20. expanding rectangle a rectangle initially has dimensions 2 cm by 4 cm. all sides begin increasing in…

Explanation:

Step1: Express length and width as functions of time

Let ( t ) be the time in seconds. The initial length ( l_0 = 4\mathrm{cm}), and the rate of change of length (\frac{dl}{dt}=1\mathrm{cm/s}). So ( l(t)=4 + t). The initial width ( w_0 = 2\mathrm{cm}), and the rate of change of width (\frac{dw}{dt}=1\mathrm{cm/s}). So ( w(t)=2 + t).

Step2: Write the formula for the area of a rectangle

The area of a rectangle (A=l\times w). Substitute ( l = 4 + t) and ( w = 2 + t) into the formula, we get (A=(4 + t)(2 + t)=8+6t+t^{2}).

Step3: Differentiate the area function with respect to time

Using the power rule (\frac{d}{dt}(t^{n})=nt^{n - 1}), (\frac{dA}{dt}=\frac{d}{dt}(8+6t+t^{2})). (\frac{dA}{dt}=6 + 2t).

Step4: Find the value of (\frac{dA}{dt}) at (t = 20)

Substitute (t = 20) into (\frac{dA}{dt}=6+2t). (\frac{dA}{dt}\mid_{t = 20}=6+2\times20). (\frac{dA}{dt}\mid_{t = 20}=6 + 40=46\mathrm{cm^{2}/s}).

Answer:

The rate at which the area of the rectangle is increasing after (20\mathrm{s}) is (46\mathrm{cm^{2}/s}).