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

A kiln starts at $20 °C$ at $t = 0$ minutes, heats at $2 °C$/min for $30$ minutes, holds constant for $10$ minutes, then cools at $1.5 °C$/min for the next $40$ minutes. Let $T(t)$ be temperature in $°C$ for $0 ≤ t ≤ 80$. Construct $T(t)$.

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 ≤ t ≤ 6. Construct V(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 ≤ x ≤ 18. Construct the piecewise linear function A(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 ≤ w ≤ 12). Model the total savings S(w) assuming continuous accumulation at the stated weekly rates within each interval.

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 ≥ 0). Construct the total cost S(d).

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 ≥ 0). Construct the total fare F(d).

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 ≥ 0). Construct the total cost S(w).

A worker is paid $20/hr for the first 8 hours in a day, $30/hr for the next 4 hours, and pay is capped at $420 beyond 12 hours. Let h be hours worked (0 ≤ h). 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 ≥ 0). Construct the total parking cost P(h).

A factory’s total production cost includes a $100 setup fee. The marginal cost is $50/unit for the first 20 units, $40/unit for the next 30 units, and $55/unit beyond 50 units. Let x be units produced (x ≥ 0). Construct the total cost C(x).

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 ≥ 0). Construct the total monthly cost C(g).

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

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 ≥ 0). Construct the total fare F(d).