Help me Please and thank you

According to the information, we know that the term 125x^5y^4 appears in the expansion when n = 6 and the coefficient of x^6y^7 in the expansion is zero, and the x^6y^7 term does not appear in the expansion.

We know that the binomial expansion of (x-y)^n is given by the formula:

where nCk represents the binomial coefficient "n choose k".

We are given that the term 125x^5y^4 appears in the expansion, so we can set up an equation using the binomial coefficient for that term:

Simplifying this equation, we get:

We can solve for n by finding the smallest value of n for which nC5 is greater than or equal to 125. We can use the formula for binomial coefficients:

to calculate the values of nCk for different values of n and k. However, it is easier to use Pascal's triangle to find the binomial coefficients quickly. The fifth row of Pascal's triangle is:

We see that the sixth term of this row is 10C5 = 252, which is greater than 125. Therefore, we know that the term 125x^5y^4 appears in the expansion when n = 6.

To find the x^6y^7 term in the expansion, we need to find the coefficient of x^6y^7 in the expansion. We can use the same formula as before, but with k = 7 instead of k = 5:

We can simplify this expression using the fact that (-1)^7 = -1:

We want to find the value of this expression, so we need to find the value of n. We know that n = 6 because the term 125x^5y^4 appears in the expansion. Therefore, we can substitute n = 6 into the expression:

We see that 6C7 is zero because we cannot choose 7 items from a set of 6. Therefore, the coefficient of x^6y^7 in the expansion is zero, and the x^6y^7 term does not appear in the expansion.

To find the x^7y^2 term in the expansion, we can use the same formula as before, but with k = 2:

We can simplify this expression using the fact that (-1)^2 = 1:

We want to find the value of this expression, so we need to find the value of n. We know that n = 6 because the term 125x^5y^4 appears in the expansion. Therefore, we can substitute n = 6 into the expression:

We can use Pascal's triangle to find that 6C2 = 15. Therefore, the coefficient of x^7y^2 in the expansion is 15, and the x^7y^2 term appears in the expansion as:

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The surface soil of the entire garden within the fencing that needs to be turned prior to planting is approximately 176.625 square feet.

Radius is a mathematical term that refers to the distance from the center of a circle, sphere, or other curved object to any point on its circumference or surface. It is typically denoted by the letter "r".

Let's call the radius of the circular garden "r". We know that the circumference of the garden (the length of the fence) is 47.1 feet, so we can set up the equation:

2πr = 47.1

Simplifying this equation by dividing both sides by 2π, we get:

r = 47.1/(2π)

Using 3.14 as a substitute for π, we can calculate:

r ≈ 7.5 feet

Now that we know the radius of the garden, we can calculate the area of the garden by using the formula for the area of a circle:

A = πr²

A = 3.14 * 7.5²

A ≈ 176.625 square feet

Therefore, the surface soil of the entire garden within the fencing that needs to be turned prior to planting is approximately 176.625 square feet.

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George's kitten is approximately 15.23 yards away from its bed

What is distance?

Distance is a measure of the length or spatial separation between two points. It is a scalar quantity, which means it has magnitude but not direction. In physics, distance is often measured in units such as meters (m), feet (ft), kilometers (km), or miles (mi).

We can use the Law of Cosines to solve this problem. Let's call the distance from the kitten's final position to the bed "d". Then, we have:

d² = 23² + 14² - 2(23)(14)cos(140°)

d² = 529 + 196 + 644cos(140°)

d² = 529 + 196 - 644(0.766)

d² = 529 + 196 - 493.304

d² = 231.696

d = sqrt(231.696)

d ≈ 15.23

Therefore, George's kitten is approximately 15.23 yards away from its bed

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Let [tex]x[/tex] be the unknown side of the triangle. Find the length of this side from the Pythagorean Theorem;

Another root of the equation is also negative, but we've eliminated it because length can't be a negative value.

To find the area of a triangle, we multiply the height of the triangle by the base length and take half.

[tex]\sqrt{363}-3\sqrt{27} ~~ \begin{cases} 363=3\cdot 11\cdot 11\\ \qquad 3\cdot 11^2\\ 27=3\cdot 3\cdot 3\\ \qquad 3\cdot 3^2 \end{cases}\implies \sqrt{3\cdot 11^2}-3\sqrt{3\cdot 3^2} \\\\\\ 11\sqrt{3}-3(3)\sqrt{3}\implies 11\sqrt{3}-9\sqrt{3}\implies 2\sqrt{3}[/tex]

An equation in slope-intercept form that models the situation is y = 1.50x + 4.

