Question

A culture of bacteria enters the exponential growth phase with 4,596 cells per mL. After 15 hours, there are 206,104,972 cells per mL. What is the instantaneous growth rate constant, with units of 1/hours, for this culture? Express your answer as an exponent rounded to two decimal places. For example 0.66666 would be entered as 6.67E-1

Answer 1

The Growth rate constant (k) will be =
`5.16 * 10^-03` per minute or 0.308 per hour.

Step-by-step explanation:

Given,

N₀ = 4596 cells / ml

N = 206104972 cells / ml

t = 15 hours

We know that,

n =
`3.3 * log \frac {b}{B}`

Where,

n = no. of generation during the period of exponential growth.

∴ n =
`3.3 * log \frac {b}{B}`

=
`3.3 * log \frac {206104972} {4596}`

= 3.3 × 44844.42

∴ n = 15.350

We know that,

g =
`\frac {t} {n}`

Where,

g = generation time

t = duration of exponential growth

∴ g =
`\frac {t} {n}` or ∴ g =
`\frac {t} {n}`

=
`\frac {15 * 60} {15.350}` = 15 / 15.350

= 900 / 15.350 ∴ g = 0.977 hours

∴ g = 58.63 minutes

We know that,

k =
`0.301 / g`

Where, k = specific growth rate

∴ k =
`0.301 / g` or ∴ k =
`0.301 / g`

= 0.301 / 58.63 = 0.301 / 0.977

∴ k =
`5.16 * 10^-03` per minutes ∴ k = 0.308 per hour