A motorboat travels kilometers in hours going upstream and kilometers in hours going downstream. What is the rate of the boat in still water and what is the rate of the current

The required center of the circle's equation is (h, k) = (-4, 1), and the radius, r = 7.

The equation of a circle with center (h,k) and radius r is :

r² = (x−h)² + (y−k)²

To convert the equation to standard form, we complete the square for both x and y. To do this, we add and subtract the square of half the coefficient of the x term and the y term, respectively:

x² + 8x + (8/2)² - 2y + 2(8/2)(1/2)y - 2y + (2/2)² = 32

Simplifying, we get:

x² + 8x + 16 - 2y + y + 1 = 32

Now we can combine like terms:

x² + 8x + 16 + y - 2y + 1 = 32

x² + 8x + y + 17 = 32 + 17

x² + 8x + y = 49

Now we have the equation in standard form:

(x + 4)² + (y - 1)² = 49

The center of the equation is (h, k) = (-4, 1), and the radius, r = 7.

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The unit rate for the skydiver falls is 200 feet per second.

The distance that an object travels in relation to the amount of time it takes to do so can be used to define speed. In other words, it is a measurement of an object's motion's speed without direction Velocities are what we get when speed and direction are combined.

Given the graph shows the distance a skydiver falls over time,

and graph titled distance a skydiver falls has time (seconds) on the x-axis, and distance (feet) on the y-axis.

The line passes from (0, 0) and (1.5, 300)

the point (1.5, 300) says that a skydiver covers 300 feet in 1.5 seconds

to find the unit rate we need to calculate the speed,

speed = distance/time

distance covered = 300 feet

time taken = 1.5 seconds

speed = 300/1.5

speed = 200 feet per second

Hence the unit rate is 200 feet per second.

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The unknown side marked with a question mark (?) is of length 5.0 approximately to one decimal place, using trigonometric ratio of sine for the angle 24.3°

The trigonometric ratios involves the relationship of an angle of a right-angled triangle to ratios of two side lengths. Basic trigonometric ratios includes; sine cosine and tangent.

sine of angle = opposite side/ hypotenuse

sin 24.3° = 2.06/?

? = 2.06/sin 24.3°

? = 5.0059

Therefore, by trigonometric ratio of sine for the angle 24.4°, the unknown side labelled with a question mark (?) has length equal to 5 rounded to one decimal place.

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The coefficient of x² is negative 3. Then the value of the constant 'k' will be negative 3.

A polynomial expression is an algebraic expression with variables and coefficients. Unknown variables are what they're termed. We can use addition, subtraction, and other mathematical operations. However, a variable is not divisible.

The coefficient of x² in the expansion of (x + 3)(x + k)(2x - 5) is -3. Then the product of the expressions is given below.

⇒ (x + 3)(x + k)(2x - 5)

⇒ (2x² + 6x - 5x - 15)(x + k)

⇒ (2x² + x - 15)(x + k)

⇒ 2x³ + x² - 15x + 2kx² + kx - 15k

⇒ 2x³ + (1 + 2k)x² + (k - 15)x + kx

Then the coefficient of x² is negative 3. Then the equation is given as,

1 + 2k = - 3

2k = - 4

k = - 2

The coefficient of x² is negative 3. Then the value of the constant 'k' will be negative 3.

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Answer:

Not similar

Not similar

Similar

Step-by-step explanation:

Similar Polygons

If two polygons are similar their:

Pair of right triangles

If the given pair of right triangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 5:3=12:4=13:5[/tex]

[tex]\implies \dfrac{5}{3}= \dfrac{12}{4}= \dfrac{13}{5}[/tex]

[tex]\implies 1.67\neq 3 \neq 2.6[/tex]

Therefore, as the corresponding sides are not in the same ratio, the right triangles are not similar.

Pair of isosceles triangles

If the given pair of isosceles triangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 10:5=10:5=6:6[/tex]

[tex]\implies \dfrac{10}{5}=\dfrac{10}{5}=\dfrac{6}{6}[/tex]

[tex]\implies 2 =2 \neq 1[/tex]

Therefore, as the corresponding sides are not in the same ratio, the isosceles triangles are not similar.

