Question

The exponential function that models cell duplication in a lab is f(t) = 2^ t+2 where f(t) is the number of cells after time t (in hours). After how many hours has the number of cells increased to 10,000?

Answer 1

12.29

Explanation:

Answer 2

ANSWER


`11.29 \: hours`

EXPLANATION

The exponential function that models cell duplication in the lab is given as


`f(t) = {2}^(t + 2)`

We want to determine the time it will take for the cells to increase to

`10,000.`

In other words, we want to find the value of

`t`
when

`f(t) = 10,000`

This gives us the equation,


`10,000 = {2}^(t + 2)`

Recall that,


`{a}^(m + n) = {a}^(m) * {a}^(n)`

We apply this property to the right hand side to obtain,


`10,000 = {2}^(t) * {2}^(2)`

This implies that,


`10,000 =4 * {2}^(t)`

We divide both sides by 4 to get,


`2500 = {2}^(t)`

We take the antilogarithm of both sides to base 10 to get,


`t = log_(2)(2500)`

This implies that,


`t = 11.29 \: hours`
to the nearest hundredth.