An airplane flies at an altitude of y = 7 miles toward a point directly over an observer (see figure). The speed of the plane is 700 miles per hour. Find the rates at which the angle of elevation π is changing when the angle is π = 30Β°,
π = 60Β°, and π = 80Β°
draw a diagram. You can see that when the plane is x feet away horizontally,
tanΞΈ = 7/x
sec^2ΞΈ dΞΈ/dt = -7/x^2 dx/dt
so plug in your numbers, knowing that dx/dt = -700
im still not getting the right answer
so show your work, and let's see what's up.
Maybe I made a mistake.
To find the rates at which the angle of elevation, π, is changing, we can use trigonometry and differentiate the given equation. Let's start with the equation of the tangent function.
The tangent of an angle is defined as the ratio of the opposite side of the triangle to the adjacent side. In this case, the opposite side would be the altitude, y, and the adjacent side would be the horizontal distance covered by the airplane, x.
Using trigonometry, we have:
tan(π) = y / x
Since the altitude, y, is given as 7 miles, we can rewrite the equation as:
tan(π) = 7 / x
To find the rates at which π is changing with respect to time, we need to differentiate both sides of this equation.
d/dt (tan(π)) = d/dt (7 / x)
Now, let's find the derivatives of both sides.
First, recall that the derivative of tan(π) is sec^2(π). Similarly, the derivative of 7 / x can be found using the quotient rule.
d/dt (tan(π)) = sec^2(π) * dπ/dt
And applying the quotient rule to the right side:
d/dt (7 / x) = (x * d(7) / dt - 7 * dx / dt) / x^2
Since dx / dt represents the rate at which x is changing, we can replace it with the speed of the plane, which is given as 700 miles per hour.
Now, substitute the values we know into the equation. For example, when π = 30Β°, the tangent of 30Β° is sqrt(3) / 3.
sec^2(30Β°) * dπ/dt = (700 * (7) - 7 * 700) / x^2
To find x^2, we can use the Pythagorean theorem.
x^2 + y^2 = h^2, where h is the hypotenuse, which in this case is given as 7 miles.
x^2 + 7^2 = 7^2
x^2 + 49 = 49
x^2 = 0
From this equation, we can see that x = 0, which means that when π = 30Β°, the airplane is directly above the observer. Therefore, the rate at which the angle of elevation is changing is also zero.
You can repeat the same process for the other values of π (π = 60Β° and π = 80Β°) to find the rates at which the angle of elevation is changing for those cases as well.