Pair of rectangles

If the given pair of rectangles are similar, their corresponding sides will be in the same ratio:

[tex]\implies 1:2=5:10[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{5}{10}[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{5 \div 5}{10 \div 5}[/tex]

[tex]\implies \dfrac{1}{2}=\dfrac{1}{2}[/tex]

Therefore, as the corresponding sides are in the same ratio, the rectangles are similar.

Each side of the park has a length of 11 units.

In mathematics, a side is a geometrical shape's aspect. Lines (line segments) and points make up the shapes we see (vertices). The vertices of these line segments connect to form a shape. These line segments are referred to as sides.

To calculate the length of each side, we can use the distance formula:  d = √((x2 - x1)2 + (y2 - y1)2). For example, to calculate the length of the side between (26.5, 12), and (38.5, 7), the equation would be d = √((38.5 - 26.5)2 + (12 - 7)2) = √(120 + 52) = √172 = 11.208. All sides have a length of 11 units.

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The amount of carrot that Fluffy the bunny had to eat at night, given the proportion eaten in the morning, is 1 / 10 of a carrot.

After Fluffy the bunny ate 1 / 2 of the carrot in the morning, the amount of the carrot that would be left is 1 / 2 of the carrot or 0. 5 of the carrot.

This means that if 2 / 5 of the carrot is left to eat at the end of the day, the amount of carrot that Fluffy the bunny had to eat at night was:

= 1 / 2 -  2 / 5

= 0. 5 - 0 . 4

= 0 . 1 of the carrot

In fractions, 0. 1 is:

= 0 . 1 / 1

= 1 / 10 of the carrot

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The total distance hiked can be written as the mixed number:

(7 + 8/13) miles.

We know that Jessie hiked all 3 trails, such that:

To find the total distance that Jessie hiked, we just need to add these mixed numbers, we will get:

(2 + 3/4) mi + (1 + 7/8) mi + (1 + 1/3) mi

We can group like terms, there we will get:

(2 + 3/4) mi + (1 + 7/8) mi + (1 + 1/3) mi

(2 + 1 + 1) mi + (3/4 + 7/8 + 1/3) mi

4mi + (13/8 + 1/3) mi

4mi + 47/13 mi

We can rewrite the fraction as:

47/13 = 39/13 + 8/13 = 3 + 8/13

Then the total distance hiked is:

(4 + 3 + 8/13)mi

(7 + 8/13) miles.

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The linear regression equation for the model is Y = ( 0.9629 )X - 2.759 where the slope of the equation is m = 0.9629

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of the line be represented as A

Now , the homework grade is represented as x

The values of x = { 71 , 72 , 62 , 80 , 85 , 74 , 90 , 83 , 62 }

And , the test grade is represented as y

The values of y = { 63 , 75 , 58 , 77 , 71 , 65 , 95 , 68 , 57 }

From the linear regression calculator , the equation of line is given as

y = mx + b where m is the slope and b is the y-intercept

Y = ( 0.9629 )X - 2.759   be equation (1)

where the slope is m = 0.9629

And , Y = Test grade ; X = homework grade

Now ,  for a student with a homework grade of 80

Substitute the value of x as 80 , we get

Y = ( 0.9629 )X - 2.759

when X = 80 , we get

Y = ( 0.9629 ) ( 80 ) - 2.759

Y = 77.032 - 2.759

On simplifying the equation , we get

Y = 74.273

Y = 74

Therefore , the test grade Y of the student is 74

Hence , the linear regression equation is Y = ( 0.9629 )X - 2.759

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The option that is the better deal between both deals for apples, is 8 apples for $9.25.

The better deal on the apples would be the deal that gives the lower amount per apple. This means that the deal that leads to each individual apple having a lower price is the deal that should be considered best.

To find the price of an apple, you would need to divide the price of the deal given, by the number of apples in the deal.

