A ladder 10 ft long rests against a vertical wall. let θ 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 θ when θ = π/3

SinØ = opp/hyp

opp is x
hyp is 10

SinØ = x/10
10SinØ = x

Sin (π/3) = 1/2
10 * 1/2 = 5

Answer: 5 ft/rad

According to you diagram I have

cos Ø = x/10
x = 10cosØ
dx/dØ = -10sinØ

so when Ø=π/3
dx/dØ = -10sin π/3
= -10 (√3/2)

To find how fast the distance x changes with respect to the angle θ, we can use trigonometry and take the derivative of x with respect to θ.

Let's set up the problem:

Given:
Length of the ladder (hypotenuse): 10 ft
Angle between the top of the ladder and the wall: θ
Distance from the bottom of the ladder to the wall: x

Using trigonometry, we can relate x and θ as follows:
sin(θ) = x/10

Let's differentiate both sides of this equation with respect to θ:

d/dθ(sin(θ)) = d/dθ(x/10)

cos(θ) = (1/10)(dx/dθ)

We want to solve for dx/dθ, which represents how x changes with respect to θ. Rearranging the equation, we get:

(10)(cos(θ)) = dx/dθ

Now, we can plug in the specific value for θ:

θ = π/3

Plugging this back into the equation, we have:

dx/dθ = (10)(cos(π/3))

Evaluating cos(π/3), which is equal to 1/2, we get:

dx/dθ = (10)(1/2)

Simplifying this, we find:

dx/dθ = 5

Therefore, when θ = π/3, the distance x changes with respect to θ at a rate of 5 ft/radian.

To find how fast x changes with respect to θ, we can use trigonometry and implicit differentiation.

Let's start by drawing a triangle to represent the situation. We have a ladder, a vertical wall, and the ground. The ladder forms a right triangle with the wall, and the angle θ is the angle between the ladder and the wall. The length of the ladder is 10 ft, and we want to find how fast the distance x changes as we change the angle θ.

First, let's label the sides of the triangle. The side opposite to the angle θ is the height of the wall, which we'll call h. The side adjacent to the angle θ is the distance from the bottom of the ladder to the wall, which we'll call x. And the hypotenuse of the triangle is the length of the ladder, which is 10 ft.

Now we can use trigonometry to relate the sides of the triangle. We know that sin(θ) = h/10 and cos(θ) = x/10.

To find how x changes with respect to θ, we need to differentiate both sides of the equation cos(θ) = x/10 with respect to θ.

Using the chain rule, we have:

d(cos(θ))/dθ = d(x/10)/dθ

To differentiate cos(θ), we get -sin(θ). And since x/10 is a constant with respect to θ, its derivative is 0.

So we have:

-sin(θ) = 0

This equation implies that sin(θ) = 0.

When θ = π/3, sin(π/3) = √3/2.

Therefore, when θ = π/3, x does not change with respect to θ since sin(π/3) = √3/2 ≠ 0.