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Chapter 44: Chapter 44

Problem 1236

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Find the height of a tree which casts a shadow 20 feet long at the same time a vertical yard stick casts a shadow 30 inches long.

Short Answer

Expert verified
The height of the tree is 24 feet, which we found by setting up a proportion using the given shadow lengths and solving for the unknown height, \(h\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Analyze the given information

We are given that the shadow of the tree is 20 feet long, and the shadow of a yardstick is 30 inches long. The height of the yardstick is 36 inches (1 yard). We will use this information to set up a proportion and solve for the height of the tree.

Step 2: Set up the proportion

Since the height and shadow length of the tree and yardstick form similar triangles, we can set up the following proportion: \[ \frac{\text{Height of tree}}{\text{Shadow of tree}} = \frac{\text{Height of yardstick}}{\text{Shadow of yardstick}} \] Let \(h\) represent the height of the tree. Then our proportion is: \[ \frac{h}{20} = \frac{36}{30} \] Notice that both the height and shadow length of the yardstick are given in inches, so we need to make sure the units are consistent by either converting everything to inches or to feet.

Step 3: Convert units (if necessary)

In our proportion, we have measurements in feet and inches. Let's convert everything to feet to simplify our calculations. We know that there are 12 inches in a foot, so we can convert the height and shadow of the yardstick to feet: Height of yardstick = 36 inches = \(3\) feet Shadow of yardstick = 30 inches = \(2.5\) feet Now our proportion looks like this: \[ \frac{h}{20} = \frac{3}{2.5} \]

Step 4: Solve the proportion

To solve this proportion for \(h\), we can cross-multiply and solve for the unknown variable: \[ 2.5h = 3 \cdot 20 \] \[ 2.5h = 60 \] Now, we can divide both sides by \(2.5\) to get the height of the tree: \[ h = \frac{60}{2.5} \] \[ h = 24 \] So, the height of the tree is 24 feet.

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