2 Heativer is working on a brochure abou hiking irails in Prince Ably National Park. One of the hills on a irall has an angle of elevation of is with a viewpoint 100 m from its base. Imagme that you walk alimg the trail from the base to the viewpoint.
a) How much alutude would you gain?
b) What horizental distance would you cover?
c) What is the grade of the trail, written as a percent?

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UpStudy AI · Accepted Answer

It looks like there's a key piece of information missing from your problem statement: the **angle of elevation** (θ). To solve parts (a), (b), and (c) of your question, we'll need the value of this angle. However, I can provide you with the general formulas and explanations for each part based on trigonometric principles. Once you have the angle, you can plug it into these formulas to get your specific answers.

### Given:
- **Horizontal distance from the base to the viewpoint (adjacent side, $$ A $$)**: 100 meters
- **Angle of elevation ($$ \theta $$)**: *This value is needed*

### a) **Altitude Gain (Opposite side, $$ O $$)**

The altitude gained when hiking up the trail can be calculated using the tangent of the angle of elevation:

$$
O = A \times \tan(\theta)
$$

**Formula:**
$$
\text{Altitude Gain} = 100 \times \tan(\theta) \quad \text{meters}
$$

### b) **Horizontal Distance Covered (Trail Length, $$ H $$)**

Assuming the trail follows the slope directly to the viewpoint, the total distance you walk is the hypotenuse of the right triangle formed by the horizontal distance and the altitude gain. This can be calculated using the cosine of the angle:

$$
H = \frac{A}{\cos(\theta)}
$$

**Formula:**
$$
\text{Trail Length} = \frac{100}{\cos(\theta)} \quad \text{meters}
$$

### c) **Grade of the Trail (Percentage)**

The grade of the trail is a measure of its steepness and is calculated as the ratio of the altitude gain to the horizontal distance, expressed as a percentage:

$$
\text{Grade (\%)} = \left( \frac{O}{A} \right) \times 100 = \tan(\theta) \times 100
$$

**Formula:**
$$
\text{Grade} = \tan(\theta) \times 100 \quad \%
$$

### **Example Calculation**

If, for example, the **angle of elevation is 30 degrees**, here's how you would calculate each part:

1. **Altitude Gain:**
   $$
   O = 100 \times \tan(30^\circ) \approx 100 \times 0.577 = 57.7 \text{ meters}
   $$

2. **Trail Length:**
   $$
   H = \frac{100}{\cos(30^\circ)} \approx \frac{100}{0.866} \approx 115.5 \text{ meters}
   $$

3. **Grade:**
   $$
   \text{Grade} = \tan(30^\circ) \times 100 \approx 0.577 \times 100 \approx 57.7\%
   $$

### **Next Steps**

Please provide the **angle of elevation ($$ \theta $$)** for the hill you're analyzing. With that information, I can help you calculate the specific altitude gain, horizontal distance covered, and the grade of the trail.
To determine the altitude gain, horizontal distance, and trail grade, you need to know the angle of elevation. Once you provide that angle, I can calculate: - **Altitude Gain**: $$ 100 \times \tan(\theta) $$ meters - **Horizontal Distance**: $$ \frac{100}{\cos(\theta)} $$ meters - **Grade**: $$ \tan(\theta) \times 100\% $$ Please provide the angle of elevation to get the specific values.

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