find the smallest value of an expression
(without using a calculator)

[tex]\bf\sqrt{(x-8)^2+(y+4)^2} +|y+3x|[/tex]

The third term of F(n)=3n-4 is 5.

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

Given,

F(n)=3n-4

F of n equal to three times of n minus four.

We need to find the third term.

To find third term we need to plug 3 in place of n.

F(3)=3(3)-4

Three times three minus four

F(3)=9-4

Nine minus four gives 5

F(3)=5

Hence, the third term of F(n)=3n-4 is 5.

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Answer:

103 N

Step-by-step explanation:

we use the equation: w = mg

where m is mass, W is weight and g is gravitational field strength

w = ?

m= 10.5 kg

g = 9.81 ms‐² (for earth)

so,

w = 10.5 × 9.81

w = 103.005

w ≈ 103 Newtons

Using the Quotient Rule, the differentiate the function with respect to x is [tex]\frac{4x^3}{9x^8+24x^4+16}[/tex] .

In the given question,

We have to differentiate the given function with respect to x.

The given function is [tex]y=\frac{x^4+1}{3x^4+4}[/tex]

We have differentiate with respect to x.

The given function is in fraction format. So solving the given expression we use Quotient Rule

According to Quotient Rule

"A method for recognizing issues where one function is divided by another is the quotient rule."

The formula of Quotient Rule is

[tex]\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x)-g'(x)f(x)}{(g(x))^2}[/tex]....................(1)

As we see in our function then f(x) = x[tex]^4+[/tex]1 and g(x) = 3x[tex]^4+[/tex]4

Before solving the function we find the value of f'(x), g'(x) and (g(x))[tex]^2.[/tex]

Now solving the f(x) for f'(x) with respect to x.

f(x) = x[tex]^4+[/tex]1

[tex]\frac{d}{dx}f(x)=\frac{d}{dx}(x^4+1)[/tex]

[tex]f'(x)=\frac{d}{dx}x^4+\frac{d}{dx}1[/tex]

We know that the differentiation of [tex]x^n=nx^{n-1}[/tex] and differentiation of constant value is zero.

So [tex]f'(x)=4x^{(4-1)}+0[/tex]

[tex]f'(x)=4x^3[/tex]

Now solving the g(x) for g'(x) with respect to x.

g(x) = 3x[tex]^4+[/tex]4

[tex]\frac{d}{dx}g(x)=\frac{d}{dx}(3x^4+4)[/tex]

[tex]g'(x)=3\frac{d}{dx}x^4+\frac{d}{dx}4[/tex]

We know that the differentiation of [tex]x^n=nx^{n-1}[/tex] and differentiation of constant value is zero.

[tex]g'(x)=3(4x^{(4-1)})+0[/tex]

[tex]g'(x)=3(4x^{3})[/tex]

[tex]g'(x)=12x^{3}[/tex]

Now finding the value of (g(x))[tex]^2.[/tex]

g(x) = 3x[tex]^4+[/tex]4

[tex](g(x))^4=(3x^4+4)^2[/tex]

Now using the formula of [tex](a+b)^2=a^2+2ab+b^2[/tex].

Now solving

[tex](g(x))^4=(3x^4)^2+2\times(3x^4)\times4+(4)^2[/tex]

Simplifying

[tex](g(x))^4=9x^8+24x^4+16[/tex]

Now putting the value of f'(x), g'(x) and (g(x))[tex]^2.[/tex] in the equation (1)

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{(3x^4+4)(4x^3)-(12x^3)(x^4+1)}{9x^8+24x^4+16}[/tex]

Simplifying

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{(3x^4(4x^3)+4(4x^3))-((12x^3)x^4+1(12x^3))}{9x^8+24x^4+16}[/tex]

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{(12x^{(4+3)}+16x^3)-(12x^{(3+4)}+12x^3)}{9x^8+24x^4+16}[/tex]

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{(12x^{7}+16x^3)-(12x^{7}+12x^3)}{9x^8+24x^4+16}[/tex]

Now solving the bracket

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{12x^{7}+16x^3-12x^{7}-12x^3}{9x^8+24x^4+16}[/tex]

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{16x^3-12x^3}{9x^8+24x^4+16}[/tex]

[tex]\frac{d}{dx}[\frac{x^4+1}{3x^4+4}]=\frac{4x^3}{9x^8+24x^4+16}[/tex]

Hence, the differentiate the function with respect to x is [tex]\frac{4x^3}{9x^8+24x^4+16}[/tex] .

