Question

45. A tree casts a shadow that is 80 ft. long when the angle of elevation of the sun is 27º. Find the height of the tree.Round your answer to the nearest tenth.Height = ft.

Answer 1

To properly answer the problem, it is best that we draw an illustration representing the problem in order for us to understand the problem, according to the problem, we have a tree facing the sun thus creating a shadow on the gorund.

In our problem it is stated that the tree cast a SHADOW of 80ft long, and has an Angle of Elevation of 27°. Therefore we can add measurements to our drawing, giving us;

Notice that the TREE, THE SHADOW, and THE LINE FORMED BY THE ANGLE OF ELEVATION creates a shape called a RIGHT TRIANGLE with a measurements of;

Since we have a right triangle we can now use the SOHCAHTOA properties of a right triangle, and SOHCAHTOA means;

(SOH) Sine of an Angle = Opposite side / Hypotheneus

(CAH) Cosine of an Angle = Adjacent side / Hypotheneus

(TOA) Tangent of an Angle = Opposite side / Adjacent side

An for our problem, the angle 27° has an opposite side which is the TREE (a), it has an adjacent side which is the SHADOW (b), and its Hypotheneus which is the Line formed by the angle of elevation (c).

Therefore in order to find the height of the TREE, we can use the CAH property since our Adjacent side is given while the Hypotheneus remains unknown. So we have;

`\begin{gathered} Tangent\text{ of an Angle = }\frac{Opposite\text{ Side}}{\text{Adjacent Side}} \\ Tan27^(\circ)=\frac{Tree}{\text{Shadow}} \\ \text{Tan 27}^(\circ)=\frac{Tree}{80\text{ ft}} \\ \text{Tre}e=(Tan27)(80ft) \\ \text{Tre}e=(0.5095)(80ft) \\ \text{Tre}e=40.7620ft \\ \text{Tre}e=40.8ft \end{gathered}`

Therefore by using the SOHCAHTOA property of a right triangle we found that the height of the TREE is 40.8 ft.