C. Polynomial and Rational Functions · Polynomial Functions and Equations

Solving Problems with Polynomial Models

C3.7 C3.4 C1.7About 75 minutes

Learning goals

I can write a polynomial equation that models a real situation.
I can solve the equation by factoring, the quadratic formula or estimation, and keep only the roots that make sense.
I can build a polynomial model from given conditions and use it to answer questions.
I can explain why some equations have no solution within a model's domain.

Words to know

mathematical model
An equation or function that describes a real situation, usually only over a limited domain.

extraneous / inadmissible root
A correct root of the equation that does not make sense in the situation (for example, a negative length), so it is rejected.

domain of a model
The x-values for which the model makes sense in the real situation.

From words to equations to answers

The last lessons gave you the tools to solve polynomial equations. Now you will use them on problems. The method is the same every time:

Define the variable (with units) and state the domain that makes sense.
Write an equation. Volume and area formulas, products of numbers, or a model built from conditions often give polynomials.
Rearrange to P(x) = 0 and solve: factor theorem, patterns, quadratic formula, or estimation.
Reject roots outside the domain and answer the question in a sentence, with units.
Check the answer in the original words of the problem.

Worked example

Example 1: The box from lesson 1, backwards

An open box is made from a 30 cm by 20 cm sheet by cutting x-by-x squares from the corners. For which values of x is the volume 1000 cm³?

V(x) = x(30 − 2x)(20 − 2x), domain 0 < x < 10.
Set 4x³ − 100x² + 600x = 1000, so 4x³ − 100x² + 600x − 1000 = 0. Divide by 4: x³ − 25x² + 150x − 250 = 0.
We already know V(5) = 1000, so x = 5 is a root. Synthetic division with 5: 1, −20, 50 | 0. The quotient is x² − 20x + 50.
Quadratic formula: x = [20 ± √(400 − 200)]/2 = 10 ± √50 = 10 ± 5√2, about 17.07 or 2.93.
17.07 is outside 0 < x < 10 (you cannot cut 17 cm squares from a 20 cm side). Reject it.

Cut squares of 5 cm or about 2.93 cm (exactly 10 − 5√2 cm).

The line V = 1000 crosses the volume graph twice inside the domain 0 < x < 10, at x ≈ 2.93 and x = 5. The third root of the equation, x ≈ 17.07, lies outside the domain.

Worked example

Example 2: An ice block for Winterlude

An ice carver at Winterlude in Ottawa wants a block shaped like a rectangular prism whose width is 10 cm more than its height and whose length is 30 cm more than its height. The volume must be 72 000 cm³. Find the dimensions.

Let x be the height in cm (x > 0). Then x(x + 10)(x + 30) = 72 000.
Expand and rearrange: x³ + 40x² + 300x − 72 000 = 0.
Look for a root near the cube root of 72 000 ≈ 41.6, but smaller, because the other two sides are longer. Try x = 30: 27 000 + 36 000 + 9000 − 72 000 = 0 ✓.
Synthetic division with 30: 1, 70, 2400 | 0. The quotient x² + 70x + 2400 has discriminant 4900 − 9600 < 0, so there are no other real roots.

The block is 30 cm high, 40 cm wide and 60 cm long. Check: 30 × 40 × 60 = 72 000 ✓.

The ice block from Example 2: height x, width x + 10, length x + 30 (in cm). Only x = 30 works.

Worked example

Example 3: Consecutive integers

The product of three consecutive integers is 2730. Find them.

Let the integers be n, n + 1, n + 2. Then n(n + 1)(n + 2) = 2730, so n³ + 3n² + 2n − 2730 = 0.
The product is close to n³ (with n the smallest number), so n is a bit less than the cube root of 2730, which is about 13.98. Try n = 13: 13 × 14 × 15 = 2730 ✓.
Synthetic division with 13 gives the quotient n² + 16n + 210, with discriminant 256 − 840 < 0: no other real roots.

13, 14 and 15

Worked example

Example 4: A model built from conditions

A section of a roller-coaster track is modelled by a cubic h(x) = kx(x − 20)(x − 40), where h is the height in metres at a horizontal distance x metres, for 0 ≤ x ≤ 20. The track is 15 m high at x = 10. (a) Find k. (b) Where else on this section is the track 15 m high?

(a) h(10) = k(10)(−10)(−30) = 3000k = 15, so k = 0.005.
(b) Solve 0.005x(x − 20)(x − 40) = 15. Multiply by 200: x(x − 20)(x − 40) = 3000, so x³ − 60x² + 800x − 3000 = 0.
x = 10 is a root (we built it in). Synthetic division with 10: 1, −50, 300 | 0, so x² − 50x + 300 = 0.
x = [50 ± √(2500 − 1200)]/2 = 25 ± √325 = 25 ± 5√13, about 6.97 or 43.03. Only 6.97 is in 0 ≤ x ≤ 20.

