A geometric progression has first term 75 and second term - 15 .
Find the common ratio.
Find the sum to infinity.
Find the area of the finite region enclosed by the curve \(y = 5 x - x ^ { 2 }\) and the \(x\)-axis.
During one day, a biological culure is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by
$$N = 20000 \times ( 1.06 ) ^ { t }$$
Using this model,
find the number of bacteria present at 11 a.m.,
find, to the nearest minute, the time when the initial number of bacteria will have doubled.