Answer:
The circle is increasing at a rate of 1.76π or about 5.5292 square centimeters per second.
Step-by-step explanation:
We want to determine the rate at which a circle's area is increasing given that its radius is increasing at a rate of 0.02 cm/s at the instant when its radius is 44 cm.
In other words, we want to find dA/dt when dr/dt = 0.02 cm/s and r = 44.
Recall that the equation of a circle is given by:
[tex]\displaystyle A = \pi r^2[/tex]
Take the derivative of both sides with respect to t:
[tex]\displaystyle \frac{d}{dt}\left[ A\right] = \frac{d}{dt}\left[ \pi r^2\right][/tex]
Implicitly differentiate:
[tex]\displaystyle \frac{dA}{dt} = 2\pi r \frac{dr}{dt}[/tex]
dr/dt = 0.02 and r = 44. Substitute and evaluate:
[tex]\displaystyle \begin{aligned} \frac{dA}{dt} & = 2\pi (44\text{ cm})\left(0.02\text{ cm/s}\right) \\ \\ & =1.76\pi \text{ cm$^2$/s} \\ \\ &\approx 5.5292 \text{ cm$^2$/s} \end{aligned}[/tex]
In conclusion, the circle is increasing at a rate of 1.76π or about 5.5292 square centimeters per second.
