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: Understanding the Question:
This is a related rates problem. We are given the rates of change of dimensions and need to find the rates of change of perimeter and area.
Step 2: Key Formula or Approach:
Perimeter \( P = 2(x + y) \implies \frac{dP}{dt} = 2 \left( \frac{dx}{dt} + \frac{dy}{dt} \right) \).
Area \( A = xy \implies \frac{dA}{dt} = x \frac{dy}{dt} + y \frac{dx}{dt} \).
Step 3: Detailed Explanation:
Given: \( \frac{dx}{dt} = -5 \text{ cm/min} \) (decrease), \( \frac{dy}{dt} = 3 \text{ cm/min} \) (increase).
Dimensions: \( x = 5, y = 2 \).
Rate of change of perimeter:
\[ \frac{dP}{dt} = 2(-5 + 3) = 2(-2) = -4 \text{ cm/min} \] Rate of change of area:
\[ \frac{dA}{dt} = 5(3) + 2(-5) = 15 - 10 = 5 \text{ cm}^2\text{/min} \] Step 4: Final Answer:
The rates are -4 and 5 respectively.