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The final solution of the inequality –2(5 – 4x) < 6x – 4 is x < 3.

Inequalities are expressions that have the <, >, ≤ and ≥ signs.

Therefore, let's solve the given inequality as follows:

–2(5 – 4x) < 6x – 4

open the brackets on the left side

–2(5 – 4x) < 6x – 4

-10 + 8x < 6x - 4

add 10 to both sides of the inequality

-10 + 8x < 6x - 4

-10 + 10 + 8x < 6x - 4 + 10

8x < 6x + 6

subtract 6x from both sides of the inequality

8x < 6x + 6

8x - 6x < 6x - 6x + 6

2x < 6

divide both sides of the inequality by 2

2x < 6

2x / 2 < 6 / 2

x < 3

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The coordinate plane that represents both lines as well as the solution of the system is shown in the diagram below.

The solution to a system of equations is the value of x and y that will make the equations of the system true if they are plugged in into either of the two equations in the system.

Thus, when the equations or functions are plotted as a line on a coordinate grid, the system intersect at a point The coordinates of this point is the solution of the system of equations.

The function for Gary, y = 2x - 5, would be a line that has a slope of 2 and cuts the y-axis at -5, while that of Genevieve y = -3x + 4 will have a slope of -3, and will intercepts the y-axis at 4.

The red line in the coordinate grid in the image below is Gary's function y = 2x - 5, while the blue line is that of Genevieve y = -3x + 4. They bot intersect at (1.8, -1.4).

The solution is: (1.8, -1.4).

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The linear equation in point slope form for the graph given below is y = (4/3)x - 1/3  .

In the question ,

a graph with the line is given

also given that the line passes through the points (1 , 1) and (4 , 5) .

in order to find the linear equation , we first need to find the slope .

slope = (5 - 1)/(4 - 1)

slope = 4/3

So , the linear equation of the line passing through the point (4 , 5) with slope 4/3 , in point slope form is

(y - 5) = (4/3)×(x - 4)

y - 5 = (4/3)x - 16/3

y = (4/3)x - 16/3 + 5

y = (4/3)x -80/15 + 75/15

y = (4/3)x -5/15

y = (4/3)x - 1/3

Therefore , The linear equation in point slope form is y = (4/3)x - 1/3  .

The given question is incomplete , the complete question is

Write a linear equation in point - slope form for the graph given below ?

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An equation of the line in point-slope form that contains the point (-2, 5) and has a slope of 1/3 is y - 5 = 1/3(x + 2).

In Mathematics, the point-slope form can be defined as a type of equation that is typically used when the slope of a straight line and one of the points on this line is given (known).

Mathematically, the point-slope form of a straight line is given by this equation:

y - y₁ = m(x - x₁)

Where:

Substituting the given parameters into the formula, we have;

y - y₁ = m(x - x₁)

y - 5 = 1/3(x - (-2))

y - 5 = 1/3(x + 2)

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The differentiation of the given function is 12[tex]x^3[/tex](5[tex]x^2[/tex] +2).

In the given question we have to differentiate the given function with respect to x.

The given function is f(x) = [tex]2x^4(5x^2+3)[/tex]

Before differentiating the function we first simplify the given expression.

To simplify the expression we multiply the 2x^4 to each term under the bracket.

Now the function;

f(x) = [tex]2x^4\cdot5x^2+3\cdot2x^4[/tex]

As we know that if the base is same en power get added.

f(x) = [tex]10x^{4+2}+6x^4[/tex]

f(x) = [tex]10x^{6}+6x^4[/tex]

Now differentiating the function with respect to x.

f'(x) = d/dx([tex]10x^{6}+6x^4[/tex])

f'(x) = 10d/dx([tex]x^6[/tex]) + 6d/dx([tex]x^4[/tex])

As we know that d/dx of x^n =nx^(n-1). So

f'(x) = 10∙6 [tex]x^5[/tex] + 6∙4 [tex]x^3[/tex]

f'(x) = 60[tex]x^5[/tex]+24[tex]x^3[/tex]

Now taking 12x^3 common

f'(x) = 12[tex]x^3[/tex](5[tex]x^2[/tex] +2)

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The radius of the circle is 3 units.

The center of the circle lies on the x-axis.

