Your parents allow you to borrow a car to get to your part-time job, but you have to pay for a tank of gas each month. Gasoline costs $2.76/gallon and the tank takes 15 gallons. You have 2 coworkers who are each willing to pay for a quarter of a tank each month to carpool with you to work. How much do you save each month? $20.70 $16.46 $18.55 $15.91

Answer:

[tex]\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx}= x - \ln(1 + e^{x}) + C\end{aligned}[/tex].

Step-by-step explanation:

The first derivative of the denominator [tex]1 + e^{x}[/tex] is [tex]e^{x}[/tex]. Rewrite the fraction to obtain that expression on the numerator.

[tex]\begin{aligned}\frac{1}{1 + e^{x}} &= \frac{1 + e^{x}}{1 + e^{x}} - \frac{e^{x}}{1+e^{x}}\\&=1-\frac{e^{x}}{1+e^{x}}\end{aligned}[/tex].

In other words,

[tex]\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx} &= \int{dx} - \int{\frac{e^{x}}{1+e^{x}}\cdot dx}\end{aligned}[/tex].

Apply [tex]u[/tex]-substitution on the integral [tex]\displaystyle \int{\frac{e^{x}}{1+e^{x}}\cdot dx}[/tex]:

Let [tex]u = 1 + e^{x}[/tex]. [tex]u > 1[/tex].

[tex]du = e^{x}\cdot dx[/tex].

[tex]\displaystyle \int{\frac{e^{x}}{1+e^{x}}\cdot dx} = \int{\frac{du}{u}} = \ln{|u|} = \ln{u} +C = \ln{(1+e^{x})}+C[/tex].

Therefore

[tex]\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx} &= \int{dx} - \int{\frac{e^{x}}{1+e^{x}}\cdot dx}\\ & = x - \ln{(1 + e^{x})}+C\end{aligned}[/tex].

Answer:

48 feet

Step-by-step explanation:

Distance = rate × time

d = 12 ft/min × 4 min

d = 48 ft

The diver will be 48 feet below the surface.

Answer:

-48 ft

Step-by-step explanation:

The unit rate of descent is

  -12 ft

-----------

  1 min

and so after 4 minutes, the diver will be 48 ft beneath the surface:

  -12 ft

----------- * 4 min = -48 ft

  1 min

Answer:

C. [tex]y=\sqrt[3]{-x} -1[/tex]

Step-by-step explanation:

Consider the eparent function [tex]y=\sqrt[3]{x}[/tex] The graph of this function is shown in the attached diagram - red curve.

Reflect this graph across the y-axis, you'll get the blue line which equation is

[tex]y=\sqrt[3]{-x}[/tex]

Translate this graph 1 unit down, then you'll get green line (that is exactly the graph from your image) and the function for this line is

[tex]y=\sqrt[3]{-x} -1[/tex]

This question is based on the reasoning. Therefore, the negative slope of the line representing the data in the table.

Given;

A team is selling discount cards as a fundraiser. The table shows the number of cards sold and the remaining amount of money needed.

We have to describe the slope of the line representing the data in the table.

Reasoning: The number of cards sold is represented by x-axis and the amount of money needed is represented  by the y-axis.

In general, as we sell more cards, we gain more money and then,  the amount of money needed goes down.

From the given graph terms, we start at the top of our graph with the amount of money needed and zero cards sold and then as we sell cards your slope of the line goes down and we need less and less money.

Therefore, the negative slope of the line representing the data in the table.

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Answer:

The height of the building is [tex]100\ ft[/tex]

Step-by-step explanation:

Let

h ----> the height of the building in feet

we know that

Using proportion

[tex]\frac{6}{4.5}=\frac{h}{75}\\ \\h=6*78/4.5\\ \\h=100\ ft[/tex]

I took the test and got 100 ft correct

The angle measure of GIH is 35 degrees. The corresponding angles theorem guarantees that when two angles are formed by intersecting lines and share the same endpoints at an arc, they will have equal measures.

Since angles HJG and GIH contain the same endpoints at arc GH, they are corresponding angles formed by intersecting lines with a transversal. According to the corresponding angles theorem, when two lines are intersected by a transversal, the corresponding angles are congruent.

Given that angle HJG measures 35 degrees, we can conclude that angle GIH also measures 35 degrees. This is because they are corresponding angles and share the same endpoints at arc GH.

Therefore, the angle measure of GIH is 35 degrees. The corresponding angles theorem guarantees that when two angles are formed by intersecting lines and share the same endpoints at an arc, they will have equal measures. In this case, both angles HJG and GIH have an angle measure of 35 degrees.

