Question

all sides of a square are increasing at a rate of 4 centimeters per second. how fast is the area changing when each side is 10 centimeters? \\(\frac{da}{dt} = 20\frac{\text{cm}^2}{\text{sec}}\\) \\(\frac{da}{dt} = 80\frac{\text{cm}^2}{\text{sec}}\\) \\(\frac{da}{dt} = 100\frac{\text{cm}^2}{\text{sec}}\\) \\(\frac{da}{dt} = 400\frac{\text{cm}^2}{\text{sec}}\\)

Explanation:

Step1: Define area formula for square

Let $s$ = side length of square. Area $A = s^2$.

Step2: Differentiate with respect to time

Differentiate $A$ using chain rule:
$$\frac{dA}{dt} = 2s \cdot \frac{ds}{dt}$$

Step3: Substitute given values

We know $\frac{ds}{dt}=4\ \text{cm/sec}$ and $s=10\ \text{cm}$:
$$\frac{dA}{dt} = 2(10)(4)$$

Step4: Calculate the final value

$$\frac{dA}{dt} = 80$$

Answer:

$\boldsymbol{\frac{dA}{dt} = 80\frac{\text{cm}^2}{\text{sec}}}$ (corresponding to the second option)