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MHT CET202523 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The length and breadth of a rectangle are x ~cm and y cm respectively. If the length decreases at the rate of 5 ~cm / minute and the breadth increases at the rate of 3 ~cm / minute, then the rates of change of the perimeter and area respectively when the length is 5 cm and breadth is 2 cm , are

Options

  1. A-4 and 5
  2. B-5 and 3
  3. C3 and 5
  4. D3 and -5

Correct answer

A. -4 and 5

Step-by-step solution

Let the rectangle have length x and breadth y , each in cm, with rates of change dx dt = -5 cm/min and dy dt = 3 cm/min. Perimeter Rate The perimeter is P = 2(x + y) . Differentiating gives dP dt = 2 ( dx dt + dy dt ) = 2(-5 + 3) = -4 cm/min. Area Rate The area is A = xy . Applying the product rule, dA dt = dx dt y + x dy dt . At x = 5 cm and y = 2 cm, this becomes dA dt = (-5)(2) + (5)(3) = 5 cm²/min. The rates of change are dP dt = -4 cm/min and dA dt = 5 cm²/min, corresponding to option A.

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