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In an experiment, there were initially 12 bacteria on a petri dish. On day 5, there were 27 bacteria, and on day 15, there were 57 bacteria.

a) If we let the function [tex]\( B(x) \)[/tex] represent the amount of bacteria present after [tex]\( x \)[/tex] days, which of the following functions models this experiment?

A. [tex]\( B(x) = 4x + 12 \)[/tex]
B. [tex]\( B(x) = 3x + 12 \)[/tex]
C. [tex]\( B(x) = 5x^2 - 22x + 12 \)[/tex]
D. [tex]\( B(x) = 4x^3 - 897x + 12 \)[/tex]
E. [tex]\( B(x) = 4x^2 - 17x + 12 \)[/tex]
F. [tex]\( B(x) = 3x^3 - 72x + 12 \)[/tex]

b) Use your choice of [tex]\( B(x) \)[/tex] to predict the amount of bacteria on day 27.

[tex]\[\square\ \square\ \text{bacteria}\][/tex]



Answer :

GinnyAnswer
To start with, we have been given three data points representing the amount of bacteria at different times:
- Initially (day 0), there are 12 bacteria.
- On day 5, there are 27 bacteria.
- On day 15, there are 57 bacteria.

We need to determine which of the given functions best models the experimental data:

A. [tex]\( B(x) = 4x + 12 \)[/tex]
B. [tex]\( B(x) = 3x + 12 \)[/tex]
D. [tex]\( B(x) = 4x^3 - 897x + 12 \)[/tex]
E. [tex]\( B(x) = 4x^2 - 17x + 12 \)[/tex]
C. [tex]\( B(x) = 5x^2 - 22x + 12 \)[/tex]
F. [tex]\( B(x) = 3x^3 - 72x + 12 \)[/tex]

We will evaluate each function using the provided data points to see which function fits all three points.

1. Function A: [tex]\( B(x) = 4x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 4(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 4(5) + 12 = 20 + 12 = 32 \)[/tex] (incorrect)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 4(15) + 12 = 60 + 12 = 72 \)[/tex] (incorrect)

2. Function B: [tex]\( B(x) = 3x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 3(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 3(5) + 12 = 15 + 12 = 27 \)[/tex] (correct)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 3(15) + 12 = 45 + 12 = 57 \)[/tex] (correct)

Since function B fits all the provided data points, we consider it the appropriate model for the experiment.

3. Function D: [tex]\( B(x) = 4x^3 - 897x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 4(0) - 897(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 4(5^3) - 897(5) + 12 = 500 - 4485 + 12 = -3973 \)[/tex] (incorrect)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 4(15^3) - 897(15) + 12 = 13500 - 13455 + 12 = 42 + 12 = 54 \)[/tex] (incorrect)

4. Function E: [tex]\( B(x) = 4x^2 - 17x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 4(0) - 17(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 4(5^2) - 17(5) + 12 = 100 - 85 + 12 = 27 \)[/tex] (correct)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 4(15^2) - 17(15) + 12 = 900 - 255 + 12 = 657 \)[/tex] (incorrect)

5. Function C: [tex]\( B(x) = 5x^2 - 22x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 5(0) - 22(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 5(5^2) - 22(5) + 12 = 125 - 110 + 12 = 27 \)[/tex] (correct)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 5(15^2) - 22(15) + 12 = 1125 - 330 + 12 = 807 \)[/tex] (incorrect)

6. Function F: [tex]\( B(x) = 3x^3 - 72x + 12 \)[/tex]

- When [tex]\( x = 0 \)[/tex]: [tex]\( B(0) = 3(0^3) - 72(0) + 12 = 12 \)[/tex] (correct)
- When [tex]\( x = 5 \)[/tex]: [tex]\( B(5) = 3(5^3) - 72(5) + 12 = 375 - 360 + 12 = 27 \)[/tex] (correct)
- When [tex]\( x = 15 \)[/tex]: [tex]\( B(15) = 3(15^3) - 72(15) + 12 = 10125 - 1080 + 12 = 9057 \)[/tex] (incorrect)

Based on the evaluations, B(x) = 3x + 12 is the correct function as it correctly models the given data points.

To predict the amount of bacteria on day 27 using [tex]\( B(x) = 3x + 12 \)[/tex]:

[tex]\( B(27) = 3(27) + 12 = 81 + 12 = 93 \)[/tex]

So, the amount of bacteria on day 27 is:
[tex]\[ \boxed{93} \][/tex] bacteria.

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