Question

PLEASE HELP ASAP

Mrs. Smith decided to take over Antelope Valley with Zombies. She started with turning into a zombie herself,
then made the zombie population grow at a rate of 1.5 times the previous amount. The population of the
Antelope Valley is 500,000. Use this information to answer the following questions:
1(a). Set up an equation and then determine the number of zombies after 5, 10, 15, 20 hours.
1(b). Make a table of this data and then use that data to make a graph.
1(c) How many hours will it take for the zombies to take over the Antelope Vallev? Explain how you came up
with your conclusion.
Now double the zombie growth rate and go back and answer questions 1(C).

Answer 1

Answer: See Below

Explanation:

y = a(b)ˣ where

• a: initial population (starting point)
• b: growth rate
• x: number of hours
• y: final population

Part 1

Given: a = 1, b = 1.5 → y = 1(1.5)ˣ

`\begin{array}c\underline{\quad a\quad}&\underline{\quad b\quad}&\underline{\quad x\quad}&\underline{\ \ a(b)^x=}&\underline{y\quad\quad}\\\1&1.5&5&1(1.5)^5=&7.6\quad\rightarrow \bold{7}\\1&1.5&10&1(1.5)^(10)=&57.7\quad\rightarrow \bold{57}\\1&1.5&15&1(1.5)^(15)=&437.9\quad\rightarrow \bold{437}\\1&1.5&20&1(1.5)^(20)=&3,325.2\quad\rightarrow \bold{3,325}\\\end{array}`

Given: a = 1, b = 1.5, y = 500,000

`500,000=1(1.5)^x\\ln(500,000)=ln(1.5)^x\\ln(500,000)=x\ ln(1.5)\\\\(ln(500,000))/(ln(1.5))=x\\\\\\32.4=x\quad \longrightarrow \quad\large\boxed{32.4\ hours}` to take over Antelope Valley

Part 2

Given a = 1, b = 2 → y = 1(2)ˣ

`\begin{array}c\underline{\quad a\quad}&\underline{\quad b\quad}&\underline{\quad x\quad}&\underline{\ \ a(b)^x=}&\underline{y\quad\quad}\\\1&2&5&1(2)^5=&\bold{32}\\1&2&10&1(2)^(10)=&\bold{1024}\\1&2&15&1(2)^(15)=&\bold{32,768}\\1&2&20&1(2)^(20)=&\bold{1,048,576}\ \rightarrow \text{exceeds population}\\\end{array}`

Given: a = 1, b = 2, y = 500,000

`500,000=1(2)^x\\ln(500,000)=ln(2)^x\\ln(500,000)=x\ ln(2)\\\\(ln(500,000))/(ln(2))=x\\\\\\18.9=x\quad \longrightarrow \quad\large\boxed{18.9\ hours}` to take over Antelope Valley