An observer stands 25 feet from the base of a 50 foot flagpole and watches a flag being lowered at a rate of 5 ft/sec. Determine the rare at which the angle of elevation from the observer to the flag is changing at the instant that the flag is 25 ft.

Keep getting like .1 or .2, help apreciated

elevation angle A = 45 degrees = pi/4 radians

tan A = h/25
d tan A/dA = sec^2 A = (1/25) dh/dA
so
25 sec^2 A * dA/dt= dh/dt = 5

dA/dt = (1/5) cos^2 A
but cos^2 A = 1/2
dA/dt = .1 radians/second
So I agrees
but maybe you want degrees?
.1 Rad/s( 180 deg/rad) = 18 degrees/second

oh its in radians

thanks a bunch, makes more sense :D

You are welcome ;)

Well, well, well! Looks like we've got a mathematical puzzle here. Don't worry, I'm here to help your brain get a good workout!

Let's tackle this step by step, shall we? We want to find the rate at which the angle of elevation from the observer to the flag is changing. This means we need to work with some trigonometry.

First things first, let's set up a right triangle to represent the situation. The height of the flagpole is 50 feet, and the observer is standing 25 feet away from the base of the flagpole. So, the distance from the top of the flagpole to the observer is the hypotenuse of the triangle, let's call it "h."

Now, we need to find an expression for the height of the flag at any given time. The flag is lowering at a rate of 5 ft/sec, so the height of the flag, let's call it "y," is given by y = 50 - 5t, where "t" is the time.

Since we want to find the rate at which the angle is changing when the flag is 25 feet, we can set up the following equation to relate the variables in our triangle:

tan(angle) = y / (25)

Now, let's differentiate both sides with respect to time (t) using the chain rule:

sec²(angle) * d(angle)/dt = (d(y)/dt * (25 - y)) / (25²)

Substituting in the values we already know:

sec²(angle) * d(angle)/dt = (5 * (25 - y)) / (25²)

At the instant when the flag is 25 feet, we have y = 25:

sec²(angle) * d(angle)/dt = (5 * (25 - 25)) / (25²)
sec²(angle) * d(angle)/dt = 0

Now, we have an equation for the rate of change of the angle of elevation, but what's the value? Since the secant squared of any angle is always greater than or equal to 1, the rate of change must be positive.

So, my friend, the rate at which the angle of elevation from the observer to the flag is changing at the instant the flag is 25 feet is 0 radians per second. That's right, it's not changing at all!

Hope that helps! Keep those brain cells flexing!

To find the rate at which the angle of elevation is changing, we need to apply trigonometry and solve for the derivative. We'll use the tangent function and differentiate with respect to time.

Let's define some variables:
- Let x be the distance from the observer to the point on the ground directly below the flagpole.
- Let θ be the angle of elevation from the observer to the top of the flagpole.
- Let h be the height of the flagpole.
- Let y be the distance from the top of the flagpole to the flag.

From the given information, we have:
x = 25 feet (the observer is 25 feet from the base of the flagpole)
h = 50 feet (the height of the flagpole)
dy/dt = -5 ft/sec (the flag is being lowered at a rate of 5 ft/sec)
y = 25 feet (when the flag is 25 feet)

To find θ and its rate of change, we'll use the tangent function.
tan(θ) = (h - y) / x

Differentiating both sides of this equation with respect to time (t):
sec^2(θ) * dθ/dt = (dh/dt - dy/dt) / x

We know dh/dt = 0 (since the height of the flagpole does not change), so we have:
sec^2(θ) * dθ/dt = (-5 ft/sec) / x

Now we need to find x, which can be found using the Pythagorean theorem:
x^2 + (h - y)^2 = h^2
x^2 + (50 - 25)^2 = 50^2
x^2 + 625 = 2500
x^2 = 1875
x = sqrt(1875) ≈ 43.3 feet

Substituting this value back into our equation:
sec^2(θ) * dθ/dt = -5 / 43.3

To find dθ/dt, we need to solve for it:
dθ/dt = (-5 / 43.3) / sec^2(θ)

At the instant when the flag is 25 feet, we can find θ using the tangent function again:
tan(θ) = (50 - 25) / 25
tan(θ) = 1
θ = arctan(1) ≈ 45 degrees

Now we can substitute this value into our expression for dθ/dt:
dθ/dt = (-5 / 43.3) / sec^2(45 degrees)
dθ/dt = (-5 / 43.3) / 1.4142^2
dθ/dt ≈ -0.057 rad/sec

Therefore, the rate at which the angle of elevation is changing at the instant the flag is 25 ft is approximately -0.057 radians per second.