Question Type 1: Creating linear models for different real-life contexts Exercises

A kiln starts at $20\,^\circ\text{C}$ at $t = 0$ minutes, heats at $2\,^\circ\text{C min}^{-1}$ for $30$ minutes, holds constant for $10$ minutes, then cools at $1.5\,^\circ\text{C min}^{-1}$ for the next $40$ minutes. Let $T(t)$ be the temperature in $^\circ\text{C}$ for $0 \le t \le 80$. Find an expression for $T(t)$.

Average cost $A(x)$ in dollars starts at $60 per unit when $x=0$, decreases by $5 per unit for the first 5 units, stays constant for the next 3 units, then increases by $2 per unit for the next 10 units. Let $x$ be the number of units produced, $0 \le x \le 18$. Construct the piecewise linear function $A(x)$.

A ride-share app charges a base fee of $3. The per-kilometer charge is subject to surge: $1.00/km for the first 5 km, $1.50/km for the next 5 km, and $2.00/km beyond 10 km. The base fee is not multiplied by surge. Let $d$ be distance ($d \ge 0$). Construct the total fare function $F(d)$.

A streaming service costs $0 for the first $14$ days, $2 per day for days $15$--$30$, and $3 per day thereafter. Let $d$ be the number of days used in a single continuous period ($d \ge 0$). Construct the total cost $S(d)$.

A worker is paid $20 per hour for the first 8 hours in a day and $30 per hour for the next 4 hours. If the worker works more than 12 hours, the total daily pay is fixed at $420. Let $h$ be the hours worked ($h \ge 0$). Construct the daily pay function $P(h)$.

A parking garage charges $0 for the first 1 hour, $3 per hour for the next 2 hours, offers a 1-hour free grace period after 3 hours, then $5 per hour beyond 4 hours. Let $h$ be hours parked ($h \ge 0$). Construct the total parking cost $P(h)$.

A shipping service charges a $5 handling fee. Weight is billed at $2/kg for the first 3 kg, $0/kg for the next 1 kg (promo), and $4/kg for any weight beyond 4 kg. Let $w$ be package weight in kg ($w \ge 0$). Construct the total cost $S(w)$.

A car is valued at $30000 at purchase ($t = 0$ years), depreciates linearly by $2000 per year for the first 3 years, remains constant for the next year, then appreciates by $500 per year for the following 2 years. Let $V(t)$ be value in dollars for $0 \le t \le 6$. Construct $V(t)$.

Find an expression for the total cost function $C(x)$.

You start with $0 in a savings account. You deposit $50 per week during weeks 1–4, $80 per week during weeks 5–8, and $20 per week during weeks 9–12. Let $w$ be the number of weeks elapsed ($0 \le w \le 12$). Model the total savings $S(w)$ assuming continuous accumulation at the stated weekly rates within each interval.

A mobile data plan charges a fixed $5 monthly fee. Data costs $10 per GB for the first 2 GB, $8 per GB for the next 3 GB, and $12 per GB beyond 5 GB. Let $g$ be GB used ($g \ge 0$). Construct the total monthly cost $C(g)$.

A taxi fare includes a $4 flagfall. Distance is charged at $1.80/km for the first 10 km, $1.20/km for the next 5 km, and $2.00/km beyond 15 km. Let $d$ be distance in km ($d \geq 0$). Construct the total fare $F(d)$.

An electricity bill has a fixed monthly charge of 10 dollars. Energy costs 0.15 dollars per kWh for the first 100 kWh, 0.12 dollars per kWh for the next 200 kWh, and 0.20 dollars per kWh beyond 300 kWh. Let $k$ be monthly kWh used. Construct the total bill $B(k)$. [6 marks]