A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After two hours the population increed to 900 cell. (a) Find an expression for the number of bacteria after t hours. P(t) = (b) Find the number of bacteria after 4 hours. (Round your answer to the nearest whole number.) P(4) = bacteria (c) Find the rate of growth after 4 hours. (Round your answer to the nearest whole number.) P'(4) = bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.) t = hr

(a) P(t) = 100 + 100(t²)

Bacteria grow at at exponential rate, which is what is being represented by the "t²" part of the expression. The number of cells initially present (100) also need to be added to the total population

(b) P(4) = 100 + 100(4²)

P(4) = 1700

replace 't' with '4' in the expression found in part (a) to calculate the number of bacterial cells after 4 hours

(c) P'(t) = 1700 + 1700(t²)

just plug in the size of bacterial population after 4 hours as has been calculated in the previous part of the question.

(d) 10000 = 100 + 100(t²)

here, we have the size of population given and need to find out the time it will take to get there, so make t the subject in the expression

= t²

solve this expression to obtain the value of t²

t² = 99

take a square root on both sides to go from t² to t

t = 9.9 hours

Hence, in just 9.9 hours, the population of bacteria would have gone from being 100 initially to being 10,000!

Hope that answers the question. Have a great day!