Consider the following problem:

A 6 meter long ladder leans against a wall when the bottom of the ladder slides away from the wall at a rate of 0.5 m/s. How fast is the angle between the top of the ladder and the wall changing, when the angle is
𝜋
6
radians?

If you wanted to solve the above problem, which of the following equations would be best to differentiate?

Hint: We want to relate the known variables with the variable we are looking for.

Question

Consider the following problem:

A 6 meter long ladder leans against a wall when the bottom of the ladder slides away from the wall at a rate of 0.5 m/s. How fast is the angle between the top of the ladder and the wall changing, when the angle is 
𝜋
6
 radians?

If you wanted to solve the above problem, which of the following equations would be best to differentiate?

Hint: We want to relate the known variables with the variable we are looking for.

BlackTom AI · Accepted Answer

To begin, let’s restate the problem in our own terms and identify the known quantities and relationships.

- The ladder has fixed length 6 m, forming a right triangle with the wall (vertical) and the ground (horizontal). Let x be the horizontal distance from the wall to the bottom of the ladder, and let θ be the angle between the ladder and the wall.
- The bottom of the ladder is sliding away from the wall at a rate dx/dt = 0.5 m/s. We want to find dθ/dt when θ = π/6.

Now, examine each option and explain why it is correct or incorrect for relating the variables so we can differentiate:

Option A: y = x tan(π/6)
- This would be a specific linear relation using the tangent of a constant angle. It presumes a fixed angle of π/6 everywhere, which is not the general relationship between y and x as the ladder moves. Since the angle is changing in the problem, this is not a useful dynamic relation for differentiation.

Option B: 0.5^2 + y^2 = 6^2
- This appears to be an incorrect reformatting of the Pythagorean relation. The correct constant-length relation is x^2 + y^2 = 6^2, where x is the horizontal leg and y is the vertical leg. The given equation mixes the known rate 0.5 in a squared form that does not reflect the actual instantaneous geometric constraint and would complicate differentiation unnecessarily.

Option C: y = x tan θ
- This would express y in terms of x and θ, but with θ defined as the angle between the ladder and the wall, the geometric relation is not simply tan θ = y/x. In fact, the angle between the ladder and the wall is complementary to the angle with the ground, so the correct trigonometric relation is different (y ≈ x cot θ). Therefore this form misrepresents the geometry and is not the best to differentiate here.

Option D: x = 6 sin θ
- This is a correct and very useful relation when θ is the angle between the ladder and the wall: the side opposite θ is the horizontal distance x, the hypotenuse is the ladder length 6. Hence sin θ = x/6, or x = 6 sin θ. This directly links x and θ and is ideal for differentiation with respect to time.

Option E: x = 6 sin(π/6)
- This evaluates to a constant (x = 3) because sin(π/6) = 1/2. It does not capture the dynamics of the moving ladder, since x is changing in time as the bottom slides. Therefore this is not suitable for differentiating to find dθ/dt.

Option F: x^2 + y^2 = 36
- This is the correct geometric constraint of a 6 m ladder, but it does not directly give a simple single-variable relation between x and θ to differentiate with respect to time. While true, differentiating this implicitly would involve dx/dt and dy/dt and would require additional relations to isolate dθ/dt. It is not as straightforward as using x = 6 sin θ for this purpose.

Putting these insights together, the most straightforward relation to differentiate to obtain dθ/dt given dx/dt is the one that directly ties x to θ through the ladder length, namely x = 6 sin θ. This provides a clean differential equation:

dx/dt = 6 cos θ · dθ/dt

From here, substitute dx/dt = 0.5 and θ = π/6 to solve for dθ/dt. The other options either misrepresent the geometry, fix θ incorrectly, or complicate the differentiation without giving a direct path to dθ/dt.

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