at a dance, there are 20 girls and 20 boys. how many ways are there to form 20 pairs consisting of 1 boy and 1 girl?

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

[tex]x-3y=9\implies -3y=-x+9\implies y=\cfrac{-x+9}{-3} \\\\\\ y=\cfrac{-x}{-3}+\cfrac{9}{-3}\implies y=\cfrac{1}{3}x-3\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]

so we're really looking for the equation of a likne whose slope is 1/3 and it passes through (-10 , 9)

[tex](\stackrel{x_1}{-10}~,~\stackrel{y_1}{9})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{9}=\stackrel{m}{ \cfrac{1}{3}}(x-\stackrel{x_1}{(-10)}) \implies y -9= \cfrac{1}{3} (x +10) \\\\\\ y-9=\cfrac{1}{3}x+\cfrac{10}{3}\implies y=\cfrac{1}{3}x+\cfrac{10}{3}+9\implies {\Large \begin{array}{llll} y=\cfrac{1}{3}x+\cfrac{37}{3} \end{array}}[/tex]

The required length of of the third side of isosceles triangle is 9 units.

An Isosceles triangle is a triangle that has two equal sides. Also, the two angles opposite the two equal sides are equal. In other words, we can say that “An isosceles triangle is a triangle which has two congruent sides“.

Let the length of Equal two sides is x.

Third side = 2x - 5

Then, perimeter is 23 cm

x + x + 2x - 5 = 23

4x - 5 = 23

4x = 28

x = 7

Then, third side is 2x - 5

= 2(7) - 5

= 14 - 5

= 9 units

Thus, required length of third side is 9 units.

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Answer: First camera: 5 Pictures per second, Second camera: 6 Pictures per second. Third camera: 7 Pictures per second. The Third camera is the fastest.

Step-by-step explanation: 40/8 = 5, 30/5 = 6, 28/4 = 7

The three (3) true statements about the quadratic function, f(x) = x^2 – 5x + 12 and its graph include the following:

A. The value of f(–10) = 82

B. The graph of the function is a parabola.

D. The graph contains the point (20, –8).

Generally speaking, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. For this quadratic function, the graph is a upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero.

Next, we would determine the other true statements about the graph of this quadratic function:

At data point (-10, 82), we have the following:

Quadratic function, f(x) = 1/5 x² – 5x + 12

Quadratic function, f(x) = x²/5 – 5x + 12

Quadratic function, f(-10) = -10²/5 – 5(-10) + 12

Quadratic function, f(-10) = 82

At data point (20, -8), we have the following:

Quadratic function, f(x) = 1/5 x² – 5x + 12

Quadratic function, f(x) = x²/5 – 5x + 12

Quadratic function, f(20) = 20²/5 – 5(20) + 12

Quadratic function, f(20) = -8

In conclusion, the graph of this quadratic function, f(x) = x^2 – 5x + 12 does not contain the data point (0, 0) as shown in the image attached below.

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The function of u that takes the input u and gives output as v is g(u) = - (8v -2)/ 12.

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Given that the equation that takes the input and output value to give the function is:

-12u + 3 = 8v + 1

Isolate the value of u.

Subtract 3 from both sides of the equation:

-12u + 3 - 3= 8v + 1 -3

-12u = 8v -2

Divide both sides of the equation with -12.

-12u / -12 = 8v - 2/ -12

u = - (8v -2)/ 12

Hence, the function of u that takes the input u and gives output as v is g(u) = - (8v -2)/ 12.

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Answer: To find the number of seconds it takes for the planes to reach the same altitude, we can use the following formula:

time = (difference in altitude) / (rate of plane B - rate of plane A)

In this case, the difference in altitude is the altitude of plane A (2457 feet) and plane B is just taking off so its altitude is 0 feet.

