A scientist writes the equation N(h) = 100e^{0.25h} to model the growth of a certain bacteria in...

Question:

A scientist writes the equation

{eq}\displaystyle\; N(h) = 100e^{0.25h} {/eq}

to model the growth of a certain bacteria in a petri dish, where {eq}N {/eq} represents the number of bacteria after {eq}h {/eq} hours.

After approximately how many hours will {eq}450 {/eq} bacteria be present? Round your answer to the nearest whole number.

(a) {eq}\; 1 {/eq} hour

(b) {eq}\; 6 {/eq} hours

(c) {eq}\; 13 {/eq} hours

(d) {eq}\; 15 {/eq} hours

Exponential Growth and Decay:

The exponential growth function is modeled by the function {eq}N(t) = N_0 e^{kt} {/eq}, where {eq}t {/eq} is time, {eq}N_0 {/eq} and {eq}N(t) {/eq} are the amount present when {eq}t=0 {/eq} and after time {eq}t {/eq}, respectively, and {eq}k>0 {/eq}.

The exponential decay function is modeled by the same function only that {eq}k<0 {/eq}.

Answer and Explanation:

We need to know time {eq}h {/eq} when {eq}N(h) =450 {/eq} provided that {eq}\displaystyle\; N(h) = 100e^{0.25h} {/eq}.

So, we'll substitute this quantity and then solve the equation for {eq}h {/eq} to obtain the time required:

{eq}\begin{align*} \displaystyle\; N(h)& = 100e^{0.25h}\\ 450& =100e^{0.25h}\\ e^{0.25h}& =4.5 \\ 0.25h& = \ln 4.5\\ h & = \frac{\ln 4.5}{0.25}\\ h & \approx 6 \ \mathrm{hours}\\ \end{align*} {/eq}

We found out that option {eq}b {/eq} is right.


Learn more about this topic:

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Exponential Growth vs. Decay

from Math 101: College Algebra

Chapter 10 / Lesson 2
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