An ant colony explores an area enclosed by a circle centeredat its nest. This area (and the circle containing it) expands as theyhave time to explore. Due to the need for ants to return to the colonyperiodically, as the radius expands, the change in the radius of the areaexplored with respect to time slows down. Specifically, if r is the radiusof the area the ants have explored in meters, then = 3 m/day. Findthe rate of change of the area explored by the ants 4 in m2/day whenthe radius of the area explored is 3m.dt

The variable r denotes the radius of the area explored by the ants and A denotes the area explored by the ants. By the formula for the area of a circle, the area enclosed by the ants = A=πr2. The rate of change of the area explored by the ants is given by dAdt. But, A is in terms of r in the formula A=πr2. Therefore, by the chain rule of differentiation, dAdt=dAdr×drdt. Note that, dAdr=ddrπr2=2πr m. Further, given that drdt=3r2 m/day.Then, dAdt=dAdr×drdt=2πr m×3r2 m/day=2πr×3r2 m2/day=6rπ m2/day When the radius of the explored area is 3 m, we have r=3. Substituting r=3 in dAdt=6rπ m2/day, we get dAdt=63π m2/day=2π m2/day Hence, the rate of change of the area explored by the ants is dAdt=2π m2/day.