Math Problem Statement

A pole 33 m tall casts a shadow 44 m long. A nearby tree casts a 1515 m shadow as shown in the diagram.

What is the height of the tree? 

A.8.328.32 m

B.11.2511.25 m

C.13.0013.00 m

D.21.0221.02 m

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Solution

To solve this problem, we can use similar triangles. The pole and the tree, along with their shadows, form two right-angled triangles. Since both the pole and the tree are standing in the same sunlight, the angle of elevation of the sun is the same for both, meaning their triangles are similar.

Step 1: Set up the proportion

Let the height of the tree be hh meters. The pole's height is 33 meters, and the pole's shadow is 44 meters long. The tree's shadow is 1515 meters long.

We can set up the proportion as follows:

Height of PoleLength of Pole’s Shadow=Height of TreeLength of Tree’s Shadow\frac{\text{Height of Pole}}{\text{Length of Pole's Shadow}} = \frac{\text{Height of Tree}}{\text{Length of Tree's Shadow}}

Substitute the known values:

3344=h1515\frac{33}{44} = \frac{h}{1515}

Step 2: Solve for hh

Cross-multiply to solve for hh:

33×1515=44×h33 \times 1515 = 44 \times h

49995=44h49995 = 44h

Now, divide both sides by 44 to find hh:

h=4999544=1136.75h = \frac{49995}{44} = 1136.75

Thus, the height of the tree is approximately 1136.75 meters.

However, there seems to be an error. I have incorrectly made the numbers seem unreasonable. Let me clarify

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportions

Formulas

Proportion formula: height of pole / length of pole's shadow = height of tree / length of tree's shadow

Theorems

Similar triangles theorem

Suitable Grade Level

Grades 7-9