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.

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.

Linda owns a field, represented by the shaded region $R$. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres.

The segments $[AB]$, $[CD]$ and $[AD]$ respectively represent the western, eastern and southern boundaries of the field. The function $f(x)$ models the northern boundary of the field between points $B$ and $C$ and is given by $f(x) = -\frac{x^2}{50} + 2x + 30$, for $0 \le x \le 70$.

A company sells $55$ cars per month for a sale price of $2000$, while incurring costs for supplies, production, and delivery of $890$ per car. Reliable market research shows that for each increase (or decrease) of the sale price by $50$, the company will sell $5$ cars less (or more).

Olava’s Pizza Company supplies and delivers large cheese pizzas. The total cost to the customer, $C$, in Papua New Guinean Kina (PGK), is modelled by the function: $C(n)=34.50n+8.50, \quad n \ge 2, \quad n \in \mathbb{Z}$ where $n$ is the number of large cheese pizzas ordered. This total cost includes a fixed cost for delivery.

Algae in a lake can grow exponentially until the lake is fully covered in algae.

In memory of the legendary musician Alex Star, tribute concerts continue to attract fans worldwide, decades after his passing in 1985. The number of attendees at these concerts, $N(t)$, can be modeled by

$N(t) = 200 \times 1.28^t$

where $t$ is the number of years since 1985.

Deserts are known for having high daily temperature ranges. Erica monitors the temperature, in $^\circ\text{C}$, 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, $T^\circ\text{C}$, during the day: $T(t) = a \cos(b(t+10)) + d, \quad 0 \leq t \leq 24$ where $t$ is the elapsed time, in hours, since midnight.

Two schools are represented by points $A(2,20)$ and $B(14,24)$ on the graph below. A road, represented by the line $R$ with equation $-x+y=4$, passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.