The ground distance a plane has traveled when it reaches a height of 2500 feet can be determined using trigonometric functions. By calculating the hypotenuse using the tangent of the 8-degree angle and the height, and then applying the cosine function, one can find the ground distance 'd' by multiplying the hypotenuse with the cosine of the angle.
To find the ground distance that a plane has traveled when it reaches a height of 2500 feet, we can use trigonometric ratios. Since the angle of elevation is 8 degrees, we're looking for the adjacent side of a right triangle (the ground distance) where we know the opposite side (the height of the plane). The trigonometric function we need is the cosine, which relates the adjacent side to the hypotenuse of a right triangle.
The cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:
cos(angle) = adjacent/hypotenuse
Let's call the ground distance 'd'. We can rearrange the formula to solve for 'd':
d = hypotenuse * cos(angle)
To find the hypotenuse (the slant distance the plane has traveled), we use the tangent function because we know the opposite side (height) and the angle:
tan(angle) = opposite/adjacent (hypotenuse in this context)
hypotenuse = opposite/tan(angle)
Substituting the values we have:
hypotenuse = 2500 / tan(8 degrees)
Then we calculate 'd' using the cosine:
d = hypotenuse * cos(8 degrees)
Replace hypotenuse with the value calculated above to get the final value of 'd'.