The population of a heard of cattle numbered was 5000 to begin with and was 10,000 after 10 years. If the population was growing exponentially, what was the growth rate? show all workA) r=2B) r=20C)r=0.69D)r=6.9

Answer:[tex]r=0.0718[/tex]. The closest value from your given choices is C)r=0.69Step-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 yearsWe 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