Question

the number N(t) of bacteria in a culture is growing exponentially. When t=0, N(t)=100, and when t=5. N(t)=300 where t is the time in hours​

Answer 1

An exponential model can represent growth or decay

(a) The rate of the function

An exponential function is represented as:

$y = ab^x$

Rewrite as:

$N(t) = N(0)b^t$

From the question, we have:

• N(0) = 100
• N(5) = 300

So, the function becomes

$N(t) = 100 \times b^t$

Substitute 5 for t

$N(5) = 100 \times b^5$

This gives

$300 = 100 \times b^5$

Divide both sides by 100

$3 = b^5$

Take 5th root of both sides

$1.25 = b$

Rewrite as:

$b =1.2457$

Hence, the rate of the model is 1.2457

(b) The function

In (a), we have:

$N(t) = 100 \times b^t$

Substitute 1.2457 for b

$N(t) = 100 \times 1.2457^t$

(c) Time the population reaches 10000

This means that:

N(t) = 10000

So, we have:

$N(t) = 100 \times 1.2457^t$

$10000 = 100 \times 1.2457^t$

Divide both sides by 100

$100 = 1.2457^t$

Take logarithm of both sides

$log(100) = log(1.2457)^t$

This gives

$log(100) = t\ log(1.2457)$

Divide both sides by log(1.2457)

$t=21$

Hence, the bacteria will reach 10000 after 21 hours

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