A ladder 41 feet long that was leaning against a vertical wall begins to slip. Its top slides down the wall while its bottom moves along the level ground at a constant speed of 4 feet per second. How fast is the top of the ladder moving when its 9 ft above the ground?
x^2+y^2 = 41^2
2 x dx/dt + 2 y dy/dt = 0
dy/dt = -(x/y) dx/dt
when y = 9
x^2 + 81 = 1681
x^2 = 1600
x = 40
so
dy/dy = - (40/9)(4)
To solve this problem, we will use related rates. Let's denote the distance between the bottom of the ladder and the wall as x feet.
We are given that the bottom of the ladder is moving at a constant speed of 4 feet per second, so dx/dt = 4 ft/s.
We want to find the rate at which the top of the ladder is moving when it is 9 feet above the ground, which means we need to find dy/dt when y = 9 ft.
Now, we can form a right triangle using the ladder as the hypotenuse, the wall as one side, and the ground as the other side. The length of the ladder is always constant at 41 ft.
Using the Pythagorean theorem, we have:
x^2 + y^2 = 41^2
Differentiating both sides with respect to time t, we get:
2x * dx/dt + 2y * dy/dt = 0
Now, we substitute the given values: x = unknown, dx/dt = 4 ft/s, y = 9 ft, and solve for dy/dt:
2(x)(4) + 2(9)(dy/dt) = 0
8x + 18(dy/dt) = 0
18(dy/dt) = -8x
(dy/dt) = -8x/18
We still need to solve for x.
Since we are given that the bottom of the ladder is moving on the level ground at a constant speed of 4 ft/s, we know that dx/dt = 4 ft/s.
We also know that x + y = 41. Rearranging this equation, we have:
x = 41 - y
Substituting y = 9 ft, we find:
x = 41 - 9
x = 32 ft
Now we substitute the value of x into our equation for (dy/dt):
(dy/dt) = (-8 * 32) / 18
(dy/dt) = -256/18 ft/s
Simplifying, we find:
(dy/dt) = -128/9 ft/s
Therefore, the top of the ladder is moving at a rate of -128/9 ft/s when it is 9 ft above the ground.
To find the rate at which the top of the ladder is moving when it is 9 feet above the ground, we can use related rates.
Let's denote the height of the ladder as "y" (measured from the ground up) and the distance from the wall to the bottom of the ladder as "x" (measured along the ground). We are given that dx/dt (the rate at which x is changing) is 4 feet per second.
We need to figure out the rate dy/dt (the rate at which y is changing) when y = 9 feet.
Since the length of the ladder is constant at 41 feet, we can use the Pythagorean theorem to relate x and y:
x² + y² = 41²
Differentiating both sides of the equation with respect to time (t), we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, we can substitute the values we know into the equation:
2x(4) + 2y(dy/dt) = 0
Simplifying the equation further, we have:
8x + 2y(dy/dt) = 0
Since we're interested in finding dy/dt when y = 9 feet, we can substitute this value into the equation:
8x + 2(9)(dy/dt) = 0
We also know that when y = 9 feet, x can be found using the Pythagorean theorem:
x = √(41² - 9²) = √(1681 - 81) = √1600 = 40 feet
Substituting x = 40 and y = 9 into the equation, we get:
8(40) + 2(9)(dy/dt) = 0
320 + 18(dy/dt) = 0
18(dy/dt) = -320
Now we can solve for dy/dt:
dy/dt = -320/18
Simplifying this expression, we have:
dy/dt ≈ -17.78 feet per second
Therefore, the top of the ladder is moving at a rate of approximately -17.78 feet per second when it is 9 feet above the ground. The negative sign indicates that the top of the ladder is sliding down the wall.