y varies directly with [tex](x-2)^2[/tex] and [tex]y=\frac{5}{3}[/tex] when x=4. Write the equation for y in terms of x.

The branch of statistics that draws conclusions about a large set of data based on a smaller set of data is often referred to as sampling statistics.

Sampling statistics involve taking a small subset of data from a larger population, and then making inferences about the population based on the sample. For example, if you wanted to determine the average height of people in a given population, you could take a sample of a few hundred people and calculate the average height of the sample. Then, you could use that information to make inferences about the average height of the entire population. The formula for calculating the sample mean, which is the average of the sample, is: X = Σx/n, where x is the sample data, Σx is the sum of the sample data, and n is the sample size. For example, if a sample of five people had heights of 165, 192, 186, 154, and 176 cm, then the sample mean would be (165+192+186+154+176)/5 = 175 cm.

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Answer:

a³ + b³ = (a+b) (a² - ab + b²)

Step-by-step explanation:

(a+b) (a² - ab + b²) =

(a+b)*a² - (a+b)*ab + (a+b)*b² =

(a³+ba²) - (a²b+ab²) + (ab²+b³) =

a³ + ba² - a²b - ab² + ab² + b³ =

a³ + ba² - a²b - ab² + ab² + b³ =

a³ + 0 - 0 + b³ = a³ + b³

Answer:

39.2

Step-by-step explanation:

Always true. The sum of the measures of the interior angles of any triangle is 180 degrees, so when one angle is 120 degrees, the remaining two angles must add to 60 degrees.

The sum of the measures of the interior angles of any triangle is 180 degrees. This is because the angles of a triangle add up to form a straight line, which is 180 degrees. Therefore, if one angle has a measure of 120 degrees, the remaining two angles must add up to the remaining 60 degrees to make 180 degrees in total. This means that the measure of angle 1 must be 60 degrees. Therefore, it is always true that when the measure of angle 4 is 120 degrees, the measure of angle 1 is 60 degrees. This is because the total sum of the interior angles of a triangle must add up to 180 degrees, so the other two angles must add up to the remaining 60 degrees.

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The room tax is 18%. Total cost fo the motel room to stay 3 days including the tax is  $361.08.

A percent represents a fraction of a hundred.

For example:

60% = 60/100

Percent is usually used to denote a portion of a whole. For example, 20% of 30, means:

20% x 30 = 20/100 x 30 = 6

In this problem:

The motel room costs $102.00 per day.

The cost for 3 days before tax is:

Cost = 3 x $102 = $306

The tax rate is 8%, it means the amount of tax is:

tax = 18% x $306 = $55.08

Total cost after tax = $306 + $55.08 = $361.08

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a system of four equations whose solution gives estimates for the temperatures T1; : : : ; T4:

T1 = 1/4(10 + 20 + T2 + T4)

T2 = 1/4(T1 + 20 + 40 + T3)

T3 = 1/4(T1 + T2 + 40 + 30)

T4 = 1/4(10 + T1 + T3 + 30). This is system of equations.

Algebra requires the simultaneous solution of two or more equations. There must be an equal number of equations and unknowns for a system to have a singular solution. The several kinds of linear equation systems are as follows:

Given that,

T1 = 1/4(10 + 20 + T2 + T4)

or, 4T1 - T2 - T4 = 30

similarly for 2nd, 3rd, and 4th nodes we have:

T2 = 1/4(T1 + 20 + 40 + T3)

or -T1 + 4T2 - T3 = 60

T3 = 1/4(T1 + T2 + 40 + 30)

or -T2 + 4T3 - T4 = 70

T4 = 1/4(10 + T1 + T3 + 30)

or -T1 - T3 + 4T4 = 40

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The area of the Circular pool will be 1256 square feet.

The area is the entire amount of space occupied by a flat (2-D) surface or an object's form. On a sheet of paper, draw a square using a pencil. It has two dimensions. The area of a form on paper is the area that it occupies.

Given, A pool that is surrounded by the Square shaped Deck.

