Question Type 3: Calculating solution to questions based around linear models Exercises

In a data plan, the price per GB $p(g)$ (for the $g^{\text{th}}$ GB) is $10 for $g=1$ and drops by $1 per GB for $g=2$ to $g=5$, remains at $6 for $g=6$ to $g=10$, then increases by $0.50 per GB for $g=11$ to $g=15$. Find $p(12)$.

A runner’s lap time $L(n)$ starts at $80\text{ s}$ on lap $1$, decreases by $3\text{ s}$ per lap for laps $2$–$5$, is constant for laps $6$–$9$, then increases by $2\text{ s}$ per lap from lap $10$ onward. Find $L(12)$.

A gadget’s selling price $P(n)$ (in dollars) after $n$ weeks follows: $P(0)=60$. It decreases by $2$ per week for $8$ weeks, stays constant for $4$ weeks, then increases by $1$ per week for the next $6$ weeks. Find $P(13)$.

A quantity $y$ varies linearly with $x$. Given $y=8$ when $x=2$ and $y=20$ when $x=8$, find $y$ when $x=5$.

A company’s revenue per unit $R(q)$ satisfies $R(0)=120$. It increases by $4 per unit for $0 < q \le 10$, then decreases by $3 per unit for $10 < q \le 25$, and remains constant thereafter. Find $R(20)$.

The function $C(q)$ is defined by: $C(q) = \begin{cases} 80 - 2q & \text{for } 0 \le q \le 12 \\ 56 & \text{for } 12 < q \le 15 \\ 56 + 1.5(q - 15) & \text{for } q > 15 \end{cases}$

Find the value of $C(18)$.

An asset’s book value $B(t)$ (dollars) is piecewise linear in time $t$ (months): $B(0)=10000$; it decreases by $300$ per month for the first $12$ months, then by $100$ per month for the next $6$ months, then increases by $150$ per month for the subsequent $18$ months, and remains constant afterward. Find $B(28)$.

A factory's average cost per unit, $C(q)$, is expressed in dollars. The cost starts at $C(0) = 50$. It falls by $4$ per unit for the first $6$ units produced, stays constant for the next $4$ units, then rises by $3$ per unit for the following $10$ units. Find $C(12)$.

A tariff's marginal price per kWh is given by the piecewise function $r(u)$ (in cents/kWh): $r(u) = \begin{cases} 20 - 0.1u, & 0 \le u \le 100 \\ 10, & 100 < u \le 150 \\ 10 + 0.08(u - 150), & 150 < u \le 200 \end{cases}$

Find $r(160)$.

Water cools from $90^\circ\text{C}$ at $5^\circ\text{C}$ per hour for the first $2$ hours, then at $2^\circ\text{C}$ per hour for the next $3$ hours, after which it is held constant. Find the temperature at $t=7$ hours.

The processing time per package $T(k)$ (in minutes) for the $k^{\text{th}}$ package is $15$ minutes for $k=1$. The time decreases by $0.5$ minutes per package for $k=2$ to $k=20$, stays constant for $k=21$ to $k=30$, and then increases by $0.2$ minutes per package for $k \ge 31$. Find $T(28)$.

A car’s speed $v(t)$ (km/h) changes linearly in stages: it increases from $0$ at $12$ km/h per minute for $5$ minutes, holds constant for $10$ minutes, then decreases at $6$ km/h per minute until it reaches $30$ km/h, after which it increases at $3$ km/h per minute. Find $v(22)$.