4.02 Lesson Check Arithmetic Sequences (9)

The highest degree οf any term in the pοlynοmial is 13, which means that the degree οf the pοlynοmial is 13.

The degree οf a term in a pοlynοmial is the sum οf the expοnents οf the variables in that term.

The degree οf the pοlynοmial w³x²-14x⁹v²w² +5−2v can be fοund by determining the highest degree οf any term in the pοlynοmial.

The degree οf the first term w³x² is 3+2 = 5

The degree οf the secοnd term -14x⁹v²w² is 9+2+2 = 13

The degree οf the third term 5 is 0

The degree οf the fοurth term -2v is 1

Therefοre, the highest degree οf any term in the pοlynοmial is 13, which means that the degree οf the pοlynοmial is 13.

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The probability that the three marbles selected are all different colors is 0.224 or 22.4%.

To find the probability that the three marbles selected are all different colors from a box that contains 4 green marbles, 9 blue marbles, and 6 red marbles, one at a time, without replacement. We can use the probability formula as follows:P(all different colors) = Number of ways to choose 3 marbles of different colors / Total number of ways to choose 3 marbles

The total number of ways to choose 3 marbles from a total of 19 marbles in the box is given by:19C3 = (19 × 18 × 17) / (3 × 2 × 1) = 969The number of ways to choose 3 marbles of different colors from a box containing 4 green marbles, 9 blue marbles, and 6 red marbles, one at a time, without replacement can be obtained as follows:4C1 × 9C1 × 6C1 = (4 × 9 × 6) = 216

Therefore, the probability of choosing 3 marbles of different colors from the box is:P(all different colors) = Number of ways to choose 3 marbles of different colors / Total number of ways to choose 3 marblesP(all different colors) = 216/969P(all different colors) = 24/107 or 0.224 or 22.4%

Therefore, the probability that the three marbles selected are all different colors is 0.224 or 22.4%.

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Answer:

Maybe like 10-15 minutes away

The value of scale factor is 18/35 and the segment PT when ST is parallel to QR is 7.71 cm.

Triangles that are similar to one another in terms of shape but differ in terms of size. In other words, the corresponding angles and sides of two comparable triangles are equal and proportionate.

Because the ratios of the respective sides of comparable triangles are identical, they may be used to solve proportional and scaling issues. As a result, if we are aware of the ratio of two sides in one triangle, we may utilise the similarity of the triangles to determine the ratio of the other triangle.

The scale factor is:

scale factor = PT/QP = (54/7)/15 = 18/35

Using similar triangles, we can set up the following proportion:

PT/PR = SQ/QR

Substituting the given values, we get:

PT/15 = 6/(6+PT)

PT = 54/7 cm

Therefore, the value of scale factor is 18/35 and the segment PT when ST is parallel to QR is 7.71 cm.

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The complete question is:

The set of points that includes all of the solutions for [tex]y = 5/2x + 3/2[/tex] is [tex](x, 5/2x + 3/2)[/tex] for all real numbers.

The set of points that includes all of the solutions for the given equation is [tex](x, 5/2x + 3/2)[/tex] for all real numbers. This is because any point on this line will satisfy the equation [tex]y = 5/2x + 3/2[/tex], which means that any solution to the equation will be included in this set of points.

Alternatively, we can find the equation of the line passing through the points (-1, -1) and (1, 4) and confirm that it is the same as the equation [tex]y = 5/2x + 3/2[/tex].

Using the two-point form of a linear equation, the slope of the line passing through the two points is:

[tex]m = (y2 - y1) / (x2 - x1) = (4 - (-1)) / (1 - (-1)) = 5/2[/tex]

Using the point-slope form of a linear equation with the slope and one of the points:

[tex]y - y1 = m(x - x1)[/tex]

[tex]y - (-1) = (5/2)(x - (-1))[/tex]

[tex]y + 1 = (5/2)(x + 1)[/tex]

[tex]y = (5/2)x + (5/2) - 1[/tex]

[tex]y = (5/2)x + (3/2)[/tex]

This is the same equation as [tex]y = 5/2x + 3/2[/tex], so we know that the set of points that includes all of the solutions is [tex](x, 5/2x + 3/2)[/tex] for all real numbers.

