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Answer:
The area of the square is increasing at a rate of 40 square centimeters per second.
Step-by-step explanation:
The area of the square ([tex]A[/tex]), in square centimeters, is represented by the following function:
[tex]A = l^{2}[/tex] (1)
Where [tex]l[/tex] is the side length, in centimeters.
Then, we derive (1) in time to calculate the rate of change of the area of the square ([tex]\frac{dA}{dt}[/tex]), in square centimeters per second:
[tex]\frac{dA}{dt} = 2\cdot l \cdot \frac{dl}{dt}[/tex]
[tex]\frac{dA}{dt} = 2\cdot \sqrt{A}\cdot \frac{dl}{dt}[/tex] (2)
Where [tex]\frac{dl}{dt}[/tex] is the rate of change of the side length, in centimeters per second.
If we know that [tex]A = 25\,cm^{2}[/tex] and [tex]\frac{dl}{dt} = 4\,\frac{cm}{s}[/tex], then the rate of change of the area of the square is:
[tex]\frac{dA}{dt} = 2\cdot \sqrt{25\,cm^{2}}\cdot \left(4\,\frac{cm}{s} \right)[/tex]
[tex]\frac{dA}{dt} = 40\,\frac{cm^{2}}{s}[/tex]
The area of the square is increasing at a rate of 40 square centimeters per second.