SL 2.5—Modelling functions Questionbank

The number of bacteria in a Petri dish is modelled by the function $N(t) = 75 \times 2^{0.5t}$, for $t \geq 0$, where $N$ is the number of bacteria and $t$ is the time in hours.

A hot winter soup has just been removed from the stove and is left outside to cool. The soup's temperature can be modelled by the function $T(t)=a+b(k^{-t})$ where $t$ is the time, in minutes, since the soup was removed from the stove.

The temperature outside is $12^{\circ} \mathrm{C}$.

In a biology lab, the population of a certain species of bacteria is observed to grow exponentially. The population $P(t)$ in number of bacteria at time $t$ hours is modeled by the equation:

$P(t) = 500e^{0.3t}$

Antibiotics $A$ and $B$ are applied to a pure culture of bacteria. The number of bacteria present initially for both antibiotics is 6000. The number of bacteria present for antibiotic $A$, $N_A$, can be modelled by the function $N_A(t) = a \times b^{-t}$, $t \geq 0$, where $t$ is the elapsed time, in hours, since the start of the experiment.

A company's profit per year was found to be changing at a rate of $\frac{dP}{dt} = 3t^2 - 8t$ where $P$ is the company's profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.

Consider the function $f(x) = ax^2 + bx + c$. The graph of $y = f(x)$ is shown in the diagram. The vertex of the graph has coordinates $(0.5, -12.5)$. The graph intersects the x-axis at two points, $(-2, 0)$ and $(p, 0)$.

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

A marathon is 42.195 kilometres.

In the $k$th training session Rosa will run further than a marathon for the first time.

Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

The amount, in milligrams, of a medicinal drug in the body $t$ hours after it was injected is given by $D(t) = 23(0.85)^t, \ t \geq 0$. Before this injection, the amount of the drug in the body was zero.