The number of bacteria in the early stages of a growing colony of bacteria can be approximated using the function: $$N(t) = 450e^{0.065t}$$ where $t$ is the time, measured in hours, and $N(t)$ is the number of bacteria in the colony at time $t$ - Leaving Cert Mathematics - Question 9 - 2020

To find the time $t$ when $N(t) = 790$, we set up the equation:

$790 = 450e^{0.065t}$

Now, divide both sides by 450:

$\frac{790}{450} = e^{0.065t}$

Taking the natural logarithm:

$\ln\left(\frac{790}{450}\right) = 0.065t$

Solving for $t$ gives:

$t = \frac{\ln\left(\frac{790}{450}\right)}{0.065} \approx 8.7$

Thus, it takes approximately 8.7 hours.

To find the average number of bacteria from $t = 3$ to $t = 12$, we use the formula:

$\text{Average} = \frac{1}{b-a} \int_a^b N(t) dt$

where $a = 3$ and $b = 12$:

$= \frac{1}{12-3} \int_3^{12} 450e^{0.065t} dt$

Calculating the integral:

$= \frac{1}{9} \left[ \frac{450}{0.065}e^{0.065t} \right]_{3}^{12}$

Evaluating gives:

$\frac{1}{9} \left( \frac{450}{0.065}(e^{0.78} - e^{0.195}) \right) \approx 743$

Thus, the average number of bacteria is approximately 743.

To find the rate of change, we first need the derivative:

$N'(t) = 450 imes 0.065 e^{0.065t}$

Evaluating at $t = 12$:

$N'(12) = 450 imes 0.065 e^{0.78} \approx 638.8$

Thus, the rate at which the colony is growing at $t = 12$ is approximately 638.8 bacteria per hour.

Setting up the inequality:

$N'(t) > 90$

Finding $N'(t)$:

$450 imes 0.065 e^{0.065t} > 90$

This simplifies to:

$e^{0.065t} > \frac{90}{29.25}$

Taking the natural logarithm:

$0.065t > \ln\left(\frac{90}{29.25}\right)$

Solving for $t$ leads to:

$t > \frac{\ln(90) - \ln(29.25)}{0.065} \approx 18$

Thus, the least value of $k$ is 18.

Setting the two equations equal:

$450e^{0.065t} = 220e^{0.17t}$

Rearranging, we have:

$\frac{450}{220} = e^{0.17t - 0.065t}$

Taking logarithms:

$\ln\left(\frac{450}{220}\right) = (0.17 - 0.065)t$

Solving for $t$ gives:

$t = \frac{\ln\left(\frac{450}{220}\right)}{0.105} \approx 7$

Thus, the time at which the number of bacteria in both colonies will be equal is approximately 7 hours.