# 3.3 Quadratic equations: applications  (Page 3/4)

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## Practice set e

A study of the air quality in a particular city by an environmental group suggests that $t$ years from now the level of carbon monoxide, in parts per million, in the air will be

$A=0.2{t}^{2}+0.1t+5.1$

(a) What is the level, in parts per million, now?

(b) How many years from now will the level of carbon monoxide be at 8 parts per million? Round to the nearest tenth.

(a). $5.1$ parts per million (b). $3.6$ years

## Sample set f

A contractor is to pour a concrete walkway around a swimming pool that is 20 feet wide and 40 feet long. The area of the walkway is to be 544 square feet. If the walkway is to be of uniform width, how wide should the contractor make it?

Step 1:  Let $x$ = the width of the walkway.
Step 2:  A diagram will help us to get the equation.

(Area of pool and walkway) — (area of pool) = (area of walkway)
$\left(20+2x\right)\left(40+2x\right)-20·40=544$
$\begin{array}{lllllll}\text{Step\hspace{0.17em}3:}\hfill & \hfill & \hfill \left(20+2x\right)\left(40+2x\right)-20·40& =\hfill & 544\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 800+120x+4{x}^{2}-800& =\hfill & 544\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 120x+4{x}^{2}& =\hfill & 544\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 4{x}^{2}+120x-544& =\hfill & 0\hfill & \hfill & \text{Divide\hspace{0.17em}each\hspace{0.17em}term\hspace{0.17em}by\hspace{0.17em}4}.\hfill \\ \hfill & \hfill & \hfill {x}^{2}+30x-136& =\hfill & 0\hfill & \hfill & \text{Solve\hspace{0.17em}by\hspace{0.17em}factoring}.\hfill \\ \hfill & \hfill & \hfill \left(x-4\right)\left(x+34\right)& =\hfill & 0\hfill & \hfill & \begin{array}{l}\text{(This\hspace{0.17em}is\hspace{0.17em}difficult\hspace{0.17em}to\hspace{0.17em}factor\hspace{0.17em}so\hspace{0.17em}we\hspace{0.17em}may\hspace{0.17em}}\\ \text{wish\hspace{0.17em}to\hspace{0.17em}use\hspace{0.17em}the\hspace{0.17em}quadratic\hspace{0.17em}formula}\text{.)\hspace{0.17em}}\end{array}\hfill \end{array}$
$\begin{array}{lllll}x-4=0\hfill & \hfill & \text{or}\hfill & \hfill & x+34=0\hfill \\ x=4\hfill & \hfill & \text{or}\hfill & \hfill & x=-34\text{\hspace{0.17em}has\hspace{0.17em}no\hspace{0.17em}physical\hspace{0.17em}meaning}\text{.}\hfill \end{array}$
Check a width of 4 feet as a solution.
$\begin{array}{lllll}\text{Step\hspace{0.17em}4:}\hfill & \hfill & \text{Area\hspace{0.17em}of\hspace{0.17em}pool\hspace{0.17em}and\hspace{0.17em}walkway\hspace{0.17em}}\hfill & =\hfill & \left(20+2·4\right)\left(40+2·4\right)\hfill \\ \hfill & \hfill & \hfill & =\hfill & \left(28\right)\left(48\right)\hfill \\ \hfill & \hfill & \hfill & =\hfill & 1344\hfill \end{array}$
Area of pool = (20)(40) = 800
Area of walkway = $\begin{array}{ccc}1344-800=544& & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\end{array}$
This solution checks.
Step 5:  The contractor should make the walkway 4 feet wide.

## Practice set f

A contractor is to pour a concrete walkway around a swimming pooi that is 15 feet wide and 25 feet long. The area of the walkway is to be 276 square feet. If the walkway is to be of uniform width, how wide should the contractor make it?

3 ft wide

## Exercises

Some of the following problems have actual applications and some are intended only as logic developers. A calculator may be helpful. The problems appear in groups and correspond to the noted Sample Set problem.

## Sample set a—type problems

The manufacturer of electronic fuel injectors determines that the number $N$ of injectors sold is related to the price $x$ per injector by $N=22x-{x}^{2}.$ At what price should the manufacturer price the injectors so that 112 of them are sold?

$8\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}14$

The owner of a stained-glass shop determines that the number $N$ of pieces of a particular type of glass sold in a month is related to the price $x$ per piece by $N=21x-{x}^{2}.$ At what price should the shop buyer price the glass so that 162 sell?

It is estimated that $t$ years from now the population of a certain city will be

$P={t}^{2}-15t+12,036$

(a) What is the population now?

(b) How many years from now will the population be 12,000?

$\text{(a)\hspace{0.17em}}12,036\text{\hspace{0.17em}\hspace{0.17em}(b)\hspace{0.17em}}3\text{\hspace{0.17em}and\hspace{0.17em}}12\text{\hspace{0.17em}years\hspace{0.17em}from\hspace{0.17em}now}$

It is estimated that $t$ years from now the population of a certain city will be

$P={t}^{2}-16t+24,060$

(a) What is the population now?

(b) How many years from now will the population be 24,000?

If an object is thrown vertically upward, its height $h$ , above the ground, in feet, after $t$ seconds is given by $h={h}_{0}+{v}_{0}t-16{t}^{2}$ , where ${h}_{0}$ is the initial height from which the object is thrown and ${v}_{0}$ is the initial velocity of the object. Using this formula and an approach like that of Sample Set A, solve this problem.

A ball thrown vertically into the air has the equation of motion $h=48+32t-16{t}^{2}.$

(a) How high is the ball at $t=0$ (the initial height of the ball)?

(b) How high is the ball at $t=1$ (after 1 second in the air)?

(c) When does the ball hit the ground? ( Hint: Determine the appropriate value for $h$ then solve for $t$ .)

$\text{(a)\hspace{0.17em}}48\text{\hspace{0.17em}feet\hspace{0.17em}\hspace{0.17em}(b)\hspace{0.17em}64\hspace{0.17em}feet\hspace{0.17em}\hspace{0.17em}(c)\hspace{0.17em}}t=3$

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