Find the value of x.

Answer:

See below for proof.

[tex]S_{500}=125250[/tex]

Step-by-step explanation:

Given arithmetic series:

Therefore:

[tex]S_n=1+2+3+...+(n-2)+(n-1)+n[/tex]

[tex]S_n=1+(1+1)+(1+2)+...+(1+n-3)+(1+n-2)+(1+n-1)[/tex]

[tex]S_n=1+(1+1)+(1+2(1))+...+(1+(n-3)(1))+(1+(n-2)(1))+(1+(n-1)(1))[/tex]

Let:

Therefore:

[tex]S_n=a+(a+d)+(a+2d)+...+(a+(n-3)d)+(a+(n-2)d)+(a+(n-1)d)[/tex]

Reverse the order:

[tex]S_n=(a+(n-1)d)+(a+(n-2)d)+(a+(n-3)d)+...+(a+2d)+(a+d)+a[/tex]

Add the two expressions for Sₙ:

[tex]2S_n=(2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)[/tex]

Therefore, the term (2a + (n – 1)d) has been repeated n times:

[tex]2S_n=n(2a+(n-1)d)[/tex]

Divide both sides by 2:

[tex]S_n=\dfrac{1}{2}n(2a+(n-1)d)[/tex]

[tex]S_n=\dfrac{1}{2}n(a+a+(n-1)d)[/tex]

Replace  a with  a₁  (first term) and a + (n – 1)d  with  aₙ (last term):

[tex]S_n=\dfrac{1}{2}n(a_1+a_n)[/tex]

To find the sum of the series 1 + 2 + 3 + ... + 500, substitute the following values into the formula:

Therefore:

[tex]\implies S_{500}=\dfrac{1}{2}(500)(1+500)[/tex]

[tex]\implies S_{500}=250(501)[/tex]

[tex]\implies S_{500}=125250[/tex]

Answer:

A = 6

B = 4

C = 8

D = 10

surface area ❀

leftmost rectangle : 8 x 4 = 32

middle rectangle : 10 x 4 = 40

rightmost rectangle : 6 x 4 = 24

top and bottom triangles : (8 x 6) = 48

total = 32 + 40 + 24 + 48 = 144cm²

Step-by-step explanation:

not sure if its correct. but i tried.

The solution is

The measure of A = 6 cm

The measure of B = 4 cm

The measure of C = 8 cm

The measure of D = 10 cm

b) The net surface area of the triangular prism = 144 cm²

What is the surface area of the prism?

The surface area of the prism is calculated as

The formula for the surface area of a prism is SA=2B+ph, where B, is the area of the base, p represents the perimeter of the base, and h stands for the height of the prism

Surface Area of the prism = 2B + ph

The area of the triangular prism is A = ph + ( 1/2 ) bh

Given data ,

Let the measure of the side A = 6 cm

The measure of side B = 4 cm

The measure of side C = 8 cm

The measure of side D = 10 cm

The surface area of the triangular prism is = 2B + ph

The surface area A = ( 6 x 8 ) + ( 6 x 4 ) + ( 10 x 4 ) + ( 8 x 4 )

The surface area A = 48 + 24 + 32 + 40

The surface area A = 144 cm²

Therefore , the value of A is 144 cm²

Hence , the area of the triangular prism is 144 cm²

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The nature of the roots of the quadratic equation x² - 6x + 9 = 0 are real and equal

Given that :

The quadratic equation x² - 6x + 9 = 0

To find :

The nature of the roots of the quadratic equation x² - 6x + 9 = 0

x² - 6x + 9 = 0

Find discriminant of the equation

Comparing the given Quadratic equation x² - 6x + 9 = 0 with the general equation ax² + bx + c = 0 we get

a = 1 , b = - 6 , c = 9

The discriminant of the quadratic equation

= b² - 4ac

= (-6)² - 4* 1*9

=36 - 36

=0

Find nature of the roots of the quadratic equation

Since the discriminant of the quadratic equation = D = 0

Hence the roots of the quadratic equation x² - 6x + 9 = 0 are real and equal

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There are four types of symbols that are used to represent linear inequalities.

What is the definition of inequality symbol?

