A ladder 30 ft. long is leaning up against a building. If the top of the ladder is being pulled up the wall of the building at a rate of 1.5 feet per minute, find the rate at which the base of the ladder is moving toward the building when it is 18 feet from the wall.
So what's next to find the rate at which the ladder is moving toward the building when it is 18 ft from the wall?
we have
x^2+y^2 = 30
so,
2x dx/dt + 2y dy/dt = 0
when x=18, y=24
2*18 dx/dt + 2*24 dy/dt = 0
. . .
36 dx/dt + 48 dy/dt = 0
48 dy/dt = -36 dx/dt
dy/dt = -.75 or -3/4
Is this right?
1.5 ft = dr/dt
2x dx/dt + 2y dy/dt = 0
2(18)(2) + 2(24)= 0
72 + 48 dy/dt = 0
48 dy/dt = -72
dy/dt = -72/48 = -3/4 feet per minute
-3/2 feet per minute I meant
You need to read the question more carefully. They gave you dy/dt, and you need to find dx/dt.
2x dx/dt + 2y dy/dt = 0
when x=18, y=24
The ladder is being pulled up the wall at 1.5 ft/min. That means dy/dt = 3/2
Plugging all that into the equation gives
2(18) dx/dt + 2(24)(3/2) = 0
36 dx/dt = -72
dx/dt = -2
As expected, the distance of the base of decreases as the top of the ladder gets pulled up.
To find the rate at which the base of the ladder is moving toward the building, we can use related rates. Let's assign some variables:
Let x be the distance of the base of the ladder from the building.
Let y be the distance of the top of the ladder from the ground.
Let t be the time.
We are given that the ladder is 30 ft. long, so we have the equation:
x^2 + y^2 = 30^2
Differentiating both sides of the equation with respect to t gives us:
2x(dx/dt) + 2y(dy/dt) = 0
Simplifying the equation, we have:
x(dx/dt) + y(dy/dt) = 0
Now, we are given that dy/dt (the rate at which the top of the ladder is being pulled up the wall) is 1.5 ft/min. We are asked to find dx/dt (the rate at which the base of the ladder is moving toward the building) when x = 18 ft.
Substituting the given values into our equation, we have:
18(dx/dt) + y(1.5) = 0
To solve for dx/dt, we need to find the value of y when x = 18 ft. We can use the Pythagorean theorem to find y:
x^2 + y^2 = 30^2
(18)^2 + y^2 = 30^2
324 + y^2 = 900
y^2 = 576
y = 24
Now we can substitute the values of x = 18 ft and y = 24 ft into our equation:
18(dx/dt) + 24(1.5) = 0
Rearranging the equation to solve for dx/dt, we get:
18(dx/dt) = -24(1.5)
dx/dt = -24(1.5)/18
dx/dt = -2 ft/min
Therefore, the rate at which the base of the ladder is moving toward the building when it is 18 feet from the wall is -2 ft/min. The negative sign indicates that the base of the ladder is moving toward the building.