Question

PLEASE HELP ASAP

Answer 1

Answer:  See Below

Step-by-step 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

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

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

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

Part 2

Given a = 1, b = 2     →   y = 1(2)ˣ

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

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

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