Question

A man is 5 feet 9 inches tall. To find the height of a tree, the shadow of the man and the shadow of the tree were measured. The length of the man's shadow was 3 feet 6 inches. The length of the tree's shadow was 10 feet 6 inches. What is the height of the tree?

Answer 1

17.25 ft

Explanation:

The figure shows two triangles X Y Z and H R Q. Side X Y has a length of h units, side Y Z has a length of 10 feet 6 inches, angle Y is a right angle. Side H R has a length of 5 feet 9 inches, side R Q has a length of 3 feet 6 inches. Angle R is a right angle.

Let h be the height of the tree.

Convert all measurements to a single unit, inches, using the fact that 1 foot is equal to 12 inches.

ZY=(10⋅12) in.+6 in.

Simplify.

ZY=126 in.

QR=(3⋅12) in.+6 in.

Simplify.

QR=42 in.

HR=(5⋅12) in.+9 in.

Simplify.

HR=69 in.

ZX¯¯¯¯¯ and QH¯¯¯¯¯¯ are parallel because the rays of the Sun that form them are parallel. Therefore, the angles they form with the ground are congruent: ∠Z≅∠Q.

∠Y≅∠R by the Right Angle Congruence Theorem.

Therefore, by the Angle-Angle Similarity △ZXY~△QHR.

Find XY. Set up a proportion with the lengths of the known sides and the unknown length (height of the tree).

ZY/QR = XY/HR

Substitute the values.

126/42=h/69

Cross multiply.

42h=69(126)

Simplify.

42h=8694

Divide both sides by 42.

h=207 in.

Therefore, the height of the tree is 207 in., or 17.25 ft.

Answer 2

17 feets 3 inches

Explanation:

Given :

Converting the parameters to inches only:

Recall:

1 feets = 12 inches

Man's height : 5 feets 9 inches = (12 *5) + 9 = 69 inches

Man's shadow : 3 feets 6 inches = (12 * 3) + 6 = 42 inches

Tree's shadow : 10 feets 6 inches = (12 * 10) + 6 = 126 inches

(Tree's height / tree's shadow) = (man's height / man's shadow)

Let tree's height = x

x / 126 = 69 / 42

Solve for x

42x = 126 * 69

42x = 8694

x = 8694 / 42

x = 207 inches

207 / 12 = 17 feets 3 inches