Option A: [tex]f(t)=10(1.4)^{t}[/tex] is the function that represents the population of the bacteria after t hours.
Explanation:
The graph shows the population of the bacteria for every t hours.
We shall determine the function using the formula,
[tex]f(t)=a(b)^t[/tex]
Let us determine the value of a and b by substituting the coordinates [tex](0,10)[/tex] and [tex](2,20)[/tex] in the above formula.
Substituting [tex](0,10)[/tex] in the formula [tex]f(t)=a(b)^t[/tex], we get,
[tex]10=a(b)^0[/tex]
[tex]10=a[/tex]
Thus, substituting [tex]a=10[/tex] and [tex](2,20)[/tex] in the formula, we have,
[tex]20=10(b)^2[/tex]
[tex]2=b^2[/tex]
[tex]\sqrt{2} =b[/tex]
[tex]1.4=b[/tex]
Thus, the value of a and b are [tex]a=10[/tex] and [tex]b=1.4[/tex]
Substituting these values in the formula [tex]f(t)=a(b)^t[/tex], we have,
[tex]f(t)=10(1.4)^{t}[/tex]
Thus, the function that represents the population of the bacteria after t hours is [tex]f(t)=10(1.4)^{t}[/tex]
Hence, Option A is the correct answer.