(a) k = 0.005 (b) at x = 25 − 5√13 ≈ 6.97 m (as well as x = 10 m).

Good habits for word problems

State the domain before you solve, so you know which roots to reject. Use an estimate (a cube root, a quick table, or a graph) to guess a first root. After you find one root, divide: the quotient is a quadratic you can always finish with the quadratic formula. And always answer in a sentence with units.

Just for fun

Why did the box refuse the answer x = 17.07 cm?

It was a perfectly good root, just too big to fit in the box.

Hands-on activity

Two boxes, same volume

You need: two sheets of letter paper, a ruler, scissors (used carefully), tape, and a cup of dry rice or cereal.

From each sheet, cut a 20 cm by 20 cm square.
The volume of an open box made by cutting x-by-x squares from the corners is V(x) = x(20 − 2x)². Solve V(x) = 512 by the factor theorem. (Hint: try x = 2.)
Build both boxes that have a volume of 512 cm³ (round the second x to the nearest millimetre).
Fill one box with rice and pour it into the other. Do they hold the same amount?

What to look for: 4x³ − 80x² + 400x − 512 = 0, or x³ − 20x² + 100x − 128 = 0. x = 2 works; dividing gives x² − 18x + 64 = 0, so x = 9 ± √17. Only 9 − √17 ≈ 4.88 cm is in the domain 0 < x < 10. The boxes are 16 × 16 × 2 cm and about 10.2 × 10.2 × 4.9 cm, and both hold about 512 cm³ (the rice test should be close, allowing for folding errors).

Your turn

For the 20 cm square sheet in the activity, which cut sizes give a volume of 512 cm³?

The product of three consecutive even integers is 480. Find them.

A box has dimensions w, w + 1 and w + 2 (cm) and volume 210 cm³. Find w.

A small cylindrical grain bin has a height 2 m more than its radius r, and a volume of 16π m³. Find r.

For the roller coaster h(x) = 0.005x(x − 20)(x − 40), find the height at x = 5 m.

Why do we reject negative roots in a volume problem?

More practice: warm-up

State the domain for x in a box made from a 30 cm by 20 cm sheet.

Write the volume of a cube with edge x + 1.

Write three consecutive integers if the smallest is n.

A root of a volume equation is x = −3. Do you keep it?

The cube root of 1000 is ...

More practice: on your own

An open box is made from a 24 cm by 24 cm sheet. Solve x(24 − 2x)² = 1024 and interpret the result.

The product of three consecutive integers is −24. Find them.

The population of fish in a lake, in hundreds, is modelled by P(t) = −t³ + 12t² + 3, where t is in years. When will the population return to its starting value?

For the roller coaster in Example 4, find every x in 0 ≤ x ≤ 20 where the height is 15 m, giving exact answers.

A rectangular prism has a square base of side x cm and height x + 4 cm. Its volume is 63 cm³. Find x.

Challenge

Explain why x(30 − 2x)(20 − 2x) = 1200 has no solution in the domain 0 < x < 10, even though the equation has a real root.

A roller-coaster section h(x) = kx(x − 20)(x − 40), 0 ≤ x ≤ 20, must be exactly 9 m high at x = 5. Find k.

A cube has edge x cm. A second box is 2 cm longer, 1 cm narrower and the same height as the cube. Its volume is 12 cm³ more than the cube's volume. Find x.

Check your understanding

Quiz

An open box from a 30 cm by 20 cm sheet has V(x) = x(30 − 2x)(20 − 2x). Which root of V(x) = 1000 must be rejected?

The product of three consecutive integers is 336. What is the smallest of them?

A box has dimensions x, x + 2 and x + 4 (cm) and volume 192 cm³. What is x?

Which equation models a cylinder whose height is 3 m more than its radius r, with volume 50π m³?

Why does the solution x = 4 of x³ − 24x² + 144x − 256 = 0 (a box from a 24 cm square) appear as a double root?

In h(x) = kx(x − 20)(x − 40), the point (10, 15) is on the graph. What is k?

For parents and teachers

Big idea (C3.7, C3.4): many real problems lead to polynomial equations. Solve them with the tools of this unit, then keep only the roots that make sense in the situation (positive lengths, x-values inside the domain).

How to help: Ask your teen to read the final answer back in a sentence with units and to check it in the original words of the problem ("Does a 30 × 40 × 60 block really have volume 72 000 cm³?"). Building a paper box from the activity makes the domain and the rejected roots very concrete.

Common difficulties:

Forgetting to state the domain, then keeping roots that make no sense.
Expanding products incorrectly when writing the equation.
Not knowing where to start the factor theorem: an estimate such as a cube root narrows the search.
Answering with x only, when the question asks for all dimensions.