Four points A, B, C, and D are coplanar, but only two at a time are collinear. Suppose a line is drawn through each pair of points. How many lines can you draw? ...?

Answer:

x=4+-isqrt(5/7

Step-by-step explanation:

Answer and Explanation :

Given : Function [tex]f(x)=\frac{x^2}{2}+2x+1[/tex]

To find : The vertex, axis of symmetry, maximum or minimum value, and the graph of the function.

Solution :

The quadratic function is in the form, [tex]y=ax^2+bx+c[/tex]

On comparing, [tex]a=\frac{1}{2}[/tex] , b=2 and c=1

The vertex of the graph is denote by (h,k) and the formula to find the vertex is

For h, The x-coordinate of the vertex is given by,

[tex]h=-\frac{b}{2a}[/tex]

[tex]h=-\frac{2}{2(\frac{1}{2})}[/tex]

[tex]h=-\frac{2}{1}[/tex]

[tex]h=-2[/tex]

For k, The y-coordinate of the vertex is given by,

[tex]k=f(h)[/tex]

[tex]k=\frac{h^2}{2}+2h+1[/tex]

[tex]k=\frac{(-2)^2}{2}+2(-2)+1[/tex]

[tex]k=2-4+1[/tex]

[tex]k=-1[/tex]

The vertex of the function is (h,k)=(-2,-1)

The x-coordinate of the vertex i.e. [tex]x=-\frac{b}{2a}[/tex] is the axis of symmetry,

So, [tex]x=-\frac{b}{2a}=-2[/tex] (solved above)

So, The axis of symmetry is x=-2.

The maximum or minimum point is determine by,

If a > 0 (positive), then the parabola opens upward and the graph has a minimum at its vertex.

[tex]a=\frac{1}{2} >0[/tex] so,  the parabola opens upward and the graph has a minimum at its vertex.

The Minimum value is given at (-2,-1)

Now, We plot the graph of the function

At different points,

x          y

-4        1

-2        -1

0         1

Refer the attached figure below.

Answer:

Cost of each student ticket is, $4.35

Step-by-step explanation:

Given the statement: Elise pays $21.75 for 5 student tickets to the fair.

Unit rate defined as the rates are expressed as a quantity of 1, such as 3 feet per second or 6 miles per hour, they are called unit rates.

[tex]unit rate = \frac{Total ticket Cost}{No of students}[/tex]

Given total cost paid by Elise = $21.75

Number of students = 5

then;

Unite rate per student = [tex]\frac{21.75}{5} = \$4.35[/tex]

Therefore, the cost of each student ticket is, $4.35

Answer:

The graph representing the solution is given below.

Step-by-step explanation:

We are given the system of inequality is,

[tex]2x + 5y\leq  9[/tex]

[tex]3x + 5y \leq  9[/tex]

Zero Test states that,

'After substituting the point (0,0) in the inequalities, if the result is true, then the solution region id towards the origin. If the result is false, the solution region is away from the origin'.

So, upon substituting (0,0) in the given inequalities, we get,

[tex]2x + 5y\leq  9[/tex] implies 0≤ 9, which is true.

[tex]3x + 5y \leq  9[/tex] implies 0≤ 9, which is true.

Thus, the solution region for both the inequalities is towards the origin.

Hence, upon plotting, the graph representing the solution set is given below.

Answer:

Give the guy above me Brainlyist

Step-by-step explanation:

Answer:

[tex]u=-x^4-2x^2y+y-y^2[/tex]

Step-by-step explanation:

We are given that

[tex](1-2x^2-2y)\frac{dy}{dx}=4x^3+4xy[/tex]

We have to solve the given differential equation

[tex](1-2x^2-2y)dy=(4x^3+4xy)dx[/tex]

[tex](1-2x^2-2y)dy-(4x^3+4xy)dx=0[/tex]

Compare with [tex]Mdx+ndy=0[/tex]

Then, we get [tex]M=-(4x^3+4xy),N=(1-2x^2-2y)[/tex]

Exact differential equation

[tex]M_y=N_x[/tex]

[tex]M_y=-4x[/tex]

[tex]N_x=-4x[/tex]

[tex]M_y=N_x[/tex]

Hence, the differential equation is an exact differential equation.

Solution of exact differential is given by

[tex]u=\int M(x,y)dx+K(y)[/tex] where K(y) is a function of y.

[tex]u=\int -(4x^3+4xy) dx+k(y)[/tex] y treated as constant

[tex]u=-x^4-2x^2y+k(y)[/tex]

[tex]u_y=N[/tex]

[tex]-2x^2+k'(y)=1-2x^2-2y[/tex]

[tex]K'(y)=1-2y[/tex]

[tex]k(y)=y-y^2[/tex]

Substitute the value then we get

Then, [tex]u=-x^4-2x^2y+y-y^2[/tex]

Final answer:

The given differential equation is exact, and by integrating the respective terms and finding the function H(y), the solution for z(x, y) for the differential equation is determined to be
     -x⁴ + y - y² + C, where C is a constant.

