For the average-cost model C(x) = (2000)/(x) + 5, what does C approach as x → ∞, and what does it mean? [1 point]

What is not eliminating the extra variable?

Use the constraint to reduce to a single-variable function before applying the model; an area written in two variables cannot be optimized directly here.

United StatesPrecalculusSyllabus dot point

How do you build an explicit polynomial or rational function from a context or data, and use it to answer questions?

Topic 1.14 Function Model Construction and Application: construct a polynomial or rational function model from a context, restricted domain or data set, and apply it to make predictions and solve problems.

A focused answer to AP Precalculus Topic 1.14, covering how to build linear, quadratic, polynomial and rational models from context or data, restrict domains appropriately, and apply the model to predictions.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this topic is asking
Building a model from a context
Restricting the domain
Applying the model
Try this

What this topic is asking

The College Board (Topic 1.14) wants you to construct an explicit polynomial or rational function from a context, a data set or a geometric situation, and then apply it to answer questions. Construction means writing the formula, choosing a sensible restricted domain, and using the model to make predictions, find extrema, or describe end behavior.

Building a model from a context

Geometric optimization (area, volume, cost) is the classic source: express the target quantity, use the constraint to eliminate a second variable, and you have a single-variable polynomial or rational model.

Restricting the domain

A constructed model usually applies only to part of the real line. Lengths and quantities must be positive; a number of items must be a non-negative integer; a rational cost model excludes the input that zeros its denominator. Stating the restricted domain is part of constructing the model, and forgetting it can produce nonsensical answers like negative areas or production of a fraction of an item.

Constructing and applying an area model

A farmer encloses a rectangular field against a straight river (no fence needed along the river) using $120$ meters of fencing for the other three sides. Construct the area as a function of the width $x$ and find the width that maximizes the area.

step 1 Define the variables and the constraint

Let $x$ be the width (the two sides perpendicular to the river) and $L$ the length (parallel to the river). The fencing covers two widths and one length: $2x + L = 120$ .

step 2 Eliminate the second variable

Solve the constraint for $L$ : $L = 120 - 2x$ . Then the area is $A(x) = x \cdot L = x(120 - 2x) = 120x - 2x^2$ .

step 3 Restrict the domain

Both $x > 0$ and $L = 120 - 2x > 0$ , so $0 < x < 60$ . The model applies only on this interval.

step 4 Apply the model to optimize

$A(x) = -2x^2 + 120x$ is a downward parabola with vertex at $x = -\frac{120}{2(-2)} = 30$ . So the width $x = 30$ meters maximizes the area, giving $A(30) = 30(120 - 60) = 1800$ square meters.

Applying the model

Once built, a model answers three kinds of question. Prediction: substitute an input to estimate an output, staying inside the restricted domain. Long-run behavior: use end behavior (Topics 1.6 and 1.7) to describe what happens as the input grows, such as an average cost approaching a limiting value. Optimization: find the maximum or minimum, often the vertex of a quadratic, to answer "what input is best".

A point that distinguishes strong answers is interpreting the mathematics back in context, with units. A vertex is not just an $x$ -value; it is "the width that maximizes area". A horizontal asymptote is not just $y = 8$ ; it is "the average cost per unit approaches eight dollars". The exam consistently awards the contextual interpretation alongside the calculation, so every numerical answer in a modelling question should be paired with a sentence saying what it means in the situation, including units and the domain on which it holds.

Try this

Q1. A box has a square base of side $x$ and a fixed volume of $32$ . Write its height $h$ as a function of $x$ . [1 point]

Cue. Volume $x^2 h = 32$ , so $h(x) = \frac{32}{x^2}$ , with domain $x > 0$ .

Q2. For the average-cost model $C(x) = \frac{2000}{x} + 5$ , what does $C$ approach as $x \to \infty$ , and what does it mean? [1 point]

Cue. $C \to 5$ ; the average cost per unit approaches the variable cost of five dollars as production grows.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part B (multiple choice, calculator allowed). A rectangular pen uses

40

meters of fencing for three sides (the fourth is a wall). If the width is

x

, which function gives the enclosed area? (A)

A(x) = x(40 - 2x)

(B)

A(x) = x(40 - x)

(C)

A(x) = 40x - x^2

(D)

A(x) = 2x(40 - x)

Show worked answer →

The correct answer is (A), $A(x) = x(40 - 2x)$ .

With a wall on one side, the fencing covers two widths and one length: $2x + L = 40$ , so $L = 40 - 2x$ . The area is width times length, $A(x) = x(40 - 2x)$ . Choice (C) would apply if the fencing covered one width and one length only.

AP 2024 (style)4 marksSection II (free response, calculator allowed). A company finds that the average cost per unit, in dollars, of producing

x

units is

C(x) = \frac{5000 + 8x}{x}

. (a) Construct a simplified expression for

C(x)

and find

C(100)

. (b) Determine the end behavior of

C

x \to \infty

and interpret it in context. (c) State an appropriate domain for the model.

Show worked answer →

A 4-point modelling question.

(a) Simplify and evaluate (1 point each): $C(x) = \frac{5000}{x} + 8$ . Then $C(100) = \frac{5000}{100} + 8 = 50 + 8 = 58$ dollars per unit.
(b) End behavior (1 point): as $x \to \infty$ , $\frac{5000}{x} \to 0$ , so $C(x) \to 8$ . The average cost approaches the per-unit variable cost of $8$ dollars as production grows, because the fixed cost is spread over more units.
(c) Domain (1 point): $x > 0$ , since the number of units produced must be positive (and the formula is undefined at $x = 0$ ).

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)