related rates?

Question

A ladder 10 ft long rests against
a vertical wall. If the bottom of
the ladder slides away from the
wall at 1ft/sec, how fast is
the top of the ladder sliding down
the wall when the -bottom of the
ladder is 6ft from the wall?

Answer 1

it's hard to see when its all words, but when once you're able to translate the words into equations, it's easier to understand:

the ladder makes a right triangle:
x^2 + y^2 = 10^2
x^2 + y^2 = 100

we want "the rate of change of y" or dy/dt, how do we get that?

differentiate with respect to time:
2x(dx/dt) + 2y(dy/dt) = 0

we want dy/dt
we're given dx/dt = 1
we're given x = 6
we need y:

to get y use pythagoras' theorem:
x^2 + y^2 = z^2
6^2 + y^2 = 10^2
y = √(100-36)
y = 8

now we just substitute:
2x(dx/dt) + 2y(dy/dt) = 0
2(6)(1) + 2(8)(dy/dt) = 0
16(dy/dt) = -12
dy/dt = -.75

the negative just means that the "side" is shrinking as oppossed to growing.

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