The first deal would therefore have a per apple price of :

= Price of deal / Number of apples

= 6. 25 / 5 apples

= $ 1. 25 per apple

The second deal would be :

= Price of deal / Number of apples

= 9. 25 / 8 apples

= $ 1. 16 per apple

The better deal is therefore 8 apples for $9.25.

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The statements are always true regarding the diagram, as follows:

1.  m∠5 + m∠6 = 180°

2. ∠ 2+ ∠ 3 = ∠ 6

3. m∠2 + m∠3 + m∠5 = 180°

An interior angle is defined as an angle produced on the inside of a polygon by extending the sides of the polygon.

Consider Δ ABC with Exterior angles as

∠ 1 , ∠ 4 , and ∠ 6

For Exterior ∠ 1 we have

∠ 1 = ∠ 5 + ∠ 3

This true due to Exterior angle Property of Triangle

Similarly,

For Exterior ∠ 4 we have

∠ 4 = ∠ 5 + ∠ 2.

This true due to the Exterior angle Property of the Triangle

Similarly,

For Exterior ∠ 6 we have

∠ 6 = ∠ 2 + ∠ 3

In a triangle sum of the measures of angles is equal to 180°

∴ m∠2 + m∠3 + m∠5 = 180°

This is true due to Triangle Sum Property

A straight angle has a measurement of 180 degrees, hence a linear pair of angles must sum up to 180 degrees.

∴ m∠5 + m∠6 = 180°

This true due to the Linear Pair.

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According to the given information in the question order from least to greatest -3.5, -3, -2, -2 1/2, 2.5.

More negative number becomes more small i.e. -3.5 will be smallest in the given question.

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Fraction data type is a numerical data type that consists of two numbers: a numerator and a denominator. It is used to represent a part of a whole or any other ratio.

Fraction data type is a numerical data type used to represent a part of a whole or any other ratio. It consists of two numbers: a numerator and a denominator. The numerator is the number of parts and the denominator is the number of equal parts that make up the whole. For example, if a pizza is cut into 4 equally sized pieces, each piece is 1/4 of the pizza. To represent this fraction in a fraction data type, the numerator would be 1 and the denominator would be 4. Fraction data type is used in mathematics, engineering, and other fields to represent relationships between different parts of a whole. It is also used to simplify calculations, such as addition and subtraction, by representing the same value in a simpler form. By using fraction data type, complex calculations can be simplified and the accuracy of results can be improved.

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The numerator and the denominator are the two numbers that make up the fraction data type, which is a numerical data type. It is employed to symbolize any ratio or a portion of a whole.

The fraction data type is a numerical data type that can be used to represent any ratio or portion of a whole. There are two figures in it: a numerator and a denominator.

The denominator is the total number of equally sized parts, while the numerator is the total number of parts. For instance, if a pizza is divided into four pieces of identical size, each one equals 1/4 of the entire pie.

The numerator and denominator of this fraction should be 1 and 4, respectively, to represent it in a fraction data type. In mathematics, engineering, and other disciplines, the fraction data type is used to indicate relationships between various components of a whole.

Additionally, it is used to make calculations like addition and subtraction simpler by expressing the same value in a more straightforward manner. Complex computations can be made simpler and results can be more accurate by using the fraction data type.

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Yes , it is possible that t for every positive integer n there is a circle in the plane which contains exactly n lattice points in its interior

A lattice point may be defined as the point of intersection of two grid lines or more than two grid lines that is placed in a regularly spaced points arrays. This is called a lattice point.

A formal power series that encodes the coefficients of a sequence of numbers is a generating function.

Here, the coefficients of the sequence {Sₙ} are encoded by the generating function S(x). where Sₙ is the number of lattice paths as described above.

Here we will use the fact that the generating function for a sequence satisfies a functional equation that relates the generating function to the sequence itself to prove that 1 + (x – 1)S(x) + xS(x)² = 0,

Here, the functional equation is

S(x) = 1 + xS(x) + xS(x)²

This equation implies that

The term 1 on the right-hand side corresponds to the fact that to reach the point (0,0) (the starting point), there is exactly one way which is to stay at the starting point.