The table has been completed below.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

Based on the information provided above, an equation in slope-intercept form that models the situation is given by;

y = mx + c

y = 1.50x + 4

When x = 2, the total cost (y) is given by;

y = 1.50(2) + 4

y = $7.

When x = 4, the total cost (y) is given by;

y = 1.50(4) + 4

y = $10.

When x = 6, the total cost (y) is given by;

y = 1.50(6) + 4

y = $13.

When x = 8, the total cost (y) is given by;

y = 1.50(8) + 4

y = $16.

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Answer:

Trapezoid

Step-by-step explanation:

It's a quadrilateral and it only has one pair of parallel sides.

Answer:

D

Step-by-step explanation:

A rotation and translation preserves the length measures of the sides, so FD is equal to 7 inches

Answer: D

Step-by-step explanation:

The expression (23,000-789) * 19 = = 421409.

An expression can contain variables, constants, and mathematical operations, and it can be used to represent a wide variety of mathematical concepts and relationships.

According to given information:

To solve the expression (23,000-789) * 19, we need to perform the subtraction operation inside the parentheses first and then multiply the result by 19.

(23,000-789) equals 22,211.

So, we have:

(23,000-789) * 19 = 22,211 * 19

Now, we can multiply the two numbers using the standard multiplication method. We multiply each digit of 19 with the digits of 22,211 starting from the right and then add the results.

22,211 x  19 =

444422  (19 x 1 = 19, write 9 and carry 1) + 420789  (19 x 1 = 19, write 9 and carry 1, 19 x 2 = 38, write 8 and carry 3)

= 421409

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Given polygons ABCD and A'B'C'D' have scale factors of 1/4 since their respective side ratios are the same.

The amount by which a shape is made smaller or larger is referred to as its scale factor. This method is employed when a 2D polygon, irregular or regular shape, such as a circle, triangle, square, or rectangle, needs to be enlarged larger. In the equation y = px, p is the scale factor for x. One may identify the scale factor between any two shapes or figures by comparing the measurements of the new shape to those of the prior shape.

Scale factor = Dimensions of new shape/Dimensions of original shape

given the original polygon's coordinates:

A(4,4) ; B(12,4) ; C(12,8) ; D(4,8)

Distance AB = X₂ - X₁

                     = 12 - 4

                     = 8 units

Distance BC = Y₂ - Y₁

                     = 8 - 4

                     = 4 units

Distance CD = X₂ - X₁

                     = 12 - 4

                     = 8 units

Distance DA = Y₂ - Y₁

                     = 8 - 4

                     = 4 units

Added coordinates for the new polygon:

A'(1,1) ; B(3,1) ; C(3,2) ; D(1,2)

Distance A'B' = X₂ - X₁

                      = 3 - 1

                      = 2 units

Distance B'C' = Y₂ - Y₁

                      = 2 - 1

                      = 1 units

Distance C'D' = X₂ - X₁

                      = 3 - 1

                      = 2 units

Distance D'A' = Y₂ - Y₁

                      = 2 - 1

                      = 1 units

Scale factor = [tex]\frac{A^{\prime} B^{\prime}}{A B}=\frac{B^{\prime} C^{\prime}}{B C}=\frac{C^{\prime} D^{\prime}}{C D}=\frac{D^{\prime} A^{\prime}}{D A}[/tex]

[tex]\frac{A B}{A^{\prime} B^{\prime}}=\frac{2}{8}=\frac{1}{4}\\\\\frac{B^{\prime} C^{\prime}}{B C}=\frac{1}{4}\\\\\frac{C^{\prime} D^{\prime}}{C D}=\frac{2}{8}=\frac{1}{4}\\\\\frac{D^{\prime} A^{\prime}}{D A}=\frac{1}{4}[/tex]

Scale factor = [tex]\frac{A^{\prime} B^{\prime}}{A B}=\frac{B^{\prime} C^{\prime}}{B C}=\frac{C^{\prime} D^{\prime}}{C D}=\frac{D^{\prime} A^{\prime}}{D A}=\frac{1}{4}[/tex]

The image was reduced using a one-fourth scale factor.

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Refer to the attachment for the graph.

a) The angular velocity = 0.4π rad/s b) The linear velocity of the hamster = 4π cm/s.

