Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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.
- 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.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