Explanation:

To solve the differential equation (1-2x² - 2y)dy/dx = 4x³ + 4xy, let's rearrange the equation in the form M(x, y)dy + N(x, y)dx = 0, which is the standard form for a first-order differential equation. We can rewrite our equation as (-4x³ - 4xy)dx + (1-2x² - 2y)dy = 0.

Now, we check if the equation is exact, that is, if ∂M/∂y = ∂N/∂x. If it is exact, there exists a function z(x, y) such that dz = M dy + N dx.

In this case, ∂M/∂y = -4x, and ∂N/∂x = -4x. Since the partial derivatives are equal, the differential is exact, meaning there exists some function z(x, y) such that dz = Ndx + Mdy. To find z(x, y), we integrate N with respect to x and M with respect to y and combine the resulting functions, making sure to include the functions of the other variable that may arise from the partial integration.

For N(x, y), we integrate -4x³dx to get -x⁴. For M(y), we integrate (1-2y)dy to get y - y². Thus z(x, y) = -x⁴ + y - y² + H(y), where H(y) is a function of y.

To find H(y), differentiate z with respect to y and equate it to M: dz/dy = 1 - 2y + H'(y) = 1 - 2y, which implies that H'(y) = 0, hence H(y) is a constant. Therefore, the function that satisfies the differential equation is z(x, y) = -x⁴ + y - y² + C, where C is a constant.

Answer:

The answer is A, 30

Step-by-step explanation:

$2,026.15 is the payment for  every two weeks

Two or more expressions with an Equal sign is called as Equation.

Given that annual salary is $52,680 and you are paid biweekly.

In a year we have 52 weeks

If you are planning your monthly budget based on a two week pay period

So there are 52/2 = 26 periods where you get paid

For every two weeks we get paid of 52680/26 = 2,026.153846

Hence, $2,026.15 is the payment for  every two weeks

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The golden ratio or golden mean is approximately 1.618 when calculated to the nearest thousandth using a calculator to evaluate (1 + √5) / 2.

The golden ratio or golden mean is a number often encountered in mathematics and art, and it is commonly represented by the Greek letter phi (φ). It can be expressed mathematically as (1 + √5) / 2. To find its decimal value to the nearest thousandth, we use a calculator to perform the operation.

First, calculate the square root of 5, then add 1 to the result. After this, divide the sum by 2. This will give us the golden ratio:

√5 ≈ 2.236
1 + √5 ≈ 3.236
(1 + √5) / 2 ≈ 1.618

So, to the nearest thousandth, the decimal value of the golden ratio is 1.618.

Answer:

[tex]f(n)=2240+(\frac{170}{Week}).n[/tex]

Step-by-step explanation:

In order to make an expression that represents the situation we need to make a function that represents Henry's savings.

This will be a function ''f(n)'' because it will depend of the variable ''n'' which is the number of weeks.

Let's start making the function by reading the problem.

''He earns $210 each week working part time after school and the weekends''

We can write :

[tex]f(n)=(\frac{210}{Week}).n[/tex]

Where ''210'' is actually $210

''Henry currently has $2240 in savings'' ⇒ We need to add this amount of money to the function.

[tex]f(n)=(\frac{210}{Week}).n+2240[/tex]

''His parents put $100 in his savings account each week and he saves one-third of his paycheck each week'' ⇒ We need to add ($100).n to the expression due to his parents and multiply by [tex]\frac{1}{3}[/tex] the expression that represents its paycheck ⇒

[tex]f(n)=(\frac{1}{3}.\frac{210}{Week}).n+2240+(\frac{100}{Week}).n[/tex]

Now, if we work with the expression :

[tex]f(n)=(\frac{70}{Week}).n+2240+(\frac{100}{Week}).n[/tex]

[tex]f(n)=2240+(\frac{170}{Week}).n[/tex]

Where the units of ''2240'' and ''170'' are $.

That is the final expression which represents the situation.

The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of  [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.

Further explanation:

Given:

The function is [tex]f\left( x \right) =\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]

Explanation:

The given function is [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}.[/tex]

Differentiate the above equation with respect to x.

[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right) &= \frac{d}{{dx}}\left( {\frac{{{x^4}}}{3} - \frac{{{x^5}}}{5}} \right)\\&= \frac{{4{x^3}}}{3} - \frac{{5{x^4}}}{5}\\&= \frac{{4{x^3}}}{3} - {x^4}\\\end{aligned}[/tex]

Again differentiate with respect to x.

[tex]\begin{aligned}\frac{{{d^2}}}{{d{x^2}}}f\left( x \right) &= \frac{{{d^2}}}{{d{x^2}}}\left( {\frac{{4{x^3}}}{3} - {x^4}} \right)\\&=\frac{{3 \times 4{x^2}}}{3} - 4{x^3}\\&= 4{x^2} - 4{x^3}\\\end{aligned}[/tex]

Substitute the first derivative equal to zero.

