a 16 foot ladder is leaning against a building. the bottom of the ladder os sliding along the pavement at 3 ft/s How fast is the top of the ladder moving down when the foot of the ladder is 2 ft from the wall
To find how fast the top of the ladder is moving down, we need to use related rates and differentiate the given information.
Let's draw a right triangle to represent the situation:
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/ | <- ladder
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pavement
Let's assign variables:
1. The distance of the foot of the ladder from the wall (x)
2. The height of the ladder (y)
3. The rate at which the ladder is sliding along the pavement (dx/dt)
Given:
dx/dt = 3 ft/s
x = 2 ft
Using the Pythagorean theorem:
x^2 + y^2 = L^2, where L is the length of the ladder
Differentiating both sides with respect to time (t):
2x(dx/dt) + 2y(dy/dt) = 0
Since we are trying to find dy/dt (the rate at which the top of the ladder is moving down), and we know dx/dt = 3 ft/s and x = 2 ft:
2(2)(3) + 2y(dy/dt) = 0
Simplifying:
12 + 2y(dy/dt) = 0
Now we can solve for dy/dt:
2y(dy/dt) = -12
dy/dt = -6/y
To find the value of y (height of the ladder), we can use the Pythagorean theorem again:
x^2 + y^2 = L^2
2^2 + y^2 = 16^2
4 + y^2 = 256
y^2 = 252
y = √252 ≈ 15.87 ft
Now that we know the height of the ladder (y ≈ 15.87 ft), we can substitute it into the equation dy/dt = -6/y:
dy/dt = -6/15.87 ≈ -0.377 ft/s
Therefore, the top of the ladder is moving down at approximately 0.377 ft/s when the foot of the ladder is 2 ft from the wall.
To solve this problem, we can use related rates and the Pythagorean theorem.
Let's define some variables:
- Let x be the distance from the bottom of the ladder to the wall.
- Let y be the distance from the top of the ladder to the ground.
- Let z be the length of the ladder (16 feet).
We are given that dx/dt = 3 ft/s (the rate at which x is changing) and we want to find dy/dt (the rate at which y is changing) when x = 2 ft.
According to the Pythagorean theorem, we have x^2 + y^2 = z^2.
Differentiating both sides of the equation with respect to time (t), we get:
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt).
Substituting the given values:
2(2 ft)(3 ft/s) + 2(y)(dy/dt) = 2(16 ft)(0).
Simplifying the equation:
4 ft/s + 2y(dy/dt) = 0.
Since the ladder is leaning against the building, y is negative. We can rewrite the equation as:
2y(dy/dt) = -4 ft/s.
Now, let's solve for dy/dt:
dy/dt = (-4 ft/s) / (2y).
When x = 2 ft, we can find y using the Pythagorean theorem:
x^2 + y^2 = z^2,
2^2 + y^2 = 16^2,
4 + y^2 = 256,
y^2 = 252,
y = sqrt(252).
Substituting this value into the equation for dy/dt:
dy/dt = (-4 ft/s) / (2 * sqrt(252)).
Calculating the value:
dy/dt ≈ -0.317 ft/s.
Therefore, when the foot of the ladder is 2 ft from the wall, the top of the ladder is moving downward at a rate of approximately 0.317 ft/s.
Let the distance of the bottom of the ladder from the wall be x, and let the distance of the top of the ladder from the floor be y.
x^2 + y^2 = 256
You want to know dy/dt when dx/dt = 3 and x = 2. At that time, y^2 = 252 and y = 15.87
2x *dx/dt + 2y*dy/dt = 0
dy/dt = -(x/y)*dx/dt = -0.126 dx/dt
Use that equation to calculate dy/dt