Answer:
To solve this problem, we can use trigonometric ratios and set up a proportion.
Let \( h \) be the height of the mountain.
From the first observation:
\( \tan(28^\circ) = \frac{h}{x} \), where \( x \) is the distance from the first observation point to the foot of the mountain.
From the second observation:
\( \tan(31^\circ) = \frac{h}{x - 1000} \), where \( x - 1000 \) is the distance from the second observation point to the foot of the mountain.
We can rearrange the equations to solve for \( h \):
\( h = x \tan(28^\circ) \) and \( h = (x - 1000) \tan(31^\circ) \).
Since both equations are equal to \( h \), we can set them equal to each other:
\[ x \tan(28^\circ) = (x - 1000) \tan(31^\circ) \]
Now, we can solve for \( x \):
\[ x \tan(28^\circ) = x\tan(31^\circ) - 1000\tan(31^\circ) \]
\[ x \tan(28^\circ) - x\tan(31^\circ) = -1000\tan(31^\circ) \]
\[ x(\tan(28^\circ) - \tan(31^\circ)) = -1000\tan(31^\circ) \]
\[ x = \frac{-1000\tan(31^\circ)}{\tan(28^\circ) - \tan(31^\circ)} \]
Now that we have found \( x \), we can substitute it into either of the original equations to find \( h \). Let's use the first one:
\[ h = x\tan(28^\circ) \]
Calculate \( x \):
\[ x = \frac{-1000\tan(31^\circ)}{\tan(28^\circ) - \tan(31^\circ)} \approx 2195.57 \text{ feet} \]
Now, substitute \( x \) into the equation for \( h \):
\[ h = 2195.57\tan(28^\circ) \approx 1089.81 \text{ feet} \]
So, the height of the mountain is approximately 1089.81 feet.
