A man is 6 feet 2 inches tall. To find the height of a tree, the shadow of the man and the shadow of the tree were measured.?

The man (as the height) and its shadow (as the base) form a right-angled triangle (triangle 1) with angle of elevation θ.
The tree (as the height) and its shadow (as the base) also form a right-angled triangle (triangle 2) with angle of elevation θ.

Let h in be the height of the tree.

In triangle 1 :
tanθ = (6 ft 2 in) / (2 ft 1 in)
tanθ = (74 in) / (25 in)
tanθ = 74/25 ...... [1]

In triangle 2 :
tanθ = (h in) / (3 ft 7 in)
tanθ = (h in) / (43 in)
tanθ = h/43 ...... [2]

[2] = [1] :
h/43 = 74/25
h = 43 × (74/25)

Height of the tree
= 43 × (74/25) in
= [43 × (74/25) in] × [(1 ft) / (12 in)]
= 10.6 ft

None of the four options is the answer.


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If the man is 6 ft 3 inches tall instead, then :

In triangle 1 :
tanθ = (6 ft 3 in) / (2 ft 1 in)
tanθ = (75 in) / (25 in)
tanθ = 3 ...... [1]

In triangle 2 :
tanθ = (h in) / (3 ft 7 in)
tanθ = (h in) / (43 in)
tanθ = h/43 ...... [2]

[2] = [1] :
h/43 = 3
h = 43 × 3
h = 129

Height of the tree
= 129 in
= (129 in) × [(1 ft) / (12 in)]
= 10.75 ft

The answer is : 10.75 ft

Are you sure on the numbers? I'm going to actually assume that the man is 6' 3" tall. That makes him exactly 3 times the length of his shadow. And then it makes the height of the tree equal to 3 times *its* shadow.

Multiply the tree's shadow by 3:
(3 feet 7 inches) x 3
= 9 feet 21 inches
= 10 feet 9 inches
= 10 9/12 feet
= 10.75 feet

Or if you want to be more formal, set up a proportion:

Tree's height ..... Man's height
--------------------- = ---------------------
Tree's shadow .. Man's shadow

Let x represent the tree's height (in inches)
The tree's shadow is 43 inches (3' 7")
The man's height is 75 inches (6' 3")
The man's shadow is 25 inches (2' 1")

Plug that into the proportion:
x/43 = 75/25
x/43 = 3

Multiply both sides by 43 to isolate x:
x = 43 * 3
x = 129 inches

To convert that back to feet, divide by 12:
129 / 12
= 10.75 feet

These form similar right triangles. The ratio of height to length of shadow is the same. Work in inches.

man :: tree
74/25 = x/43
x = 43(74/25) = 127.28" = 10' 7.28"
= 10.6'

Suppose the man is 6'3" instead of 6'2".

75/25 = x/43
x = 3(43) = 129" = 10'9" = 10 3/4 ft

Explanation: The sun is so far away that its rays are assumed to be parallel. Thus the angle from the end of the shadow to the top of the object is the same. The objects are assumed perpendicular to the ground. Thus similar by two corresponding angles.

74 / 25 = x / 43
x = 43 x 74 / 25 ins
x = 127.28 ins
x = 10.6 ft _______height of tree

To work in whole numbers, I'd convert to inches.
man's height = 6x12 + 2 = 74 inches
man's shadow = 2x12 + 1 = 25 inches
tree's shadow = 3x12 + 7 = 43 inches

The heights and shadows are in the same propoportions, with the tree's height as the unknown value x.

(tree's height)/(tree's shadow) = (man's height)/(man's shadow)
x / 43 = 74 / 25
x = 43 x 74 / 25 = 3182 / 25 inches

Divide that by 12 to get 3182/300 feet. About 10.61 feet, and that's not on your list.

Lazy do it yourself...