A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the trie as . How accurately must the angle be
measured if the percent errar in estimating the height of the tree is to be less than ? (Round your answer to three decimal places.)
radians
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Ask by Wheeler Griffiths. in the United States

Mar 31,2025

Question

A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the trie as . How accurately must the angle be
measured if the percent errar in estimating the height of the tree is to be less than ? (Round your answer to three decimal places.)
radians
Need Help? Readit

Ask by Wheeler Griffiths. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

Let the height of the tree be given by

$$
h = 50 \tan \theta,
$$

where $$\theta = 65.3^\circ$$. We wish to know how accurately the angle must be measured so that the percent error in $$h$$ is less than $$3\%$$.

Step 1. Differentiate $$h$$ with respect to $$\theta$$.

$$
\frac{dh}{d\theta} = 50 \sec^2 \theta.
$$

Step 2. Using linear approximation for small errors, a small error $$d\theta$$ in $$\theta$$ produces an error in $$h$$ of

$$
dh \approx 50 \sec^2 \theta\, d\theta.
$$

Step 3. The relative error in $$h$$ is approximately

$$
\frac{dh}{h} \approx \frac{50 \sec^2 \theta\, d\theta}{50 \tan \theta} = \frac{\sec^2 \theta}{\tan \theta}\, d\theta.
$$

Step 4. We require that

$$
\left|\frac{dh}{h}\right| \leq 0.03.
$$

Thus,

$$
\left|\frac{\sec^2 \theta}{\tan \theta}\, d\theta\right| \leq 0.03.
$$

Step 5. Solving for $$|d\theta|$$ gives

$$
|d\theta| \leq 0.03\, \frac{\tan \theta}{\sec^2 \theta}.
$$

Step 6. Now, substitute $$\theta = 65.3^\circ$$. We first compute:

$$
\tan 65.3^\circ \approx 2.174, \quad \text{and} \quad \sec^2 65.3^\circ = 1 + \tan^2 65.3^\circ \approx 1 + (2.174)^2 \approx 1 + 4.727 \approx 5.727.
$$

Step 7. Substitute these values:

$$
|d\theta| \leq 0.03\, \frac{2.174}{5.727} \approx 0.03 \times 0.380 \approx 0.0114 \text{ radians}.
$$

Rounded to three decimal places, the required accuracy is

$$
|d\theta| \leq 0.011 \text{ radians}.
$$
The angle must be measured with an accuracy of $$ |d\theta| \leq 0.011 $$ radians to ensure the height estimation error is less than 3%.

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