a plane flying at 33,000 feet is 13o miles away from the airport when it begins to descent.find the angleof depression of the plane to the starting point of the airport
Draw the diagram. It should be clear that if the 130 miles is along the ground, then
tanθ = (33000/5280) / 130
Otherwise, if the 130 miles is line-of-sight,
sinθ = (33000/5280) / 130
To find the angle of depression of the plane to the starting point of the airport, we need to use the tangent function. The angle of depression is the angle between the line of sight from the observer (in this case, the starting point of the airport) to the object (the plane) and a horizontal line.
Let's assume the starting point of the airport is at point A, and the plane is at point P. We have a right triangle formed with the vertical side AP, the horizontal side AB, and the hypotenuse BP.
We know the distance AP is 130 miles, and the height BP is 33,000 feet. However, we need to convert the height to miles to match the units. Since 1 mile is approximately 5280 feet, the height BP in miles is:
BP = 33,000 feet / 5280 feet/mile
BP ≈ 6.25 miles
Now we can apply the tangent function:
tan(angle) = opposite/adjacent
tan(angle) = AP/AB
Substituting the given values:
tan(angle) = 130 miles/6.25 miles
Dividing numerator and denominator:
tan(angle) ≈ 20.8
Now we need to find the inverse tangent (or arctan) of 20.8 to get the angle:
angle ≈ arctan(20.8)
Using a calculator or trigonometric table, we find:
angle ≈ 89.4 degrees
Therefore, the angle of depression of the plane to the starting point of the airport is approximately 89.4 degrees.