The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

The standard equation of a circle is:

[tex]x^{2} +y^{2} +2gx+2fy+c=0[/tex]

Given circle equation:

[tex]x^{2} +y^{2} -2x-8=0[/tex]

on comparing

2gx = -2x

g = -2

2fy = 0

f = 0

Center = (2,0)

radius r = [tex]\sqrt{g^{2}+f^{2}-c }[/tex]

= [tex]\sqrt{9}[/tex]
r = 3 units.

Hence the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

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The future value of the account in 3 years is $15,918.

What is a future value?

The worth of a current asset at some point in the future based on an estimated rate of growth is known as future value (FV). For investors and financial planners, the future value is crucial because they use it to predict how much an investment made now will be worth in the future.

Here, we have

FV  =C×[ ((1+i)ⁿ −1)/i]

where:

C = cash flow per period

i = interest rate

n = number of payments

By applying the future value formula, we get

= ​$5,000×[ ((1+0.06)³ −1)/0.06 ]

= $5,000×3.1836

= $15,918

​Hence, the future value of the account in 3 years is $15,918.

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Of the part, the whole, and the percent, an information which is still unknown to Ms. Morgan is the number of non-athletes and it is equal to 84 students.

Of the part, the whole, and the percent, an information which is still unknown to Mr. Gregory is the total number of students and it is equal to 180 students.

Since the Ms. Morgan found that​ 30% of her 120 students are athletes, the known and unknown values include the following:

Known value (whole) = 120 students.

Known value (percent) = 30%

Unknown value (part) = ?

Next, we would determine the unknown value as follows:

Number of athletes = 30/100 × 120

Number of athletes = 30 × 1.2

Number of athletes = 36 students.

Unknown value (non-athletes) = 120 students - 36 students

Unknown value (non-athletes) = 84 students.

Since Mr. Gregory found that​ 27, or​ 15%, of his student are athletes, the known and unknown values include the following:

Known value (part) = 27 students.

Known value (percent) = 15%

Unknown value (whole) = ?

For the unknown value, we have:

Let the variable T represent the total number of students.

15/100 × T = 27

0.15T = 27

Total number of students, T = 27/0.15

Total number of students, T = 180 students.

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Complete Question:

Ms. Morgan is a science teacher. She found that​ 30% of her 120 students are athletes. Mr. Gregory is a math teacher. He found that​ 27, or​ 15%, of his student are athletes. Complete parts a and b below.

a. Of the part, the whole, and the percent, which information is still unknown to Ms. Morgan? Find this unknown value.  

b. Of the part, the whole, and the percent, which information is still unknown to Mr. Gregory? Find this unknown value.​

Answer: Estimate : 1000    Exact : 942.83

Step-by-step explanation:

The estimate is 1000 because 96.8 is rounded to 100 and 9.74 is rounded to 10, so 100 times 10 is 1000. If you wanted the exact, then the exact is 942.83200, If you simplify this total, then the final answer is 942.83.

Answer:

y=3x+2

Step-by-step explanation:

Divide by 5 on all sides.

3x-y=-2

Step-by-step explanation:

The probability that the sample proportion of the Republicans will differ from the population by more than 4% is 95.56%

The probability is calculated using z score, this is used to determine the amount of difference a sample, X is from the standard deviation

The z score is given by the formula

z = (X - μ) / σ

Definition of variables

population mean, μ  = p = 50% = 0.5

sample size, n = 629

standard deviation, σ

= √{(p(1 - p)) / n}

= √{(0.50(1 - 0.50)) / 629}

= √{(0.50 * 0.50) / 629}

= 0.0199

Differ by more than 4% is differ by 50% + 4%

X = 50% + 4% = 54% = 0.54

The probability of more than 54% P = P(p > 54%) = P(p > 0.54)

P(p > 0.54) = 1 - P(p < 0.54)

Z score for z < 0.54

z = (X - μ) / σ

= (0.54 - 0.50) / 0.0199

= 2.01

p value for z < 2.01 = 0.0444

P(p > 0.54) = 1 - P(p < 0.54)

= 1 - 0.0444

= 0.9556

= 95.56%

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Answer:

-----------------------------

According to the remainder theorem, if we subtract 68 from the given trinomial it should be divisible by x - 4. In other words, one of zero's will be x = 4.

Proved that if a quadrilateral is a square then the quadrilateral is also a rectangle.

A quadrilateral is a closed shape created by uniting four points, any three of which cannot be collinear. Four sides, four angles, and four vertices make up a quadrilateral.

