A ladder 14 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?
X = 14 sin theta
dX/d(theta) = 14 cos theta
when theta = pi/3, cos theta = 1/2
Therefore the answer is 7 feet per radian
To find how fast x changes with respect to theta, we can use the chain rule of differentiation.
Let's set up the equation based on the given information:
We have a right triangle formed by the ladder, the wall, and the distance x. The hypotenuse of this triangle is the ladder, and the angle between the ladder and the wall is theta.
Using trigonometry, we can say that sin(theta) = x / 14ft.
Differentiating both sides of the equation with respect to t (time), we get:
d/dt(sin(theta)) = d/dt(x/14ft)
Let's simplify the equation further:
cos(theta) * d(theta)/dt = (1/14ft) * dx/dt
Since we are interested in finding dx/dt (how fast x changes with respect to t) when theta = pi/3, we can substitute this value into our equation:
cos(pi/3) * d(pi/3)/dt = (1/14ft) * dx/dt
cos(pi/3) = 1/2, and d(pi/3)/dt = 0 (since it's a constant)
Substituting the values, we get:
(1/2) * 0 = (1/14ft) * dx/dt
Simplifying the equation, we find:
0 = dx/dt / 14ft
Multiplying both sides of the equation by 14ft:
0 = dx/dt
Therefore, we conclude that when theta = pi/3, the distance x is not changing with respect to time. In other words, x remains constant.
So, dx/dt = 0 when theta = pi/3.
To find how fast x changes with respect to θ, we can use implicit differentiation. Let's start by drawing a diagram to visualize the situation:
|\
| \
x | \
| \ 14 ft
| \
|_____\
wall
As the bottom of the ladder slides away from the wall, the distance x changes, but the length of the ladder (14 ft) remains constant. We want to find the rate of change of x with respect to θ when θ = π/3.
To begin, let's use trigonometric ratios to relate x and θ:
From the right triangle formed by the ladder, the wall, and the ground, we have:
sin(θ) = x/14
To differentiate implicitly, we can differentiate both sides of the equation with respect to θ. Remember, we treat x as a function of θ because it is changing with respect to θ:
d(sin(θ))/dθ = d(x/14)/dθ
Now, let's use the chain rule to differentiate each term:
cos(θ) = (1/14) * dx/dθ
Now, we have an equation that relates the rates of change of x and θ. Specifically, we have:
dx/dθ = 14 * cos(θ)
To find dx/dθ when θ = π/3, we substitute θ = π/3 into the equation:
dx/dθ = 14 * cos(π/3)
= 14 * (1/2)
= 7
Therefore, when θ = π/3, dx/dθ = 7. Thus, the distance x is changing at a rate of 7 units per unit change in θ when θ is π/3.