Question

the statue of liberty is approximately 305 feet tall. if the angle of elevation of a ship to the top of the statue is 21.6°, how far, to the nearest foot, is the ship from the statues base? (round to the nearest foot)

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle situation where the height of the Statue of Liberty is the opposite side ($y = 305$ feet) and the distance from the ship to the base of the statue is the adjacent side ($x$) with respect to the angle of elevation $\theta=21.6^{\circ}$. We use the tangent function $\tan\theta=\frac{y}{x}$.

Step2: Solve for $x$

Rearranging the formula $\tan\theta=\frac{y}{x}$ gives $x = \frac{y}{\tan\theta}$. Substitute $y = 305$ and $\theta = 21.6^{\circ}$ into the formula. So $x=\frac{305}{\tan(21.6^{\circ})}$.

Step3: Calculate the value of $x$

We know that $\tan(21.6^{\circ})\approx0.400$. Then $x=\frac{305}{0.400}=762.5$.

Answer:

763 feet