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The solution set for the quadratic expression given is : (-8, 5)

Using the expression given :

divide through by 2

y = x² + 3x - 40

As a quadratic expression ;

x² + 3x - 40 = 0

factors whose product gives -40 and addition gives 3

x² + 8x - 5x - 40

factorize

x(x + 8) -5(x + 8)

(x + 8) = 0 or (x - 5) = 0

x = -8 or x = 5

Hence, the solution set is (-8 or 5)

Answer:

so you can simplify the ratio

Step-by-step explanation:

Answer:

U = -1/6

How to solve:

Multiply both sides by 2U to get rid of the denominator and you should end up with

[tex] - 5 = 15(2u)[/tex]

then multiply and devide to get your U by itself.

you should then get

[tex]u = - \frac{1}{6} [/tex]

after symplifying.

Answer:

Answer:

u=−10

Step-by-step explanation:

Let's solve your equation step-by-step.

u− 5 /2 u=15

Step 1: Simplify both sides of the equation.

u− 5/ 2 u=15

u+ −5 /2 u=15 (u+ −5 2 u)=15

(u+ −5 2 u)=15(Combine Like Terms)

−3 /2 u=15

−3 /2 u=15

Step 2: Multiply both sides by 2/(-3).

( 2 −3 )*( −3 2 u)=( −3 )*(15)

u=−10

Answer:

30

Step-by-step explanation:

60% of the 50 students are 12 years old

60% * 50

Changing to decimal form

.60 * 50

30

30 of the students are 12

Answer:

Step-by-step explanation:

T = time in seconds

H = distance

Tground = time to return to ground

Tmax = time at maximum height

H = -16T^2 + 672T...eqn 1

projectile returns to ground at H =0,

subs for H in eqn 1...

0 = -16T^2 + 672T

solving for T we get...

16T^2 = 672T

=> Tground = 42secs

Tmax = 0.5 Tground = 21secs

Answer:

The slope is m=-5

Step-by-step explanation:

we know that

The equation of the line into slope intercept form is equal to

y=mx+b

where

m is the slope

b is the y-intercept

In this problem we have

y=-(2/3)-5x

so

the slope is equal to

m=-5

the y-intercept is equal to

b=-2/3

Answer:

$5600

Step-by-step explanation:

Original Cost = x

Total Cost = $5824

Sales Tax = 4% or 0.04

Problem Setup

(Original Cost) + 4% (Original Cost) = Total Cost

x + 0.04x = 5824

1x+ 0.04x = 5824

1.04x = 5824

1.04x/1.04 = 5824/1.04

x=5600 Original Cost

To Check math multiply 5600 x 0.04 = 224.

Then add $5600 + $224 = $5824

The price of the car before sales tax is $5600.  

To determine the cost of the car before sales tax, we can use the equation provided:

[tex]x + 0.04x = 5824[/tex]

Here, [tex]x[/tex] represents the original price of the car before sales tax, and [tex]0.04x[/tex] represents the amount of sales tax, which is 4% of the car's price.

First, combine the terms on the left side of the equation:

[tex]1x + 0.04x = 1.04x[/tex]  

Therefore, the equation simplifies to:

[tex]1.04x = 5824[/tex]

Next, to solve for [tex]x[/tex], divide both sides of the equation by 1.04:

[tex]x = \frac{5824}{1.04}[/tex]

Calculating the right side:

[tex]x = 5600[/tex]

The probability that the next post will have 0 comments is P = 4/9.

The probability will be equal to the quotient between the number of posts with no comments and the total number of posts.

We know that there are 9 posts, and by looking at the table, we can see that 4 out of these 9 posts have 0 coments.

Then the probability that the next post will have 0 comments is:

P = 4/9

So the correct option is B.

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The probability that a person will order dessert given that they ordered a steak for dinner, this is an example of conditional probability.

Given that another event (B) has previously occurred, the likelihood of an event (A) occurring.

Event A in Option(C) - A person will order dessert

Event B in Option(C) - A person will order steak for dinner

Event A will only occur if the event B has already completed, hence according to the definition this is an example of conditional probability.