A quick way to compare distinct quantities is to utilise inequality symbols. "Greater than," "less than," "greater than or equal to," and "less than or equal to" are the four inequality symbols. Thus, the phrase "5 is greater than 2" may be represented as "5>2."

What does inequality simply means?

Inequality is a relationship between two expressions or values that is not equal to each other.

There are four symbols of inequalities:

>: greater than

<: less than

≥: greater than or equal to

≤: less than or equal to

≠: not equal to

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The correct solution of the given function is g'(t)=20 for some t in the interval. The correct option is (B).

Definition of function- Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in sciences.

Given that g(10)= 220, g(0)=20

Explanation: A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

For a given function, y = F(x), if the value of y is increasing on increasing the value of x, then the function is known as an increasing function and if the value of y is decreasing on increasing the value of x, then the function is known as a decreasing function.

So putting values in condition of differentiation for increasing function we get:

=[tex]\frac{g(10)-g(0)}{10-0}[/tex]=[tex]\frac{220-20}{10}[/tex]=[tex]\frac{200}{10}[/tex]=20

The correct solution for given function is g'(t) = 20 for some t in the interval.

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We come to the conclusion that there are three different kinds of parameters that can be used with functions in Python. Default, keyword, and arbitrary arguments are these.

a number that can be any number it is defined to be, but for which no particular value is chosen. Since it may represent any number while also having the value of any number, it is frequently employed in proofs since it allows the proof to be applied to several scenarios.

here

Python parameters are variables that serve as stand-ins for the real values a function needs.

They are supplied as arguments when the function is called.

An illustration would be the function's arguments.

Default, keyword, and arbitrary arguments are these.

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Answer:

(0,2)

Step-by-step explanation:

The equations must be derived from the graph:

First, find the slope of f(x):

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex] let [tex](x_1,y_1)=(-3,3)[/tex] and [tex](x_2,y_2)=(3,1)[/tex]

[tex]m=\frac{(1)-(3)}{(3)-(-3)}\\m=\frac{-2}{6}\\m=-\frac{1}{3}[/tex]

Then use [tex]y-y_1=m(x-x_1)[/tex]

[tex]y-3=-\frac{1}{3}[x-(-3)]\\y-3=-\frac{1}{3}x-1\\y=-\frac{1}{3}x+2\\f(x)=-\frac{1}{3}x+2[/tex]

For g(x), let:

[tex](x_1,y_1)=(-3,0)\\(x_2,y_2)=(0,2)[/tex]

[tex]m=\frac{(2)-(0)}{(0)-(-3)}\\m=\frac{2}{3}[/tex]

[tex]y-0=\frac{2}{3}[x-(-3)]\\y=\frac{2}{3}x+2\\g(x)=\frac{2}{3}x+2[/tex]

Now, to solve for x, allow the two expressions written in terms of x equal each other.

[tex]-\frac{1}{3}x+2=\frac{2}{3}x+2\\(-\frac{1}{3}x+2)+\frac{1}{3}x=(\frac{2}{3}x+2)+\frac{1}{3}x\\2=x+2\\(2)-2=(x+2)-2\\x=0[/tex]

To solve for y, substitute 0 for x in f(x):

[tex]y=-\frac{1}{3}x+2\\y=-\frac{1}{3}(0)+2\\y=0+2\\y=2[/tex]

So, the solution to this system of equations would be the ordered pair (0,2).  This solution is seen on the graph as the intersection of the two lines.

All pairs of an integer array whose sum is equal to a given number in Python using the algorithm based on below three steps:

Step 1: Initialize array and its values

Step 2: Initialize value of sum

Step 3: Call the function find

Programming in Python for all pairs of given number : We will get an array as input from user. We need to find all possible pairs from the given array whose sum is same as given sum. Example:

Array : [5, 2, 0, 4, 1, 8, 7]

Sum = 7

Possible pairs: [5, 2], [0,7],

Algorithm based on below three steps:

Step 1: Initialize array and its values

Step 2: Initialize value of sum

Step 3: Call the function find

Algorithm for function find

Step 1: Iterate on the elements of array with variable i, from 0 to length of array as you want.

Thus, we get the required results.

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The real estate agency made $24,375 on the house. The agent earned $6,093.75 in commission.

