SL 2.5—Modelling functions Questionbank

The number of bacteria in a Petri dish is modelled by the function $N(t)=75 \times 2^{0.5 t}, t \geq 0$ where $N$ is the number of bacteria and 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 $T(t)=a+b\left(k^{-t}\right)$ where $t$ is the time, in minutes, since the soup was removed from the stove. The temperature outside is $12^{\circ} \mathrm{C}$.

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.

Joe is kayaking to an island located 7 km from where he starts. The distance, $D$, in kilometres, that Joe has travelled from his original position can be modelled by the exponential function $D(t)=a+b\left(k^{-t}\right), t \geq 0$ where $t$ is the time in minutes since Joe started kayaking.

A remote-controlled sailboat's velocity is dependent on the wind speed. The sailboat's velocity is lower during very high and very low wind speeds. The sailboat's velocity can be modelled by the function $V(w)=0.0025 w(2-w)(w-35), 2 \leq w \leq 35$ where $V$ is the sailboat's velocity, in km per hour, and $w$ is the wind speed, in km per hour.

A rectangular sheet of cardboard 60 cm by 100 cm has square sides of $x \mathrm{~cm}$ cut from each corner.

The temperature, $T$, of a cake, in degrees Celsius, ${ }^{\circ} C$, can be modelled by the function $T(t)=a \times 1.17^{-\frac{t}{4}}+18, t \geq 0$ where $a$ is a constant and $t$ is the time, in minutes, since the cake was taken out of the oven.

A factory produces cardboard boxes in the shape of a cuboid, with a fixed height of 25 cm and a base of varying area. The area, $A$, of each base can be modelled by the function $A(x)=x(50-x), 10 \leq x \leq 40$ Where is the width of the base of the cardboard box in centimetres. Cardboard box $M$ has a width of 12 cm .

A Ferris wheel rotates at a constant speed, the height of a particular seat above the ground is modelled by the function $H(t)=-14 \cos \left(10^{\circ} \times t\right)+16, t \geq 0$ where $H$ is the height of the seat above the ground, in metres, and $t$ is the elapsed time, in seconds, since the start of the ride. Write down :

The downward speed, $V$, in metres per second, of a bird making a dive into the water to catch a fish can be modelled by the function $V(t)=62-62 \times 3.5^{-\frac{t}{2}}, t \geq 0$ where $t$, in seconds, is the time the bird is diving.