A ladder 54 feet tall is place against a tall building. The bottom of the ladder is sliding away from the building horizontally at a rate of 5 feet/sec. Find the rate at which the indicated angle, θ , is changing when the angle is 26 degrees . Express your answer in radians per sec and round to four decimals.
If the foot of the ladder is x feet from the wall, we have
cosθ = x/54
-sinθ dθ/dt = 1/54 dx/dt
when θ=26°, you have
dθ/dt = -5/(54 sin26°) = -.21 = -12°/s
thank you so much i understand it now!
To find the rate at which the indicated angle θ is changing, we can use the concept of related rates.
Let's label the distance from the building to the bottom of the ladder as x and the distance from the top of the ladder to the ground as y.
We are given that the bottom of the ladder is sliding away from the building horizontally at a rate of 5 feet/sec. This means that dx/dt = 5.
We need to find dθ/dt when θ = 26 degrees.
Using the properties of a right triangle, we can relate x, y, and the ladder height h:
x^2 + y^2 = h^2
Differentiating both sides of this equation with respect to t:
2x * (dx/dt) + 2y * (dy/dt) = 2h * (dh/dt)
Since the ladder height h is constant, dh/dt = 0.
Plugging in the given values dx/dt = 5, h = 54, and θ = 26 degrees (which is approximately 0.4536 radians), we get:
2x * (5) + 2y * (dy/dt) = 2(54) * 0
Simplifying the equation:
10x + 2y * (dy/dt) = 0
Now we need to find the value of y when θ = 26 degrees.
Using trigonometry, we know that y = h * sin(θ):
y = 54 * sin(26) = 23.0389
Plugging this value into the equation:
10x + 2(23.0389) * (dy/dt) = 0
Simplifying:
10x + 46.0779 * (dy/dt) = 0
Solving for dy/dt:
(dy/dt) = -10x / 46.0779
To find the value of x, we can use the fact that x = h * cos(θ):
x = 54 * cos(26) = 47.9441
Plugging this value into the equation:
(dy/dt) = -10(47.9441) / 46.0779
Simplifying:
(dy/dt) = -10.3390
Therefore, the rate at which the indicated angle θ is changing when θ = 26 degrees is approximately -10.3390 radians per second.
To solve this problem, we need to use trigonometry and related rates. Let's break it down step by step:
1. First, let's define the variables:
- Let "h" be the distance between the bottom of the ladder and the tall building.
- Let "θ" be the angle between the ladder and the ground.
- Let "dh/dt" be the rate at which the ladder is sliding away from the building horizontally (given as 5 feet/sec).
2. We know that the ladder is 54 feet tall, so h = 54.
3. We want to find dθ/dt, the rate at which the angle θ is changing. To do this, we need to relate the variables using trigonometry.
4. We can use the tangent function:
tan(θ) = h / x
where x is the distance along the ground from the building to the ladder's base.
5. We differentiate both sides of the equation with respect to time:
d(tan(θ))/dt = d(h/x)/dt
6. Using the quotient rule, we can rewrite the equation as:
sec^2(θ) * dθ/dt = (x * dh/dt - h * dx/dt) / x^2
7. Rearranging the equation, we get:
dθ/dt = [(x * dh/dt - h * dx/dt) / (x^2)] * (1 / sec^2(θ))
8. Now, we need to determine the values of x and sec(θ) when θ = 26 degrees.
9. From the given information, we know that x is changing at a rate of 5 feet/sec (dx/dt = 5 ft/sec).
10. To find sec(θ), we can use the Pythagorean theorem:
sec(θ) = sqrt(1 + tan^2(θ))
11. Plugging in the given value of θ = 26 degrees, we have:
sec(θ) = sqrt(1 + tan^2(26))
12. Now we have all the values needed to calculate dθ/dt:
dθ/dt = [(x * dh/dt - h * dx/dt) / (x^2)] * (1 / sec^2(θ))
= [(x * 0 - 54 * 5) / (x^2)] * (1 / (1 + tan^2(26)))
13. Plugging in the known values, we get:
dθ/dt = (-270) / (x^2 * (1 + tan^2(26)))
14. To find the rate of change of the angle in radians per second, we need to convert degrees to radians. There are π/180 radians in 1 degree.
15. Plugging in this conversion factor, we get:
dθ/dt = (-270) / (x^2 * (1 + tan^2(26))) * (π/180)
16. Round the result to four decimal places.
Following these steps, you can now calculate the rate at which the angle θ is changing when it is 26 degrees.