Answer:

18 2/3

Step-by-step explanation:

Rigid transformation is a type of geometric transformation in which the shape of an object is preserved. It includes translations, rotations, and reflections, but does not include any scaling, shearing, or any other kind of distortion. In other words, rigid transformations preserve the distances between points in an object.

Four distinct approaches to modify the position and/or shape of a point, line, or geometric figure are collectively referred to as transformations. The Pre-Image of the object is its initial shape, while the Image, after transformation, is the final shape and location of the thing.

A dramatic change in look or form is referred to as a transformation. An important event, such as acquiring your driver's license, starting college, or getting married, may cause your life to alter. A metamorphosis is a significant, abrupt change.

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The domain and range are equal to: D. domain is x is all real numbers where x does not equal 1 comma range is y is all real numbers where y does not equal 7.

In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined. This ultimately implies that, a domain represents the input values (x-values) to a function.

In Mathematics, a range can be defined as the set of all real numbers that connects with the elements of a domain. This ultimately implies that, a range simply refers to the set of all possible output numerical values (real numbers), which are shown on the y-coordinate (y-axis) of a graph.

Translating the word sentence into an algebraic expression (function), we have;

Function = (3x² + x - 4)/(x - 1)

Function = (3x² + 4x - 3x - 4)/(x - 1)

Function = [(3x + 4)(x - 1)]/(x - 1)

Function = 3x + 4

Note: The domain is all real numbers except one (1) because one (1) would make the function to become undefined.

When x = 1, the range of this function are all real numbers except;

Range = 3x + 4

Range = 3(1) + 4

Range = 7

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The graph of a slope of 3 is a  straight line.

        Let us assume the co-ordinates

              [tex](x _1, y_1) = (2,1)\\(x _2, y_2) = (5,10)\\\\\text{We know} ,\\\\m =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\m = \frac{10-1}{5-2} \\[/tex]

              m = 9/ 3

              m = 3 (Refer image for graph)

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Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.

Definition of accuracy-The accuracy formula provides accuracy as a difference of error rate from 100%. To find accuracy we first need to calculate the error rate. And the error rate is the percentage value of the difference of the observed and the actual value, divided by the actual value.

Given: A study of the accuracy of fast food drive-through orders, one restaurant had 37 orders that were not accurate among 333 orders observed. Significance level-0.05

Explanation: Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.

Let a be the population proportion of the orders that were not accurate.

Then according to the claim we have,

[tex]H_{o} :[/tex]  [tex]p[/tex] = [tex]0.10[/tex]

[tex]H{a} :p[/tex] ≠ [tex]0.10[/tex]

Since the alternative hypothesis is two-tailed so the hypothesis test is a  two-tailed test.

For example,

n = 333

The proportion of the orders that were not accurate =  [tex]p{'}[/tex][tex]=\frac{37}{333}[/tex] ≈[tex]0.112[/tex]

Test statistics for population proportion:-

Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.

Let a be the population proportion of the orders that were not accurate.

Then according to the claim we have,

[tex]H_{o} :p=0.10[/tex]

[tex]H_{a} :p[/tex] ≠ [tex]0.10[/tex]

Since the alternative hypothesis is two-tailed so the hypothesis test is a  two-tailed test.

For example,

n = 333

The proportion of the orders that were not accurate =  [tex]p^{'} =\frac{37}{333}[/tex] ≈ [tex]0.112[/tex]

Test statistics for population proportion:-

[tex]z=\frac{p-p'}\sqrt{p(1-p)/\sqrt{n}[/tex][tex]=\frac{0.112-0.10}\sqrt{0.10(0.90)}/ {333}[/tex] ≈ [tex]0.75[/tex]

By using the standard normal distribution table,

The p-value : [tex]2(z > 0.75)=2.55[/tex]

Since the p-value is greater than the significance level (0.05), so we do not reject the null hypothesis.

Hence, we conclude that the accuracy rate appears to be​ acceptable and satisfactory.

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When a 180kg man is standing at 1m distance and a boy is standing 4m away from a fulcrum at opposite ends, the mass of boy so that it is balanced is 45kg.

Therefore the answer is 45kg.

Let the mass of the boy be Wb, the mass of the man is Wm = 180kg. The distance of boy from fulcrum is Xb = 4m and the distance of the man from fulcrum is Xm = 1m.