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The surface area of the triangular prisms are:

1. 153 cm²

2. 87 cm²

3. 12 cm²

4. 123 cm²

5. 118 cm²

To find the surface area of the triangular prism is the sum of all faces of the net of the triangular prism.

Problem 1:  

Area of the rectangle 1 = 9 * 5 = 45 cm²

Area of the rectangle 2 = 9 * 6 = 54 cm²

Area of the rectangle 3 = 9 * 4 = 36 cm²

Area of the two identical triangles = (6 * 3) = 18 cm²

The surface area = 45 + 54 + 36 + 9 = 153 cm²

Problem 2:  

Area of the rectangle 1 = 8 * 3 = 24 cm²

Area of the rectangle 2 = 6 * 3 = 18 cm²

Area of the rectangle 3 = 5 * 3 = 15 cm²

Area of the two identical triangles = 6 * 5 = 30 cm²

The surface area = 24 + 18 + 15 + 30 = 87 cm²

Problem 3:  

Area of the rectangle 1 = 5 * 5 = 25 cm²

Area of the rectangle 2 = 5 * 4 = 20 cm²

Area of the rectangle 3 = 5 * 3 = 15 cm²

Area of the two identical triangles = 4 * 3 = 12 cm²

The surface area = 25 + 20 + 15 + 12 = 72 cm²

Problem 4:  

Area of the rectangle 1 = 5 * 4 = 20 cm²

Area of the rectangle 2 = 7 * 5 = 35 cm²

Area of the rectangle 3 = 8 * 5 = 40 cm²

Area of the two identical triangles = 7 * 4 = 28 cm²

The surface area = 20 + 35 + 40 + 28 = 123 cm²

Problem 5:  

Area of the rectangle 1 = 7 * 5 = 35 cm²

Area of the rectangle 2 = 7 * 4 = 28 cm²

Area of the rectangle 3 = 7 * 5 = 35 cm²

Area of the two identical triangles = 4 * 5 = 20 cm²

The surface area = 35 + 28 + 35 + 20 = 118 cm²

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The required solution to the given equation is x = 106.04, which is the correct answer that would be option (A).

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

The equation is given in the question following:

x - 69.80 = 36.24

We have to determine the solution to the given equation

As per the question, we have

x - 69.80 = 36.24

Add 69.80 on both sides of the equation, and we get

x - 69.80 + 69.80 = 36.24 + 69.80

x = 106.04

Hence, the correct answer would be an option (A).

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Answer:$48.80

Step-by-step explanation:

An angle is a measurement of how much of a "turn" there is between two lines or line segments. Consider two sticks meeting at a point. The place where they intersect is known as the vertex, and the lines or line segments are known as the angle's sides. When we turn one of the sticks so that it is no longer directed against the other, we have established an angle.

For kids, think of it as the space between two rays with the same terminus. It expresses how "open" or "closed" the space between the two rays is. The amount of rotation around the vertex can be measured in degrees. From 0 to 180, or 0 to 360. A complete rotation is 360 degrees or 2 radians.

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The compound inequality for the weight of the ball is represented as 14 > w< 16.

When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.

Given that the ball used in a soccer game may not weigh more than 16 ounces or less than 14 ounces at the start of the match. After 1.5 ounces of air was added to a ball, the ball was approved for use in a game.

The compound inequality will be written as,

14 > w< 16.

Here the weight of the ball varies between 14 ounces to 16 ounces means that the minimum weight should be 14 ounces and the maximum is 16 ounces.