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For this case we have the following functions:

[tex]f (x) = 2x ^ 2-8x + 6\\g (x) = 3x-1[/tex]

We must find f (x) + g (x):

We have to:

[tex]f (x) + g (x) = 2x ^ 2-8x + 6 + (3x-1)\\f (x) + g (x) = 2x ^ 2-8x + 6 + 3x-1[/tex]

We add similar terms:

[tex]f (x) + g (x) = 2x ^ 2-5x + 5[/tex]

Answer:

[tex]f (x) + g (x) = 2x ^ 2-5x + 5[/tex]

OPTION C

Try to put in 9 and 10

Answer:

[tex]y=x^{2} -\frac{48}{7}x+\frac{64}{7}[/tex]

Step-by-step explanation:

step 1

Find the roots of the quadratic equation

we know that

The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to

[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]

in this problem we have

[tex]x^{2} -6x+7=0[/tex]  

so

[tex]a=1\\b=-6\\c=7[/tex]

substitute in the formula

[tex]x=\frac{6(+/-)\sqrt{-6^{2}-4(1)(7)}} {2(1)}[/tex]

[tex]x=\frac{6(+/-)2\sqrt{2}} {2}[/tex]

[tex]x=3(+/-)\sqrt{2}[/tex]

[tex]x1=3(+)\sqrt{2}[/tex]

[tex]x2=3(-)\sqrt{2}[/tex]

The roots of the equation are a and b

so

[tex]a=3(+)\sqrt{2}[/tex]

[tex]b=3(-)\sqrt{2}[/tex]

step 2

Find a quadratic equation with roots a+1/b and b + 1/a

so

[tex]a+\frac{1}{b} =(3+\sqrt{2})+\frac{1}{3-\sqrt{2}} =\frac{9-2+1}{3-\sqrt{2}} =\frac{8}{3-\sqrt{2}}=\frac{8}{3-\sqrt{2}}*(\frac{3+\sqrt{2}}{3+\sqrt{2}})=\frac{8}{7}(3+\sqrt{2})[/tex]

[tex]b+\frac{1}{a} =(3-\sqrt{2})+\frac{1}{3+\sqrt{2}} =\frac{9-2+1}{3+\sqrt{2}} =\frac{8}{3+\sqrt{2}}=\frac{8}{3+\sqrt{2}}*(\frac{3-\sqrt{2}}{3-\sqrt{2}})=\frac{8}{7}(3-\sqrt{2})[/tex]

The quadratic equation is equal to

[tex]y=(x-\frac{8}{7}(3+\sqrt{2}))(x-\frac{8}{7}(3-\sqrt{2}))[/tex]

[tex]y=x^{2} -\frac{8}{7}(3-\sqrt{2})x-\frac{8}{7}(3+\sqrt{2})x+\frac{64}{49}(7)[/tex]

[tex]y=x^{2} -\frac{48}{7}x+\frac{64}{7}[/tex]

Answer:

-1,1,3

Step-by-step explanation:

The zeros,roots and the x-intercept are the same, so see where the graph touches the x axis and that or thosepoints is/are the zeros

Step-by-step explanation:

all work is shown and pictured

The GCF of 12x^2y and 27xy^3z is 3xy.

The full sort of GCF is greatest divisor. it's also referred to as highest  common factor.

we have to seek out GCF of

We can write them as:

1)2*2*3*x*x*y

2)3*3*3*x*y*y*y*z

GCF=3*x*y

=3xy

Hence the best common measure of 12x^2y and 27xy^3z is 3xy.

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Answer:

40320

Step-by-step explanation:

There are 8 different people we could assign to the first task.  Now there are 7 tasks and 7 people left.  There are 7 different people we could assign to the next task.    And so on

8 *7*6*5*4*3*2*1

8!

40320

Final answer:

The professor can assign eight different tasks to her eight teaching assistants in 40,320 different ways, calculated by the factorial of eight (8!).

Explanation:

The question pertains to the number of ways to assign eight different tasks to eight teaching assistants. This problem can be solved using the concept of permutations, which is a fundamental principle in combinatorics, a branch of mathematics.

To find the number of different permutations of eight distinct items (the tasks), we would calculate 8!, which signifies factorial of eight. This represents the product of all positive integers up to 8:

Therefore, there are 40,320 different ways the professor can make the assignments to her eight teaching assistants.

Answer: Ok, this have come up with. Here are 4 different definitions theorem could possiably mean :)

Hope that helps you out :D

First divide 5 to both sides to isolate p. Since 5 is being multiplied by p, division (the opposite of multiplication) will cancel 5 out (in this case it will make 5 one) from the right side and bring it over to the left side.

20 ≥  5p

20 ÷ 5 ≥ 5p ÷ 5

4 ≥  1p

4 ≥  p

For the graph will you have a empty or colored in circle?

If the symbol is ≥ or ≤ then the circle will be colored in. This represents that the number the circle is on is included.

If the symbol is > or < then the circle will be empty. This represents that the number the circle is on is NOT included.

Which direction will the ray go?

If the variable is LESS then the number then the arrow will go to the left of the circle.

If the variable is MORE then the number then the arrow will go to the right of the circle.

In this case your inequality is:

4 ≥ p OR p ≤ 4

aka 4 is greater then p OR p is less then 4

This means that the graph will have an colored circle and the arrow will go to the left of 4. Look at image below.

