Modeling with Polynomial & Rational Functions (College Board AP® Precalculus): Exam Questions

On the initial day of sales ($t=0$) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later ($t=91$), there were 76 thousand units of the game sold that day.

The number of units of the video game sold on a given day can be modeled by the function $G$ given by $G\left(t\right)=40+7.961\ln \left(t+1\right)$, where $G\left(t\right)$ is the number of units sold, in thousands, on day $t$ since the initial day of sales.

(i) Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from $t=0$ to $t=91$ days. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in (i) to estimate the number of units of the video game sold, in thousands, on day $t=50$. Show the work that leads to your answer.

(iii) Let $A_t$ represent the estimate of the number of units of the video game sold, in thousands, using the average rate of change found in (i). For $A_{50}$, found in (ii), it can be shown that $A_{50}<G\left(50\right)$. Explain why, in general, $A_t<G\left(t\right)$ for all $t$, where $0<t<91$.

A water tank is being filled. For the first 5 minutes after filling begins, water enters the tank at a constant rate of 8 liters per minute. Beginning at $t=5$ minutes, a valve is partially closed, and from that moment onward the volume of water in the tank can be modeled by the function

$g\left(t\right)=-0.5t^2+10t+12.5$

in liters. Initially (at $t=0$) the tank contains 10 liters of water.

Write a piecewise-defined function $L\left(t\right)$ that gives the volume of water (in liters) in the tank as a function of time $t$ (in minutes) for $0\le t\le 9$.

Determine the value of $L\left(5\right)$. Verify that the second piece of your function from part (a) gives the same value at $t=5$, and explain why it is appropriate, in the context of this problem, for the two pieces to give the same value at $t=5$.

The table below gives values of a function $h$ at selected values of $x$.

Use the method of successive differences to identify the type of polynomial function (linear, quadratic, cubic, etc.) that is most appropriate to model $h$. Show the work that leads to your answer.

Given that $h\left(x\right)=a{x}^3+b{x}^2+cx+d$ models the data exactly, determine the value of $d$.

A scientist is monitoring the temperature of a chemical reaction. The temperature, in degrees Celsius, $t$ minutes after the reaction begins is modeled by the polynomial function $T$ given by $T\left(t\right)=-2.7t^2+24t+18$, for $0\le t\le 9$.

Find the value of $T\left(4.2\right)$ as a decimal approximation. Show the work that leads to your answer.

An electric utility company begins a phased installation of smart meters. At time $t=0$, the company begins tracking the total number of smart meters installed, in thousands.

The table gives the total number of smart meters installed, in thousands, for selected times $t$ months after tracking began.

(i) Use the given data to find the average rate of change of the total number of smart meters installed, in thousands per month, from $t=0$ to $t=2$ months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in part B (i) to estimate the total number of smart meters installed, in thousands,for $t=1.8$ months. Show the work that leads to your answer.

The function $f$ is increasing and is defined for $x\ge 0$. The table gives values of $f\left(x\right)$ at selected values of $x$.

(i) Based on the table, which of the following function types best models function $f$: linear, quadratic, exponential or logarithmic?

(ii) Give a reason for your answer in part C (i) based on the relationship between the change in the output values of $f$ and the change in the input values of $f$. Refer to the values in the table in your reasoning.

The intensity $I$ (in watts per square meter) of a sound at a distance $d$ (in meters) from its source is inversely proportional to the square of the distance. When $d=2$ meters, the intensity is 75 watts per square meter.

Write an equation that gives $I$ as a function of $d$.

Determine the distance $d$ at which the intensity equals 18 watts per square meter. Express your answer as a decimal approximation.

A startup company's monthly profit, in thousands of dollars, can be modeled by a cubic polynomial function $P$, where $P\left(t\right)$ gives the profit in month $t$, $t$ months after launch ($t\ge 0$).

The graph of $y=P\left(t\right)$ has a zero at $t=1$ where the graph crosses the $t$-axis, and a zero at $t=5$ where the graph is tangent to the $t$-axis.

In the launch month (at $t=0$), the company recorded a profit of $-10$ thousand dollars.

Use the given information to construct an expression for $P\left(t\right)$ in factored form. Show the work that leads to your answer.

A small online retailer's monthly profit, in thousands of dollars, $t$ months after launch is modeled by the function $P$ given by

$P\left(t\right)=-0.5t^2+6t-11$

for $t\ge 0$.

The company will continue operations as long as it does not make a monthly loss (i.e. as long as $P\left(t\right)\ge 0$).

Find the contextually valid domain of $P$. Express the endpoints of the domain as decimal approximations. Show the work that leads to your answer.

Use $P$ to find the maximum monthly profit the company can earn during the contextually valid operating period, in thousands of dollars.

The number of subscribers, in thousands, to a new streaming service $t$ months after launch is modeled by the function $S$ given by $S\left(t\right)=a{t}^2+bt+c$, where $a$, $b$, and $c$ are constants. The data set below gives the number of subscribers at three selected values of $t$.

(i) Use the given data to write three equations that can be used to find the values for constants $a$, $b$, and $c$ in the expression for $S\left(t\right)$.

(ii) Find the values for $a$, $b$, and $c$ as decimal approximations.

A pharmaceutical study models the concentration of a drug in a patient's bloodstream as the drug is administered through a continuous IV infusion. The concentration, in milligrams per liter, can be modeled by a rational function $C$ of the form

$C\left(t\right)=\frac{at}{t+b}$

where $t$ is the time, in hours, since the infusion began ($t\ge 0$), and $a$ and $b$ are positive constants.

Two hours after the infusion begins, the concentration is 4 milligrams per liter, and six hours after the infusion begins, the concentration is 6 milligrams per liter.

Construct the function $C$. Show the work that leads to your answer.

The graph of $C$ has a horizontal asymptote. Identify the equation of this horizontal asymptote and explain its meaning in the context of the scenario.

Identify one underlying assumption of this model in the context of the scenario.

A company's monthly revenue, in thousands of dollars, is modeled by the function $R$ given by $R\left(t\right)=-2t^2+30t+5$ for $0\le t\le 15$, where $t$ is the number of months since the model began.

(i) Use the model to find the average rate of change of the revenue, in thousands of dollars per month, from $t=2$ to $t=7$ months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in part (i) to estimate the monthly revenue, in thousands of dollars, at $t=4$ months. Show the work that leads to your answer.

(iii) Let $A_t$ represent the estimate of the monthly revenue, in thousands of dollars, using the average rate of change found in part (i). For $A_4$ found in part (ii), it can be shown that $A_4<R\left(4\right)$. Explain why, in general, $A_t<R\left(t\right)$ for all $t$, where $2<t<7$. Your explanation should include a reference to the graph of $R$ and its relationship to $A_t$.

A small business's total cumulative sales, in thousands of dollars, since the launch of a new product is modeled by the quadratic function $S$ given by $S\left(t\right)=-0.5t^2+18t+4$ for $0\le t\le T$, where $t$ is the number of months since launch and $T$ is a positive constant.

The quadratic function model $S$ has exactly one absolute minimum or one absolute maximum. That minimum or maximum can be used to determine a domain restriction for $S$. Based on the context of the problem, explain how that minimum or maximum can be used to determine a boundary for the domain of $S$.