The variable a is the length of the ladder. The variable h is the height of the ladder's top at time t, and x is the distance from the wall to the ladder's bottom. Suppose that the length of the ladder is 7.0 meters and the top is sliding down the wall at a rate of 0.4 m/s. Calculate dx/dt when h = 5.4. (Round your answer to three decimal places.)

Question

Addisyn Franklin

Mathematics

3 June, 17:34

The variable a is the length of the ladder. The variable h is the height of the ladder's top at time t, and x is the distance from the wall to the ladder's bottom. Suppose that the length of the ladder is 7.0 meters and the top is sliding down the wall at a rate of 0.4 m/s. Calculate dx/dt when h = 5.4. (Round your answer to three decimal places.)

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Gwendolen · Answer 1

0.485 m/s

Step-by-step explanation:

The Pythagorean theorem tells you ...

a² = h² + x²

Differentiating with respect to time, we find ...

0 = 2h·h' + 2x·x'

Solving for x', we get ...

x' = - h' (h/x)

To evaluate this, we need to find the value of x when h=5.4. We can do this using the original Pythagorean relation.

7.0² = 5.4² + x²

x = √ (49-29.16) ≈ 4.454

Then the desired rate of change is ...

x' = - (-0.4 m/s) (5.4/4.454) ≈ 0.485 m/s