5.8 Modeling using variation  (Page 6/14)

 Page 6 / 14

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

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 exercises, 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 $\text{\hspace{0.17em}}\left(2,6\right).$

For the following exercises, complete the task.

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

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

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

4 with multiplicity 1

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

$\frac{1}{2}\text{\hspace{0.17em}}$ with multiplicity 1, 3 with multiplicity 3

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}-5x+1$

answer and questions in exercise 11.2 sums
what is a algebra
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI