Q49. - Algebra 1

Question

The population of bacteria in a petri dish increases according to the model$\mathit{p}\mathbf{=}\mathbf{550}{\mathbf{(}\mathbf{2}\mathbf{.7}\mathbf{)}}^{\mathbf{0}\mathbf{.008}\mathbf{t}}$, where $\mathit{t}$ is the number of hours and $\mathit{t}\mathbf{=}\mathbf{0}$corresponds to$\mathbf{1}\mathbf{:}\mathbf{00}\mathbf{ }\mathit{P}\mathbf{.}\mathit{M}$. Use this model to estimate the number of bacteria in the dish at $\mathbf{5}\mathbf{:}\mathbf{00}\mathbf{ }\mathit{P}\mathbf{.}\mathit{M}$.

Step-by-Step Solution

Answer

The numbers of bacteria on the dish at $5:00 PM$ are $\mathbf{567}\mathbf{.}\mathbf{76}$.

Step 1. State the concept of Exponential function.

An exponential function is a function of the form $y=a{b}^x$ where $a\ne 0$, $b>0$, and $b\ne 1$.

Step 2. Estimate the value of t .

The equation $p=550\times {\left(2.7\right)}^{0.008t}$, where $t$ is the number of hours since $1:00$ PM

$t$ will be the time between $1:00$ P.M to $5:00$ P.M.

Therefore, $t = 5.00 PM – 1.00 PM = 4$ hours.

Step 3. Simplify for p .

Substitute 4 for $t$ into the equation $p=550\times {\left(2.7\right)}^{0.008t}$ and simplify for $p$.

$\begin{array}{c}p=550\times {\left(2.7\right)}^{0.008t}\\ =550\times {\left(2.7\right)}^{0.008\times 4}\\ =567.76\end{array}$       

 The numbers of bacteria on the dish at $5:00$ PM are $567.76$.