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Answer:

[tex]125cm^{3}[/tex]

Step-by-step explanation:

TSA of a Cuboid=  2LW x 2LH x 2WH
250 = 2 ( L X W X H)
250 / 2 = L X W X H
125 = L X W X H

The volume of a cuboid = L X W X H
125 = L X W X H
Volume = 125cm3

The system of equations solved by elimination has its solution to be (2, 2)

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

-3x + 4y = 2

3x + 6y = 18

When the above equations are added, the variable x is eliminated

Using the above as a guide, we have the following:

4y + 6y = 2 + 18

Evaluate

10y = 20

So, we have

y = 2

Recall that

-3x + 4y = 2

So, we have

-3x + 8 = 2

Evaluate

-3x = -6

Divide

x = 2

Hence, the solution is (2, 2)

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Isaac can fit 27 boxes inside the carton.

To determine how many boxes will fit inside the carton, we need to calculate the volume of each box and the volume of the carton. The volume of each box is 4 x 6 x 2 = 48 cubic inches. The volume of the carton is 12 x 18 x 6 = 1296 cubic inches. To find the number of boxes that will fit, we divide the volume of the carton by the volume of each box: 1296 / 48 = 27. Therefore, 27 boxes can fit inside the carton. It's important to note that this calculation assumes the boxes will be packed in a way that utilizes all available space without any wasted gaps.

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As per the given question, the length of the square shed is 8 meters.

Length of the sheet = 4m

Width of the sheet = 16m

A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. Since a rectangle's opposite sides are both equal and parallel, a parallelogram is another name for this geometric shape.

Using the formula for the area of the rectangle -

Area = Length x Width

Now,

Let the length of the square shed be x, therefore -

x² = 4(16) = 64

x = √64

= 8

Complete Question:

Two storage sheds are to have the same area. One is square and one is rectangular. The rectangular she is 4 meters wide and 16 meters long. What is the length of the square shed?

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Use the formula A = one-half b h to find the area of the base. The area of the base, A, is One-half (8) (14) = 56 meters. The volume of the prism, V is (56) (7) = 392 meters cubed.

A prism is a solid figure having two bases that are the same in size and shape, flat sides, and a continuous cross-section. The two bases of a prism, which might be triangles, rectangles, or any other polygon, determine the name of the prism in question. Prisms having triangle bases, for instance, are referred to as triangular prisms, whereas those with square bases are referred to as square prisms, and so on. Three rectangle lateral sides and two triangular bases make up a triangular prism.

Given that the figure is a right triangular prism with, base 8 meters and 14 m, also the height of the prism is 7 m.

Thus, the correct steps to calculate the volume of the right triangular prism are:

Use the formula A = one-half b h to find the area of the base.

The area of the base, A, is One-half (8) (14) = 56 meters.

The volume of the prism, V is (56) (7) = 392 meters cubed.

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A right triangle at the base of the a spherical structure with such a height of twenty centimeters (cm) has a lateral area of [tex]= 416 cm^2[/tex].

There are 6 vertices, 9 edges, and 5 faces in a triangular prism. It contains 3 rectangle faces as addition to a pair of triangle-shaped faces. It is a polyhedron. The two triangle-shaped bases are parallel to one another.

A three-dimensional form with flat sides is referred to as a prism. It features two identically shaped and sized ends. From one end of the form to the other, it shares a comparable cross-section.

The triangle base's perimeter must be determined. The length of all 3 sides are added to determine the perimeter:

[tex]8 + 15 + 17 = 40[/tex]

The lateral area is,

[tex]2(8 * 20) + 2(15 * 20) + 2(17 * 20)[/tex]

[tex]= 416 cm^2[/tex]

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Answer:

v = 110

Step-by-step explanation:

The big line is a 180 degree line. We need to find the value of the missing side so we take 70 degrees and subtract it from the total angle 180 degrees. That leaves us 110 degrees, so v = 110 degrees.

The volume of the hemisphere with diameter 9 is approximately 76.0435 cubic units.

What is hemisphere ?

