An observer 2.5 miles from the launch pad of a space shuttle measures the angle of elevation to the base of the shuttle to be 25 degrees soon after lift off. How high is the shuttle at that instant? (Assume that the shuttle is still moving vertically)
tan 25° = height/2.5
height = 2.5tan25° = .....
To determine the height of the shuttle at that instant, we can use trigonometry and the concept of the tangent function.
Let's consider a right-angled triangle with the observer, the base of the shuttle, and the height of the shuttle. The angle of elevation, in this case, is the angle between the line of sight from the observer to the base of the shuttle and the horizontal ground.
Given that the angle of elevation is 25 degrees, we can define the tangent of the angle as:
tan(25°) = height of the shuttle / distance from the observer to the base of the shuttle
We know that the distance from the observer to the base of the shuttle is 2.5 miles. However, we need to convert this distance into the same unit as the height of the shuttle.
Next, we rearrange the equation to solve for the height of the shuttle:
height of the shuttle = tan(25°) * distance from the observer to the base of the shuttle
Now, let's calculate the height using this equation:
Step 1: Convert the distance from miles to a more appropriate unit such as feet. Since 1 mile is equal to 5280 feet, we have:
distance from the observer to the base of the shuttle = 2.5 miles * 5280 feet/mile
Step 2: Calculate the height using the tangent of 25°:
height of the shuttle = tan(25°) * distance from the observer to the base of the shuttle
Substituting the values:
height of the shuttle ≈ tan(25°) * (2.5 miles * 5280 feet/mile)
Now you can simplify and calculate the height using a scientific calculator or an online trigonometric calculator.
Note: Make sure your calculator is set to degrees mode when evaluating trigonometric functions.
By following these steps, you will be able to determine the height of the space shuttle at that instant.