Yolanda owns 4 rabbits. She expects the number of rabbits to double every year.
a) After how many years will she have 64 rabbits?
b) Write an equation to model this situation.
Show work please.
Answer (3)
It is geometric sequence. Therefore: 4"> a 1 = 4 q = 2 S = a 1 ∗ 1 − q 1 − q n 64 = 4 ∗ 1 − 2 1 − 2 n 16 = − 1 1 − 2 n − 16 = 1 − 2 n 16 + 1 = 2 n 2 4 + 1 = 2 n n > 4
After 4 yours she will have over 64 rabbits.
[a) -after\ how\ many\ years\ will\ she\ have\ 64\ rabbits\ \4\cdot \frac{1-2^n}{1-2} =64\ /:4\ \\frac{1-2^n}{-1} =16\ \ \ \Leftrightarrow\ \ \ 2^n-1=16\ \ \ \Leftrightarrow\ \ \ 2^n=17>16\ \2^n>2^4\ .\ \ \ \ \ \ \ \ \ \ \ \ \ \ a^b>a^c\ \ \wedge\ \ \ a>1\ \ \ \Rightarrow\ \ \ b>c\
4\ \Ans.\ Yolanda\ have\ 64\ rabbits\ after\ 4\ years]
b ) b − h o w m u c h r abbi t s w a s in t h e b e g innin g t − h o w man y t im es w i ll b e an in cre a se in t h e n u mb er o f r abbi t s . in t h e ye a rs e − h o w man y r abbi t s a re e x p ec t e d Y o l an d a b ⋅ 1 − t 1 − t n = e
Yolanda will have 64 rabbits after 4 years. The equation modeling this growth is P = 4 × 2 n , where P is the number of rabbits after n years. This reflects the initial amount of rabbits and the doubling effect each year.
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