So, substituting these values into the formula:

time = (2457) / (70.75 - 25.25)

time = 2457 / 45.5

time = 53.7 seconds

So it takes 53.7 seconds for the planes to reach the same altitude. To find out what altitude they will be at when they reach the same altitude, we can use the formula:

Altitude = (rate of plane A x time) + altitude of plane A

Substituting the values we have:

Altitude = (25.25 x 53.7) + 2457

Altitude = 1351.6 + 2457

Altitude = 3809.6 ft

So, both planes will be at 3809.6 ft. when they reach the same altitude.

Step-by-step explanation:

AB = BA, the first pair of matrices does not satisfy [tex]AB \neq BA[/tex].

AB = BA in general, even if A and B are both square. We say that A and B commute if AB = BA. We cannot argue that AB = AC provides B = C for a generic matrix A. (However, since A is invertible, we may multiply both sides of the equation AB = AC to the left by A1 to yield B = C.)

For matrices, A and B, the product AB is not necessarily equal to BA. In fact, matrix multiplication is not commutative in general. Therefore, we need to check each pair of matrices to see if their products are equal in order to determine which pair satisfies [tex]$AB \neq BA$[/tex].

[tex]$$\begin{align*}AB &= \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] \\\&= \left[\begin{array}{cc}1 \cdot 7 + 0 \cdot 3 & 1 \cdot 0 + 0 \cdot 4 \ 3 \cdot 7 - 2 \cdot 3 & 3 \cdot 0 - 2 \cdot 4\end{array}\right] \\\&= \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]\end{align*}[/tex]

[tex]$$\begin{align*}BA &= \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \\\&= \left[\begin{array}{cc}7 \cdot 1 + 0 \cdot 3 & 7 \cdot 0 + 0 \cdot (-2) \ 3 \cdot 1 + 4 \cdot 3 & 3 \cdot 0 + 4 \cdot (-2)\end{array}\right] \\\&= \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]\end{align*}[/tex]

Since AB = BA, the first pair of matrices does not satisfy [tex]AB \neq BA[/tex]. We can proceed in the same manner to check the other pairs of matrices:

[tex]$$\begin{align*}\\AB &= \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \left[\begin{array}{cc}8 & 0 \ 11 & -3\end{array}\right] \\\&= \left[\begin{array}{cc}1 \cdot 8 + 0 \cdot 11 & 1 \cdot 0 + 0 \cdot (-3) \ 3 \cdot 8 - 2 \cdot 11 & 3 \cdot 0 - 2 \cdot (-3)\end{array}\right] \\\&= \left[\begin{array}{cc}8 & 0 \ 2 & 6\end{array}\right]\end{align*}[/tex]

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As per the given least-square regression line the residual for the player is 12.6.

The term least square regression line in math is defined as  the line that makes the vertical distance from the data points to the regression line as small as possible.

Here we have the least square regression line as y = 280. 3 + 85. 7x.

As we all know that the residual is given by the difference between the observed point and the predicted point.

While we looking into the given problem the expected score over x hours is written as,

=> f(x) = 280.3 + 85.7x.

Here we need to find the residual for the player, then we have to apply the value of x as 3, then we get,

=> f(3) = 280.3 + 85.7 x 3 = 537.4.

As here we know that the observed score was of 550, hence the residual is calculated as,

=> R = 550 - 537.4 = 12.6.

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First, we should find the slope of the line we're starting with.

3y = 6x + 5 can be put into slope-intercept form by dividing both sides by 3.

 y = 2x + 5/3

The slope of this line is 2.

A parallel line has to have a slope of 2 as well, so we know we're looking for a line with a slope of 2.

Options A and B have that.  Options C and D do not.

Now if (3,3) is a point on the line, then (3,3) must also be a solution for the equation.

Checking Option A:

    3 = 2(3) - 1  is not true.  3 ≠ 6 - 1

Checking Option B:

    3 = 2(3) - 3 is true.  3 = 6 - 3

Option B is the answer, since it has the right slope and works for the point (3,3).

Answer:

B) y = 2x - 3

Step-by-step explanation:

3y = 6x + 5  To put in the slope intercept form.  Divide all the way through by 3

y = 2x + 5/3

When lines are parallel, they have the same slope.

