Answer:To find the vertical change in feet when a car travels up a slight grade with an angle of elevation of 2 degrees for 1 mile, you can use trigonometry. The vertical change (height) can be calculated using the formula:
Vertical Change (in feet) = Horizontal Distance (in miles) * Tan(Angle of Elevation)
In this case, the horizontal distance is 1 mile, and the angle of elevation is 2 degrees. First, you'll need to convert the angle from degrees to radians since most calculators use radians for trigonometric functions:
Angle in radians = Angle in degrees * (π / 180)
Angle in radians = 2 degrees * (π / 180) = 0.0349 radians (approximately)
Now, you can calculate the vertical change:
Vertical Change (in feet) = 1 mile * Tan(0.0349 radians)
Vertical Change (in feet) ≈ 1 mile * 0.0619
Vertical Change (in feet) ≈ 0.0619 miles
To convert the vertical change from miles to feet, multiply by the number of feet in a mile (1 mile = 5280 feet):
Vertical Change (in feet) ≈ 0.0619 miles * 5280 feet/mile
Vertical Change (in feet) ≈ 326.832 feet
So, when the car travels up the slight grade with an angle of elevation of 2 degrees for 1 mile, the vertical change is approximately 326.832 feet.
Step-by-step explanation:Certainly, here's a step-by-step explanation of how to calculate the vertical change in feet when a car travels up a slight grade with an angle of elevation of 2 degrees for 1 mile using trigonometry:
Step 1: Convert the Angle to Radians
The angle of elevation is given in degrees (2 degrees), but trigonometric functions typically use radians. To convert from degrees to radians, you use the following formula:
Angle in radians = Angle in degrees * (π / 180)
So, in this case:
Angle in radians = 2 degrees * (π / 180) = 0.0349 radians (approximately)
Step 2: Use the Tangent Function
The tangent (tan) function relates the angle of elevation to the vertical change and horizontal distance. The formula for finding the vertical change is:
Vertical Change (in feet) = Horizontal Distance (in miles) * Tan(Angle in radians)
Step 3: Plug in the Values
You have the following values:
- Horizontal Distance (in miles) = 1 mile
- Angle in radians = 0.0349 radians (from Step 1)
Plug these values into the formula:
Vertical Change (in feet) = 1 mile * Tan(0.0349 radians)
Step 4: Calculate the Tangent
Calculate the tangent of the angle using a calculator:
Tan(0.0349 radians) ≈ 0.0619
Step 5: Calculate the Vertical Change
Now, use the calculated tangent value in the formula to find the vertical change:
Vertical Change (in feet) = 1 mile * 0.0619 ≈ 0.0619 miles
Step 6: Convert Miles to Feet
To get the vertical change in feet, you need to convert from miles to feet. There are 5280 feet in 1 mile, so:
Vertical Change (in feet) ≈ 0.0619 miles * 5280 feet/mile ≈ 326.832 feet
So, the vertical change when the car travels up the slight grade with an angle of elevation of 2 degrees for 1 mile is approximately 326.832 feet.