A 5-meter ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at 2 m/s, how fast is the top sliding down when the bottom is 3 meters from the wall?
Answer and explanation
Correct answer: D. 1.5 m/s
- Step 1: By the Pythagorean theorem, x^2 + y^2 = 5^2 = 25.
- Step 2: When x = 3, y = sqrt(25 - 9) = 4.
- Step 3: Differentiate with respect to time t: 2x(dx/dt) + 2y(dy/dt) = 0, so x(dx/dt) + y(dy/dt) = 0.
- Step 4: Substitute known values: 3(2) + 4(dy/dt) = 0, so 6 + 4*(dy/dt) = 0, giving dy/dt = -1.5 m/s.
- Step 5: The speed at which the top slides down is 1.5 m/s.
Answer
1.5 m/s.
Why not the others: 2.0 m/s assumes equal rates; 1.2 m/s computes 6/5; 2.5 m/s computes 5/2.
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