# 3.9 Modeling using variation  (Page 6/14)

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The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

2.88 inches

The kinetic energy $\text{\hspace{0.17em}}K\text{\hspace{0.17em}}$ of a moving object varies jointly with its mass $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and the square of its velocity $\text{\hspace{0.17em}}v.\text{\hspace{0.17em}}$ If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

## Chapter review exercises

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

## Complex Numbers

Perform the indicated operation with complex numbers.

$\left(4+3i\right)+\left(-2-5i\right)$

$2-2i$

$\left(6-5i\right)-\left(10+3i\right)$

$\left(2-3i\right)\left(3+6i\right)$

$24+3i$

$\frac{2-i}{2+i}$

Solve the following equations over the complex number system.

${x}^{2}-4x+5=0$

${x}^{2}+2x+10=0$

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.

$f\left(x\right)={x}^{2}-4x-5$

$f\left(x\right)=-2{x}^{2}-4x$

For the following problems, find the equation of the quadratic function using the given information.

The vertex is $\left(–2,3\right)$ and a point on the graph is $\text{\hspace{0.17em}}\left(3,6\right).$

$f\left(x\right)=\frac{3}{25}{\left(x+2\right)}^{2}+3$

The vertex is $\text{\hspace{0.17em}}\left(–3,6.5\right)\text{\hspace{0.17em}}$ and a point on the graph is $\text{\hspace{0.17em}}\left(2,6\right).$

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

300 meters by 150 meters, the longer side parallel to river.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, $\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ in terms of horizontal distance traveled, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ given by $\text{\hspace{0.17em}}h\left(x\right)=\frac{-32}{{\left(120\right)}^{2}}{x}^{2}+x.\text{\hspace{0.17em}}$ Find the maximum height the object attains.

## Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

$f\left(x\right)=4{x}^{5}-3{x}^{3}+2x-1$

Yes, degree = 5, leading coefficient = 4

$f\left(x\right)={5}^{x+1}-{x}^{2}$

$f\left(x\right)={x}^{2}\left(3-6x+{x}^{2}\right)$

Yes, degree = 4, leading coefficient = 1

For the following exercises, determine end behavior of the polynomial function.

$f\left(x\right)=2{x}^{4}+3{x}^{3}-5{x}^{2}+7$

$f\left(x\right)=4{x}^{3}-6{x}^{2}+2$

$\text{As}\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)=2{x}^{2}\left(1+3x-{x}^{2}\right)$

## Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

$f\left(x\right)={\left(x+3\right)}^{2}\left(2x-1\right){\left(x+1\right)}^{3}$

–3 with multiplicity 2, $-\frac{1}{2}$ with multiplicity 1, –1 with multiplicity 3

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich