Question

each side of a square is increasing at a rate of 7 cm/s. at what rate (in cm²/s) is the area of the square increasing when the area of the square is 9 cm²?

Explanation:

Step1: Find the side - length of the square

Let the side - length of the square be $s$ and the area be $A$. We know that $A = s^{2}$. Given $A = 9\ cm^{2}$, then $s=\sqrt{A}=\sqrt{9}=3\ cm$.

Step2: Differentiate the area formula with respect to time $t$

Differentiate $A = s^{2}$ with respect to $t$ using the chain - rule. $\frac{dA}{dt}=2s\frac{ds}{dt}$.

Step3: Substitute the known values

We are given that $\frac{ds}{dt}=7\ cm/s$ and $s = 3\ cm$. Substitute these values into the equation $\frac{dA}{dt}=2s\frac{ds}{dt}$. Then $\frac{dA}{dt}=2\times3\times7 = 42\ cm^{2}/s$.

Answer:

$42$