Quadrilaterals have a few characteristics. The list is as follows.

Each side has four sides.

Each of them has four vertices.

Two diagonals are present.

360 degrees is the sum of all interior angles.

Given, to prove if a quadrilateral is a square, then the quadrilateral is a rectangle.

The Square has four equal sides, and all four angles are right-angled.

Rectangle has equal opposite sides, and all four angles are the right angle.

Here, the two sides of the square can be increased to form a rectangle.

Hence, proved that if a quadrilateral is a square then the quadrilateral is also a rectangle.

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Answer:

I'm not sure if its right but I got

6/9

Step-by-step explanation:

What I did was:

I took the 2 points:

(-4,4) and (5,-2)

Then did rise over run

Which left me with 6/9

The vertex of the parabola is (-1,-1)

What is vertex of parabola?

The vertex of a parabola is the highest point or the lowest point, also known as the maximum or minimum of the parabola. The vertex is the point of intersection of the parabola and its line of symmetry.

The vertex is the maximum or minimum point of a parabola.

The vertex is the point where the parabola changes direction.

Here, the point where the parabola changes it direction is (-1,-1).

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The amount that should be invested to the nearest ten dollars is $930.

When an amount is compounded daily, it means that the investment increases in value everyday over the investment period.

In order to determine how much should be invested today, the present value of $1890 has to be determined.

The formula for determining the present value is:

PV = FV ÷ (1 + r)^nm

Where:

PV = 1890  ÷ (1.00016)^(12 x 365) = 930

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The number of downloads that were standard version were 560 songs.

The first step is to form a system of equations:

2.9s + 4.5h = 5629 equation 1

s + h = 1450  equation 2

Where:

Multiply equation 2 by 4.5:

4.5s + 4.5h = 6525 equation 3

Subtract equation 3 from equation 1:

896 = 1.6s

Divide both sides of the equation by 1.6:

s = 560

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Answer:

  x = 2

Step-by-step explanation:

You want x such that dividing by 5 and then adding 2 gives the same outcome as adding 22 and then dividing by 10.

  (x/5) +2 . . . . the value after dividing by 5 and adding 2

  (x +22)/10 . . . . the value after adding 22, then dividing by 10

These two outcomes are the same:

  x/5 +2 = (x +22)/10

Multiplying the equation by 10 gives ...

  2x +20 = x +22

  x = 2 . . . . . . . . . . . . subtract x+20

The number is x = 2.

__

Additional comment

The outcome in each case is 2.4.

The probability that the marble picked is not yellow is:

10/12

Given:

we have a bag containing exactly 2 red, 2 yellow, 4 green,

3 white, and 1 blue marble.

we are asked to determine the probability that the marble is not yellow = ?

Favorable outcomes for yellow marbles = 2

Total number of outcomes = 12

Probability, P(E) = Number of outcomes favorable to E/Number of all possible outcomes of the experiment.

P(picking an yellow marble) = 2/12

therefore,

P(not picking a yellow marble) = 1 - 2/12

= 12-2/12

= 10/12

Hence the probability that the marble is not yellow is 10/12

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By using addition, subtraction, multiplication and division of integers,

it can be calculated that

There are 3 groups of 9 and 21 groups of 10

What is addition, subtraction, multiplication and division of integers?

Addition of integers is used to find the sum total of two or more integers.

Subtraction of integers is used find the difference between two integers.

Repeated addition is called multiplication. Multiplication is used to find the product of two or more integers.

Division is the process in which the value of single unit can be calculated from the value of multiple unit

The number to be divided is known as dividend, the number by which the dividend is divided is the divisor, the result obtained is the quotient and the remaining part is the remainder.

There is a well known formula for division

Divisor x Quotient + Remainder = Dividend.

Here, addition, subtraction, multiplication and division of integers will be used

Total number of boys = 121

Total number of girls = 116

Total number of students = 121 + 116 = 237

Now,

The leader first makes groups of nine students each and then assigns any remaining students to make some groups of 10 students

So the group of 9 is to be made in such a way that on multiplying the number by 9, unit digit of 7 is obtained

So the number is 3 as 9 [tex]\times[/tex] 3 = 27

Remaining number of students = 237 - 21 = 210

They will enter into a group of 10

Number of group of 10 = 210[tex]\div[/tex] 10 = 21

There are 3 groups of 9 and 21 groups of 10

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Answer:   $11.50

==================================================

Explanation:

x = cost of one burger

y = cost of one hot dog

3 burgers = 3x dollars

2 hot dogs = 2y dollars

3 burgers + 2 hot dogs = 3x+2y = 8 dollars

The first equation to set up is 3x+2y = 8 which is based on the info of what Ricky purchased.

