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Chapter 39: Chapter 39

Problem 1166

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From a point \(5 \mathrm{ft}\). above the horizontal ground, and $30 \mathrm{ft}$. from the trunk of a tree, the line of sight to the top of the tree is measured as \(52^{\circ}\) with the horizontal. Find the height of the tree.

Short Answer

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The height of the tree is approximately 38 feet.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Define the variables

Let's denote the height of the tree as \(h\), the distance from the trunk of the tree to the point as \(d\), the angle of elevation as \(\theta\), and the height of the point as \(p\). In this problem, we have \(d = 30\) ft, \(\theta = 52^\circ\), and \(p = 5\) ft.

Step 2: Find the total length of the tree using tangent formula

Using the tangent formula, we can express the height of the tree, h as: \[\tan{\theta} = \frac{h - p}{d}\] Substitute the given values into the formula: \[\tan{52^\circ} = \frac{h - 5}{30}\]

Step 3: Solve for h

Now, isolate h in the equation: \[h = 30 \cdot \tan{52^\circ} + 5\] Calculate the value of \(\tan{52^\circ}\) and multiply it by 30 and then add 5: \[h = 30 \cdot 1.2799 + 5\] \[h \approx 38.397\]

Step 4: Round the result

Round the result to the nearest whole number: \[h \approx 38\]

Step 5: State the height of the tree

The height of the tree is approximately 38 feet.

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