For balancing, the moment about the fulcrum should be zero. That is,

Wb × g × Xb = Wm × g × Xm

Wb × 4 = 180 × 1

Wb = 180/ 4

Wb = 45

--The question is incomplete without figure, answer to the attached figure--

"An 180-kg man balances the boy teeter-totter as shown: Note: Ignore the weight of the board. What is the approximate mass of the boy?"

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Step-by-step explanation:

12 boxes * 8 candies = 96 total

divided among 30 kids:

each kid gets 3 pieces with (96 - (30 *3)) = 6 remaining.

The number of golf balls that are in each bucket would be = bucket X = 62, bucket y = 34

Division is one of the major arithmetic operation that is used to know the number of groups that can be placed under data sets when shared.

The number of golf balls that are picked up at driving range = 96

The number of buckets present= 2( X and y)

Bucket X = 28 + a

Bucket y = a

To calculate for each bucket first find a.

But 96 = 28+ a+ a

96 = 28 + 2a

2a = 96- 28

2a = 68

a = 68/2 = 34

Bucket X = 28+34 = 62

Bucket y = 34.

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The value of k is ±4 so that the roots of the quadratic equation 9x² + 3kx +4 are equal

A second-degree polynomial equation, which must contain at least one squared component, is what is known as a quadratic. Quadratic equations is another name for them. The quadratic equation's general form is as follows: ax² + bx + c = 0

Quadratic equation,  ax2 + bx + c = 0 has equal roots if (b2 - 4ac) = 0

Hence in quadratic equation, 9x2 + 3kx +4 = 0 has equal roots if

(3k)² - 4×9×4  = 0

Hence 9k² = 144   or  k² = 16   or  k = ±4

In order for the roots of the quadratic equation 9x2 + 3kx + 4 to be equal, the value of k must be 4.

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The 10 percent rule is used to approximate the independence of trials where sampling is taken without replacement.

Why do we check the 10% condition in stats?

It's important to check the 10% condition before calculating probabilities involving x because we want to ensure that the observations in the sample are close to independent.

According to the premise of independence, your data are not related in any manner (at least not in a way that your model hasn't taken into consideration). Actually, there are two assumptions: Since the groups should have diverse compositions, independent observations should be performed between them.

The frequency with which students ask to leave the room during crucial informative periods is reduced thanks to this technique. The 10/10 rule, which prohibits pupils from leaving the classroom during the beginning or last ten minutes of class, is used to achieve this.

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The gradient of y =  3x - 1 is 3.

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all. Any two different points on a line can be used to calculate the slope of any line. The ratio of "vertical change" to "horizontal change" between two different locations on a line is calculated using the slope of a line formula. A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes.

Given

Equation

y = 3x - 1

Using slope intercept form,

y = mx + b

m = slope

Slope = 3

The gradient of y =  3x - 1 is 3.

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Assuming a 10% rate of orders that are incorrect. The accuracy rate seems to be satisfactory.

Let a be the population proportion of the orders that were not accurate.

Then according to the claim we have,

[tex]H_{o} : p=0.10\\H_{a} : p\neq 0.10[/tex]

Since the alternative hypothesis is two-tailed so the hypothesis test is a  two-tailed test.

For example,

n = 333

The proportion of the orders that were not accurate = [tex]p'=\frac{37}{333}[/tex] ≈ [tex]0.112[/tex]

Test statistics for population proportion:-

[tex]z=\frac{p'-p}{\sqrt{\frac{p(1-p)}{n} } }\\= \frac{0.112-0.10}{\sqrt{\frac{0.10(0.90)}{333} } }[/tex] ≈[tex]0.75[/tex]

By using the standard normal distribution table,

The p-value : [tex]2(z > 0.75)= 2.55[/tex]

Since the p-value is greater than the significance level (0.05), so we do not reject the null hypothesis.

Hence, we conclude that the accuracy rate appears to be​ acceptable.