Let " 20 feet" be the radius of the pool. that makes the side of the deck will be 20 + 20 = 40 feet

From the formula of the area of a circle:

Area of circle= pi * radius^2

In our case,

Area of the pool = 3.14 * 20 * 20

Area of the pool = 1256 Feet square

Area of the square made by deck = Side * side

Area of the square made by deck =  40 * 40 = 1600

Area of deck = area of Square - the area of the circle

Area of deck = 1600 - 1256

Area of deck = 344 Square feet

therefore, The area of the Circular pool will be 1256 square feet.

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The next term in the given pattern is 36.

A recurring arrangement of numbers, forms, colours, and other elements constitutes a pattern in mathematics. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series.

The number of squares in the given pattern is as follows is 9, 16, 25,....

So, the pattern is 3², 4², 5², 6², 7², 8²,.....

9, 16, 25, 36, 49, 64,....

Hence, the next term in the given pattern is 36.

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The value of the given expression is 12167.

Exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself.

The given expression is (14+√81)³.

= (14+9)³

= (23)³

= 12167

Therefore, the value of the given expression is 12167.

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Answer:

NO

Step-by-step explanation:

In the pythagorean theorem,

[tex]{a}^{2} + {b}^{2} = {c}^{2} [/tex]

So it should be

[tex] {10}^{2} + {24}^{2} = {28}^{2} [/tex]

But,

[tex]100 + 576 = 676 \: and \: {28}^{2} = 784[/tex]

Answer:

28.5

Brainly is probably going to take this down because of the picture but here you go anyway

Step-by-step explanation:

You're welcome.

a) No, a single spray of 10 grams of butter would not be enough to entirely cover the area of 4 slices of toast at 2 meters. The area of 4 slices of toast is larger than the area of 1 slice of toast, meaning that it would require more butter than 10 grams to cover the entire area.

b) A single spray of 10 grams of butter would be enough to cover the area of 8 slices of toast at 3 meters.

c) These changes would affect the amount of butter on each piece of toast in that there would be more butter on each piece with a larger rack and further distance from the "butter gun".

d) At 5 meters from the gun, our up-and-coming chef would be able to spray 16 slices of toast with a single burst of 10 grams of butter. Each piece of toast would also have more butter than it would if it was sprayed at a closer distance.

e) To duplicate the original single-slice toast at 5 meters from the gun, the chef would need to reduce the amount of butter being sprayed. He could do this by simply reducing the distance between the gun and the toast rack.

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Complete question :

A harried chef can't keep up with his breakfast orders and decides to automate the process with a "butter gun." The gun provides a 10 gram burst of butter spray with each hit of the trigger. He starts with a rack that holds one piece of toast 1 meter away. The demand again exceeds supply and he realizes he can use the same gun to produce more toast by placing a larger rack at a greater distance. He finds that a four-slice rack works at a distance of 2 meters. Pleased with profits he carries the idea a step further and puts a holder 3 meters away, maintaining a 10 gram spray. See figure below.a) If 1 spray entirely covers the area of a single slice at 1 meter, can it entirely cover the area of 4 slices at 2 meters? Why or why not?b) How many slices can one spray cover with a 3 meter arragnement?c) How do these changes affect the amount of butter on each piece of toastd) Our up-and-coming chef moves the fun back to 5 meters. Predict how much toast he can spray and how much better each piece gets.e) Explain how he could duplicate the original single-slice toast at 5 meters from the gun.

The quadrilateral ABCD has the vertices B (2, 5) and D (0,-3). If ABCD is a square, the coordinates of the other two vertices are (5, 0) and (-3, 2). So option C is correct

Distance is a numerical measurement of how far apart two or more points are in space, typically in one, two, or three dimensions. It is used to describe the separation between objects, locations, or events.

Given that,

The quadrilateral ABCD,

whose two opposite vertices B (2, 5) and D (0,-3)

other two vertices = ?

