Question Type 1: Recognizing and choosing appropriate models given descriptions or graphs Exercises

A delivery service charges a flat fee of $50 plus $2 per kilometre. Define a linear model $C(d)$ for the total cost given distance $d$ in kilometres, and state the reasonable domain if deliveries are within 100 km.

A rectangular swimming pool has a shallow end 1 m deep, which increases linearly to a deep end 3 m deep over a horizontal distance of 20 m. Identify a linear model for the depth $d(x)$ in terms of the horizontal distance $x$ from the shallow end, and specify the reasonable domain for $x$.

A water tank is being filled so that the water height increases uniformly from 0 m to 2 m in 50 minutes. Propose a linear model $H(t)$ for the height after $t$ minutes of filling, and determine the reasonable domain for $t$.

A car’s value depreciates by 15% per year. If its initial value is $20,000, propose an exponential model $V(t)$ for its value after $t$ years and specify the domain for $t$.

A bacteria culture starts with 100 individuals and doubles every hour. Propose an exponential model $P(t)$ for the population after $t$ hours, and state the reasonable domain for $t$.

Carbon-14 decays with a half-life of 5730 years. Starting with 10 g of C-14, formulate an exponential decay model $M(t)$ for the mass after $t$ years and give its domain.

The height of a ball thrown vertically is given by $h(t)=-5t^2+20t+1$, where $h$ is in meters and $t$ in seconds. Identify the model type and determine the domain for $t$ until the ball hits the ground.

A pool has three sections: from $x=0$ to $x=5$ the depth is constant 1 m; from $x=5$ to $x=25$ the depth increases linearly from 1 m to 3 m; from $x=25$ to $x=30$ the depth is constant 3 m. Determine a piecewise linear function $h(x)$ for the depth and state its domain.

A company’s profit $P$ (in thousands of dollars) as a function of unit price $x$ (in dollars) is modeled by $P(x)=-2x^2+80x-600$. Identify the type of model, and specify a reasonable domain for $x$ if the unit price must be between 0 and 40 dollars.

A coffee cup cools from 90 °C to 60 °C in 10 minutes following Newton’s law of cooling. Assume an exponential model $T(t)=90e^{kt}$. Determine the constant $k$ and specify the practical domain for $t$.

A logistic growth model for a population $P(t)$ is given by $P(t)=\frac{1000}{1+19e^{-0.8t}},$ where $t$ is in years. Identify the model type and state the reasonable domain for $t$.