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
[tex]\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}[/tex]
Given: a = 1, b = 1.5, y = 500,000
[tex]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}[/tex] to take over Antelope Valley
Part 2
Given a = 1, b = 2 → y = 1(2)ˣ
[tex]\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}[/tex]
Given: a = 1, b = 2, y = 500,000
[tex]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}[/tex] to take over Antelope Valley
