Q.The length $x$ of a rectangle is decreasing at the rate of $5 \text{ cm/minute}$ and the width $y$ is increasing at the rate of $4 \text{ cm/minute}$. When $x = 8 \text{cm}$ and $y = 6 \text{cm}$, find the rates of change of

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

Step 1: Given rates and variables

$\frac{dx}{dt} = -5$ cm/min (decreasing), $\frac{dy}{dt} = +4$ cm/min.

At the instant: $x = 8$ cm, $y = 6$ cm.

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

Differentiate: $\frac{dP}{dt} = 2\left(\frac{dx}{dt} + \frac{dy}{dt}\right) = 2(-5 + 4) = 2(-1) = -2$ cm/min.

Step 3: Area $A = xy$ …

The Core Idea: Related Rates

When two quantities change with time, their rates of change are linked through the geometry that relates them. Here, the rectangle's perimeter $P$ and area $A$ are both functions of $x$ and $y$, and we know $\frac{dx}{dt} = -5$ cm/min (decreasing, so negative) and $\frac{dy}{dt} = +4$ cm/min (increasing, so positive). The trick is to differentiate $P$ and $A$ implicitly with respect to $t$, then plug in the known values.

Step-by-step solution

1. Write the formulas

Perimeter: $P = 2x + 2y$

Area: $A = x \cdot y$

2. Differentiate both with respect to time $t$

Since $x$ and $y$ are functions of $t$, we use the chain rule:

(The area uses the product rule: derivative of $x y$ is $x' y + x y'$.)

3. Substitute the given rates

We have $\frac{dx}{dt} = -5$, $\frac{dy}{dt} = 4$.

For the perimeter:

So the perimeter is decreasing at 2 cm per minute — and notice it doesn't depend on $x$ or $y$ at all. That makes sense: the perimeter formula is linear, so its rate is constant.

Method: Related Rates With Two Independently-Varying Quantities

Use this pattern when TWO separate dimensions (not one dimension driving a single derived quantity) each change with time at their own given rate, and the target quantity depends on both.

Steps

Step 1: Assign signed rates to each changing quantity

Represent "decreasing at rate $k$" as $\frac{dx}{dt}=-k$ and "increasing at rate $k$" as $\frac{dy}{dt}=+k$ — the sign is part of the given data and must be carried through every subsequent step.

Step 2: Write the formula for each target quantity in terms of both variables

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

Step 3: Differentiate each formula with respect to time

For the perimeter (a sum, so term-by-term differentiation):

$\frac{dP}{dt} = 2\frac{dx}{dt} + 2\frac{dy}{dt}$ …

Common Mistakes

Mistake 1: Dropping the negative sign on the decreasing quantity

"Decreasing at $5\text{ cm/min}$" means $\frac{dx}{dt}=-5$, not $+5$ — using a positive value for a quantity that is explicitly shrinking flips the sign of both the perimeter and area rates.

Mistake 2: Forgetting the product rule for the area

Since $A=xy$ with both $x$ and $y$ changing in time, $\frac{dA}{dt}$ needs BOTH terms $x\frac{dy}{dt}$ and $y\frac{dx}{dt}$ (product rule) — differentiating only one variable while treating the other as a fixed constant misses half the rate.

Mistake 3: Assuming a decreasing side must mean a decreasing area …