Linear velocity is the rate of change of an object's position with respect to time. It measures how fast an object is moving in a straight line, and is often expressed in units of distance per time, such as meters per second or kilometers per hour.

a) The angular velocity of the wheel can be calculated as follows:

Angular velocity = (2π * number of revolutions) / time

Here, the wheel completes one revolution in 5 seconds, so the number of revolutions is 1 and the time is 5 seconds. Therefore,

Angular velocity = (2π * 1) / 5

Angular velocity = 0.4π rad/s (or approximately 1.26 rad/s)

b) The linear velocity of the hamster can be calculated as follows:

Linear velocity = radius * angular velocity

Here, the radius of the wheel is 10 cm, and the angular velocity we found in part (a) is 0.4π rad/s. Therefore,

Linear velocity = 10 cm * 0.4π rad/s

Linear velocity = 4π cm/s (or approximately 12.57 cm/s)

So the hamster is running at a linear velocity of approximately 12.57 cm/s.

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If a researcher is standing in a corner (point c) of a triangle plot. the angle to point a is S76E .  the distance between point A and point B is 210 ft.

To find the distance between points A and B, we can use the Law of Cosines. Let's first label the angles of the triangle:

   Angle ACB = 180 - (76 + 32) = 72 degrees

   Angle BAC = 76 degrees

   Angle ABC = 32 degrees

Using the Law of Cosines, we have:

AB^2 = AC^2 + BC^2 - 2(AC)(BC)cos(ACB)

where;

AB is the distance between points A and B

AC is the distance between points A and C (210 ft

BC is the distance between points B and C (150 ft)

ACB is the angle between AC and BC (72 degrees)

Plugging in the values, we get:

AB^2 = 210^2 + 150^2 - 2(210)(150)cos(72)

AB^2 = 44100

AB = sqrt(44100)

AB = 210 ft

Therefore, the distance between point A and point B is 210 ft.

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One per 2,500 can be used to represent the rate of 80cm to 2km.

To write 80cm to 2km as a rate, we need to convert the units to the same system. We can convert 80cm to kilometers by dividing by 100,000 (since there are 100,000 centimeters in a kilometer):

80 cm/100,000 = 0.0008 km

Now we can express the rate as:

0.0008 km per 2 km

Or we can simplify it by dividing both the numerator and denominator by 0.0008:

1 per 2,500

Therefore, 80cm to 2km can be expressed as a rate of 1 per 2,500.

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In the given right triangle, by using the Pythagorean theorem, the value of x is 5√2 and the value of y is 5√2

From the question, we are to determine the value of x and y in the given diagram

The diagram shows a right triangle with a hypotenuse of 10.

The right is a special type of right triangle. The base angles of the triangle are equal.

Thus,

The right triangle is an isosceles right triangle.

This means the legs of the triangle are equal.

That is,

x = y

From the Pythagorean theorem, we can write that

x² + y² = 10²

Since x = y, we can write that

x² + x² = 10²

2x² = 100

x² = 50

x = √50

x = 5√2

Hence,

y = 5√2

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In a a case whereby factory that has 6 X 10³ workers makes 2.4 X 105 products each day, the average amount of products made per worker is 4 *10^1 products per worker

The term "average" in mathematics typically refers to a measure of central tendency, which is a way to describe the typical or common value in a set of numbers. There are several types of averages.

We can calculate the required value as

( 2.4 X 10^5 )/ (6 X 10³)

=4 *10^1 products per worker

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The AA criterion for the similarity between two triangles means that if two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.

The steps to prove the AA criterion for similarity using GeoGebra is as follows:

Open GeoGebra and select the "Geometry" option to create a new geometry file.

From the toolbar on the left, select the "Triangle" tool and create two triangles by clicking on three points on the screen.

Use the "Measure" tool to measure the angles in both triangles. To do this, select the "Measure" tool from the toolbar, click on one of the angles in the first triangle, and then click on the corresponding angle in the second triangle.

Record the measurements of the two corresponding angles and calculate their values.

Use the "Move" tool to adjust the position of one of the triangles. Make sure that the two corresponding angles are still the same size.

Observe the triangles to see if they still appear to be similar, even after the movement.

Repeat the process several times, each time moving one of the triangles and observing if they are still similar.

Draw a conclusion based on your observations. Can you prove that the two triangles are similar based on the AA criterion?

