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

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 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 $P(13)$.

Define $C(q)$ by: $C(q)=80-2q$ for $0\le q\le 12$; $C(q)=56$ for $12<q\le 15$; $C(q)=56+1.5(q-15)$ for $q>15$. Compute $C(18)$.

A factory’s average cost per unit $C(q)$ 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 $C(12)$.

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.

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 tariff’s marginal price per kWh is piecewise: $r(u)=20-0.1u$ cents/kWh for $0\le u\le 100$; $r(u)=10$ for $100<u\le 150$; $r(u)=10+0.08(u-150)$ for $150<u\le 200$. Find $r(160)$.

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

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

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 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 $R(20)$.

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 $v(22)$ minutes.