# Find the rate of change of area, perimeter and the lengths of the diagonals of the rectangle?

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The square of the diagonal is as you say: $$D^2=W^2+L^2$$ Now use implicit differentiation to get: $$2D \frac{dD}{dt}=2W \frac{dW}{dt}+2L \frac{dL}{dt}$$ At the instant of interest you are given $W,\ L,\ \frac{dW}{dt}, \ \frac{dL}{dt}$, and know how to calculate $D$. You substitute all of these into the above equation and solve for $\frac{dD}{dt}$.

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Updated on August 15, 2022

The length $l$ of a rectangle is decreasing at the rate of $2 \space cm/sec$ while the width $w$ is increasing at the rate of $2\space cm/sec$. When $L=12cm$ and $W=5cm$. Find the rate of change of

1. The area.

2. The perimeter.

3. The lengths of the diagonals of the rectangle.

My attempt:

If I want to calculate the rate of change of the diagonal, of course I am going to use the relation $D^2=L^2 + W^2$ Which is $D=\sqrt{(L^2+ W^2)}$, at this instant $L=2.4\times W$, so $L^2=5.76\times W^2$, or $(L^2)/5.76=W^2$

Can I substitute any of these in the diagonal equation and use it to find the rate of change?

Thanks.

• almagest over 7 years
You differentiate first, then substitute the values.