Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

A culture of bacteria has an initial population of 9300 bacteria and doubles every 3
hours. Using the formula P = Po 25, where Pt is the population after thours, P
is the initial population, t is the time i hours and d is the doubling time, what is the
population of bacteria in the culture after 10 hours, to the nearest whole number

Sagot :

Answer:

The approximate population of bacteria in the culture after 10 hours is 93,738.  

Step-by-step explanation:

General Concepts:

  • Exponential Functions.
  • Exponential Growth.
  • Doubling Time Model.
  • Logarithmic Form.

BPEMDAS Order of Operations:

  1. Brackets.
  2. Parenthesis.
  3. Exponents.
  4. Multiplication.
  5. Division.
  6. Addition.
  7. Subtraction.

Definitions:

We are given the following Exponential Growth Function (Doubling Time Model), [tex]\displaystyle\mathsf{P_{(t)}\:=\:P_0\cdot2^{(t/d)}}[/tex] where:

  • [tex]\displaystyle\sf{P_t\:\:\rightarrow}[/tex] The population of bacteria after “t ” number of hours.
  • [tex]\displaystyle\sf{P_0 \:\:\rightarrow}[/tex] The initial population of bacteria.
  • [tex]\displaystyle{t \:\:\rightarrow}[/tex]  Time unit (in hours).
  • [tex]\displaystyle{\textit d \:\:\rightarrow}[/tex]  Doubling time, which represents the amount of time it takes for the population of bacteria to grow exponentially to become twice its initial quantity.  

Solution:

Step 1: Identify the given values.

  • [tex]\displaystyle\sf{P_0\:=}[/tex] 9,300.
  • t = 10 hours.
  • d = 3.  

Step 2: Find value.

1. Substitute the values into the given exponential function.

  [tex]\displaystyle\mathsf{P_{(t)} = P_0\cdot2^{(t/d)}}[/tex]

  [tex]\displaystyle\mathsf{\longrightarrow P_{(10)} = 9300\cdot2^{(10/3)}}[/tex]

2. Evaluate using the BPEMDAS order of operations.

  [tex]\displaystyle\mathsf{P_{(10)} = 9300\cdot2^{(10/3)}\quad \Longrightarrow BPEMDAS:\:(Parenthesis\:\:and\:\:Division).}[/tex]

  [tex]\displaystyle\sf P_{(10)} = 9300\cdot2^{(3.333333)}\quad\Longrightarrow BPEMDAS:\:(Exponent).}[/tex]

  [tex]\displaystyle\sf P_{(10)} = 9300\cdot(10.079368399)\quad \Longrightarrow BPEMDAS:(Multiplication).}[/tex]

 [tex]\boxed{\displaystyle\mathsf{P_{(10)} \approx 93,738.13\:\:\:or\:\:93,738}}[/tex]

Hence, the population of bacteria in the culture after 10 hours is approximately 93,738.  

Double-check:

We can solve for the amount of time (t ) it takes for the population of bacteria to increase to 93,738.

1. Identify given:

  • [tex]\displaystyle\mathsf{P_{(t)} = 93,738 }[/tex].
  • [tex]\displaystyle\mathsf{P_0 = 9,300}[/tex].
  • d = 3.

2. Substitute the values into the given exponential function.

  [tex]\displaystyle\mathsf{P_{(t)} = P_0\cdot2^{(t/d)}}[/tex]

  [tex]\displaystyle\mathsf{\longrightarrow 93,378 = 9,300\cdot2^{(t/3)}}[/tex]

3. Divide both sides by 9,300:

  [tex]\displaystyle\mathsf{\longrightarrow \frac{93,378}{9,300} = \frac{9,300\cdot2^{(t/3)}}{9,300}}[/tex]

  [tex]\displaystyle\mathsf{\longrightarrow 10.07936840 = 2^{(t/3)}}[/tex]

4. Transform the right-hand side of the equation into logarithmic form.

  [tex]\boxed{\displaystyle\mathsf{\underbrace{ x = a^y}_{Exponential\:Form} \longrightarrow \underbrace{y = log_a x}_{Logarithmic\:Form}}}[/tex]    

  [tex]\displaystyle\mathsf{\longrightarrow 10.07936840 = \bigg[\:\frac{t}{3}\:\bigg]log(2)}[/tex]  

5. Take the log of both sides of the equation (without rounding off any digits).  

  [tex]\displaystyle\mathsf{log(10.07936840) = \bigg[\:\frac{t}{3}\:\bigg]log(2)}[/tex]

  [tex]\displaystyle\mathsf{\longrightarrow 1.003433319 = \bigg[\:\frac{t}{3}\:\bigg]\cdot(0.301029996)}[/tex]

6. Divide both sides by (0.301029996).

  [tex]\displaystyle\mathsf{\frac{1.003433319}{0.301029996} = \frac{\bigg[\:\frac{t}{3}\:\bigg]\cdot(0.301029996) }{0.301029996}}[/tex]

  [tex]\displaystyle\mathsf{\longrightarrow 3.3333333 = \frac{t}{3}}[/tex]

7. Multiply both sides of the equation by 3 to isolate "t."

  [tex]\displaystyle\mathsf{(3)\cdot(3.3333333) = \bigg[\:\frac{t}{3}\:\bigg]\cdot(3)}[/tex]

  [tex]\boxed{\displaystyle\mathsf{t\approx10}}[/tex]

Hence, it will take about 10 hours for the population of bacteria to increase to 93,378.    

__________________________________

Learn more about Exponential Functions on:

https://brainly.com/question/18522519            

View image djtwinx017