A ladder 20 ft long rests against a vertical wall. Let \theta be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to \theta when \theta = pi / 3
To find the rate of change of x with respect to θ, we need to consider the relationship between x and θ given by trigonometry.
Let's label the right triangle formed by the ladder, the wall, and the ground. The hypotenuse of the triangle represents the ladder, and the vertical side represents the wall.
We have the following trigonometric ratio:
sin(θ) = opposite/hypotenuse = x/20
To find how x changes with respect to θ, we can differentiate both sides of this equation with respect to θ:
d(sin(θ))/dθ = d(x/20)/dθ
Using the chain rule, we can rewrite the left-hand side:
cos(θ) = (1/20)(dx/dθ)
Now, we can solve for dx/dθ, which gives us the rate at which x changes with respect to θ:
dx/dθ = 20cos(θ)
To find the value of dx/dθ when θ = π/3, substitute this value into the equation:
dx/dθ = 20cos(π/3)
dx/dθ = 20(1/2)
dx/dθ = 10
Therefore, when θ = π/3, the rate at which x changes with respect to θ is 10 ft/radian.