Final answer:
To relate the area of a circle to its radius, we use the equation A = πr². Differentiating this equation, we get dA/dt = 2πr * dr/dt, where dA/dt is the rate of change of the area and dr/dt is the rate of change of the radius. We can solve for dr/dt using the given information in each part of the question.
Step-by-step explanation:
To write an equation relating the area of a circle (A) and the radius of the circle (r), we can use the formula for the area of a circle: A = πr². This equation states that the area of a circle is equal to pi (approximately 3.14) times the radius squared.
When the area of the circle is increasing at a rate of 3 cm²/s, we can differentiate the equation with respect to time to find the rate of change of the radius. Taking the derivative of A = πr², we get dA/dt = 2πr * dr/dt. Since we are given that dA/dt = 3 cm²/s, we can substitute this value into the equation to solve for dr/dt.
For part A, when the radius is 1 cm, we can substitute r = 1 cm into the equation and solve for dr/dt.
For part B, when the circumference is 2 cm, we can use the formula for the circumference of a circle: C = 2πr. We know that C = 2 cm, so we can substitute this value into the equation and solve for the radius. Once we have the radius, we can use the same equation as in part A to find dr/dt.