This problem can be solved by hit and trial method,

It is known that,

all the sides of square has equal length,

lets take,

A =  (5, 0)

C =  (-3, 2)

Now, by calculating the length of each side, we can check

Distance between A & B

AB = √[(-3-0)²+(2+3)]

     = √34

BC = √34

CD = √34

DA = √34

Hence, A & B are respectively,  (5, 0) and (-3, 2)

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The solution to the system of equations is x = -( 1/3 ) and y = -( 4/3 )

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the system of equations be A and B

Now , the first equation is

y = 4x   be equation (1)

And , the second equation is

y = x - 1   be equation (2)

Substituting the value of y in equation (1) , we get

4x = x - 1

Subtracting x on both sides of the equation , we get

3x = -1

Divide by 3 on both sides of the equation , we get

x = -1/3

Substituting the value of x in equation (1) , we get

y = -4/3

Hence , the equations are solved

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force f as a Cartesian vector: F = (59.40i - 88.18j - 83.18k) lb

A three-dimensional plane may be used to represent the geometric objects in the cartesian plane; this representation is known as the cartesian form. The X, Y, and Z axes are used to depict it in three dimensions. A point, a line, or a plane can also be shown in a cartesian form in addition to this.

The co-ordinates of pints A and B is: (0,0,0) ft

A(-10cos70°sin30°, 10cos70°cos30°, 10sin70°)ft

B(5, -7)ft

so, the position vector from origin to the point A:

r(OA) = A - O

= (-10cos70°sin30°i + 10cos70°cos30°j + 10sin70°k) - (0i + 0j + 0k)

= (-10cos70°sin30°i + 10cos70°cos30°j + 10sin70°k)

the position vector from origin to the point B:

r(OB) = B - O

= (5i - 7j) - (0i + 0j)

= (5i - 7j)

next, the position vector from A to B:

r(AB) = r(OB) - r(OA)

r(AB) = (5i - 7j) - (-10cos70°sin30°i + 10cos70°cos30°j + 10sin70°k)

r(AB) = (6.710i - 9.962j -9.937k) ft

magnitude of the position vector:

|r(AB)| = √(6.710)² + (-9.962)² + (-9.937)²

|r(AB)| = 15.25 ft

the force vector acting from the point A to B:

F = F r(AB) / |r(AB)|

F = 135 × {(6.710i - 9.962j -9.937k)}/  15.25

F = (59.40i - 88.18j - 83.18k) lb

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The complete question is as follows:

This is the equation of the line with a slope of 1/3 and a point of (12, 5).

The slope of a line is a measure of its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope can be positive (the line rises from left to right), negative (the line falls from left to right), or zero (the line is horizontal).

The point-slope formula for a line is given by:

y - y1 = m(x - x1)

where m is the slope of the line, (x1, y1) is a point on the line, and (x, y) is any other point on the line.

In this case, the slope is 1/3 and the point is (12, 5). Substituting these values into the formula:

y - 5 = 1/3(x - 12)

Expanding the right-hand side:

y - 5 = 1/3x - 4

Adding 5 to both sides:

y = 1/3x - 4 + 5

Simplifying:

y = 1/3x + 1

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statistics is the study of statistics is the science of conducting studies to collect, , , analyze, and draw conclusions from data. It involves using mathematical concepts such as probability and regression analysis to analyze data and make predictions.

Statistical methods are used in a wide variety of fields, including economics, medicine, engineering, and social sciences. Statisticians use data to make decisions, test hypotheses, and evaluate the effectiveness of policies and programs. Statistical techniques are also widely used in business, finance, and marketing to help make decisions and understand customer behavior.

1. Statistics is the science of collecting, organizing, analyzing, and interpreting data to draw conclusions.

2. Statisticians use mathematical concepts such as probability and regression analysis to analyze data and make predictions.

3. Statistical methods are used in a variety of fields, including economics, medicine, engineering, and social sciences.

4. Statistical techniques are used in business, finance, and marketing to understand customer behavior and make decisions.

5. Statisticians use data to test hypotheses, evaluate the effectiveness of policies and programs, and make decisions.

Statistics is a science that uses mathematical concepts such as probability and regression analysis to analyze data and make predictions. It is used in a variety of fields, including economics, medicine, engineering, and social sciences, as well as in business, finance, and marketing to understand customer behavior and make decisions. Statistical methods are used to test hypotheses, evaluate the effectiveness of policies and programs, and make decisions.

