A bacteria culture is grown in a petri dish. The table gives the area of the culture, in square
centimeters, at various times, in hours, since the beginning of an experiment. The area of the
bacteria culture is proportional to the actual number of bacteria in the culture.

The area of the bacteria culture can be modeled by the function C given by C(t)= ab',
where C (t) is the area, in square centimeters, at time t hours since the beginning of the
experiment

Part A
(1) Use the given data to write two equations that can be used to find the values for constants
a and b in the expression for C(t).

(2)Find the values for a and b.

Part B
(1) use the given data to find the average rate of change of the area, in square centimeters per hour, from t=0 to t=2 hours. Express your answer as a fraction or decimal. Show the computations that lead to your answer.

(2) interpret the meaning of your answer from (1) in the context of the problem

(3) use the rate of change found
in part (1) to estimate the area of the bacteria culture, in square centimeters, at time t=3 hours. Show the work that leads to your answer

(4) does the model found in Part A demonstrate exponential growth or exponential decay? Give a reason for your answer.

To write two equations that can be used to find the values for the constants a and b, we can substitute two points from the given table into the function. Let's use the points (0, 0.4) and (1, 0.6):

[tex]ab^0=0.4[/tex]

[tex]ab^1=0.6[/tex]

Part B

To find the values of a and b, first solve for the first equation for a:

[tex]ab^0=0.4\\\\a(1)=0.4\\\\a=0.4[/tex]

Now, substitute the value of a = 0.4 into the second equation and solve for b:

[tex]0.4b^1=0.6\\\\0.4b=0.6\\\\b=\dfrac{0.6}{0.4}\\\\b=1.5[/tex]

Therefore, the values of a and b are a = 0.4 and b = 1.5.

So, the function is:

[tex]C(t) = 0.4 \cdot 1.5^t[/tex]

Part B

To find the average rate of change of the area C(t) from t = 0 to t = 2, we can use the average rate of change formula:

[tex]\boxed{\begin{array}{c}\underline{\textsf{Average rate of change of function $f(x)$}}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\\textsf{over the interval $[a,b]$}\end{array}}[/tex]

In this case:

a = 0
b = 2

From the table:

C(0) = 0.4
C(2) = 0.9

Therefore:

[tex]\textsf{Average rate of change}=\dfrac{C(2)-C(0)}{2-0}\\\\\\\textsf{Average rate of change}=\dfrac{0.9-0.4}{2-0}\\\\\\\textsf{Average rate of change}=\dfrac{0.5}{2}\\\\\\\textsf{Average rate of change}=0.25\; \sf cm^2/h[/tex]

This means that with each additional hour of time that passes, the area of the bacteria culture is increasing by 0.25 square centimeters.

To estimate the area at t = 3 hours using the found average rate of change, add the average rate of change for one more hour to the value of C(t) at t = 2:

[tex]C(3) = C(2) + \textsf{Average rate of change} \times 1\; \sf hour[/tex]

[tex]C(3)=0.9+0.25\; \textsf{cm$^2$/hour} \times 1\; \sf hour[/tex]

[tex]C(3)=0.9+0.25[/tex]

[tex]C(3)=1.15\; \sf cm^2[/tex]

Therefore, the estimated area of the bacteria culture at t = 3 hours is 1.15 cm².

The model found in Part A demonstrates exponential growth because the value of the base (b) is greater than 1, and the coefficient (a) is positive.