A hemisphere is a three-dimensional geometric shape that consists of half of a sphere. It is formed by slicing a sphere into two equal parts along a plane that passes through its center. The term "hemisphere" comes from the Greek words "hemi," meaning half, and "sphaira," meaning sphere.

A hemisphere has a curved surface that is a portion of a sphere's surface, as well as a flat circular base that is formed by the plane of the cut. The diameter of the base is equal to the diameter of the original sphere, and the radius is half of the diameter.

To find the volume of the hemisphere, we first need to know its formula. A hemisphere is half of a sphere, so its volume is half of the volume of a sphere with the same radius.

The formula for the volume of a sphere is:

V = (4/3)πr³

where V is the volume and r is the radius.

Since the hemisphere has diameter 9, its radius is half of the diameter, which is:

r = 9/2 = 4.5

Substituting the value of r in the formula for the volume of a sphere, we get:

V = (4/3)π(4.5)³

V = (4/3)π(91.125)

V = 152.087

Since the hemisphere is half of a sphere with radius 4.5, its volume is half of the volume of that sphere, which is:

V_hemisphere = (1/2)V_sphere

V_hemisphere = (1/2)(4/3)π(4.5)³

V_hemisphere = 76.0435

Therefore, the volume of the hemisphere with diameter 9 is approximately 76.0435 cubic units.

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Answer: 3π

Step-by-step explanation:

360 - 90 = 270°

Perimeter of full circle = πd = 4π

Arc length = [tex]\frac{270}{360}[/tex] × 4π = 3π

These matrices have various dimensions, making it difficult to determine their total. Because of this, the sum of these matrices is either undefinable or unattainable.

Due to their various dimensions, it is difficult to find the sum of these matrices.

One row and three columns make up the second matrix, whereas two rows and one column make up the first matrix. Two matrices can only be added if they share the same dimensions. To be more precise, two matrices can only be added if they have the same number of rows and columns.

The matrices [3 -8] and [4 -5 -6] cannot be added together since they have different numbers of rows and columns. Because of this, the sum of these matrices is either undefinable or unattainable.

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(a) The exploratory factor analysis (EFA) can be useful in two ways: Dimensional Reduction: The dimensionality of a set of data can be reduced by exploratory factor analysis. Rather than dealing with several variables, it identifies the basic factors that describe the data set. This can be achieved by extracting factor information that shows the covariance structure of the variables that underlie the observations. Investigation of latent variables: Exploratory factor analysis can be used to determine whether an underlying dimension, such as depression or anxiety, underlies a collection of observable variables, such as feelings of depression, nervousness, or irritability. Latent variables are those that cannot be observed directly. (b) Factor analysis and principal component analysis are two techniques for data reduction that are frequently confused. The principal component analysis, on the other hand, is a technique that creates uncorrelated variables to replace the original variables. It can be viewed as a special case of factor analysis in which all of the extracted factors are used. It includes the raw data's total variance, which is defined by a few components. Principal component analysis is a method of data reduction that extracts uncorrelated factors from the original variables. The correlation structure of the original variables is utilized in factor analysis to define the underlying factors.

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Using the discriminant of the given equation 5x - 9x² + 6 = 0 which is 241, we can say that the the equation has two real and distinct solutions.

The mentioned equation is 5x - 9x² + 6 = 0. Now we have to find the discriminant of the equation.

Discriminant: ax² + bx + c = 0 is b² - 4ac. The discriminant is used to determine the nature of the roots of the quadratic equation. It can be categorized as the following: If b² - 4ac > 0, the quadratic equation has two real and distinct roots.If b² - 4ac = 0, the quadratic equation has one real and repeated root.If b² - 4ac < 0, the quadratic equation has no real roots or two complex roots.

In the given equation 5x - 9x² + 6 = 0, the coefficients are as follows:

a = -9, b = 5, and c = 6.

Now, we can find the discriminant: Discriminant = b² - 4ac= (5)² - 4(-9)(6)= 25 + 216= 241.

Thus, the discriminant of the given equation is 241. As the discriminant is greater than 0, the quadratic equation has two real and distinct roots.

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The total number of students is calculated by dividing the total number of chess club sign ups by the proportion of students who signed up Therefore, 210/0.6 = 840 students in the school.

15 students out of 25 signed up for chess club.