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The values of x, y and z are 33 degrees, 35 degrees and 109 degrees

From the question, we have the following parametes that can be used in our computation:

The parallelogram

By the alternate interior angle theorem, we have

x = 33 degrees

Also, we have

z = 109 degrees

The sum of angles in a triangle is 180

So, we have

33 + y + 109 = 180

Evaluate

y = 38 degrees

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Answer:

28 m

Step-by-step explanation:

r = l/a

 = 16/4/7

 = 16*7/4

  = 28

The formula r = (arc length / central angle) is used to calculate the radius of a circle. Consequently, r = (16/4/7) = 4.57 m

1. Multiply the central angle (4/7 rad) by the arc length (16 m).

2. Determine the outcome to determine the circle's radius (r): 16 / 4/7 = 4.57 m

The equation r = (arc length / central angle) can be used to calculate the radius of a circle. With the help of this formula, we can determine the radius of a circle given the length of an arc and its central angle. In this instance, the centre angle is 4/7 rad, and the arc length is 16 m. Divide the arc length by the central angle to get the radius: 16 / 4/7 = 4.57 m. The circle's radius is 4.57 m as a result.

Finding other circle measures is made easier by knowing the radius of the circle. For instance, using the formula circumference = 2r, we can determine the circumference of a circle if we know its radius. Any circle's circumference can be determined using this formula given its radius. The circumference in this instance would be 2 x 4.57 m, or 28.7 m. Furthermore, we may utilise the radius to get the circle's area by applying the formula area = r2. Any circle's area can be determined using this formula given its radius. The area in this instance would be x 4.572 = 66.3 m2.

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The total number of two-person tents taken on the camping trip was 0, and the total number of four-person tents taken on the camping trip was 50.

The total number of tents taken on the camping trip was 60, and the total number of people was 202. We want to determine how many of each type of tent were taken. Let x represent the number of two-person tents and y represent the number of four-person tents. We can use the following equation to solve for x and y:

2x + 4y = 202

After rearranging the equation to solve for x, we get x = (202 - 4y)/2. Substituting this into the equation, we get 2((202 - 4y)/2) + 4y = 202. Simplifying, we get (202 - 4y) + 4y = 202. Then, 0 + 4y = 202. Dividing both sides by 4, we get y = 50.5. Plugging this value into the equation x = (202 - 4y)/2, we get x = (202 - 202)/2, which simplifies to x = 0.

Therefore, the total number of two-person tents taken on the camping trip was 0, and the total number of four-person tents taken on the camping trip was 50.

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The x-intercept is x = -7/2

The y-intercept is when y = 7.

The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.

The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.

We have,

The equation of the graph is -2x + y = 7.

Now,

The x-intercept is when y = 0.

The y-intercept is when x = 0.

So,

-2x + y = 7

Substitute x = 0 and y = 0.

-2 x 0 + y = 7

y = 7

And,

-2x + 0 = 7

-2x = 7

x = -7/2

Thus,

x = -7/2 is the x-intercept.

y = 7 is the y-intercept.

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The ending inventory amount is p68,000: (200 x 65) + (300 x 68) + (150 x 70) - (500 x 68) = 68,000.

1. Calculate the total cost of the beginning inventory: 200 units x p65 = p13,000

2. Calculate the total cost of the first purchase: 300 units x p68 = p20,400

3. Calculate the total cost of the second purchase: 150 units x p70 = p10,500

4. Calculate the total cost of the units sold: 500 units x p68 = p34,000

5. Calculate the total cost of the ending inventory amount : (13,000 + 20,400 + 10,500) - 34,000 = p68,000

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Calculating the probability of a mean time watching television falling between 2 and 3 hours requires knowledge of the population's distribution of time spent watching television.

The probability that a random sample of adults will result in a mean time watching television on a weekday of between 2 and 3 hours can be determined by using the standard normal distribution. This requires knowledge of the population's mean and standard deviation for time spent watching television. If these values are not known, they can be estimated using a sample from the population. The calculated probability assumes that the population's distribution of time spent watching television is normal. If this assumption is not met, alternative methods may be used to estimate the probability.

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The double integral of the given equation is -0.983

What is meant by integral?

