Answer :
Certainly! Let's find the angle the road makes with the horizontal step by step.
### Step 1: Understand the Problem
We are given:
- A 5-mile-long straight segment of a road.
- A hill that is 4000 feet high.
We need to determine the angle the road makes with the horizontal.
### Step 2: Convert Units
Since the hill height is given in feet and the road length is in miles, we should convert the road's length to feet for consistency.
- 1 mile = 5280 feet
- Road length in feet = 5 miles × 5280 feet/mile = 26400 feet
### Step 3: Use Trigonometry
To find the angle, we can use trigonometry. Specifically, we use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{hill height}}{\text{road length}} \][/tex]
- Opposite side (hill height) = 4000 feet
- Adjacent side (road length) = 26400 feet
Calculate [tex]\(\theta\)[/tex] by finding the arctangent:
[tex]\[ \theta = \tan^{-1}\left(\frac{4000}{26400}\right) \][/tex]
### Step 4: Convert the Angle to Degrees
The arctangent function will give the angle in radians. We convert radians to degrees since degrees are a more common unit for measuring angles:
[tex]\[ \text{Angle in degrees} = \text{Angle in radians} \times \left(\frac{180}{\pi}\right) \][/tex]
### Step 5: Conclusion
After performing these calculations, we determine:
- The road makes an angle of approximately [tex]\( 8.62 \)[/tex] degrees with the horizontal.
This is the angle at which the road climbs the hill.
### Step 1: Understand the Problem
We are given:
- A 5-mile-long straight segment of a road.
- A hill that is 4000 feet high.
We need to determine the angle the road makes with the horizontal.
### Step 2: Convert Units
Since the hill height is given in feet and the road length is in miles, we should convert the road's length to feet for consistency.
- 1 mile = 5280 feet
- Road length in feet = 5 miles × 5280 feet/mile = 26400 feet
### Step 3: Use Trigonometry
To find the angle, we can use trigonometry. Specifically, we use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{hill height}}{\text{road length}} \][/tex]
- Opposite side (hill height) = 4000 feet
- Adjacent side (road length) = 26400 feet
Calculate [tex]\(\theta\)[/tex] by finding the arctangent:
[tex]\[ \theta = \tan^{-1}\left(\frac{4000}{26400}\right) \][/tex]
### Step 4: Convert the Angle to Degrees
The arctangent function will give the angle in radians. We convert radians to degrees since degrees are a more common unit for measuring angles:
[tex]\[ \text{Angle in degrees} = \text{Angle in radians} \times \left(\frac{180}{\pi}\right) \][/tex]
### Step 5: Conclusion
After performing these calculations, we determine:
- The road makes an angle of approximately [tex]\( 8.62 \)[/tex] degrees with the horizontal.
This is the angle at which the road climbs the hill.
