- A circular stain is growing.
The rate of increase of its radius is inversely proportional to the square of the radius. At time \(t\) seconds the circular stain has radius \(r \mathrm {~cm}\) and area \(A \mathrm {~cm} ^ { 2 }\).
a. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { k } { \sqrt { A } }\).
Given that
- the initial area of the circular stain is \(0.09 \mathrm {~cm} ^ { 2 }\).
- after 10 seconds the area of the circular stain is \(0.36 \mathrm {~cm} ^ { 2 }\).
b. Solve the differential equation to find a complete equation linking \(A\) and \(t\).