Q.The length $x$ of a rectangle is decreasing at the rate of $3$ cm/minute and the width $y$ is increasing at the rate of $2$ cm/minute. When $x = 10$ cm and $y = 6$ cm, find the rates of change of

Concept: Related Rates — we differentiate the geometric formulas with respect to time, using the chain rule.

Step 1: Given rates

$\frac{dx}{dt} = -3$ cm/min (decreasing), $\frac{dy}{dt} = +2$ cm/min (increasing).

Step 2: Perimeter $P = 2x + 2y$

Differentiate: $\frac{dP}{dt} = 2\frac{dx}{dt} + 2\frac{dy}{dt} = 2(-3) + 2(2) = -6 + 4 = -2$ cm/min.

Step 3: Area $A = xy$ …

Given. $\dfrac{dx}{dt}=-3$ cm/min (length decreasing), $\dfrac{dy}{dt}=+2$ cm/min (width increasing); at the instant $x=10$ cm, $y=6$ cm.

1. Perimeter. $P=2(x+y)$, so

   $\frac{dP}{dt}=2\!\left(\frac{dx}{dt}+\frac{dy}{dt}\right)=2(-3+2)=-2\ \text{cm/min}.$

   The negative sign means the perimeter is decreasing at $2$ cm/min.
2. Area. $A=xy$, so by the product rule …

Method: Related Rates for Several Quantities at Once (Sum and Product Rules)

This method handles a related-rates problem that asks for more than one derived rate (here, both perimeter and area) from the same pair of given rates, using the sum rule for one formula and the product rule for the other.

Steps

Step 1: Record the given rates with their correct signs

A quantity described as "decreasing" gets a negative rate and one "increasing" gets a positive rate — write both explicitly, e.g. $\frac{dx}{dt}=-3$ (length decreasing) and $\frac{dy}{dt}=+2$ (width increasing), before doing anything else.

Step 2: Write the formula for each quantity you need a rate for

Perimeter: $P=2x+2y$. Area: $A=xy$.

Step 3: Differentiate each with the appropriate rule

For the perimeter, a sum of terms each linear in one variable just needs the sum rule:

For the area, since it's a product of the two changing variables, the product rule is required — not a simple sum:

Step 4: Substitute the given instantaneous values and rates …

Common Mistakes

Mistake 1: Dropping the negative sign on the decreasing rate

A student writes $\frac{dx}{dt}=3$ instead of $-3$ because the problem states the magnitude "3 cm/minute" without repeating the word "decreasing" at the point of substitution. Why it's wrong: the sign carries the actual physical meaning (shrinking vs. growing), and using $+3$ instead of $-3$ flips the sign of every downstream result, including whether the perimeter is found to be increasing or decreasing. Correct approach: assign the sign to each given rate the moment you record it — decreasing is always negative, increasing always positive — before any substitution.

Mistake 2: Using the sum rule for the area instead of the product rule …