Question Type 2: Determining mathematical formulations for contextualized models such as the function, domain, range. Exercises

The cost $P(x)$ (in dollars) of a taxi ride is given by $P(x)=3+2.5x,$ where $x$ is the distance traveled in km and $0\le x\le50$. Determine the domain and range of $P$.

The depth $D(x)$ (in metres) of a swimming pool from the shallow end ($x=0$ m) to the deep end ($x=20$ m) is modelled by $D(x)=1+0.04x \text{.}$ Determine the domain and range of $D$.

A rectangular pool is 25 m by 10 m. A uniform walkway of width $x$ (in meters) is built around it. The total area (pool plus walkway) is given by $A(x)=(25+2x)(10+2x)\text{.}$ Determine the domain of $x$ so that the walkway is physically possible (width at most half the shorter side) and find the corresponding range of $A(x)$.

The depth $D(t)$ (in meters) of water in a pool evaporates linearly as $D(t) = 2 - 0.1t$, where $t$ is time in days and evaporation continues until the pool is empty.

Determine the domain and range of $D$.

A swimming pool leaks so that its volume (in litres) after $t$ hours is given by

$V(t) = 10000\text{e}^{-0.05t}$

Determine the domain and range of $V$.

A printing company charges $C(n)=0.05n+2$ dollars for $n$ pages printed, where $n$ is a non-negative integer. Determine the domain (in context) and the range of $C$.

The temperature of a cooling coffee cup (in °C) is modeled by $T(t)=5+75e^{-0.1t},$ where $t$ is time in minutes since pouring. Determine the domain and range of $T$.

Determine the domain and range of $V$ when the depth varies from 0 to 2.5 meters.

The height (in meters) of a ball thrown upward is modeled by $h(t)=-4.9t^2+19.6t+1$ where $t$ is time in seconds. Determine the domain of $t$ for which the ball is on or above the ground and find the corresponding range of $h$.

Two pumps fill a pool at a combined rate of $500 \text{ L/min}$, so the volume filled in $t$ minutes is given by $V(t) = 500t$ If the pool capacity is $25\,000 \text{ L}$, determine the domain and range of $V$.

A water tower has height $h(t) = 8 - 0.2t$ (in meters) of water at time $t$ hours after midnight, until it is empty. Determine the practical domain of $t$ and the corresponding range of $h$.