Question

Organisms A and B start out with the same population size.

Organism A's population doubles every day. After 5 days, the population stops growing and a virus cuts it in half every day for 3 days.

Organism B's population grows at the same rate but is not infected with the virus. After 8 days, how much larger is organism B's population than organism A's population? Answer the questions to find out.

1. By what factor does organism A's population grow in the first five days? Express your answer as an exponential expression. (2 points)















2. The expression showing organism A's decrease in population over the next 3 days is (\small {\frac{1}{2}})^3(
2
1
​
)
3
. This can be written as (2–1)3.

Write (2–1)3 with the same base but one exponent. (2 points)















3. By combining the increase and decrease, find an exponential expression for the total change in organism A's population after 8 days. Show your work. (2 points)















4. Write an exponential expression showing organism B's increase in population over the same 8 days. (2 points)















5. Use your answers to questions 3 and 4 to write an expression for how many times greater organism B's population is than organism A's population after 8 days.
Simplify your expression, then write it as a number that is not in exponential form. Show your process. (2 points)

Answer 1

`\textsf{1.} \quad y=a \cdot 2^5\;\;\textsf{(where $a$ is the initial population)}`

`\textsf{2.} \quad 2^(-3)`

`\textsf{3.} \quad y=a \cdot 2^(2)\;\;\textsf{(where $a$ is the initial population)}`

`\textsf{4.} \quad y=a \cdot 2^(8)\;\;\textsf{(where $a$ is the initial population)}`

`\textsf{5.} \quad 64`

Explanation:

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

Question 1

Given that organism A's population doubles every day for the first 5 days:

• b = 2
• x = 5

Therefore, an exponential expression to express organism A's population growth in the first 5 days is:

`\boxed{y=a \cdot 2^5}`

(where a is the initial population).

Question 2

Given expression showing organism A's decrease in population over the next 3 days:

• 
  `\left(2^(-1)\right)^3`

To rewrite the expression with the same base but one exponent,

`\textsf{apply the exponent rule} \quad (a^b)^c=a^(bc):`

`\implies 2^(-3)`

Question 3

Combine the increase and decrease in organism A's population after 8 days by multiplying the rates. So the exponential expression for the total change in organism A's population after 8 days is:

`\implies y=a \cdot 2^5 \cdot 2^(-3)`

`\textsf{Apply the exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`\implies y=a \cdot 2^(5-3)`

`\implies y=a \cdot 2^(2)`

Question 4

As organism B's population grows at the same rate as organism A, but is not infected with the virus, its population doubles each day. Therefore, an exponential expression showing its increase in population over the same 8 days is:

`\boxed{y=a \cdot 2^8}`

where a is the initial population.

Question 5

To calculate how many times greater organism B's population is than organism A's population after 8 days, divide the expression for organism B by the expression for organism A:

`\implies (a \cdot 2^8)/(a \cdot 2^2)`

`\implies (2^8)/(2^2)`

`\textsf{Apply the exponent rule} \quad (a^b)/(a^c)=a^(b-c):`

`\implies 2^(8-2)`

`\implies 2^6=64`