The count in a bacteria culture was 800 after 15 minutes and 6443 after 25 minutes. assume the growth can be modelled exponentially by a function of the form q(t) = a e^{r t}, where t is in minutes.

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The count in a bacteria culture was 700 after 20 minutes and 1000 after 40 minutes. What was the initial size of the culture?

490 microorganisms.

Explanation:

I will assume exponential growth for bacteria. This means that we can model the growth with an exponential function:

f

(

t

)

=

A

0

e

k

t

where

k

is the growth constant and

A

0

is the initial amount of bacteria.

Sub the two known values into the function to get two equations:

700

=

A

0

e

20

k

(1)

1000

=

A

0

e

40

k

(2)

Divide (2) by (1) to find

k

:

1000

700

=

A

0

e

40

k

A

0

e

20

k

10

7

=

e

40

k

−

20

k

=

e

20

k

Take the natural log of both sides to isolate

k

:

ln

(

10

7

)

=

ln

e

20

k

ln

(

10

7

)

=

20

k

k

=

ln

(

10

7

)

20

Now that we have the growth constant,

k

, we can substitute one of the points in to solve for the initial amount,

A

0

:

(

40

,

1000

)

1000

=

A

0

e

ln

(

10

7

)

20

⋅

40

A

0

=

1000

e

0.0178

⋅

40

=

490

Estimate the following (by rounding off to nearest hundred (a)6,941 (b)6,320 (c) 416 (d)126 (e)109 (f)202 (g) 146 (h)7,699 (I)8,311 (j)986

Step-by-step explanation: