Final answer:
When the area of a square is 36 cm², the rate at which the area is increasing is found to be 36 cm²/s using the concepts of related rates in calculus and the given rate of increase for the sides of the square.
Step-by-step explanation:
To find the rate at which the area of the square is increasing when the area is 36 cm2, we can use the concept of related rates from calculus. Let's denote s as the length of a side of the square and A as the area of the square. Since A = s2, we can differentiate both sides with respect to time t to find the rates of change.
Given that each side of a square is increasing at a rate of 3 cm/s, we have
ds/dt = 3 cm/s. We can use the chain rule to differentiate the area with respect to time:
dA/dt = 2s · ds/dt
When the area is 36 cm2, the length of each side of the square is 6 cm (since 62 = 36). Now, we plug in this value and the rate at which the side length is changing into the equation:
dA/dt = 2 · 6 cm · 3 cm/s = 36 cm2/s
Therefore, the area of the square is increasing at a rate of 36 cm2/s when the area of the square is 36 cm2.