Question

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a woman standing on a hill sees a flagpole that she knows is 70 ft tall. the angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. find her distance x from the pole. (round your answer to one decimal place.)
x =

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sprecalc7 6.2.060

Explanation:

Step1: Set up tangent - based equations

Let the height from the horizontal line of the woman's eye - level to the bottom of the pole be $h_1$ and to the top of the pole be $h_2$. We know that $\tan14^{\circ}=\frac{h_1}{x}$ and $\tan18^{\circ}=\frac{h_2}{x}$, and $h_1 + h_2=70$.

Step2: Express $h_1$ and $h_2$ in terms of $x$

From $\tan14^{\circ}=\frac{h_1}{x}$, we get $h_1 = x\tan14^{\circ}$. From $\tan18^{\circ}=\frac{h_2}{x}$, we get $h_2 = x\tan18^{\circ}$.

Step3: Substitute $h_1$ and $h_2$ into the height - sum equation

Substitute $h_1$ and $h_2$ into $h_1 + h_2 = 70$, we have $x\tan14^{\circ}+x\tan18^{\circ}=70$.

Step4: Factor out $x$

Factor out $x$ from the left - hand side: $x(\tan14^{\circ}+\tan18^{\circ}) = 70$.

Step5: Solve for $x$

$x=\frac{70}{\tan14^{\circ}+\tan18^{\circ}}$. We know that $\tan14^{\circ}\approx0.2493$ and $\tan18^{\circ}\approx0.3249$. Then $x=\frac{70}{0.2493 + 0.3249}=\frac{70}{0.5742}\approx121.9$.

Answer:

$121.9$