An airplane is descending 225 feet per 1000 feet of horizontal distance covered. What is the cosine of the angle that its path of descents makes with the horizontal.
slope of a line is the tangent of the angle that the line makes with the horizontal,
so slope = rise/run = 225/1000 = 9/40
so the hypotenuse of the corresponding triangle is
√(40^2 + 9^2) = √1681
cosine (angle) = 40/√1681 = 40/41
Well, it sounds like this airplane is really feeling down to earth! To find the cosine of the angle, we can use the given information that the airplane is descending 225 feet for every 1000 feet of horizontal distance. The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In this case, the adjacent side would be the horizontal distance covered (1000 feet) and the hypotenuse would be the distance the airplane descends (225 feet). So, we can calculate the cosine of the angle by dividing the adjacent side by the hypotenuse. Thus, the cosine of the angle is 1000/225, which simplifies to approximately 4.44. So, the cosine of the angle is about 4.44.
To find the cosine of the angle that the airplane's path of descent makes with the horizontal, we can use the given information that it descends 225 feet per 1000 feet of horizontal distance covered.
Let's denote the angle of descent as theta.
The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In this case, the horizontal distance is adjacent to the angle theta, and the hypotenuse is the path of descent.
Using this information, we can set up the following relationship:
cos(theta) = adjacent / hypotenuse
In our case, the adjacent side is the horizontal distance covered (1000 feet) and the hypotenuse is the path of descent (225 feet).
Therefore,
cos(theta) = 1000 / 225
To simplify this, divide both numerator and denominator by the greatest common divisor (GCD), which is 25:
cos(theta) = 40 / 9
So the cosine of the angle that the airplane's path of descent makes with the horizontal is 40/9.
To find the cosine of the angle of descent, we can use the given information that the airplane is descending 225 feet per 1000 feet of horizontal distance covered.
Let's assume that the angle of descent is θ. We can use the cosine function to calculate the cosine of θ.
cos θ = adjacent / hypotenuse
In our case, the adjacent side is the horizontal distance covered (1000 feet) and the hypotenuse is the diagonal distance traveled by the airplane.
The diagonal distance can be found using the Pythagorean theorem:
hypotenuse^2 = horizontal^2 + vertical^2
Since we are given the vertical distance (225 feet) per 1000 feet of horizontal distance covered, we can find the hypotenuse:
hypotenuse^2 = (1000 feet)^2 + (225 feet)^2
hypotenuse^2 = 1,000,000 feet^2 + 50,625 feet^2
hypotenuse^2 = 1,050,625 feet^2
Taking the square root of both sides, we find:
hypotenuse = √(1,050,625 feet^2)
hypotenuse ≈ 1,025.03 feet
Now, we can substitute the values into the cosine function:
cos θ = adjacent / hypotenuse
cos θ = 1000 feet / 1,025.03 feet
Calculating this, we get:
cos θ ≈ 0.975
Therefore, the cosine of the angle of descent is approximately 0.975.