A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 31 From a point 1000 feet closer to the mountain along the plain, they find that the angle of elevation is \( 35^{\circ} \). How high (in feet) is the mountain?

To find the height of the mountain, we can use trigonometry and the information provided. Let \( h \) be the height of the mountain and \( d \) be the distance from the first observation point to the base of the mountain. From the first observation point: \[ \tan(31^{\circ}) = \frac{h}{d} \implies h = d \cdot \tan(31^{\circ}) \] From the second observation point, which is 1000 feet closer: \[ \tan(35^{\circ}) = \frac{h}{d - 1000} \implies h = (d - 1000) \cdot \tan(35^{\circ}) \] Setting the two expressions for \( h \) equal to each other: \[ d \cdot \tan(31^{\circ}) = (d - 1000) \cdot \tan(35^{\circ}) \] Now, we can solve for \( d \): Expanding the equation: \[ d \cdot \tan(31^{\circ}) = d \cdot \tan(35^{\circ}) - 1000 \cdot \tan(35^{\circ}) \] Rearranging gives us: \[ d \cdot \tan(31^{\circ}) - d \cdot \tan(35^{\circ}) = -1000 \cdot \tan(35^{\circ}) \] Factoring out \( d \): \[ d (\tan(31^{\circ}) - \tan(35^{\circ})) = -1000 \cdot \tan(35^{\circ}) \] Now solve for \( d \): \[ d = \frac{-1000 \cdot \tan(35^{\circ})}{\tan(31^{\circ}) - \tan(35^{\circ})} \] Calculating \( d \) using the approximate values \( \tan(31^{\circ}) \approx 0.6018 \) and \( \tan(35^{\circ}) \approx 0.7002 \): \[ d = \frac{-1000 \cdot 0.7002}{0.6018 - 0.7002} = \frac{-700.2}{-0.0984} \approx 7110.40 \text{ feet} \] Now calculate \( h \) using one of our expressions for \( h \): \[ h = d \cdot \tan(31^{\circ}) \approx 7110.40 \cdot 0.6018 \approx 4271.62 \text{ feet} \] Thus, the height of the mountain is approximately \( 4272 \) feet.