"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?"
I get that you use a^2 + b^2 = c^2 but when you differentiate it, would you get 2a(da/dt) + 2b(db/dt) = 2c(dc/dt)? Then plug numbers in?
yes
5^2 + 12^2 = 13^2 (you should know that by the way)
so base b at 12 and height a at 5
c is 15 feet forever so dc/dt = 0
2 a (da/dt) + 2 b (db/dt) = 0
ao
5 da/dt = - 12 (6)
da/dt = - 72/5
Thanks!
When it asks to find the rate the area is changing at that point in time, did you get the answer -357/5? Because I did, but it doesn't seem right...
I assume you mean the triangle that the ladder makes with the ground and the wall.
Area = (1/2)ab
d(Area)/dt = (1/2)a db/dt + (1/2)b da/dt
so when a= 5, b=12 , db/dt = 6, da/dt = -72/5
d(Area)/dt = (1/2)(5)(6) +(1/2)(12)(-72/5)
= 15 - 86.4
= -71.4
To solve this problem, you can indeed use the relationship between the sides of a right triangle (a^2 + b^2 = c^2) and differentiate both sides with respect to time.
Let's assign variables to the different quantities involved:
- Let a represent the distance between the base of the ladder and the building.
- Let b represent the distance between the top of the ladder and the ground.
- Let c represent the length of the ladder.
Given that a = 12 ft and da/dt = 6 ft/s, we can also identify db/dt, the rate at which the top of the ladder is sliding down the wall.
Differentiating the equation a^2 + b^2 = c^2 with respect to time (t), we have:
2a(da/dt) + 2b(db/dt) = 2c(dc/dt)
Substituting the given values, we have:
2(12)(6) + 2b(db/dt) = 2c(dc/dt)
Simplifying, we get:
144 + 2b(db/dt) = 2c(dc/dt)
Since we are interested in db/dt, the rate at which the top of the ladder is sliding down the wall, we can solve for it.
We know that c is a constant as it represents the length of the ladder, which remains fixed at 13 ft. Thus, dc/dt = 0.
Plugging in the values, we can solve for b(db/dt):
144 + 2b(db/dt) = 2(13)(0)
Simplifying further, we have:
144 + 2b(db/dt) = 0
Now, solving for b(db/dt):
2b(db/dt) = -144
Dividing through by 2b, we find:
db/dt = -72/b
To determine the rate at which the top of the ladder is sliding down the wall at the given point in time, you need to find the value of b. You can use the Pythagorean theorem, a^2 + b^2 = c^2, with the given values to calculate b.
Once you have the value of b, substitute it back into db/dt = -72/b to get the specific rate. This will allow you to determine how fast the top of the ladder is sliding down the wall at that point in time.