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The Laplace transform of the given integral equation is:

[tex]F(s) = L{f(t)} = ∫(0 to ∞) e^(-st)f(t) dt\\f(t) = L^-1{F(s)} = ∫(0 to ∞) e^(st)F(s) ds[/tex]

Given the equation

[tex]f(t) = 4t + 16sin(t) + ∫(0 to t) e^(t-u)f(u) du[/tex],

we can take the Laplace transform of both sides:

[tex]F(s) = L{f(t)} = L{4t + 16sin(t)} + L{∫(0 to t) e^(t-u)f(u) du}[/tex]

Using the properties of the Laplace transform, we have:

[tex]L{4t} = 4/s^2\\L{16sin(t)} = 16/s^2 + 16s\\L{∫(0 to t) e^(t-u)f(u) du} = F(s)L{e^(t-u)} \\= F(s)e^(-u)(1/s-1)^u|(0,t) \\= F(s)(e^(-t)/s - 1)[/tex]

By substituting these results into the equation, we get:

[tex]F(s) = (4/s^2 + 16/s^2 + 16s) + (F(s)(e^(-t)/s - 1))[/tex]

Solving for F(s) we get:

[tex]F(s) = (4/s^2 + 16/s^2 + 16s)/(1 - e^(-t)/s)[/tex]

To find f(t) we need to take the inverse Laplace transform of F(s)

[tex]f(t) = L^-1{F(s)} = L^-1{(4/s^2 + 16/s^2 + 16s)/(1 - e^(-t)/s)}[/tex]

This is the solution of the integral equation in terms of the Laplace transform.

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On solving the equations x+2y = 14 and -x+3y = 26 , by equal value method , we get that the value of y is  8 .

The System Of Equations is given as : x+2y = 14 and -x+3y=26 ;

In the Equal value method , we equate the value of any one variable and simplify further .

in the first equation x+2y = 14 ; the value of x can be written as :

[tex]x = 14 - 2y[/tex] ; ....equation(1) ;

the second equation , [tex]-x+3y=26[/tex] , can be expressed as :  

[tex]x=3y-26[/tex] ;  ....equation(2) ;

equating both equation(1) and equation(2) ,

we get ;

⇒ 14 - 2y = 3y - 26 ;

⇒ 3y + 2y = 26 + 14 ;

⇒ 5y = 40 ;

⇒ y = 8 ;

Therefore , the value of y is = 8 .

The given question is incomplete , the complete question is

How do you solve for y in the system of equations x+2y = 14 and -x+3y=26  by  equal value method ?

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By solving the given expression , The resultant could be x >= 48 or

x >= -24 .

What is Algebraic expression ?

Algebraic expression can be defined as the combination of constants , variables and values .

Given expression,

| 1/4 x  - 3 | - 5 >= 4

shifting 5 to RHS

we get,

| 1/4 x  - 3 | >= 9

case 1 :

1/4 x  - 3 >= +9

1/4 x >= 12

x >= 48

Case 2 :

1/4 x  - 3 >= -9

1/4 x >= -9+3

1/4 x >= -6

x >= -24

Hence , by solving the given expression , we get x >= 48 or x >= -24 .

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The probability:  P(A, A) if two cards are randomly selected with replacement is one ninth. Option B

You should be aware that probability is the chance of occurance of an event.  the probability of an event is written thus

P(E) = Number of required outcome divided by the total number of possible outcomes

The possible outcomes are the spelling the word ADD,

The probabilities are 1/3, 1/3, 1/3 respectively.

So, P(A, A) if two cards are randomly selected with replacement will be

P(A, A) = 1/3 * 1/3

Therefore the probability of the event is 1/9

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Answer: one ninth i took the test

Step-by-step explanation:

Sixteen teachers ordered a book to be used in their classrooms. The book costs $350, which is to be shared equally. The most appropriate representation of how much each teacher will pay is $350 / 16 = $ 21.88.

Since there are a total of sixteen teachers who placed an order for a book and the cost of the book is $ 350.

Thus, in order to determine the amount to be paid by each teacher equally is calculated by dividing the total cost of the book by the number of teachers, as follows

=  $ 350 / 16

=  $  21.875

After rounding off, it is

=   $ 21.88

Thus, each teacher will pay $ 21.88 for the book.

