A 10-ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec, how fast, in ft/sec, is the top of the ladder sliding down the wall, at the instant when the bottom of the ladder is 6 ft from the wall? Answer with 2 decimal places. Type your answer in the space below.
If the base of the ladder is x from the wall, and the top of the ladder is y high, then
x^2+y^2 = 10^2
x dx/dt + y dy/dt = 0
So, figure y when x=6 and just plug in the numbers to find dy/dt.
dy/dt=-y dy/dt/x
To solve this problem, we can use the concept of related rates. We are given that the bottom of the ladder is sliding away from the wall at a rate of 1 ft/sec. We are asked to find how fast the top of the ladder is sliding down the wall.
Let's assume that the bottom of the ladder is point A, the top of the ladder is point B, and the point on the wall where the ladder touches is point C. We are given that the distance AC is 6 ft and we are asked to find the rate of change of the distance BC, which is the height of the ladder.
We are given that d(AC)/dt = 1 ft/sec. We need to find d(BC)/dt, which represents how fast the height of the ladder is changing.
The ladder forms a right triangle ABC, where AB is the hypotenuse, and AC and BC are the legs. We can use the Pythagorean theorem to relate the variables:
AB^2 = AC^2 + BC^2
Differentiating both sides of this equation with respect to time t, we get:
2AB(d(AB)/dt) = 2AC(d(AC)/dt) + 2BC(d(BC)/dt)
Since we are interested in finding d(BC)/dt, we can rearrange the equation and substitute the given values:
2AB(d(AB)/dt) = 2AC(d(AC)/dt) + 2BC(d(BC)/dt)
2(10 ft)(d(AB)/dt) = 2(6 ft)(1 ft/sec) + 2BC(d(BC)/dt)
Simplifying the equation further:
10(d(AB)/dt) = 6 + BC(d(BC)/dt)
At the instant when the bottom of the ladder is 6 ft from the wall, AC = 6 ft. From the Pythagorean theorem, we can calculate that BC = sqrt(10^2 - 6^2) = sqrt(28) = 2sqrt(7) ft.
Substituting the values into the equation:
10(d(AB)/dt) = 6 + 2sqrt(7)(d(BC)/dt)
To find d(BC)/dt, we can solve for it:
d(BC)/dt = (10(d(AB)/dt) - 6) / (2sqrt(7))
Now, to find the value of d(BC)/dt, we need to know the value of d(AB)/dt, which is not given in the question. Without additional information, we cannot provide an exact numerical answer.