Therefore, 15/25 = 0.6 of the students signed up for chess club.

Therefore, 0.6 x total number of students = 210 (the number of students who signed up for chess club).

Therefore, 210/0.6

= 840 students in the school.

In one class, 15 out of 25 students signed up for the chess club. This means that 0.6 (or 60%) of the students in the class decided to join. If this rate stays constant across the entire school, then it is possible to calculate the total number of students in the school by using the proportion of students who signed up for the chess club. To calculate the total number of students, we use the equation: 0.6 x total number of students = 210 (the number of students who signed up for chess club). By dividing both sides of the equation by 0.6, we get 210/0.6 = 840 students in the school. Therefore, there are 840 students in the whole school if 210 signed up for chess club.

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Answer:

Step-by-step explanation:

The value of r(s(4)) is -33.

What is a function?

In mathematics, a function is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain) with the property that each input is related to exactly one output.

To find the value of r(s(4)), we first need to find the value of s(4), and then use that as the input for the function r.

To find s(4), we substitute x = 4 into the expression for s(x):

s(4) = 3(4) - 1 = 12 - 1 = 11

Now that we have the value of s(4), we can use it as the input for the function r:

r(s(4)) = r(11) = -3(11) = -33

Therefore, the value of r(s(4)) is -33.

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The volume of the triangular prism is 32.8 m³.

What is volume?

A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder, or a sphere. Different shapes have different volumes.

To find the volume of the triangular prism, we need to use the formula:

Volume = (Area of base) x Height

We are given the surface area of the prism, which includes the area of all five faces. However, we only need the area of the two triangular faces, since the base of the prism is a triangle.

The area of a triangle is given by the formula:

Area = (base x height) / 2

Substituting the given values, we have:

Area of each triangular face = (4.1 m x 2 m) / 2 = 4.1 m²

The total area of the two triangular faces is therefore:

2 x 4.1 m² = 8.2 m²

Now, we can use the formula for the volume of a prism:

Volume = (Area of base) x Height

Substituting the given values, we have:

Volume = (8.2 m²) x 4 m = 32.8 m³

Therefore, the volume of the triangular prism is 32.8 m³.

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The angle of elevation of the sun is approximately 71.56 degrees when a 42-ft mast casts a 15-ft shadow.

We can use trigonometry to determine the angle of elevation of the sun. Let's call the angle we're looking for "theta".

We know that the tangent of theta is equal to the opposite side (the height of the mast) divided by the adjacent side (the length of the shadow):

tan(theta) = height of mast / length of shadow

tan(theta) = 42 / 15

Now we can use a calculator to find the inverse tangent of both sides of the equation:

theta = [tex]tan^{-1}(42/15)[/tex]

theta ≈ 71.56 degrees

So the angle of elevation of the sun is approximately 71.56 degrees.

If you want to verify the solution, you can use the following diagram to better understand the problem and see how the angle of elevation is related to the height of the mast and the length of the shadow:

In this diagram, the angle theta represents the angle of elevation of the sun, the height of the mast is 42 ft, and the length of the shadow is 15 ft. We can see that the tangent of theta is equal to the height of the mast divided by the length of the shadow, which is what we used in our calculation:

tan(theta) = height of mast / length of shadow

tan(theta) = 42 / 15

We can then solve for theta using inverse tangent:

theta = [tex]tan^{-1}(42/15)[/tex]

theta ≈ 71.56 degrees

As a result, the sun is elevated at a about 71.56 degree angle.

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Answer:

C) trapezoid

Step-by-step explanation:

See attached image.  The shape formed from the lines is a trapezoid, for the reasons listed.

The midpoint of the line segment is (-2, 2). The correct answer is F. (-2, 2).

What is the equation of the line?

A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

((x1 + x2)/2, (y1 + y2)/2)

Using this formula, we can find the midpoint of the line segment that has endpoints (-10,-4) and (6,8):

Midpoint = ((-10 + 6)/2, (-4 + 8)/2) = (-2, 2)

Therefore, the midpoint of the line segment is (-2, 2). The correct answer is F. (-2, 2).

