To solve this problem, we need to rely on proportional reasoning using the similar triangles created by the trees and their shadows.

Step-by-Step Breakdown:

1. Determine the Ratios: Since the triangles formed by the trees and their shadows are proportional, the ratio of the shadow length to the tree height for both the young tree and the full-size tree must be the same.

Let:

• $h_y$ be the height of the young tree
• $h_f$ be the height of the full-size tree
• $s_y$ be the shadow length of the young tree
• $s_f$ be the shadow length of the full-size tree

We can set up the proportional relationship as:

$\frac{s_y}{h_y} = \frac{s_f}{h_f}$

Given that both ratios are equal, we can use this to solve for the unknown heights, provided we have some measurements.

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Example Solutions:

1. Unsimplified Ratios:

• The unsimplified ratios are the raw fractions of shadow length over tree height, based on measurements taken by Christianne.

$\text{Ratio of Young Tree: } \frac{s_y}{h_y}$ $\text{Ratio of Full-Size Tree: } \frac{s_f}{h_f}$

2. Simplified Ratios:

• The simplified ratios are the fractions reduced to their simplest form by dividing the numerator and denominator by their greatest common divisor (GCD).

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3. Find the Height of the Full-Size Tree:

Suppose Christianne measured the young tree to be $h_y$ feet and we know the shadow lengths $s_y$ and $s_f$ (shadow of the full-size tree). We can solve for $h_f$ using the proportion:

$\frac{s_y}{h_y} = \frac{s_f}{h_f}$

Solving for $h_f$ (the height of the full-size tree):

$h_f = \frac{s_f \cdot h_y}{s_y}$

Plug in the values for $s_y$, $h_y$, and $s_f$ to find $h_f$, and then round the answer to two decimal places.

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4. Find the Height of the Young Tree:

If Christianne determines that the full-size tree (or house) is $h_f$ feet tall, and we know the shadow lengths, we can use the same proportion to solve for the height of the young tree:

$h_y = \frac{s_y \cdot h_f}{s_f}$

Substitute the known values and round the result to two decimal places.

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Let me know if you have the specific measurements, and I can help calculate the exact ratios and heights for you.

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5 Related Questions:

1. What is the significance of proportional reasoning in solving geometric problems?
2. How does the concept of similar triangles apply to real-life situations like measuring tree heights?
3. Can you explain how to simplify a ratio?
4. How do changes in the shadow length affect the height calculations of the tree?
5. How can you check if two triangles are similar using proportional reasoning?

Tip: Always remember that in proportional reasoning, corresponding sides of similar triangles must have equal ratios!