of the mountain at different angles. We are given the distance between the two points, 5 miles, and we need to determine the height of the mountain in feet.

Key information:

• Angle of elevation from point A = $13.7^\circ$
• Angle of elevation from point B = $11.2^\circ$
• Distance between points A and B = 5 miles (horizontal distance)
• We need to find the height $h$ of the mountain.

We will use the tangent function from trigonometry:

$\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}$

Where:

• $\theta$ is the angle of elevation,
• The "height" is the height of the mountain,
• The "horizontal distance" is the distance from the point of observation (either point A or point B) to the point on the ground directly beneath the top of the mountain.

We need to first set up two equations for the height based on the two observation points, A and B.

Step 1: Set up the equations

Let $h$ be the height of the mountain, and $x$ be the horizontal distance from point B to the point on the ground directly beneath the top of the mountain.

• From point A, the tangent of the angle of elevation is: $\tan(13.7^\circ) = \frac{h}{x + 5}$

• From point B, the tangent of the angle of elevation is: $\tan(11.2^\circ) = \frac{h}{x}$

Step 2: Solve the system of equations

We now have the system of two equations:

1. $\tan(13.7^\circ) = \frac{h}{x + 5}$
2. $\tan(11.2^\circ) = \frac{h}{x}$

Step 3: Solve for $h$

First, solve for $h$ in both equations:

$h = (x + 5) \tan(13.7^\circ)$ $h = x \tan(11.2^\circ)$

Now, set the two expressions for $h$ equal to each other:

$(x + 5) \tan(13.7^\circ) = x \tan(11.2^\circ)$

Now, solve for $x$.

Step 4: Compute

Let’s perform the calculations step-by-step:

1. $\tan(13.7^\circ) \approx 0.242$
2. $\tan(11.2^\circ) \approx 0.196$

Now, substitute these values into the equation:

$(x + 5) \times 0.242 = x \times 0.196$

Expanding both sides:

$0.242x + 1.21 = 0.196x$

Now, isolate $x$:

$0.242x - 0.196x = -1.21$ $0.046x = -1.21$ $x = \frac{-1.21}{0.046} \approx 26.3 \text{ miles}$

Step 5: Find the height $h$

Now that we know $x \approx 26.3$ miles, we can substitute this into the equation for $h$ from point B:

$h = x \times \tan(11.2^\circ) = 26.3 \times 0.196 \approx 5.15 \text{ miles}$

To convert the height from miles to feet, multiply by 5280 (since 1 mile = 5280 feet):

$h = 5.15 \times 5280 \approx 27,192 \text{ feet}$

Final Answer:

The height of the mountain is approximately 27,192 feet.

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Would you like further explanation or assistance with this problem? Here are some related questions you might find interesting:

1. How does the tangent function relate to angles of elevation and depression?
2. How do we convert between miles and feet in distance problems?
3. How can the law of sines or cosines be applied in elevation problems?
4. What are the possible errors when measuring angles for trigonometric problems?
5. How would the answer change if the distance between A and B were longer or shorter?

Tip: When solving elevation problems, it’s always helpful to carefully label and visualize the points and angles involved to avoid mistakes in setting up the equations.