I haven't done this type of problem in a while, so if someone can just show me how to set it up, I can figure out the rest. (:
"A 13-foot ladder is leaning against a building when its base begins to slide away from the base of the building. By the time the base is 12 feet from the building, the base is moving at the rate of 6 ft/sec. How fast is the top of the ladder sliding down the wall at that point in time?"
a^2+b^2=c^2
ada/dt + bdb/dt = cdc/dt
(6)da/dt+12(6) =13(0)
da/dt =−12ft/sec
To solve this problem, we can use related rates, which involves finding the rate of change of one quantity with respect to another quantity. In this case, we need to find the rate at which the top of the ladder is sliding down the wall (dz/dt) when the base is 12 feet from the building and moving at 6 ft/sec.
Let's assign variables to the different quantities involved:
- Let z represent the distance between the top of the ladder and the ground.
- Let x represent the distance between the base of the ladder and the building.
- Let y represent the length of the ladder.
From the problem statement, we are given:
- dz/dt: The rate at which z is changing (what we need to find).
- dx/dt = 6 ft/sec: The rate at which x is changing.
- y = 13 ft: The given length of the ladder.
We can use the Pythagorean theorem to relate these variables:
z^2 + x^2 = y^2
Differentiating both sides of this equation with respect to time (t), we get:
2z * dz/dt + 2x * dx/dt = 0 (since the length of the ladder, y, is constant)
Now, we can plug in the values we know:
z = ?
x = 12 ft
dx/dt = 6 ft/sec
y = 13 ft
We can solve for dz/dt:
2z * dz/dt + 2x * dx/dt = 0
2z * dz/dt = -2x * dx/dt
dz/dt = -x * dx/dt / z
Substituting the known values:
dz/dt = -12 ft * 6 ft/sec / z
Now, we need to find the value of z. To do that, we can use the Pythagorean theorem again:
z^2 + 12^2 = 13^2
z^2 + 144 = 169
z^2 = 169 - 144
z^2 = 25
z = 5 ft
Substituting z = 5 ft into our equation for dz/dt:
dz/dt = -12 ft * 6 ft/sec / 5 ft
Simplifying the expression:
dz/dt = -72 ft/sec
Therefore, the top of the ladder is sliding down the wall at a rate of 72 ft/sec at that point in time. The negative sign indicates that the ladder is sliding downwards.