Question

a tree standing vertically on level ground casts a 47 foot long shadow. if the angle of elevation from the end of the shadow to the top of the tree is 46.5°, find the height of the tree. round your answer to one decimal place. feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the shadow length (adjacent side) is 47 feet, the height of the tree (opposite side) is \( h \), and the angle of elevation is \( 46.5^\circ \). We use the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).
So, \( \tan(46.5^\circ)=\frac{h}{47} \).

Step2: Solve for \( h \)

Multiply both sides by 47: \( h = 47\times\tan(46.5^\circ) \).
Calculate \( \tan(46.5^\circ)\approx1.057 \).
Then \( h\approx47\times1.057 = 49.679 \approx 49.7 \) (rounded to one decimal place).

Answer:

49.7