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The difference in the average rate of change of Chris' savings and Lily's savings in the period from Month 3 to Month 12 is given as follows:

A. $5.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.

Chris savings increased from 35 to 170 in the period from Month 3 to Month 12, hence the average rate of change is given as follows:

(170 - 35)/9 = $15.

Lily's Savings increased from 30 to 120 in the period from Month 3 to Month 12, hence the average rate of change is given as follows:

(120 - 30)/9 = $10.

Hence the difference is of:

15 - 10 = $5.

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One coterminal angle to 30 degrees is -330 degrees. 150 degrees and 510 degrees are coterminal angles. Similarly, any angle that differs by a multiple of 360 degrees from the given angle is coterminal with it.

Coterminal angles are angles that have the same initial side and the same terminal side. They can be either negative or positive, and they can have any degree measure. Coterminal angles are determined by adding or subtracting 360 degrees (or 2π radians) from the given angle as many times as necessary until the resulting angle falls between 0 and 360 degrees.

Here are the steps to determine coterminal angles: Add or subtract 360 degrees from the given angle. Keep adding or subtracting 360 degrees as many times as necessary to obtain an angle between 0 and 360 degrees. For example, consider the given angle of 150 degrees.150 degrees + 360 degrees = 510 degrees510 degrees - 360 degrees = 150 degreesThe angle 510 degrees is coterminal with 150 degrees. However, it is a very large angle that is difficult to visualize on the coordinate plane.

Therefore, we can find the smallest positive coterminal angle to 150 degrees by subtracting 360 degrees from 510 degrees as many times as necessary until we get an angle between 0 and 360 degrees.510 degrees - 360 degrees = 150 degrees (already between 0 and 360 degrees). Any angle that differs by a multiple of 360 degrees from the given angle is coterminal with it. For example, 870 degrees, 1230 degrees, and -330 degrees are all coterminal with 150 degrees.

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Answer:

9.6 days

Step-by-step explanation:

100 chickens, 12 days

find the days for 10 chickens: 1.2 days (to get 10 you have to divide 100 by ten, so divide 12 by ten too)

times by 8 to get the days for 80 chickens: 9.6 days

Answer: Let's assume that the bag of corn contains "C" units of corn. We know that this amount of corn can feed 100 chickens for 12 days.

To determine how long the same bag of corn would feed 80 chickens, we can use the following proportion:

(Corn units)/(chickens * days) = (Corn units)/(chickens * days)

We can set up two ratios, one for the original situation and one for the new situation, and then cross-multiply to solve for the unknown value:

C/100 * 12 = C/80 * d

where "d" is the number of days that the same bag of corn would feed 80 chickens.

Simplifying the equation, we get:

C/10 = C/80 * d

Multiplying both sides by 80, we get:

8C = Cd

Dividing both sides by "C", we get:

8 = d

Therefore, the same bag of corn would feed 80 chickens for 8 days.

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Answer:

  66

Step-by-step explanation:

You want the remainder from (x^10 +2x^9 +3x^8 +...+10x +11)/(x -1).

The remainder from dividing polynomial f(x) by (x-a) is the value f(a). Here, that means we can find the remainder by evaluating (x^10 +2x^9 +3x^8 +...+10x +11) for x=1. When we do that, we see the value is simply the sum of the coefficients:

  f(1) = (1 + 2 + 3 + ... + 10 + 11) = (11)(11 +1)/2 = 66

The remainder from dividing by (x-1) is 66.

__

Additional comment

The sum of integers 1..n is (n)(n+1)/2.

The volume of the rectangular prism is 105.57

What is the volume of a rectangular prism?

A rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces.

The volume of a rectangular prism=Length X Width X Height

Where l, h and w represent the length, height and width respectively.

Given that;

Length of the prism is 2 times the width L = 2w

the width is 4 times the height  w = 4h

Now let’s plug these values;

the surface area equation, it becomes;

A = 2(2w(w) + w(w/4) + w/4(2w))

A = 2(2w^2 + w^2/4 + w^2/2)

A = 1/2(7w^2)

792 x2 = 7w^2

w = 15.04

The volume of a rectangular prism=Length X Width X Height

V = 2w x w x w/4

V x 2 = w³

V = 15.04³/2

V = 105.57

Thus, the volume is 105.57.

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