https://file.aiquickdraw.com/mathbot/file/1726222322998fa391ywc.jpg

Let's solve each problem step-by-step:

### Problem 13
**Given:**
- Sun’s elevation angle: \( 65^\circ \)
- Height of the boy: \( 2.0 \, \text{m} \)

**To Find:**
- Length of the shadow cast by the boy.

Using the trigonometric relation:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height of the boy}}{\text{length of the shadow}}
\]

\[
\tan(65^\circ) = \frac{2.0 \, \text{m}}{\text{length of the shadow}}
\]

Solving for the length of the shadow:

\[
\text{length of the shadow} = \frac{2.0 \, \text{m}}{\tan(65^\circ)}
\]

Now, calculating the length:

\[
\text{length of the shadow} = \frac{2.0}{\tan(65^\circ)} \approx \frac{2.0}{2.1445} \approx 0.93 \, \text{m}
\]

So, the length of the shadow is approximately **0.9 m**, matching the given answer.

### Problem 14
**Given:**
- One town is \( 35.0 \, \text{km} \) south.
- The other town is \( 72.0 \, \text{km} \) west.

**To Find:**
- Shortest length of the highway.
- Direction of the highway.

Using the Pythagorean theorem:

\[
\text{length} = \sqrt{(35.0)^2 + (72.0)^2}
\]

Calculating the shortest distance:

\[
\text{length} = \sqrt{1225 + 5184} = \sqrt{6409} \approx 80.1 \, \text{km}
\]

To find the angle of the highway, we use:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{35.0}{72.0}
\]

Calculating the angle:

\[
\theta = \tan^{-1}\left(\frac{35.0}{72.0}\right) \approx 26.1^\circ
\]

The highway's direction would be \( 90^\circ - 26.1^\circ = 64.1^\circ \) towards the southwest.

So, the shortest length of the highway is approximately **80.1 km**, directed **64.1° SW**, matching the given answers.

---

Would you like any further details or have any questions?

### Related Questions:
1. How would the shadow length change if the sun's elevation angle was lower?
2. What is the formula for calculating the angle of elevation or depression in trigonometry?
3. How would the highway's length and direction change if one town were further south or west?
4. Can you calculate the length of the shadow for different heights of objects?
5. What would be the effect on the shortest highway length if the distance between the towns changed?

**Tip:** Always make sure to double-check the units and make sure they are consistent throughout your calculations!

13. If the sun’s elevation angle above the ground is 65°, estimate the length of the shadow cast by a 2.0 m tall boy. [0.9m] 14. A highway is to be built between two towns, one of which lies 35.0 km south and 72.0 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed? [80.1km, 64.1°SW]

Explore how to estimate the length of a shadow cast by a boy and determine the shortest path and angle for a highway using trigonometry. This guide covers the use of the tangent function and the Pythagorean theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation