Question

dr. gonzales recorded a count of 160 bacteria present in a culture. after 6 hours, they counted 394 bacteria present. assuming the bacteria culture grows at a continuous rate, write an exponential model for this situation in terms of t, the number of hours passed. round to two decimal places, when necessary. show your work here hint: to add an exponent (x^b), type exponent or press ^

Explanation:

Step1: Recall continuous growth formula

The continuous exponential growth model is $N(t) = N_0 e^{kt}$, where:

• $N_0$ = initial bacteria count,
• $k$ = growth rate,
• $t$ = time in hours,
• $N(t)$ = bacteria count at time $t$.

Here, $N_0 = 160$, $t=6$, $N(6)=394$.

Step2: Substitute known values

Substitute into the formula:
$394 = 160 e^{k \times 6}$

Step3: Isolate the exponential term

Divide both sides by 160:
$\frac{394}{160} = e^{6k}$
Simplify left side: $2.4625 = e^{6k}$

Step4: Solve for $k$ using natural log

Take natural logarithm of both sides:
$\ln(2.4625) = 6k$
Calculate $\ln(2.4625) \approx 0.901$
Then $k = \frac{0.901}{6} \approx 0.15$

Step5: Write final model

Substitute $N_0=160$ and $k\approx0.15$ into the original formula.

Answer:

$N(t) = 160e^{0.15t}$