So the slope will be 2.  We will use the point to find the y intercept

m = 2

x = 3   This is from the point (3,3)

y = 3   this is from the point (3,3)

y = mx + b

3 = 2(3) + b

3 = 6 + b   Subtract 6 from both sides

3-6 = 6- 6 + b

-3 = b

Now that we have the slope (2) and the y intercept (-3) we can write the equation.

y = mx + b

y = 2x -3

The pressure on the snow on the boy is P=1379.83 N.

A person weighing 180 pounds stands on snowshoes.

If the total area of the 2 snowshoes is 900 square inches

The mass of person on the snowshoes is given by,

m=180 pounds=180×0.454=81.6455≅81.65 kg

Force acting on a body due to gravity is given by, f = mg

The weight force of the person is given by,

The area of the snowshoes is given by,

⇒A=900 square inches

⇒1inch=0.025 m

Thus, Area in [tex]m^{2}[/tex] given by A=900×[tex]0.0254^{2}[/tex]

=900×0.000645

=0.5805[tex]m^{2}[/tex]

The expression of the pressure in the fluid containing tube is given by FA

; where F is the force exerted by the liquid and A is the area of the tube.

From the expression of pressure, the resulting pressure value is given by FA.

=800.900.5805

=19.82≈19.8pa

The pressure on the snow on the boy is =1379.83 N.

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The length of the hammer is given as follows:

30.36 cm.

The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.

The sides in this problem are given as follows:

9 cm and 29 cm.

The length of the hammer is the hypotenuse, hence it is obtained as follows:

l² = 9² + 29²

[tex]l = \sqrt{9^2 + 29^2}[/tex]

l = 30.36 cm.

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Answer: To find the number of cars that should be produced to incur minimum cost, we need to find the minimum value of the cost function C(x). Since C(x) is a quadratic function, it will have a minimum value, which is either a relative minimum or an absolute minimum. To find the number of cars that should be produced to incur the minimum cost, we will find the derivative of C(x) and set it equal to zero.

To find the derivative of C(x), we can use the power rule, which states that the derivative of x^n is n*x^(n-1).

The derivative of C(x) = 21000 - 90x + 0.1x^2 is

C'(x) = -90 + 0.2x

To find the minimum cost, we set C'(x) equal to zero and solve for x:

-90 + 0.2x = 0

0.2x = 90

x = 450

So, producing 450 cars will incur the minimum cost.

To verify that this is indeed a minimum, we can find the second derivative of C(x) which is 0.2, since the second derivative is positive, we can confirm that x = 450 is a relative minimum.

Alternatively, we can substitute x = 450 into the cost function and find the minimum value, which is C(450) = 21000 - 90(450) + 0.1(450)^2, which is less than any other value of C(x) for x≠450.

Therefore, to incur the minimum cost, 450 cars should be produced.

Step-by-step explanation:

Answer:

x=4.5

Step-by-step explanation:

Isolate the variable by dividing by each side by factors that don't contain the variable.

Answer: The surface area of a square pyramid with a side length of 2m and slant height of 3m is 12 square meters.

Step-by-step explanation:

The surface area of a square pyramid can be calculated using the formula:

Surface area = Base area + 4 * (Area of one of the lateral faces)

The base area of the pyramid is the area of the square base, which is the side length squared: 2m * 2m = 4m^2

To find the area of one of the lateral faces, we can use the Pythagorean theorem to find the length of the face, which is the hypotenuse of a right triangle with the slant height and half of the side length as the other two sides.

so, (1/2 * 2m)^2 + 3^2 = x^2

x= (1/2 * 2m)^2 + 3^2

x = 2

Therefore, the surface area of the pyramid is:

Surface area = 4m^2 + 4 * 2m^2 = 4m^2 + 8m^2 = 12m^2

So, the surface area of a square pyramid with a side length of 2m and a slant height of 3m is 12 square meters.

Answer: SOME1 ANSWER QUICK

Step-by-step explanation:

The price paid by school district for each typewriter after applying 18% quantify discount is equal to  $106.56.