[tex]\begin{aligned}\frac{d}{{dx}}f\left( x \right)&= 0\\\frac{{4{x^3}}}{3} - {x^4}&= 0\\\frac{{4{x^3}}}{3} &= {x^4}\\\frac{4}{3}&= \frac{{{x^4}}}{{{x^3}}}\\\frac{4}{3}&= x\\\end{aligned}[/tex]

The value of x is [tex]\boxed{\frac{4}{3}}[/tex] for which the derivative of [tex]f\left( x \right)=\dfrac{{{x^4}}}{3} - \dfrac{{{x^5}}}{5}[/tex] attains the maximum.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Application of derivatives

Keywords: Derivative, attains, maximum, value of x, function, differentiate, minimum value.

Answer:

[tex]y=(x-1)^{2}+4[/tex]

Step-by-step explanation:

To write this quadratic function in vertex form, which is the explicit form of the parabola, we have to complete the square in the expression.

First, we have to take the coefficient of the linear term and find the squared power of its half:

[tex](\frac{b}{2} )^{2}=(\frac{2}{2} )^{2}=1[/tex]

Then, we add and subtract this number in the quadratic expression:

[tex]y=x^{2}-2x+5+1-1[/tex]

Now, we use the three terms that can be factorize as the squared power of a binomial expression:

[tex]y=(x^{2}-2x+1)+5-1[/tex]

Then, we find the square root of the first term and third term, and we form the squared power:

[tex]y=(x-1)^{2}+4[/tex]

Now, this vertex form is explicit, because it says from the beginning what's the coordinates of the vertex, which is: [tex](1;4)[/tex], as minimum, because the parabola is concave up.

In vertex form, the quadratic equation is written as;

⇒ y = (x - 1)² + 4

An algebraic equation with the second degree of the variable is called an Quadratic equation.

We have to given that;

The quadratic equation is,

⇒ y = x² - 2x + 5

Now, We can write in vertex form as;

⇒ y = x² - 2x + 5

⇒ y = x² - 2x + 1 + 4

⇒ y = (x - 1)² + 4

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Since the value of m is -17/6, equation c has a negative solution.

An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.

It is given that,

7 1/2 + m = 4 2/3

We have to apply the arithmetic operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.

15/2+m=14/3

Convert the fraction value and rearrange the equation as follows,

m=14/3-15/2

m=-17/6

Thus, the value of m is -17/6, equation c has a negative solution.

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Answer:

y=3x+2 is in slope intercept form. If you have a graph to go with this problem, if this equation is correct there should be points on (0,2) and (1,3)

Step-by-step explanation:

Answer:

The answer is the option A

[tex]\frac{1}{2}(\sqrt{[(y2)^{2}+(x2)^{2}]*[(y1)^{2}+(x1)^{2}]})[/tex]

Step-by-step explanation:

see the attached figure with letters to better understand the problem

we know that

The area of the triangle is equal to

[tex]A=\frac{1}{2}bh[/tex]

where

b is the base

h is the height

In this problem

[tex]b=AB, h=AC[/tex]

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

Find the distance AB

[tex]A(0,0), B(x2,y2)[/tex]

substitute

[tex]dAB=\sqrt{(y2-0)^{2}+(x2-0)^{2}}[/tex]

[tex]dAB=\sqrt{(y2)^{2}+(x2)^{2}}[/tex]

Find the distance AC

[tex]A(0,0), C(x1,y1)[/tex]

substitute

[tex]dAC=\sqrt{(y1-0)^{2}+(x1-0)^{2}}[/tex]

[tex]dAC=\sqrt{(y1)^{2}+(x1)^{2}}[/tex]

Find the area of the triangle

we have

[tex]b=\sqrt{(y2)^{2}+(x2)^{2}}, h=\sqrt{(y1)^{2}+(x1)^{2}}[/tex]

substitute  

[tex]A=\frac{1}{2}(\sqrt{(y2)^{2}+(x2)^{2}})(\sqrt{(y1)^{2}+(x1)^{2}})[/tex]

[tex]A=\frac{1}{2}(\sqrt{[(y2)^{2}+(x2)^{2}]*[(y1)^{2}+(x1)^{2}]})[/tex]

Answer:

A for plato users

Step-by-step explanation:

Answer:

The factors of the  7x² - 12x - 4 = 0 are (7x + 2)(x -2) .

Option (b) is correct i.e none of the above .

Step-by-step explanation:

As the expression given in the question.

7x² - 12x - 4 = 0

7x² - 14x + 2x - 4 = 0

7x (x - 2)  + 2(x - 2) = 0

(7x + 2)(x -2) = 0

Thus the factors of the  7x² - 12x - 4 = 0 are (7x + 2)(x -2) .

Option (b) is correct i.e none of the above .

Answer:

0.00000146

Step-by-step explanation:

Your welcome, and you better call saul

The standard form of 1.46e - 6 is 0.00000146.

To convert the number 1.46e-6 to standard form, we need to move the decimal point to the left as many times as indicated by the exponent of the scientific notation, which is -6 in this case.

This means moving the decimal point six places to the left.

Start with the number in scientific notation:

1.46e - 6.

Move the decimal point six places to the left (because of the negative exponent):

0.00000146.