Part 3
An exponential growth function has the general form:
[tex]f(t)=a\cdot(1+r)^t[/tex]
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:
[tex]f(0)=a(1+r)^0=a\cdot1=a[/tex]
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:
[tex]C(h)=C(0)\cdot(1+r)^h[/tex]
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:
[tex]\begin{gathered} C(1)=10\cdot(1+r)^1 \\ \\ 12=10+10r \\ \\ 12-10=10r \\ \\ 10r=2 \\ \\ r=0.2 \end{gathered}[/tex]
Thus, the number of cells C(h) is given by:
[tex]C(h)=10\cdot(1.2)^h[/tex]
Notice that this is valid for C(15) = 154:
[tex]C(15)=10\cdot(1.2)^{15}\cong154.07\cong154_{}[/tex]
Part 1
Then, using this formula, we find:
[tex]\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}[/tex]
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):
[tex]C(24)=10(1.2)^{24}\cong795[/tex]
Therefore, the answer is 795 cells.