Regular selling price of each typewriter = $129.95

Number of typewriter required by school district  = 50

Total cost of 50 type writers = $129.95 × 50

                                               = $6497.5

Discount percent on whole quantity = 18%

18% of $6497.5

= ( 18 / 100 ) × $6497.5

= $1169.55

Total price paid by school district for 50 typewriters

= $6497.5 - $1169.55

= $5327.95

Price paid by school district for 1 typewriters

= $5327.95 / 50

= $106.559

= $106.56

Therefore, the price paid school district for each typewriter is equal to

$106.56.

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Required equation to represent the translation 6 units left and reflected over x-axis is y = - (x + 6 ).

Let us consider the original equation be  y = x.

Translation to the left by k unit is f(x) = f ( x + k )

Now the equation gets translated 6 units to the left represents as

y = ( x + 6 )

Now the reflected over the x-axis is represented by :

Here y -coordinates change the sign.

If ( x, y ) after reflection over x -axis it is  ( x , -y ).

y = - ( x + 6 )

Equation to represents the reflected over x-axis and translated 6 units left of the original function y = x is written as y = - ( x + 6 ).

Therefore, the equation to represent the required translation and reflection is given by y = - ( x + 6 ).

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3x-y=2 and 4x+9y=36 make up the graph equation in option F. next we receive [tex]x=\frac{2+y}{3}$$[/tex] and [tex]x=\frac{36-9 y}{4}$$[/tex] .

solve for [tex]$x, 3 x-y=2 \quad: \quad x=\frac{2+y}{3}$[/tex]

Steps

3 x-y=2

Turn y to the right.

3 x=2+y

Divide both sides by 3

[tex]\frac{3 x}{3}=\frac{2}{3}+\frac{y}{3}$$[/tex]

Simplify

[tex]x=\frac{2+y}{3}$$[/tex]

solve for x, 4 x+9 y=36:  x=[tex]\frac{36-9 y}{4}[/tex]

Steps

4 x+9 y=36

Turn 9 y to the right.

4 x=36-9 y

By dividing both sides by 4,

[tex]\frac{4 x}{4}=\frac{36}{4}-\frac{9 y}{4}$$[/tex]

Simplify

[tex]x=\frac{36-9 y}{4}$$[/tex]

Therefore the correct answer is option F ) 3x-y=2 , 4x+9y=36 .

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Set of all natural numbers which divide 24 in tabular form {1,2,3,4,6,8,12,24} and set-builder form is  {x ∈ ℕ | x ∣ 24} ℕ denotes the set of natural numbers and x ∣ 24 means x divides 24.

Normative Data We utilize the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 to count or arrange objects in a particular order. Complete numbers are those that fall between the zero and artificial number range. not a decimal or fraction.

Decimal numbers and fractions are not integers. Although all natural whole and numbers are integers (as well as all whole numbers), not all integer are whole or natural numbers. 5 is an integer even though it is not a whole number or a natural number.

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The probability that less than 7 of the 30 students are left-handed is given as follows:

0.9742 = 97.42%.

The mass probability formula, giving the probability of x successes, is of:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are given by:

The parameter values for this problem are given as follows:

n = 30, p = 0.1.

The probability that less than 7 of the 30 students are left-handed is given as follows:

P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6).

Using a binomial distribution calculator with the parameter, finding each of the probabilities and adding them, we have that:

P(X < 7) = 0.9742 = 97.42%.

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Answer:

20

Step-by-step explanation:

A, B, C, D, E is a set of 5 letters.

A letter can only be used once.

You have 5 choices for the first letter.

5

Since the letter that was selected to go first cannot be used again, now you have 4 choices for the second letter.

5 × 4

5 × 4 = 20

Answer: 20

True -  the statement says that a basic variable corresponds to a pivot column in the coefficient matrix. From the definition of basic variables, basic variables correspond to the columns that have a leading 1's (pivot columns).

A mathematical system model based on the use of a linear operator is known as a "linear system" in systems theory. In contrast to nonlinear cases, linear systems often display considerably simpler traits and attributes.

