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Problem 1166

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.

Expert verified
The height of the tree is approximately 38 feet.
See the step by step solution

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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