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Chapter 13: Chapter 13

Problem 248

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A boy knows that his height is \(6 \mathrm{ft}\). and his shadow is $4 \mathrm{ft}\(. long. At the same time of day, a tree's shadow is \)24 \mathrm{ft}$. long. How high is the tree?

Short Answer

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The height of the tree is \(36 \, \mathrm{ft}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Identify the ratio of heights and shadows

Since the shadow is formed by the similar triangles of the boy and the tree, we can write the ratio of their heights to the lengths of their shadows as follows: \[ \frac{\text{Height of Boy}}{\text{Length of Boy's Shadow}} = \frac{\text{Height of Tree}}{\text{Length of Tree's Shadow}} \]

Step 2: Fill in the known values

We know the height of the boy is 6 ft, the length of the boy's shadow is 4 ft, and the length of the tree's shadow is 24 ft. Let the height of the tree be \(h\). We can now fill in the known values in the equation obtained in Step 1: \[ \frac{6}{4} = \frac{h}{24} \]

Step 3: Solve for the height of the tree

To find the height of the tree, \(h\), we can solve the equation obtained in Step 2: \[ \frac{6}{4} = \frac{h}{24} \Rightarrow h = \frac{6}{4} \times 24 \] \[ h = \frac{3}{2} \times 24 = 3 \times 12 = 36 \] So the height of the tree is \(h = 36 \, \mathrm{ft}\).

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