Question

23.) a hiker stands 80 ft from a giant redwood tree and sights the top with an angle of elevation of 75°. how tall is the tree to the nearest foot? 23.)

Explanation:

Step1: Use the tangent function

In a right - triangle (where the distance from the hiker to the tree is the adjacent side \(x = 80\) ft and the height of the tree \(h\) is the opposite side), the tangent of an angle of elevation \(\theta\) is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 75^{\circ}\) and \(x = 80\) ft. So, \(\tan(75^{\circ})=\frac{h}{80}\).

Step2: Solve for \(h\)

We know that \(\tan(75^{\circ})=\tan(45^{\circ}+ 30^{\circ})=\frac{\tan45^{\circ}+\tan30^{\circ}}{1 - \tan45^{\circ}\tan30^{\circ}}=\frac{1+\frac{\sqrt{3}}{3}}{1-(1)\times\frac{\sqrt{3}}{3}}=\frac{3 + \sqrt{3}}{3-\sqrt{3}}\). Rationalizing the denominator: \(\frac{(3 + \sqrt{3})(3+\sqrt{3})}{(3-\sqrt{3})(3+\sqrt{3})}=\frac{9 + 6\sqrt{3}+3}{9 - 3}=\frac{12+6\sqrt{3}}{6}=2+\sqrt{3}\approx2 + 1.732=3.732\). Then, since \(h = 80\times\tan(75^{\circ})\), substituting \(\tan(75^{\circ})\approx3.732\), we get \(h=80\times3.732 = 298.56\).

Answer:

\(299\) ft