Question

The rates of change of the perimeter and area of a rectangle respectively when length $x=5$ cm and breadth $y=2$ cm, if $dx/dt = -5$ cm/min and $dy/dt = 3$ cm/min, are

• A. -4 and 5
• B. -5 and 3
• C. 3 and 5
• D. 3 and -5

Hint:

Remember to apply the product rule for area ($xy$) and sum rule for perimeter ($2x+2y$).

Answer 1

The Correct Option is A

Step 1: Perimeter Change
$P = 2(x+y) \implies \frac{dP}{dt} = 2(\frac{dx}{dt} + \frac{dy}{dt})$.
$\frac{dP}{dt} = 2(-5 + 3) = 2(-2) = -4$ cm/min.
Step 2: Area Change
$A = xy \implies \frac{dA}{dt} = x\frac{dy}{dt} + y\frac{dx}{dt}$.
Step 3: Calculation
$\frac{dA}{dt} = (5)(3) + (2)(-5) = 15 - 10 = 5$ cm$^2$/min.
Step 4: Conclusion
Perimeter rate = -4, Area rate = 5.
Final Answer:(A)