In linear systems, the variables are really only multiplied with constants and then added together; they are never multiplied by each other. In order to express both static and dynamic relationships between variables, linear systems are used.

Something relating to a line is linear. To build a line, all of the linear equations are used. Any equation that doesn't result in a straight line is considered as non. It has a variable slope value and appears as a curve on a graph.

A linear system is one in which both the superposition and homogeneity principles hold true.

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Answer:[tex]-\frac{3}{8} x +7y=-1[/tex]

Step-by-step explanation:

Standard form is ax+by=c, where a,b,c are coefficients.

We are given [tex]7y=\frac{3}{8} x-1[/tex]. All we have to do is move some terms around so that the X is on the left side.

[tex]7y=\frac{3}{8} x-1[/tex]             [subtract both sides by 3/8x]
[tex]-\frac{3}{8} x +7y=-1[/tex]

This gives out final answer of [tex]-\frac{3}{8} x +7y=-1[/tex].

For given linear equation x will be 5.

What is a linear equation ?

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form

Ax + B = 0

e.g. x-10=0. Here, x is a variable, A is a coefficient and B is constant.

The standard form of a linear equation in two variables is of the form

Ax + By = C

e.g. 2x-4y=10. Here, x and y are variables, A and B are coefficients and C is a constant.

Now,

As given linear equation is

15-x=10

x=15-10

x=5

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The required product of the same 500 consecutive integers whose sum is equal to 250 is zero.

Let us consider consecutive integers represented by x, x+ 1, x+ 2 , ..... so on.

Sum of 500 integers = 250

x + x + 1 + x+ 2 + ....+ x + 499 = 250

⇒ 500x + 1 + 2 + ...+499 = 250

⇒ 500x + ( 500 × 499 )/2 = 250

⇒ 500x + 250 × 499 = 250

⇒ 250 ( 2x + 499 ) = 250

⇒ 2x + 499 = 1

⇒x = -249

Required 500 numbers are :

-249 , -248 , -247 , ......, 0, 1, ....... 250

Here zero is also a part of 500 integers.

Product of 499 number with zero is zero.

⇒( -249 ×-248 ×.....× 250 ) × 0 = 0

Therefore, the product of 500 consecutive integers with sum 250 is equal to zero.

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Each of the statements should be completed as follows;

In Mathematics, a horizontal asymptote simply refers to a horizontal line (y = b) where the graph of a function approaches the line as the input values approach negative infinity (-∞) to positive infinity (∞).

In this context, the purple line on the graph of this exponential growth function represents a horizontal asymptote and it has an equation of y = 1.

Additionally, the base value of this exponential growth function f(x) = 3^x + 1 is equal to 1 and its domain includes all real numbers while its range includes all real numbers that are greater than 1 i.e 1 < y < ∞.

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The terminal side of a 130° angle in standard position lies in the second quadrant. Two angles that are co-terminal to 130° are 490° and -230°.

When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.

Given angle is 130°.

The first quadrant, second quadrant, third quadrant, and fourth quadrant make up the whole graph plane.

Angles in the first quadrant vary from 0° to 90°.

Angles in the second quadrant vary from 90° to 180°.

The angles in the third quadrant can be found between 180° and 270°.

Angles in the fourth quadrant can be found between 270° and 360°.

Therefore, the plot's beginning side is at zero degrees and its terminal side is at 130° in the second quadrant, since the specified angle is 130°.

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Answer: 2nd, -230, 490

Step-by-step explanation:

Edge 2023

The process of evaluate the integral by interpreting it in terms of areas is by apply the summation rule on it.

The term called integral is defined as a mathematical object that can be interpreted as an area or a generalization of area.

Here we need to find the way to evaluate the integral by interpreting it in terms of areas.

As we all know that the integration symbol ∫ is an elongated S, suggesting sigma or summation.

Here on a definite integral, we have to use the above and below the summation symbol are the boundaries of the interval, [a, b].

Where the numbers a and b are x-values and are called the limits of integration and it specifically a is the lower limit and b is the upper limit.

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