Angie bought a shirt that was priced at $16.50. She paid $18.82 at the counter. Find the tax rate.

The lengths of the segments into which the bisector of each angle divides the opposite side: 12.7, 20.4 and 9.8 respectively

The median of a triangle is defined as the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. A triangle can have three medians, all of which will intersect at the centroid (the arithmetic mean position of all the points in the triangle) of the triangle

The medians of the triangle are represented by the line segments ma, mb, and mc. The length of each median can be calculated as follows:

Ma = √(2b²) + (2c²) - a²

. Mb = √(2a²) + (2c²) - b²

Mc = √(2a²) + (2b²) - c²

Applying the formulas

⇒Ma = √(2*3²) + (2*11²) - 10²

Ma = √(18) + (242) - 100

Ma = √160

Ma = 12.7 units

Mb = √(2a²) + (2c²) - b²

Mb = √(2*10²) + (2*11²) - 3²

Mb = √(200) + (242) - 9

Mb = √415

Mb = 20.4 units

Mc = √(2*10²) + (2*3²) - 11²

Mc = √(200) + (18) - 121

Mc = √97

Mc = 9.8

The bisector of each angle divides the opposite side into: 12.7, 20.4 and 9.8 units respectively

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The expansion of 1/4(x - 12) is x/4 - 3

In order to expand and simplify an expression, we need to multiply out the parentheses and then simplify the resulting expression by collecting the like terms. Expanding brackets (or multiplying out) is the process by which we remove parentheses. It is the reverse process of factorisation.

For example 4(3-y) can be expanded by multiplying the factor 4 by 3 and y = 12- 4y

Similarly the expression 1/4(x - 12) can be expanded by firstly multiply the factor ¼ by x and 12

therefore 1/4(x - 12) = 1/4×x -12×1/4

= x/4 - 3

Therefore the expansion of 1/4(x - 12) is x/4 -3

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The given system of equations can be written as a matrix equation and solved by using the inverse coefficient matrix. The inverse coefficient matrix is 1/10 [4 -2], [3  5], and when multiplied by the vector of variables [x, y] the vector of constants [4, 10] is obtained. This gives the solution x = 6 and y = 2.

Given system:

x + 2y = 4

3x + 4y = 10

Matrix equation:

[x  y]   [1  2]   [4]

[    ] = [    ] x [  ]

[3  4]   [3  4]   [10]

Inverse coefficient matrix:

1/10 [4 -2]

    [3  5]

Solution:

1/10 [4 -2] [x]   [4]

    [3  5] [y] = [10]

x = 6

y = 2

The given system of equations can be written as a matrix equation which can be solved by using the inverse coefficient matrix. This inverse coefficient matrix is found by calculating the inverse of the matrix of coefficients of the variables. In this case, the matrix of coefficients is [1, 2], [3, 4], and the inverse of this matrix is 1/10 [4 -2], [3  5]. Multiplying this inverse coefficient matrix by the vector of variables [x, y] will give the vector of constants [4, 10]. This allows us to solve for the variables, giving x = 6 and y = 2. Therefore, the solution of the given system of equations is x = 6 and y = 2.

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Answer:

36 members paid the total money that would be made would be $180

Step-by-step explanation:

48*75%=36

36*5=180

The length of the side BC of right triangle ABC can be found by applying the Pythagorean theorem with AC = 12 and DC = 4. The length of BC can be expressed in simplest radical form.

Using the Pythagorean Theorem, BC^2 = AC^2 - DC^2

= 12^2 - 4^2

= 144 - 16

= 128

Therefore, BC = [tex]\sqrt{128}[/tex]= 2[tex]\sqrt{32}[/tex] = 8[tex]\sqrt{2}[/tex]

The length of the side BC of a right triangle ABC can be found by applying the Pythagorean theorem. The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In this case, AC = 12 and DC = 4, so BC^2 = AC^2 - DC^2 = 144 - 16 = 128. Therefore, which is the length of \overline{BC}BC in its simplest radical form. This result can be verified by calculating the length of the hypotenuse AC and confirming that it is equal to the square root of the sum of the squares of BC and DC.

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The equation for the circle is (x-h)^2 + (y-k)^2 = r^2. Substituting the values, we get (x-0)^2 + (y-5)^2 = 25.

Substituting y=1.5x+5 into the equation, we get (x-0)^2 + (1.5x+5-5)^2 = 25

Simplifying, we get 1.5x^2 +15x+25=25

Solving, we get x=1.

Therefore, the point of intersection is (1,1.5).

