brainteaser there are 5 people sitting arund a table with different birthdays, what is the probability they sit in the right order both clockwise and counter clockwise

Knowing merely angle-angle-angle (AAA) does not work, because it can result in triangles that are similar but not congruent.

A set of triangles must have equal sides and angles in order for them to be congruent. The sides of the triangles may or may not be equal in the case of a triangle with all of the corresponding angles equal, or the AAA condition. Only if the two triangles are similar would the congruence be true for two triangles with identical respective angles.

So, knowing merely angle-angle-angle (AAA) does not work, because it can result in triangles that are similar but not congruent.

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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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Both Carlos and Irene offered answers equivalent to the original expression.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)

Given expression is

28x − 8.

The prime factors of 28 is 2 and 7.

Rewrite 28 as the product prime factors:

28 = 7 × 2 × 2.

The prime factors of 8 is 2.

Rewrite 8 as the product prime factors:

8 = 2 × 2 × 2.

Replace 28 = 7 × 2 × 2 and 8 = 2 × 2 × 2 in the given expression:

28x − 8

= (7 × 2 × 2) x − (2 × 2 × 2)

Taking common 2

= 2[ (7 × 2)x − (2 × 2)] = 2(14x − 4)

Again taking common 2:

= 2× 2[ 7x − 2]

= 4(7x − 2)

Thus 2(14x − 4) and 4(7x − 2) is the equivalent expression of 28x − 8.

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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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To find the median of a set of numbers, you first need to put the numbers in order from smallest to largest. Once the numbers are ordered, the median is the middle number in the list.

In this case, the ordered list of numbers is: 2, 3, 3, 4, 5, 6, 7.

The median of this list is 4.

Sorting all the data points and selecting the central one are required to determine the middle number (or if there are two middle numbers, taking the mean of those two numbers). Example: The number 4 is in the middle when the numbers are grouped (1, 4, 7), making it the median of the numbers (4, 1 and 7).

By summing up the center values and dividing them by two, you may obtain the mean and, from there, the median.

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

Set is a collection of unique objects. The objects are unique, that is there is no repetition in set, making it distinct from a group. A set is well defined.

In mathematics, the objects belonging to a set are called the elements of that particular set.

These elements will be listed in a pair of curly brackets. For example, A is a set of first ten natural numbers, that is its elements are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. According to the notation

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

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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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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

From the given options ASA is not a test for similarity. Hence option D is the correct answer.

Theorem for similarity of triangles states that we can say two triangles are similar if their corresponding angles are same and the corresponding sides are same. And a result of that their corresponding remaining sides and angles will be also equal.

Hence the possible similarity postulates for two triangles to be similar are AAA(angle angle angle) SAS(side angle side),SSS(side side side)

Therefore the correct option is D. angle side angle (ASA) is the one which does not fall under similarity test.

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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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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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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 word or phrase no which best describes this kind of advertising is inaccurate.

The correct answer option is option B

Mode refers to a statistical data which occurs most in a given set of data.

Advertisement can be defined as a commercial solicitation by a producer to the public in order to buy his product.

Given data:

$7, $8, $8, $15, $15, $16, $17, $18, and $20

Hence, it can be said that the mode of the most popular meals is not $8

Therefore, using $8 as a kind of advertisement for the mode of most popular meals is inaccurate

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The required median of the first ten natural numbers is 5.5.

The median is the number in the middle of a list of numbers in ascending order, from lowest to highest.

The median is the middlemost element if the total number of elements in the list is odd, and the median is the average of the middlemost two values if the total number of elements in the list is even.

The list of the first 10 natural numbers is: 1,2,3,4,5,6,7,8,9,10

Here is the total number of elements in the list is even.

And the two middlemost elements are 5 and 6.

The required median is: 5+6 /2 = 11/2 = 5.5

Hence, the required median of the first ten natural numbers is 5.5.

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Answer:

  (c)  two right triangles, each with both legs 40 inches long

Step-by-step explanation:

You want to know which pair of triangles are congruent, given their descriptions.

There are basically 4 theorems that allow you to claim congruence of triangles:

  AAS, ASA, SAS, SSS

A fifth one applies to right triangles only: HL.

These tell you that you need to have three matching parts: 3 sides, 2 sides and the angle between (except right triangle), or 1 side and 2 angles.

An isosceles triangle with only one or two sides defined is not sufficient.

A right triangle with only the hypotenuse defined is not sufficient.

A right triangle with two legs and the (right) angle between them has a sufficient number of matching parts to be able to claim congruence to another triangle similarly defined.

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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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:

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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Answer:

Step-by-step explanation:

Translations, rotations, and reflections can prove that figures are congruent because these transformations preserve angle measures and distances of the figures. These transformations are also called rigid motions.

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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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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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:

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]

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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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.

Answer:

2.5n ≤ 15

Step-by-step explanation:

Let n = number of times

Each toll is $2.50, so the toll for n trips is n times $2.50, or 2.5n.

The total amount of money available is $15, so 2.5n must be less than or equal to 15.

Inequality:

2.5n ≤ 15

(In case you're asked next, here is the solution of the inequality:

Divide both sides by 2.5.

2.5n/2.5 ≤ 15/2.5

n ≤ 6

A person can drive up to 6 times with $6.)