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Answer:

[tex]5\sqrt{2}+\sqrt{10}[/tex]

Step-by-step explanation:

[tex]\sqrt{5}(\sqrt{10}+\sqrt{2}) \\ \\ =\sqrt{50}+\sqrt{10} \\ \\ =5\sqrt{2}+\sqrt{10}[/tex]

Answer:

108 yards for 27 students

Step-by-step explanation:

12/3

Step 1: to figure out how many yards it takes to make one costume you have to divide 12 and 3.

12/3= 4

It takes 4 yards to make one costume.

Step 2: to figure out the amount of yards used for 27 students you take 4 yards times 27 students.

4x27= 108

b) A bar graph is a horizontal or vertical representation of the frequency or relative frequency of the categories. The height of each rectangle represents the frequency category's frequency or relative frequency.

A C. A Pareto chart is a bar graph whose bars are drawn in decreasing order of frequency or relative frequency t of each rectangle represents the category's frequency or relative frequency

A bar graph is used to represent data that is categorical or numerical. It is typically used to compare different categories of data, such as sales figures for different products or the number of students in different grades in a school.

The data is typically represented in a horizontal or vertical bar chart, with the length of each bar representing the quantity of the data. The categories are usually arranged from highest to lowest in a bar graph.

A Pareto chart is a specific type of bar graph. It is used to show the distribution of data into different categories and the relative importance or frequency of each category.

The categories are typically arranged in order of importance, with the most important or frequent category first and the least important or frequent category last. The bars in a Pareto chart are usually drawn in descending order, with the highest category on the left.

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Complete question:

What is a bar graph? What is a Pareto chart? What is a bar graph? O A. A bar graph is a circular graph that uses bars to divide it into segments corresponding to each category The segments are proportional to the frequency of its category O B. A bar graph is a horizontal or vertical representation of the frequency or relative trequency of the categories. The height of each rectangle represents the of the frequency category's frequency or relative frequency o c. A bar graph represents the frequencies of values either from left to-right or right-to-left. The different segments of the bar are used to indicate the different frequencies of each category What is a Pareto chart O A. A Pareto chart is a combination of a pie chart and a bar graph O B. A Pareto chart is a display of two data sets side ty side where the heigh O C. A Pareto chart is a bar graph whose bars are drawn in decreasing order of trequency or relative trequency t of each rectangle represents the category's frequency or relative frequency

The amount of wood needed to rebuild the frame in meters is 24.58 meters.

The whole number of boards supplied is 13

Given the dimension of the long jump pit as shown in the image attached.

In order to determine the amount of wood needed to rebuild the frame in meters, you need to calculate the perimeter of the long jump pit.

The perimeter of a rectangle is given by:

P = 2(L + W)

where L is the length and W is the width

From the image: L = 9.54 meters and W = 2.75 meters

P = 2(9.54+2.75)

P = 2(12.29)

P = 24.58 meters

Thus, the amount of wood needed to rebuild the frame in meters is 24.58 meters

How many boards did the volunteers supply round your calculations to the nearest hundredth:

No. of boards = 24.58 / 1.8287 = 13.44

Thus, the whole number of boards supplied is 13

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Complete Question

The figure in the attached image makes the question complete

Willow's equation is written using the greatest common factor and the distributive property as follows

= 9(4 + 3)

The distributive property states that multiplying the total of two or more addends by a number produces the same outcome as multiplying each addend separately by the number and adding the products together.

Additive inverse involves adding which involves two numbers that has opposite sign. the addition lead to zero

The factors of 27 are 1, 3, 9, 27 and 36  1, 2, 3, 4, 6, 9, 12, 18, 36,

the greatest common factor is 9.

writing the equation gives

= 36 + 27

= 9 * (4 + 3)

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Answer:

Step-by-step explanation:

The greatest common factor to rewrite 36 + 27 as (4 + 3 ) is 9.

What is an expression?

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

36 + 27

Taking out 9 commons.

9 (4 + 3)

Thus,

The greatest common factor to rewrite 36 + 27 as (4 + 3 ) is 9.

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