In mathematics, an integral lends numerical values to functions to represent concepts like volume, area, and displacement that result from combining infinitesimally small amounts of data. Integration is the action of locating integrals.

The integrals of a function in two variables over a region in R², or the real number plane, are referred to as double integrals in mathematics.

The surface area of a 2D figure can be determined primarily using the double integral, which is indicated by the symbol "∫∫". By using double integration, we may quickly determine the area of a rectangular region. Double integration challenges will be easier to address if we are familiar with simple integration.

We have to find the double integral of ∫∫(7x + 6y) dA over region D.

D is bounded by y = x and y = x²

From the graph below, we can see the region is bounded between x = 0 and 1

So the equation becomes

[tex]\int\limits^1_0\int\limits^{x}_{x^2} (7x+6y)dydx[/tex]

The lower limit of y is x² because the lower part of the region is x² and the upper part is x.

Evaluating,

[tex]\int\limits^1_0\int\limits^{x}_{x^2} (7x+6y)dydx = \int\limits^1_0[7xy+6y^2/2]^x_x^2dx= \int\limits^1_0[(7x^3+6x^4/2) - ( 7x^2+6x^2/2]dx[/tex]

                                               = [tex]\int\limits^1_0[(7x^3+6x^4/2) -10x^2]dx[/tex]

              =[tex][(7x^4/4+6x^5/10 -10x^3/3]^1_0 = 7/4 + 6/10-10/3 =[/tex]  

   = -0.983

Therefore the double integral of the given equation is -0.983.

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Note that in the above scenario, the expert in Geometry of the two students is Miracle. All angles across from each other are NOT vertical angles; NOT all adjacent angles form linear pairs.

Vertical angles have two properties: 1) their sides are defined by the same lines; and 2) they share a vertex but not a side. Vertical angles are equivalent, according to the vertical angle theorem.

A vertex is a place on a polygon where the object's sides or edges meet, or where two rays or line segments intersect. Vertices is the plural of vertex.

In the original image, you can see visually that the angles formed by the lines "intersection of the lines" are not equal because they do not share the same vertex.

See the updated image to see the adjusted intersections and show how it is that the angles are now vertical. Note as well that the newly added angles x and y are also vertical and therefore congruent.

So in the new image:

1 ≅ 3

2≅ 4; and

x ≅ y

With regard to the question, if whether adjacent angles always form linear pairs, as indicated above, this statement is false. This is evidenced by ∠2 and ∠3 in the original image which are adjacent but non-linear.

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For a uniform distribution, the probability of any given value is the same across the entire range of the distribution. Therefore, for the given random variable Y, the probability of Y being less than or equal to 0.4 (P(Y <= 0.4)) is 0.4. The probability of Y being less than 0.4 (P(Y < 0.4)) is also 0.4.

For the third probability, P(0.1 < Y <= 0.15 or 0.77 ≤ Y < 0.88), the probability is the sum of the probabilities of the two ranges. The probability of Y being between 0.1 and 0.15 is 0.05, and the probability of Y being between 0.77 and 0.88 is 0.11. Therefore, the total probability is 0.05 + 0.11 = 0.16.

In conclusion, the probabilities of the given random variable Y are P(Y <= 0.4) = 0.4, P(Y < 0.4) = 0.4, and P(0.1 < Y <= 0.15 or 0.77 ≤ Y < 0.88) = 0.16.

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The approximation of √0.96 using the linearization of f(x) = √(1/x) at x = 1 is 0.6836.

To find the linearization of f(x) = √(1/x) at x = 0  use the tangent line of f(x) at x = 0. To do this, we first need to find the derivative of f(x) at x = 0, which is f'(0).

Using the power rule for derivatives, we have:

f(x) = √(1/x)

f'(x) = -1/([tex]2x^{(3/2)[/tex])

Now, we can use f'(0) to find the tangent line at x = 0. The equation for the tangent line is given by:

y = f'(0)(x - 0) + f(0)

Since f(0) is undefined (since the square root of zero is undefined), we can't use the equation above. However, we can still find the slope of the tangent line at x = 0. The slope is given by f'(0), which is:

f'(0) = -1/(2(0)[tex]^{(3/2)[/tex]) = undefined

So, we can conclude that the linearization of f(x) = √(1/x) at x = 0 is undefined.

