Question

A biologist just discovered a new strain of bacteria that helps defend the human body against the flu virus. To know the dosage that should be given to someone, the doctor must first know if the bacteria can multiply fast enough to combat the virus. To find the rate at which the bacteria multiplies, she puts 10 cells in a petri dish. In an hour, she comes back to find that there are now 12 cells in the dish.

Answer 1

Part 3

An exponential growth function has the general form:

`f(t)=a\cdot(1+r)^t`

where r is the rate of growth, t is the time, and a is a constant. Notice that if calculate f(t) for t = 0, we have (1 + r)º = 1 (any number with exponent 0 equals 1). So, we obtain:

`f(0)=a(1+r)^0=a\cdot1=a`

Thus, the constant a is the initial value of the function.

Now, the rate at which a bacteria grows is exponential. So, the function C(h) is given by:

`C(h)=C(0)\cdot(1+r)^h`

Notice that we represented the time by the letter h instead of t.

Since C(0) = 10 and C(1) = 12, we can replace h by 1 to find:

`\begin{gathered} C(1)=10\cdot(1+r)^1 \\ \\ 12=10+10r \\ \\ 12-10=10r \\ \\ 10r=2 \\ \\ r=0.2 \end{gathered}`

Thus, the number of cells C(h) is given by:

`C(h)=10\cdot(1.2)^h`

Notice that this is valid for C(15) = 154:

`C(15)=10\cdot(1.2)^(15)\cong154.07\cong154_{}`

Part 1

Then, using this formula, we find:

`\begin{gathered} C(2)=10(1.2)^2\cong14 \\ \\ C(3)=10(1.2)^3\cong17.3\cong17 \\ \\ C(4)=10(1.2)^4\cong20.7\cong21 \\ \\ C(5)=10(1.2)^5\cong24.9\cong25 \\ \\ C(6)=10(1.2)^6\cong29.9\cong30 \\ \\ C(7)=10(1.2)^7\cong35.8\cong36 \\ \\ C(8)=10(1.2)^8\cong43 \\ \\ C(9)=10(1.2)^9\cong51.6\cong52 \\ \\ C(10)=10(1.2)^(10)\cong61.9\cong62 \\ \\ C(11)=10(1.2)^(11)\cong74.3\cong74 \\ \\ C(12)=10(1.2)^(12)\cong89.2\cong89 \\ \\ C(13)=10(1.2)^(13)\cong107 \\ \\ C(14)=10(1.2)^(14)\cong128.4\cong128 \end{gathered}`

Part 2

Now, plotting the points, rounded to the nearest whole cell, on the graph, we obtain:

Part 4

Using a calculator, we obtain the following graph of the function C(h):

Comparing the graph to the plot of the data, we see that they match.

Part 5

After a full day, it has passed 24 hours. So, we need to use h = 24 in the function C(h):

`C(24)=10(1.2)^(24)\cong795`

Therefore, the answer is 795 cells.