3.3 Quadratic equations: applications  (Page 4/4)

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A woman’s glasses accidently fall off her face while she is looking out of a window in a tall building. The equation relating $h$ , the height above the ground in feet, and $t$ , the time in seconds her glasses have been falling, is $h=64-16{t}^{2}.$

(a) How high was the woman’s face when her glasses fell off?

(b) How many seconds after the glasses fell did they hit the ground?

Sample set b—type problems

The length of a rectangle is 6 feet more than twice its width. The area is 8 square feet. Find the dimensions.

$\text{length}=8;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{width}=1$

The length of a rectangle is 18 inches more than three times its width. The area is 81 square inches. Find the dimensions.

The length of a rectangle is two thirds its width. The area is 14 square meters. Find the dimensions.

$\text{width}=\sqrt{21}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{length}=\text{\hspace{0.17em}}\frac{2}{3}\sqrt{21}$

The length of a rectangle is four ninths its width. The area is 144 square feet. Find the dimensions.

The area of a triangle is 14 square inches. The base is 3 inches longer than the height. Find both the length of the base and height.

$b=7;\text{\hspace{0.17em}}\text{\hspace{0.17em}}h=4$

The area of a triangle is 34 square centimeters. The base is 1 cm longer than twice the height. Find both the length of the base and the height.

Sample set c—type problems

The product of two consecutive integers is 72. Find them.

$-9,-8\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}8,9$

The product of two consecutive negative integers is 42. Find them.

The product of two consecutive odd integers is 143. Find them. ( Hint: The quadratic equation is factorable, but the quadratic formula may be quicker.)

$-13,-11\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}11,13$

The product of two consecutive even integers is 168. Find them.

Three is added to an integer and that sum is doubled. When this result is multiplied by the original integer the product is 20. Find the integer.

$n=2,-5$

Four is added to three times an integer. When this sum and the original integer are multiplied, the product is $-1.$ Find the integer.

Sample set d—type problems

A box with no top and a square base is to be made by cutting out 2-inch squares from each corner and folding up the sides of a piece of cardboard.The volume of the box is to be 25 cubic inches. What size should the piece of cardboard be?

$4+\sqrt{12.5}\text{\hspace{0.17em}}\text{inches}$

A box with no top and a square base is to made by cutting out 8-inch squares from each corner and folding up the sides of a piece of cardboard. The volume of the box is to be 124 cubic inches. What size should the piece of cardboard be?

Sample set e—type problems

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, will be $A=0.1{t}^{2}+0.1t+2.2.$

(a) What is the level, in parts per million, of carbon monoxide in the air now?

(b) How many years from now will the level of carbon monoxide be at 3 parts per million?

(a) carbon monoxide now $2.2$ parts per million
(b) $2.37\text{\hspace{0.17em}}\text{years}$

A similar study to that of problem 21 suggests $A=0.3{t}^{2}+0.25t+3.0.$

(a) What is the level, in parts per million, of carbon monoxide in the air now?

(b) How many years from now will the level of carbon monoxide be at 3.1 parts per million?

Sample set f—type problems

A contractor is to pour a concrete walkway around a wading pool that is 4 feet wide and 8 feet long. The area of the walkway and pool is to be 96 square feet. If the walkway is to be of uniform width, how wide should it be?

$x=2$

Astrophysical problem

A very interesting application of quadratic equations is determining the length of a solar eclipse (the moon passing between the earth and sun). The length of a solar eclipse is found by solving the quadratic equation

${\left(a+bt\right)}^{2}+{\left(c+dt\right)}^{2}={\left(e+ft\right)}^{2}$

for $t$ . The letters $a,b,c,d,e,$ and $f$ are constants that pertain to a particular eclipse. The equation is a quadratic equation in $t$ and can be solved by the quadratic formula (and definitely a calculator). Two values of $t$ will result. The length of the eclipse is just the difference of these $t$ -values.

The following constants are from a solar eclipse that occurred on August 3, 431 B.C.

$\begin{array}{ccccccc}a& =& -619& & b& =& 1438\\ c& =& 912& & d& =& -833\\ e& =& 1890.5& & f& =& -2\end{array}$
Determine the length of this particular solar eclipse.

Exercises for review

( [link] ) Find the sum: $\frac{2x+10}{{x}^{2}+x-2}+\frac{x+3}{{x}^{2}-3x+2}.$

$\frac{3x+14}{\left(x+2\right)\left(x-2\right)}$

( [link] ) Solve the fractional equation $\frac{4}{x+12}+\frac{3}{x+3}=\frac{4}{{x}^{2}+5x+6}.$
( Hint: Check for extraneous solutions.)

( [link] ) One pipe can fill a tank in 120 seconds and another pipe can fill the same tank in 90 seconds. How long will it take both pipes working together to fill the tank?

$51\frac{3}{7}$

( [link] ) Use the quadratic formula to solve $10{x}^{2}-3x-1=0.$

( [link] ) Use the quadratic formula to solve $4{x}^{2}-3x=0.$

$x=0,\frac{3}{4}$

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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