Answer:
[tex]r=0.0718[/tex]. The closest value from your given choices is C)r=0.69
Step-by-step explanation:
To solve this we are using the standard exponential growth equation:
[tex]f(t)=A(1+r)^t[/tex]
where
[tex]f(t)[/tex] is the final population after [tex]t[/tex] years
[tex]A[/tex] is the initial population
[tex]r[/tex] is the growth rate in decimal form
[tex]t[/tex] is the time in years
We know from our problem that the initial population is 5000, the final population is 10000, and the time is 10 years, so [tex]A=5000[/tex], [tex]f(t)=10000[/tex], and [tex]t=10[/tex].
Let's replace the values and solve for [tex]r[/tex]:
[tex]f(t)=A(1+r)^t[/tex]
[tex]10000=5000(1+r)^{10}[/tex]
Divide both sides by 5000
[tex]\frac{10000}{5000} =(1+r)^{10}[/tex]
[tex]2=(1+r)^{10}[/tex]
Take root of 10 to both sides
[tex]\sqrt[10]{2} =\sqrt[10]{(1+r)^{10}}[/tex]
[tex]\sqrt[10]{2} =1+r[/tex]
Subtract 1 from both sides
[tex]\sqrt[10]{2}-1=r[/tex]
[tex]r=\sqrt[10]{2}-1[/tex]
[tex]r=1.0718-1[/tex]
[tex]r=0.0718[/tex]
We can conclude that the growth rate of our exponential equation is [tex]r=0.0718[/tex]. The closest value from your given choices is C)r=0.69
