A 10m ladder leans against a wall at an angle theta with the horizontal. The top of the ladder is x metres above the ground. If the bottom of the ladder is pushed towards the wall, find the rate at which x changes with respect to theta when theta equals 60 degrees. Express your answer in units ofmetres/degrees.
To find the rate at which x changes with respect to theta (dx/dθ), we can use trigonometry and take the derivative of the equation relating the ladder, the wall, and the ground.
Let's break down the problem:
1. We have a right triangle formed by the ladder, the wall, and the ground.
2. The length of the ladder is given as 10 meters.
3. The top of the ladder is x meters above the ground.
4. The angle between the ladder and the ground is θ (theta).
5. We need to find the rate at which x changes with respect to θ (dx/dθ) when θ equals 60 degrees.
To connect x, the ladder length, and the angle θ, we can use the trigonometric function sine.
In a right triangle, the sine of an angle θ is defined as the ratio of the length of the side opposite to θ (in this case, x) to the hypotenuse (the ladder, 10 meters). So we have:
sin θ = x / 10
To isolate x, we can rearrange the equation:
x = 10 * sin θ
Now, we can take the derivative of both sides of this equation with respect to θ to find the rate at which x changes with respect to θ:
d(x) / d(θ) = d(10 * sin θ) / d(θ)
Using the chain rule, we can differentiate the equation:
d(x) / d(θ) = 10 * cos θ
Finally, we substitute θ = 60 degrees into the equation to find the rate at which x changes with respect to θ:
d(x) / d(θ) = 10 * cos 60 = 10 * 0.5 = 5
Therefore, the rate at which x changes with respect to θ when θ equals 60 degrees is 5 meters/degree.