# 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

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4