To approximate √0.96, we can use the linearization of f(x) = √(1/x) at x = 1, since 0.96 is close to 1. The linearization of f(x) = √(1/x) at x = 1 is given by:

y = f'(1)(x - 1) + f(1)

f'(1) = -1/([tex]2(1)^{(3/2)[/tex]) = -1/√2

y = -1/√2(x - 1) + √1/1 = -1/√2x + √2

Now, we can use this linearization to approximate √0.96:

y = -1/√2 * 0.96 + √2 = 0.96 * -1/√2 + √2

y = 0.96 * -1/√2 + √2 = 0.96 * -0.7071 + 1.41

y ≈ 0.6836

So, the approximation of √0.96 using the linearization of f(x) = √(1/x) at x = 1 is 0.6836.

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The distance between the two points is 5.4 meters.

What is coordinate ?

A coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

The distance between two points in a rectangular coordinate system can be calculated using the Pythagorean theorem, which states that the distance between two points is the square root of the sum of the squares of the differences of the x-coordinates and the y-coordinates.

Given the two points (1.8, 3.0) and (4.5, -1.7), the distance can be calculated as follows:

d = √((4.5 - 1.8)^2 + (-1.7 - 3.0)^2) = √(2.7^2 + 4.7^2) = √(7.29 + 22.09) = √29.38 = 5.4

So, the distance between the two points is 5.4 meters.

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The distance between the two points is 5.4 meters.

A coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

The distance between two points in a rectangular coordinate system can be calculated using the Pythagorean theorem, which states that the distance between two points is the square root of the sum of the squares of the differences of the x-coordinates and the y-coordinates.

Given the two points (1.8, 3.0) and (4.5, -1.7), the distance can be calculated as follows:

d = √((4.5 - 1.8)^2 + (-1.7 - 3.0)^2) = √(2.7^2 + 4.7^2) = √(7.29 + 22.09) = √29.38 = 5.4

So, the distance between the two points is 5.4 meters.

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The terms of the following arithmetic series are thus:

1)a₁=6

2)a₂=15

3)a₃=24

4)a₄=33

The following math series is given:

[tex]S_4=[/tex][tex]^4_k_=_1(-3+9k)[/tex]

A list of integers is considered an arithmetic series if there is a consistent difference between any two terms that follow one another in the series.

The total of the terms in an arithmetic sequence with a set number of terms is known as an arithmetic series. Here is a straightforward formula for calculating the sum: in Formula 1 If S denotes the sum of a term-filled arithmetic sequence, then The first and last term values, as well as the total number of terms, are necessary for this formula.

Put k=1 in the given series for a1:

So,

[tex]a_1=-3+9*1\\a_1=6[/tex]

Similarly,

[tex]a_2=-3+9*2\\a_2=15\\a_3=-3+9*3\\a_3=24\\a_4=-3+9*4\\a_4=33[/tex]

The terms of the following arithmetic series are thus:

1)a₁=6

2)a₂=15

3)a₃=24

4)a₄=33

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The entire space or area occupied by an isosceles triangle's sides in a two-dimensional space is known as the triangle's area. Thus the area will be "112.5 cm²".

what is the maximum possible area of an isosceles triangle if the two equal sides are each 15 cm long?

The area is a function of the apex angle β according to

Area = (1/2)(2 cm)²×sin(β)

This will be a maximum for β = 90°. The corresponding length of the third side is  2√2 cm ≈ 2.2828 cm.

The entire space or area occupied by an isosceles triangle's sides in a two-dimensional space is known as the triangle's area. A triangle with two equal sides, which also means two equal angles, is referred to as an isosceles triangle. The following characteristics of an isosceles triangle set it apart from other kinds of triangles: The vertex angle, also known as the apex angle, is the angle formed by the two equal sides of an isosceles triangle. The base is the side that the vertex angle faces, and the base angles are equal.

The vertex angle and the base are both divided in half by the perpendicular from the vertex angle.

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