ANSWER
[tex]11.29 \: hours[/tex]
EXPLANATION
The exponential function that models cell duplication in the lab is given as
[tex]f(t) = {2}^{t + 2} [/tex]
We want to determine the time it will take for the cells to increase to
[tex]10,000.[/tex]
In other words, we want to find the value of
[tex]t[/tex]
when
[tex]f(t) = 10,000[/tex]
This gives us the equation,
[tex]10,000 = {2}^{t + 2} [/tex]
Recall that,
[tex] {a}^{m + n} = {a}^{m} \times {a}^{n} [/tex]
We apply this property to the right hand side to obtain,
[tex]10,000 = {2}^{t} \times {2}^{2} [/tex]
This implies that,
[tex]10,000 =4 \times {2}^{t} [/tex]
We divide both sides by 4 to get,
[tex]2500 = {2}^{t} [/tex]
We take the antilogarithm of both sides to base 10 to get,
[tex]t = log_{2}(2500) [/tex]
This implies that,
[tex]t = 11.29 \: hours[/tex]
to the nearest hundredth.
