10. A culture of “diseased” bacteria grows at a rate that is inversely proportional to the square root of the number present. If there are initially 9 units and 16 units are present after 2 days, after how many days there will be 36 units?

Let's say N(t) be the number of units of bacteria after t days. We're given that the rate of growth is inversely proportional to the square root of the number of bacteria present. This can be written as: N'(t) = -k / sqrt(N(t)) where k is a constant. We can multiply both sides of the equation by sqrt(N(t)) and separate variables to get: sqrt(N(t)) N'(t) = -k int sqrt(N(t)) N'(t) dt = int -k dt (2/3) sqrt(N(t))^3 = -kt + C We can use the fact that there are initially 9 units of bacteria to find C : (2/3) sqrt(9)^3 = -k * 0 + C C = 54 This now gives us the following equation: (2/3) sqrt(N(t))^3 = -kt + 54 We're also given that there are 16 units of bacteria after 2 days. We can use this information to find k: (2/3) sqrt(16)^3 = -2k + 54 k = 6 This gives us the following equation for N(t) : (2/3) sqrt(N(t))^3 = -6t + 54 We can solve this equation for the number of days it takes for N(t) to reach 36: (2/3) sqrt(36)^3 = -6t + 54 t = 8 Therefore, it will take 8 days for the number of bacteria to reach 36.