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Answer:

225 cm 2

Step-by-step explanation:

Answer:

Surface Area = 6* (5 inches) 2 = 6* (25 square inches)

Step-by-step explanation:

Answer:

  (d)  25 feet and 20 feet​

Step-by-step explanation:

You want the possible dimensions of a pool with an area of 500 square feet and a perimeter of 90 feet.

The area is the product of the length and width. The area of the pools offered in the answer choices are ...

  (a)  15·30 = 450 . . . square feet

  (b)  10·35 = 350 . . . square feet

  (c)  50·10 = 500 . . . square feet

  (d)  25·20 = 500 . . . square feet

The area requirement eliminates answer choices A and B.

The perimeter is twice the sum of length and width. The perimeters of the possible pools are ...

  (c) 2(50 +10) = 120 . . . feet

  (d) 2(25 +20) = 90 . . . feet

The perimeter requirement eliminates answer choice C.

The pool's possible length and width are 25 feet and 20 feet, choice D.

__

Additional comment

You could write a quadratic equation for the pool dimensions, but doing that will generally involve more work than checking the given answer choices.

If x is the width, then 45-x is the length, and the area is ...

  x(45 -x) = 500

  x² -45x +506.25 = -500 +506.25 . . . multiply by -1, complete the square

  x = 22.5 -√6.25 = 20 . . . . take the square root; width is the smaller dimension

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The appropriate measurement unit for force is  kg.m/s²

A force is simply referred to as either a push or pull of an object resulting from the object's interaction with another object.

From Newton's Second Law, force is expressed as;

F = m × a

Where is mass of object and a is the acceleration

One Newton is defined as the force required to accelerate a mass of one kilogram at a rate of one meter per second squared.

Since F = m × a

Plug in m = kg and a = m/s²

F = kg × m/s²

F = kg.m/s²

Therefore, the unit of measurement is kg.m/s².

Option B is the correct answer.

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Answer: Area 6 Perimeter 12

Step-by-step explanation:

First, you have to find the perimeter. 4+5+3=12. Then, you need to find the area, which is base*height/2. The base is 3 and the height is 4. 4*3=12. Then you need to do 12/2=6. The area is 6

Answer:

The answer for P=12cm,A=6cm²

Step-by-step explanation:

Perimeter of triangle=a+b+c

P=5+3+4

P=12cm

Area of triangle =1/2bh

A=1/2×3×4

A=2×3

A=6cm²

The requried quantity of drug that remains in the body, in the long run, is 220 mg.

Let's denote the amount of drug in the body just before taking the nth tablet by [tex]D_n[/tex]. We know that [tex]D_0 = 0[/tex](since the patient hasn't taken any tablets yet) and that 5% of the drug is present in the body just before taking a tablet remains in the body. Therefore, we can write:

[tex]D_1 = 110\\D_2 = 0.95 * D_1 + 110 = 203.5\\D_3 = 0.95 * D_2 + 110 = 292.225\\D_4 = 0.95 * D_3 + 110 = 376.81375[/tex]

We can see that the formula for [tex]D_n[/tex] is:

[tex]D_n = 110 * (1 - 0.05)^n + 110 * (1 - (1 - 0.05)^n) / 0.05[/tex]

Simplifying this formula, we get:

[tex]D_n = 220 * (1 - 0.95^n)[/tex]

Now, we want to find the amount of drug that remains in the body in the long run. This means we want to find the limit of [tex]D_n[/tex]  as n goes to infinity. Using the formula above, we can see that:

[tex]lim_ {n- > oo} D_n = lim_ {n- > oo} 220 * (1 - 0.95^n) = 220[/tex]

Therefore, the quantity of drug that remains in the body, in the long run, is 220 mg.

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Answer:

5.4

Step-by-step explanation:

square of 29 = 5.385164807

5.385164807 estimated is 5.4

The second polygon appears to be the same as the first polygon but shifted 7 units to the right. Therefore, the correct answer is: 7 units to the right.

A closed, two-dimensional shape with straight sides is called a polygon.

Triangles, quadrilaterals, pentagons, hexagons, and other shapes are examples of polygons.

Usually, a polygon's name indicates how many sides it has.

For instance, a quadrilateral has four sides, a triangle has three, and so on.

To create the image, the second polygon A' B' C' D' was moved from its original position to its new position.

Observing the coordinates of the corresponding vertices of the polygons, we see that each vertex of the second polygon was moved 7 units to the right to obtain its new position.