Through similar logic, the other equation is x+2y = 6 based on what Betty ordered.

This is the system of equations

[tex]\begin{cases}3x+2y = 8\\x+2y = 6\end{cases}[/tex]

There are a few approaches we could take. I'll use substitution.

Let's isolate x in the 2nd equation.

x+2y = 6

x = -2y+6

Then plug this into the 1st equation. Solve for x.

3x + 2y = 8

3(-2y+6) + 2y = 8

-6y+18+2y = 8

-4y+18 = 8

-4y = 8-18

-4y = -10

y = -10/(-4)

y = 2.50 dollars is the cost of one hot dog

Then,

x = -2y+6

x = -2*2.50+6

x = -5+6

x = 1 dollar is the cost of one burger

-----------------------

Check:

The x and y values are confirmed correct.

------------------------

The last thing to find out is how much is spent if Billy bought 4 burgers and 3 hot dogs

4x+3y = 4*1 + 3*2.50 = 4 + 7.50 = 11.50 dollars is the final answer

This assumes that Billy is paying for himself and the other people pay for themselves as well. If Billy is paying for everyone, then the total comes to 11.50+8+6 = 25.50 dollars.

The value of t using straight line angle is 24°.

What is straight line angle ?

  A straight angle in geometry is one that has a length of 180 degrees. Because it seems to be a straight line, it is known as a straight angle. In other words, it is an angle whose sides extend in the same straight line in directions opposite to the vertex.

In mathematics, a straight angle is defined as an angle with a degree value of 180. It looks to be a straight line, hence the name "straight." In essence, two rays linked end to end create an angle according to the mathematical concept of angles. As a result, the two rays of the 180-degree angle are subtended in the direction of their endpoints, which are in the opposite direction.

Here we know that sum of angle in straight line at any point is 180° .

Then,

=> (4t-7)°+83°=180°

=> 4t-7+83 = 180

=> 4t+76 = 180

=> 4t = 180-76 = 104

=> t = 104/4 = 26°

Hence the value of t=24°

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Answer: 2.5>9.5

Step-by-step explanation: maybe we should both pay attention to math

The table that represents this relationship regarding the volume is table D.

From the information, the approximate volume in milliliters, m, for a volume of f fluid ounces is equal to 29.57 times the value of f.

Therefore, when f is 1, the millimeters will be:

= 1 × 29.57

= 29.57

When f is 2, the millimeters will be:

= 2 × 29.57

= 59.14

When f is 3, the millimeters will be:

= 3 ×29.57

= 88.71

When f is 4, the millimeters will be:

= 4 × 29.57

= 118.28

Therefore the correct option is table D.

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The turning point is at x = 5, and this means that the substance changes from liquid to gas when x = 5.

Here we have the temperature function:

T(x) = (x - 5)^3 + 9

The turning point is the point where the curvature of the function changes,  by differentiating the function two times and finding the zero, we can get the turning points.

T'(x) = 3*(x - 5)^2

T''(x) = 3*2*(x - 5) = 6*(x - 5)

This is zero when x = 5, then:

T''(5) = 6*(5 - 5) = 0

So the turning point is there.

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William needs to detail his parents at least 12 times to buy the golf club.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Different types of inequalities are represented by a variety of notations.

The symbols we use in Inequality Expressions are

Not equal ' ≠ '  

Greater than  ' > '

Less than ' < '

Less than or equal  ' ≤ '

Greater than or equal ' ≥'

Here we have

William plans to buy the new Callaway Rogue for at least $ 550

The amount that William saved = $ 325

The amount he earns each time he details his parents = $ 20

Here we don't know how many times he detailed his parents,

So represent it with a variable x

After detailing x times the amount he earned will equal $ 20x

The total amount that William will have = $ 325 + $ 20x

William wants at least $550

Therefore,

The inequality that will represent the above situation is

=> 325 + 20x ≥ 550 ------(1)

Now solve the above inequality to get the value of x

=> 65 + 4x ≥ 110

=> 4x ≥ 45

=> x ≥ 11.25

Here we can say 11.25 times so x = 12 times

Therefore,

William needs to detail his parents at least 12 times to buy the golf club.

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