Question

the measurement from the base of a tree to the tip of its shadow is 100 ft. the angle of inclination is 60°. given: sin 60° = 0.866, cos 60° = 0.5, tan 60° = 1.732. how tall is the tree? a. 50 ft, b. 86.6 ft, c. 115.5 ft, d. 173.2 ft, e. 200 ft

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the \(60^\circ\) angle is 100 ft (the length of the shadow), and the height of the tree is the opposite side (\(h\)) to the \(60^\circ\) angle. We use the tangent function, which is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Apply the tangent formula

Given \(\theta = 60^\circ\), adjacent \(= 100\) ft, and \(\tan(60^\circ)=1.732\). So, \(\tan(60^\circ)=\frac{h}{100}\).

Step3: Solve for \(h\)

Rearrange the formula: \(h = 100\times\tan(60^\circ)\). Substitute \(\tan(60^\circ)=1.732\), we get \(h = 100\times1.732 = 173.2\) ft. Wait, no, wait. Wait, the given \(\sin(60^\circ)=0.866\), \(\cos(60^\circ)=0.5\), \(\tan(60^\circ)=1.732\). Wait, but maybe I made a mistake. Wait, the tree is vertical, so the triangle is right-angled, with the shadow as the adjacent side (100 ft), the tree height as opposite side (\(h\)), and the angle of elevation is \(60^\circ\). So \(\tan(\theta)=\frac{opposite}{adjacent}\), so \(h = adjacent\times\tan(\theta)=100\times\tan(60^\circ)=100\times1.732 = 173.2\)? But wait, the options have D as 173.2. But wait, maybe I confused the sides. Wait, no, the distance from the base of the tree to the tip of the shadow is 100 ft, which is the adjacent side. Wait, but let's check again. Wait, the angle of inclination is from the tip of the shadow to the top of the tree, so the adjacent side is 100 ft, opposite is the tree height. So \(\tan(60^\circ)=\frac{h}{100}\), so \(h = 100\times1.732 = 173.2\). But wait, the given \(\sin(60^\circ)=0.866\), maybe using sine? Wait, no, sine is opposite over hypotenuse. If we use sine, we need the hypotenuse. But we know the adjacent side. Wait, maybe the problem is using the wrong function? Wait, no, let's re-express. Wait, the distance from the base of the tree to the tip of the shadow is 100 ft (adjacent), angle is \(60^\circ\), so \(\tan(60^\circ)=\frac{h}{100}\), so \(h = 100\times1.732 = 173.2\), which is option D. But wait, wait, the given \(\sin(60^\circ)=0.866\), maybe the problem is using sine? Wait, no, if we use sine, we need the hypotenuse. The hypotenuse would be \(\frac{100}{\cos(60^\circ)}=\frac{100}{0.5}=200\) ft (since \(\cos(60^\circ)=\frac{adjacent}{hypotenuse}\), so hypotenuse \(=\frac{adjacent}{\cos(60^\circ)}=\frac{100}{0.5}=200\)). Then, using sine, \(\sin(60^\circ)=\frac{opposite}{hypotenuse}\), so opposite \(= hypotenuse\times\sin(60^\circ)=200\times0.866 = 173.2\)? Wait, no, that's the same result. Wait, but maybe I made a mistake. Wait, the problem says "the measurement from the base of a tree to the tip of its shadow is 100 ft". So that's the adjacent side (100 ft), angle of inclination (elevation) is \(60^\circ\). So the height of the tree is the opposite side. So \(\tan(60^\circ)=\frac{h}{100}\), so \(h = 100\times1.732 = 173.2\) ft, which is option D.

Wait, but wait, the given \(\sin(60^\circ)=0.866\), \(\cos(60^\circ)=0.5\), \(\tan(60^\circ)=1.732\). So let's recalculate. If we use \(\tan(60^\circ)=\frac{h}{100}\), then \(h = 100\times1.732 = 173.2\) ft. So the answer should be D.

Answer:

D. 173.2 ft