Therefore, the translation used to create the image is 7 units to the right.

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One such tool that enables us to visualise the given math issue using various-sized rectangles or bars is the bar model.

A bar model in mathematics is a visual representation of a subject or problem where the known and unknown quantities are represented by bars or boxes. To answer issues involving the four operations of addition, subtraction, multiplication, and division, bar models are most frequently utilised.

To help kids "see" mathematical structure, the bar model is utilised in teaching for mastery. It is not a method for solving issues, but rather a strategy to uncover the mathematical underpinnings of a problem and obtain knowledge and clarity to aid in its resolution.

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Image attached below,

Answer:

1/8

Step-by-step explanation:

If "over" refers to (4 * 6) as a whole,

[tex]\frac{3}{4*6}\\\\=\frac{3}{24}\\\\=\frac{1}{8}[/tex]

If "over" refers to division,

[tex]3 \div 4 \times 6\\= 0.75 \times 6\\= 4.5[/tex]

I would suppose it's the first one, so, ignore the second one if you haven't learnt BODMAS yet.

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Answer:

20

Step-by-step explanation:

17+8+36+32+9+11+20+18+29+13+16+31

12

=240

12

=20

Finally, the number is divisible by 9 because the sum of its digits is 27, which is a multiple of 9 is 7,983.

In mathematics, a number is a quantity or value used to represent a quantity or a position in a sequence. Numbers are used to quantify things and to perform mathematical operations such as addition, subtraction, multiplication, and division. There are many different types of numbers, including natural numbers (1, 2, 3...), integers (..., -2, -1, 0, 1, 2, ...), rational numbers (numbers that can be expressed as a ratio of two integers), irrational numbers (numbers that cannot be expressed as a ratio of two integers), and real numbers (all rational and irrational numbers). Numbers can also be classified as positive, negative, or zero. Positive numbers are greater than zero, negative numbers are less than zero, and zero represents a neutral or null value. Numbers are used extensively in many fields of study, including science, engineering, economics, and statistics. They are used to quantify measurements, represent data, and make predictions based on patterns and relationships.

Here.

It is odd and between 3,000 and 8,000.

The largest digit in our number system is 9, which is in the one's place.

Two of the digits in the number are the same digit, which is 8 in the thousand's and ten's place.

The hundred's digit and ten's digit are both 8, so the difference is zero.

The digit in the thousand's place is 4 less than the one's place, which is 5.

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Using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.

In the Euclidean plane, a rectangle is a quadrilateral with four right angles.

A parallelogram with a right angle or an equiangular quadrilateral, where equiangular means that all of its angles are equal, are other ways to define it.

A rectangle with four equal-length sides is a square.

Squares are not always rectangles, but rectangles are always squares.

So, the pool and walkway together have a 360 square foot total space.

Assume that the walkway is w feet in width.

Since the pool has an additional w feet of width on both sides, the total area's measurements can be expressed as (8 + 2w) by (10 + 2w).

Thus, we can construct the following equation:

(8 + 2w) x (10 + 2w) = 360

By enlarging and condensing the left side, we obtain:

4w² + 36w - 80 = 0

Using the quadratic formula, we can find w:

w = (-b ± √(b² - 4ac)) / 2a

Here, an equals 4, b equals 36, and c equals -80. When these values are added to the formula, we obtain:

w = (-36 ± √(36² - 4(4)(-80))) / 8

w = (-36 ± √(1872)) / 8

w ≈ -5.6 or w ≈ 3.6

We can ignore the first option because the width cannot be negative. Consequently, the pathway should be around 3.6 feet wide.

Therefore, using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.

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The depreciation expense for each year is 1,000,000 shs.

Accumulated depreciation up to year 6 is 6,000,000 shs.

The depreciation using the straight line method is given by the formula:

Depreciation expense per year = (Initial cost - Salvage value) / Useful life

Initial cost = 5,000,000 + 800,000 + 1,500,000 = 7,300,000 shs.

Salvage value = 1,300,000 shs.

Depreciation expense per year = (7,300,000 - 1,300,000) / 6

                                                    = 6,000,000 / 6

                                                    = 1,000,000 shs.

Thus, the depreciation expense for each year is 1,000,000 shs.

Accumulated depreciation up to year 6 = Depreciation expense per year * Number of years

Accumulated depreciation up to year 6 = 1,000,000 * 6 = 6,000,000 shs.

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