Question

# The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x=8 cm and y=6 cm find the rates of change of (a) perimeter, and (b) the area of the rectangle.

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Hint: - Here in this question we go through by derivative rule because the decreasing and increasing rates are given i.e. $\dfrac{{dx}}{{dt}} =$-5cm/min and $\dfrac{{dy}}{{dt}} =$4cm/min. We just use these terms in the derivatives formula of perimeter and area for finding the answers.

It is given that length (x) is decreasing at the rate of 5 cm/min and the width (y) is increasing at the rate of 4 cm/min, i.e. $\dfrac{{dx}}{{dt}} =$-5cm/min (here we use minus sign because the rate is decreasing). And $\dfrac{{dy}}{{dt}} =$4cm/min
(i)
We know that the perimeter of the rectangle is 2(x+y) where x is length and y is width.

For finding the change in perimeter we just differentiate the formula of perimeter with respect to time.

Here p=2(x+y)

After differentiating with respect to time that is t is,
$\dfrac{{dp}}{{dt}} = \dfrac{d}{{dt}}\left( {2(x + y)} \right)$
$\Rightarrow \dfrac{{dp}}{{dt}} = 2\left( {\dfrac{{dx}}{{dt}} + \dfrac{{dy}}{{dt}}} \right)$ ………… (1)

Now put the values of $\dfrac{{dx}}{{dt}}$and $\dfrac{{dy}}{{dt}}$ in equation (1) we get,
$\Rightarrow \dfrac{{dp}}{{dt}} = 2\left( { - 5 + 4} \right) = - 2$

Hence the perimeter is decreasing at the rate of 2 cm/min because we get the answer negative.

(ii)
We know that the area of the rectangle is xy where x is length and y is width.

For finding the change in area we just differentiate the formula of area with respect to time.
Here, A=xy.

After differentiating with respect to time that is t is,
$\dfrac{{dA}}{{dt}} = \dfrac{d}{{dt}}(xy)$
$\Rightarrow \dfrac{{dA}}{{dt}} = \left( {y\dfrac{{dx}}{{dt}} + x\dfrac{{dy}}{{dt}}} \right)$ ……….. (2)
Here we apply product rule for differentiation.

Now put the values of $\dfrac{{dx}}{{dt}}$and $\dfrac{{dy}}{{dt}}$ and for the area the points are also given in the question i.e. x=8cm and y=6cm. put these values in the equation (2) we get,
$\Rightarrow \dfrac{{dA}}{{dt}} = \left( {6( - 5) + 8(4)} \right) = 2$

Hence the area is increasing at the rate of $2c{m^2}/\min$ because we get the answer positive.

Note: - Whenever we face such a type of question the key concept for solving the question is first write the data that is given in the question in mathematical terms and always keep in mind the write decreasing differentiation with a negative sign. Then apply the differentiation on that formula whose rates are asked in the question then by putting the value in that formula after differentiation you will get the answer.