Question 137·Medium·Nonlinear Functions

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A microbiologist observes that a bacterial culture contains $2,400$ cells at time $t = 0$ hours. Eight hours later, the culture contains $9,600$ cells. The growth of the culture is modeled by an exponential function $N$ , where $N(t)$ is the number of cells $t$ hours after the initial observation.

Which equation could approximately define $N$ , with the hourly growth factor rounded to the nearest hundredth?

For an exponential-modeling question, use $t=0$ first because it immediately identifies the initial-value coefficient. Then compare the later value with the initial value to find the total growth factor, and take the appropriate root to find the growth factor per time unit. If the choices use decimals, check whether the prompt indicates that the factor is rounded.

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Identify the parts of the model

An exponential model has the form $N(t)=a\cdot b^t$ . Consider what $a$ represents when $t=0$ .

Use the initial observation

Substitute $t=0$ into $N(t)=a\cdot b^t$ . This identifies the coefficient that the correct equation must have.

Compare the two population counts

Determine how many times as large $9,600$ is as $2,400$ . Then use the fact that this change occurs over $8$ hours to determine the approximate hourly growth factor.

Desmos Guide

Enter the candidate functions

In Desmos, enter the four answer choices as separate functions, such as $f(t)=9,600(1.19)^t$ , $g(t)=2,400(0.19)^t$ , $h(t)=9,600(0.19)^t$ , and $k(t)=2,400(1.19)^t$ .

Check the initial value

Evaluate each function at $t=0$ by entering $f(0)$ , $g(0)$ , $h(0)$ , and $k(0)$ . Eliminate functions whose output is not $2,400$ .

Verify the rounded growth factor

Evaluate the remaining function at $t=8$ . Its output is close to $9,600$ because the factor $1.19$ is the hourly growth factor rounded to the nearest hundredth.

Step-by-step Explanation

Write the general form of an exponential model

For exponential growth, the number of cells can be modeled as

N(t)=a\cdot b^t,

where $a$ is the initial amount and $b$ is the hourly growth factor.

Use the initial value

At $t=0$ , the culture contains $2,400$ cells. Since

N(0)=a\cdot b^0=a,

the initial value is $a=2,400$ . Therefore, the model must have the form

N(t)=2,400\cdot b^t.

Find the hourly growth factor

After $8$ hours, the culture has quadrupled:

\frac{9,600}{2,400}=4.

Thus,

b^8=4,

b=4^{1/8}\approx1.19.

With the growth factor rounded to the nearest hundredth, the model is

N(t)=2,400(1.19)^t.

Question 137·Medium·Nonlinear Functions

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Step-by-step Explanation

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Step-by-step Explanation