SL 2.6—Modelling skills Questionbank

The average temperature of a city, $C$, in degrees Celsius, fluctuates throughout a year and can be modelled by the function $C(t)=a \sin (k t)+b$ where $t$ is the elapsed time, in weeks, since the start of the year. The average temperature of the city in week 4 is 27 degrees Celsius and in week 28 it is 12 degrees Celsius.

A company sells $L$ litres of water per month and their total monthly profit, $P$, can be modelled by the function $P(x)=(x-0.45) \times N(x)$ where $x$ is the sale price of each litre sold, in dollars, at and $N$ is the linear function for the number of litres the company can sell per month at each given sale price.

The number of German words, $W$, Helen remembers after completing a German language course decreases exponentially over time when she does not practice her German. This decrease can be modelled by the function $W(t)=a \times b^{-t}+320, t \geq 0$ Where $a$ and $b$ are positive constants and is the time in years since Helen completed the German language course. Helen can remember 2400 German words as soon as she completes the German language course.

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modelled by $y = a x^{2} + c$.

Point $P$ has coordinates $(-4, 4)$, point $O$ has coordinates $(0, 0)$ and point $Q$ has coordinates $(4, 4)$.

A local bakery offers fresh baguettes for delivery. The total cost to the customer, $C$ in Euros (€), is modeled by $C(n) = 3.2n + 5$, where $n$ is the number of baguettes ordered and includes a fixed delivery fee.

Deserts are known for having high daily temperature ranges. Erica monitors the temperature, in ${ }^{\circ} C$, on a particular day in a desert. The table below shows some of the information she recorded.

Erica uses her observations to form the following model for the temperature, $T^{\circ} C$, during the day $T(t)=a \cos (b(t+10))+d, 0 \leq t \leq 24$ where $t$ is the elapsed time, in hours, since midnight.

1. Calculate the value of $t$ when the maximum temperature occurs and fill in the time in the table above in am/pm format.
2. Find the value of $a, b, d$.

Erica goes exploring in the desert at $6: 30$ am and leaves once the temperature reaches $32^{\circ} \mathrm{C}$. 3. Calculate the temperature range Erica experiences whilst in the desert.

Algae in a lake can grow exponentially until the lake is fully covered in algae. Find the number of days it takes for a lake to be fully covered in algae when $\frac{1}{2048}$ of the lake is covered today and the covered area doubles once every five days.

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modeled by $y=a x^{2}+c$

Point $P$ has coordinates $(-4,4)$, point $O$ has coordinates $(0,0)$ and point $Q$ has coordinates $(4,4)$.

A potato top pie is removed from the oven and is left to cool. The pie's temperature, $T$, in ${ }^{\circ} C$, can be modelled by the function $T(t)=a+b\left(k^{-t}\right), t \geq 0$ Where $t$ is the time, in minutes, since the pie was removed from the oven. The temperature of the kitchen is $24^{\circ} \mathrm{C}$.

A fence of length $L$, in metres, is made to form a rectangle around a house that borders a forest on one side. The fence does not run along the side next to the forest. The cost of the fence is \ 22.20$per metre. The total cost of the fence is$$ 2250$. Calculate :