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 25°. from a point 1500 feet closer to the mountain along the plain, they find that the angle of elevation is 29°. how high (in feet) is the mountain? question help: video

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Accepted Answer

# Explanation:
## Step1: Set up equations using tangent function
Let the height of the mountain be $ h $ feet and the distance from the second - closer point to the base of the mountain be $ x $ feet.
From the angle of elevation of $ 29^{\circ}$, we have $\tan29^{\circ}=\frac{h}{x}$, so $h = x\tan29^{\circ}$.
From the angle of elevation of $25^{\circ}$, we have $\tan25^{\circ}=\frac{h}{x + 1500}$, so $h=(x + 1500)\tan25^{\circ}$.

## Step2: Equate the two expressions for $h$
Since $x\tan29^{\circ}=(x + 1500)\tan25^{\circ}$
$$
\begin{align*}
x\tan29^{\circ}&=x\tan25^{\circ}+1500\tan25^{\circ}\
x(\tan29^{\circ}-\tan25^{\circ})&=1500\tan25^{\circ}\
x&=\frac{1500\tan25^{\circ}}{\tan29^{\circ}-\tan25^{\circ}}
\end{align*}
$$
We know that $\tan25^{\circ}\approx0.4663$, $\tan29^{\circ}\approx0.5543$
$$
\begin{align*}
x&=\frac{1500\times0.4663}{0.5543 - 0.4663}\
&=\frac{699.45}{0.088}\
&=7948.2955
\end{align*}
$$

## Step3: Calculate the height $h$
Since $h = x\tan29^{\circ}$, substituting $x = 7948.2955$
$$
\begin{align*}
h&=7948.2955\times0.5543\
&\approx4406
\end{align*}
$$

# Answer:
$4406$

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QUESTION IMAGE

Question

Explanation:

Step1: Set up equations using tangent function

Step2: Equate the two expressions for \(h\)

Step3: Calculate the height \(h\)

Answer: