TS EAMCET · Maths · Application of Derivatives

A rod of length 41 m with an end \(A\) on the floor and another end \(B\) on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of \(3 \mathrm{ft} / \mathrm{min}\). When the end B is at the height of 9 ft from the floor, then the rate at which the area of the triangle formed by the rod with wall and floor changes at that instant is (in \(\mathrm{ft} / \mathrm{min}\))

A \(-\frac{1519}{6}\)
B \(\frac{1618}{3}\)
C \(-\frac{1600}{3}\)
D \(\frac{1509}{6}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(-\frac{1519}{6}\)

Step-by-step Solution

Detailed explanation

\(x^2 + y^2 = L^2\) \(x^2 + 9^2 = 41^2 \Rightarrow x^2 = 1681 - 81 = 1600 \Rightarrow x = 40\) \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0\) \(2(40)(3) + 2(9)\frac{dy}{dt} = 0 \Rightarrow 240 + 18\frac{dy}{dt} = 0 \Rightarrow \frac{dy}{dt} = -\frac{240}{18} = -\frac{40}{3}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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