assume a circle is expanding with time where r= t+3. What is the formula for rate of change of an area of the circle with reswpect to time?
To find the formula for the rate of change of the area of the circle with respect to time, we can use the formula for the area of a circle: A = πr², where A is the area and r is the radius.
Given that r = t + 3, we substitute this expression for r into the area formula:
A = π(t + 3)².
To determine the rate of change of the area (dA) with respect to time (dt), we differentiate the area formula with respect to time:
dA/dt = d/dt [π(t + 3)²].
To simplify the derivative, we apply the chain rule:
dA/dt = 2π(t + 3) * d/dt(t + 3).
The derivative of (t + 3) with respect to t is just 1, since the derivative of t with respect to itself is 1.
Therefore, the formula for the rate of change of the area of the circle with respect to time is:
dA/dt = 2π(t + 3).