Suppose there is a 10 meter ladder leaning against the wall of a building and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 8 meters from the wall?
The ladder is the hypotenuse of a right triangle, with the sides being the ground and the wall.
By Pythagorean Theorem, a^2 + b^2 = c^2. c, the length of the ladder, is constant. Take b to be the ladder's height on the wall, and a the distance on the ground from the wall.
Differentiate with respect to time.
2a(da/dt) + 2b(db/dt) = 0
db/dt = -(a/b)(da/dt)
When the base of the ladder is 8m from the wall, c = 10 so b = 6 by Pythagorean Theorem. da/dt is given as 3m/s; find db/dt.
Well, this is quite a "laddering" question! Let's "step up" and tackle it together.
To find the rate at which the top of the ladder is sliding down the wall, we can use a little trigonometry. Let's call the distance between the base of the ladder and the building "x" (which is 8 meters in this case). The height of the ladder (the distance between the top of the ladder and the ground) can be called "y".
From Pythagoras' theorem, we have x^2 + y^2 = 10^2 (since the ladder is 10 meters long).
Differentiating both sides with respect to time (t), we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, we need to find dy/dt (the rate at which the top of the ladder is sliding down the wall) when x = 8, and we know that dx/dt (the rate at which the base of the ladder is sliding away) is 3 meters per second.
Substituting the given values, we have:
2(8)(3) + 2y(dy/dt) = 0
Simplifying, we have 48 + 2y(dy/dt) = 0
To find dy/dt, we can solve for it:
2y(dy/dt) = -48
Dividing both sides by 2y, we get:
dy/dt = -48 / (2y)
Since we know that x = 8 and y is the height of the ladder, substituting these values we have:
dy/dt = -48 / (2y) = -24 / y
Since the height of the ladder is not given, we can't determine the exact value. However, what we can say is that when the base of the ladder is 8 meters from the wall, the top of the ladder is sliding down the wall at a rate of -24/y meters per second. Remember, "y" stands for the height of the ladder.
So, "ladder" hope this answer helps you reach new "heights" of understanding!
To find the rate at which the top of the ladder is sliding down the wall, you can use the concept of related rates. We'll start by labeling the given information:
Let x be the distance from the base of the ladder to the wall (in meters).
Let y be the distance from the top of the ladder to the ground (in meters).
We are given:
dx/dt = 3 m/s (the rate at which the base of the ladder is sliding away from the wall)
we need to find:
dy/dt (the rate at which the top of the ladder is sliding down the wall) when x = 8 meters.
Now, let's write down the relationship between x and y using the Pythagorean theorem:
x^2 + y^2 = 10^2
Differentiating implicitly with respect to time (t), we get:
2x(dx/dt) + 2y(dy/dt) = 0
Since our goal is to find dy/dt, we can rearrange the equation to solve for it:
2y(dy/dt) = -2x(dx/dt)
dy/dt = (-2x(dx/dt)) / (2y)
Now we can substitute the known values:
dx/dt = 3 m/s
x = 8 m
To find y, we can solve the Pythagorean equation:
8^2 + y^2 = 10^2
64 + y^2 = 100
y^2 = 36
y = 6 m
So we have:
dy/dt = (-2(8)(3)) / (2(6))
Now we can calculate:
dy/dt = -48 / 12
dy/dt = -4 m/s
Therefore, the top of the ladder is sliding down the wall at a rate of 4 meters per second when the base of the ladder is 8 meters from the wall.
To solve this problem, we can use the concept of related rates. Let's consider two quantities: the distance from the base of the ladder to the wall (x) and the distance from the top of the ladder to the ground (y). We want to determine dy/dt, the rate at which y is changing with respect to time.
Given:
dx/dt = 3 m/s (the rate at which the base is sliding away from the wall)
x = 8 m (the distance from the wall to the base of the ladder)
Since we have a right triangle formed by the ladder, we can use the Pythagorean theorem to relate x, y, and the length of the ladder (l):
l^2 = x^2 + y^2
Differentiating both sides of the equation with respect to time (t), we get:
2l(dl/dt) = 2x(dx/dt) + 2y(dy/dt)
Simplifying the equation and substituting the known values, we have:
10^2 (dl/dt) = 8^2 (3) + y (dy/dt)
Since we want to find dy/dt when x = 8 m, we need to find y. Using the Pythagorean theorem:
10^2 = 8^2 + y^2
100 = 64 + y^2
y^2 = 36
y = 6
Substituting the values of x, dx/dt, l, and y into the equation, we have:
100 (dl/dt) = 64 (3) + 6 (dy/dt)
Simplifying further:
100 (dl/dt) = 192 + 6 (dy/dt)
Since dl/dt is the rate at which the ladder is sliding down the wall, we can set dl/dt = -6 (negative since the ladder is sliding down). Now we can solve for dy/dt:
100 (-6) = 192 + 6 (dy/dt)
-600 = 192 + 6 (dy/dt)
-600 - 192 = 6 (dy/dt)
-792 = 6 (dy/dt)
Dividing by 6 to isolate dy/dt:
(dy/dt) = -792 / 6
(dy/dt) = -132
Hence, the top of the ladder is sliding down the wall at a rate of 132 meters per second.