4.4 A car rides a slope of $$ \frac{1}{8} $$ for a distance of 100 m . Calculate the perpendicular height with which the car rose and the angle of elevation forming.

Question

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To solve this problem, we need to find two things:

1. The perpendicular height (h) that the car rose.
2. The angle of elevation (θ) of the slope.

### Step 1: Calculate the Perpendicular Height

The slope of $$ \frac{1}{8} $$ means that for every 8 units of horizontal distance, the vertical height increases by 1 unit.

Given that the car rides a distance of 100 m along the slope, we can use the slope ratio to find the height:

$$
\text{slope} = \frac{\text{height}}{\text{horizontal distance}} = \frac{1}{8}
$$

Let $$ d $$ be the horizontal distance. The relationship can be expressed as:

$$
h = \frac{1}{8} \cdot d
$$

To find $$ d $$, we can use the Pythagorean theorem, where the slope distance (100 m) is the hypotenuse:

$$
d^2 + h^2 = 100^2
$$

Substituting $$ h $$ into the equation gives us:

$$
d^2 + \left(\frac{1}{8}d\right)^2 = 100^2
$$

### Step 2: Solve for $$ d $$

Let's simplify and solve for $$ d $$:

$$
d^2 + \frac{1}{64}d^2 = 10000
$$

Combining the terms:

$$
\left(1 + \frac{1}{64}\right)d^2 = 10000
$$

$$
\frac{65}{64}d^2 = 10000
$$

Now, solving for $$ d^2 $$:

$$
d^2 = 10000 \cdot \frac{64}{65}
$$

Now, let's calculate $$ d^2 $$ and then find $$ h $$.

### Step 3: Calculate $$ h $$

After finding $$ d $$, we can calculate $$ h $$ using the relationship $$ h = \frac{1}{8} \cdot d $$.

### Step 4: Calculate the Angle of Elevation

The angle of elevation can be calculated using the tangent function:

$$
\tan(\theta) = \frac{h}{d}
$$

Thus,

$$
\theta = \tan^{-1}\left(\frac{h}{d}\right)
$$

Now, let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$10000\times \frac{64}{65}$$
- step1: Reduce the numbers:
  $$2000\times \frac{64}{13}$$
- step2: Multiply:
  $$\frac{2000\times 64}{13}$$
- step3: Multiply:
  $$\frac{128000}{13}$$
Calculate or simplify the expression $$ \sqrt(10000 * (64/65)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{10000\times \frac{64}{65}}$$
- step1: Multiply the numbers:
  $$\sqrt{\frac{128000}{13}}$$
- step2: Use the properties of radicals:
  $$\frac{\sqrt{128000}}{\sqrt{13}}$$
- step3: Simplify the expression:
  $$\frac{160\sqrt{5}}{\sqrt{13}}$$
- step4: Multiply by the Conjugate:
  $$\frac{160\sqrt{5}\times \sqrt{13}}{\sqrt{13}\times \sqrt{13}}$$
- step5: Multiply the numbers:
  $$\frac{160\sqrt{65}}{\sqrt{13}\times \sqrt{13}}$$
- step6: Multiply the numbers:
  $$\frac{160\sqrt{65}}{13}$$
Calculate or simplify the expression $$ (1/8) * \sqrt(10000 * (64/65)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{8}\sqrt{10000\times \frac{64}{65}}$$
- step1: Multiply the numbers:
  $$\frac{1}{8}\sqrt{\frac{128000}{13}}$$
- step2: Simplify the root:
  $$\frac{1}{8}\times \frac{160\sqrt{65}}{13}$$
- step3: Reduce the numbers:
  $$1\times \frac{20\sqrt{65}}{13}$$
- step4: Multiply:
  $$\frac{20\sqrt{65}}{13}$$
Calculate or simplify the expression $$ \arctan(12.403473 / 99.227788) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\arctan\left(\frac{12.403473}{99.227788}\right)$$
- step1: Divide the terms:
  $$\arctan\left(\frac{12403473}{99227788}\right)$$
### Final Calculations

1. **Horizontal Distance $$ d $$**:
   $$
   d \approx 99.23 \text{ m}
   $$

2. **Perpendicular Height $$ h $$**:
   $$
   h \approx 12.40 \text{ m}
   $$

3. **Angle of Elevation $$ \theta $$**:
   $$
   \theta \approx 0.1244 \text{ radians} \quad \text{(which is approximately } 7.13^\circ\text{)}
   $$

### Summary of Results
- The perpendicular height with which the car rose is approximately **12.40 m**.
- The angle of elevation forming is approximately **7.13 degrees**.
The car rose approximately 12.40 meters and the angle of elevation is about 7.13 degrees.

4.4 A car rides a slope of for a distance of 100 m . Calculate the perpendicular
height with which the car rose and the angle of elevation forming.

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Solución

Step 1: Calculate the Perpendicular Height

Step 2: Solve for

Step 3: Calculate

Step 4: Calculate the Angle of Elevation

Final Calculations

Summary of Results

Bonus Knowledge

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4.4 A car rides a slope of for a distance of 100 m . Calculate the perpendicular height with which the car rose and the angle of elevation forming.