Hope this helped!

~Just a girl in love with Shawn Mendes

A. It is a many-to-one function.

Hello! It will be a pleasure to help to figure out what's the correct answer to this problem. First of all, we have the following function:

[tex]f(x) = 2x^3 + 2x^2-x[/tex]

When plotting this function, we get the red graph of the function shown below. So let's solve this as follows:

True

A function is said to be many-to-one there are values of the dependent variable (y-values) that corresponds to more than one value of the independent variable (x-values). To test this, we need to use the Horizontal Line Test. So let's take the horizontal line [tex]y=0.5[/tex], and you can see from the first figure below that [tex]y=0.5[/tex] is mapped onto [tex]x=-1.241 \ x=-0.344 \ and \ x=0.585[/tex]. so this is a many-to-one function.

Since this is a many-to-one function, it can't be a one-to-one function.

False

Indeed, this is a function

False

It passes the vertical line test because any vertical line can intersect the graph of the function at most once. An example of this is shown in the second figure below.

(2x^2)(3x^3)(4x)^2

=6x^5(16x^2)

=96X^7

Answer:

24 x^7

Step-by-step explanation:

(2 x^2)(3 x^3)(4 x)^2 = 2*3*4 * x² * x³ * x² = 24 x^7

Answer:

Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter, and is the longest chord.

Step-by-step explanation:

Answer:

Diameter

Step-by-step explanation:

Answer:

AB = CD = 3√5

Step-by-step explanation:

We use Pythagoras' Theorem.

AB² = CD² = 3²+6² = 9+36 = 9·(1+4) = 9·5

=> AB = CD = √(9·5) => AB = CD = 3√5

Answer:

[tex]\large\boxed{-5\dfrac{2}{5}=-\dfrac{27}{5}}[/tex]

Step-by-step explanation:

[tex]\bold{Method\ 1}:\\\\5=\dfrac{5\cdot5}{5}=\dfrac{25}{5}\\\\-5\dfrac{2}{5}=-\left(5+\dfrac{2}{5}\right)=-\left(\dfrac{25}{5}+\dfrac{2}{5}\right)=-\dfrac{25+2}{5}=-\dfrac{27}{5}\\\\\bold{Method\ 2:}\\\\\text{Multiply the integer by the denominator and add it to the numerator.}\\\\-5\dfrac{2}{5}=-\dfrac{(5)(5)+2}{5}=-\dfrac{25+2}{5}=-\dfrac{27}{5}[/tex]

Answer:

5 x 5 + 2

= -27/5 (5 is denominator)

Answer:

Step-by-step explanation:

The length is described as 4(-9 +w/2), which simplifies to 2w-36. The area is the product of length and width, so is ...

  Area = width×length

  126 = w(2w -36) = 2w² -36w

__

This equation can be divided by 2, then the square completed to give ...

  w² -18w = 63

  w² -18w +81 = 144 . . . . add 81 to complete the square

  (w -9)² = 12² . . . . . . . . . show as squares

  w -9 = 12 . . . . . . . . . . . . take the positive square root

  w = 12 +9 = 21

The width of the placemat is 21 inches.

Answer:

[tex]126in^2=2w^2-36w[/tex] and w= 21 inches.

Step-by-step explanation:

The placemat has an area of [tex]126in^2[/tex]. Now, let's call w to the width, the exercise says that the length of the placemat is four times the quantity of nine less than half its width, that is

length = 4(w/2-9).

Now, the area is length*width, so

[tex]A=w*4(\frac{w}{2}-9)= \frac{4w^2}{2}-36w = 2w^2-36w.[/tex]

and we know that [tex]A=126in^2[/tex], so

[tex]126in^2=2w^2-36w.[/tex]

Now, let's find the solution.

[tex]2w^2-36w-126=0[/tex] is a quadratic formula of the form [tex]ax^2+bx+c[/tex] with a=2, b= -36 and c = -126. The solutions will be

[tex]w=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

[tex]w=\frac{-(-36)\pm\sqrt{36^2-4(2)(-126)}}{2a}[/tex]

[tex]w=\frac{36\pm\sqrt{1296+1008}}{2*2}[/tex]

[tex]w=\frac{36\pm\sqrt{2304}}{4}[/tex]

[tex]w=\frac{36\pm48}{4}[/tex]

[tex]w=\frac{36+48}{4}[/tex] or [tex]w=\frac{36-48}{4}[/tex]

[tex]w=\frac{84}{4}[/tex] or [tex]w=\frac{-12}{4}[/tex]

as we are searching a distnace, we are going to use the positive solution, that is

[tex]w=\frac{84}{4}= 21.[/tex]

Then, the width is 21 inches.