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The nature of the roots of a given quadratic equation √3x²+10x+7√3=0 is real and distinct because of the value of b²-4ac>0.

where a, b, and c are the coefficients in the standard form of the quadratic equation ax²+bx+c=0

Given quadratic equation is  √3x²+10x+7√3=0   on comparing with the standard form of a quadratic equation ax²+bx+c=0. Hence a=√3,b=10,

and c=7√3

Now  b²-4ac=10²-4×√3×7√3=100-28×3=16>0.

here the value of b²-4ac is greater than 0.

We know if the value of b²-4ac then the roots of the given equations are real and distinct.

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On observing the provided question, we can say that - In Korea, 7th grades = 13 or 14 year old, 10th grade is = 16 or 17 year old ,  11th grade is = 17 or 18 and  12th grade is = 18 or 19.

Grading may be done using letters, numbers, percentages, points, or descriptions. The most prevalent grades are in the form of letters, and they usually correspond to a variety of percentages or points. Many people consider "A" and "B" grades to be "good." D and F grades are referred to as "poor grades." Commonly regarded as average is a "C."

We can say that

In Korea,

7th grades =  13 or 14 year old

10th grades = 16 or 17 year old ,

11th grade is = 17 or 18 and  12th grade is = 18 or 19.

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The median of the first 10 natural numbers is 5.5.

What are the mean, median, and example?

The most frequent number, or the one that happens the most frequently, is known as the mode.A data collection is ordered from least to largest, and the median is the midpoint number. A data set's mode is the number that appears the most frequently.

                     Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.

x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10/10

 x = 55/10

 x = 5.5

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The complete question is-

How do you find the mean of first 10 natural numbers ?

Answer:

There are 6 outcomes: 1, 2, 3, 4, 5, or 6 could appear on the top face of the number cube.

Step-by-step explanation:

How many outcomes does a 6 sided die have? 36 For each of 6 outcomes for the first die the second die may have any of 6 outcomes, so the total is 6+6+6+6+6+6=36, or more compactly, 6⋅6=36. How many possible outcomes are there when a six-sided die is rolled and a coin is flipped at the same time? 6 possible outcomes

Answer:

(B) In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.

(C) Vertical angles prove that Angle 3 is congruent to Angle 6.

(D) The triangles are similar because alternate interior angles are congruent.

Step-by-step explanation:

Alternate Interior Angles Theorem

If a line intersects a set of parallel lines in the same plane at two distinct points, the alternate interior angles that are formed are congruent.

As line q has intersected the set of parallel lines c and d, angles 1 and 4 are alternate interior angles, and are therefore congruent.

As line p has intersected the set of parallel lines c and d, angles 2 and 5 are alternate interior angles, and are therefore congruent.

Angle-Angle similarity states that if any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.  Therefore, the triangles are similar because alternate interior angles are congruent.

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

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

The intersection of lines p and q created two opposite vertical angles: angle 3 and angle 6 are opposite vertical angles.  Therefore:

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

Triangle similarity

Two triangles are similar if their corresponding angles are the same size.

As angles 1 and 4 are alternate interior angles and are therefore congruent, they are corresponding angles.  Similarly, as angles 2 and 5 are alternate interior angles and are therefore congruent, they are corresponding angles

Therefore this proves that the two triangles are similar and Angle 1 and Angle 4 are corresponding angles.

After rotating 90 degrees anticlockwise, B's new coordinates are (y, -x).

By switching the x and y coordinates and negating the new x coordinate, we can rotate a point (x, y) 90 degrees clockwise around the origin.

After rotating the triangle 90 degrees anticlockwise around the origin, note the new coordinates of B.

Let (x, y) be point B's coordinates.

Fllowing a 90° anticlockwise turn:

B'x, the modified x coordinate, will now be y.

B'y, the modified y coordinate, will now be -x.

Therefore, B's new coordinates are (y, -x).

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Answer:

1. 38 = 88 - (10 + x + 25)

38 = 88 - 35 - x

x = 53 - 38

x = 15

2. Spanish only = 65 - (5 + 15 + 10)

= 65 - 30

= 35

3. x = 180 - (65 + 25 + 38 + 15 + 5 + 10 + 35)

x = 180 -  193

x = -13

4. 65 + 38 + 35

= 138

Step-by-step explanation:

Answer:

$0.52

Step-by-step explanation:

the pear is 0.90 and the total for the 5 bananas and one pineapple is 3.50

3.50-0.90=2.6 2.6/5=$0.52