Math Problem Statement

Zachary is 1.85 meters tall. At 1 p.m., he measures the length of a tree's shadow to be 37.25 meters. He stands 33 meters away from the tree, so that the tip of his shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

Solution

To solve this problem, we can use the concept of similar triangles. Zachary and the tree, along with their respective shadows, form two similar right triangles.

Step 1: Understand the Setup

  • Zachary's height: 1.85 meters.
  • The length of Zachary's shadow: We can find this using the fact that the tip of his shadow meets the tip of the tree's shadow. The total shadow length of the tree is 37.25 meters, and Zachary is standing 33 meters away from the tree.

Let’s define:

  • hh = the height of the tree.
  • The total length of the tree's shadow: 37.25 meters.
  • Zachary's shadow length = Total length of the tree's shadow – Zachary's distance from the tree = 37.2533=4.2537.25 - 33 = 4.25 meters.

Step 2: Set Up the Proportions

Since the two triangles are similar, the ratio of height to shadow length for Zachary is the same as the ratio of height to shadow length for the tree:

htreeshadow length of tree=hZacharyshadow length of Zachary\frac{h_{\text{tree}}}{\text{shadow length of tree}} = \frac{h_{\text{Zachary}}}{\text{shadow length of Zachary}}

Substituting the known values:

h37.25=1.854.25\frac{h}{37.25} = \frac{1.85}{4.25}

Step 3: Solve for hh (the height of the tree)

To find hh, we solve the equation:

h=1.854.25×37.25h = \frac{1.85}{4.25} \times 37.25

h=0.4353×37.25h = 0.4353 \times 37.25

h16.23 metersh \approx 16.23 \text{ meters}

Final Answer:

The height of the tree is approximately 16.23 meters to the nearest hundredth of a meter.

Would you like further clarification or details? Here are some related questions you might find interesting:

  1. How would the tree's height change if Zachary's distance from the tree were different?
  2. What other real-world applications use similar triangles in problem-solving?
  3. Can you calculate the length of the shadow of any object given its height and the length of a shadow for another object under the same conditions?
  4. How would the problem change if Zachary’s shadow were twice as long as the tree’s shadow?
  5. Can the concept of similar triangles be used in three-dimensional geometry?

Tip: When solving real-world problems with geometry, always try to visualize the scenario with a diagram to better understand the relationships between the different elements!

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Math Problem Analysis

Mathematical Concepts

Similar Triangles
Proportions
Ratios

Formulas

\frac{h_{\text{tree}}}{\text{shadow length of tree}} = \frac{h_{\text{Zachary}}}{\text{shadow length of Zachary}}

Theorems

Similarity of Triangles

Suitable Grade Level

Grades 7-9