The line y=1.5x+5 intersects the circle with radius 5 and centre (0,5) at the point (1,1.5) in the first quadrant.

The equation for a circle with centre (0,5) and radius 5 is (x-0)^2 + (y-5)^2 = 25. When this equation is combined with the line equation y=1.5x+5, it can be solved to find that the line intersects the circle at the point (1,1.5), which is located in the first quadrant.

The equation for the circle is (x-h)^2 + (y-k)^2 = r^2. Substituting the values, we get (x-0)^2 + (y-5)^2 = 25.

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The correct answer among the following option is , the distance between point a and b of segment ab.

What are Points?

A point is denoted by a dot(.). A point represents position only. It has no size.

Points given in the question are a and b.

What is Line Segment?

A line segment is a part of line with a definite start point and a definite end point.

Their are infinite number of point between the end and start point of the segment.

For the construction,

To copy the segment ab, The distance between the two points a nd b is required.

To calculate the distance, Use the distance formula if the co-ordinates of point a and b are given.

The distance can also be measured using a measuring instrument like

a ruler, A measuring tape if the co-ordinates are not given.

As the distance between both the points a and b is known, A new segment can be drawn of the same distance as ab using two new points and can be named cd or a'b'.

To copy the segment ab, The distance between the point a and b is required.

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Answer:the distance between points a and b

Step-by-step explanation: took the test and got it right

You will need 280 cups of cranberry juice and 56 cups of Sprite to make 56 cups of the famous punch.

What is the ratio definition?

A ratio is a mathematical comparison of two or more values or quantities. It is usually written as a fraction, with the first value or quantity being the numerator and the second value or quantity being the denominator.

To make 56 cups of famous punch using the ratio of 5 cups of cranberry juice to 2 cups of Sprite, you can use the following method:

Write the ratio of cranberry juice to Sprite as a fraction: 5 cups cranberry juice / 2 cups sprite

Multiply the fraction by the desired number of cups of punch: (5 cups cranberry juice / 2 cups sprite) x 56 cups punch

Simplify the equation by cross-multiplying: 5 cups cranberry juice x 56 cups punch = 2 cups sprite x 56 cups punch

Solve for the number of cups of cranberry juice: 5 cups cranberry juice x 56 cups punch = 280 cups of cranberry juice

To find the amount of Sprite use the ratio 5:2, meaning you have to use 2 cups of sprite for every 5 cups of cranberry juice. So use 280/5*2 = 56 cups of sprite

Hence, you will need 280 cups of cranberry juice and 56 cups of Sprite to make 56 cups of the famous punch.

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The distance of the fire from each tower is:

i. From Pine Knob: 41 kilometers

ii. From Colt Station: 15 kilometers

Bearing as a topic in mathematics can be used to determine the exact position or location of an object considering its angle of position and distance to a reference point.

Considering the given question; let the distance of the fire (F) from Pine knob (PK) be represented by x, and that from Colt Station (CS) be represented by y.

Angle at PK = 80 - 65

                    = 15^o

Angle at CS = 65 + 70

                    = 135^o

Angle at F can be determined as;

15 + 135 + F = 180   (sum of angle in a triangle)

150 + F = 180

F = 30^o

So that applying the Sine rule to determine the distances of the fire from PK and CS, we have;

a/Sin A = b/Sin B = c/Sin C

29/Sin 30 = y/Sin 15 = x/ Sin 135

29/Sin 30 = y/Sin 15

y = (29*Sin 15)/ Sin 30

  = 7.5052/ 0.5

  = 15.0104

y = 15 kilometers

29/Sin 30 = x/Sin 135

x = (29*Sin 135)/ Sin 30

  = 20.5059/ 0.5

  = 41.0118

x = 41 kilometers

Therefore, the distance from Pine Knob to the fire is 41 kilometers. While the distance from Colt Station to the fire is 15 kilometers.

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Answer:

x = 25 cm

Step-by-step explanation:

20% =

20/100 = 0.20

20% longer =

20% + 100% =

120% =

120/100 = 1.20

Hence, x+5 is 1.20 times that of x:

x * 1.20 = x + 5

1.2x = x + 5

1.2x - x = 5  ==> subtract x on both sides to move x to one side of the

                      equation

1.2x - 1x = 5  ==> 1x = x

(1.2 - 1)x = 5  ==> distribution property

0.2x = 5

x = 25  ==> multiply both sides by 5 ( 0.2 * 5 = 1 )

The triangles ∆AEC ≅ ∆AFC by Side-Angle-Side theorem

The complete question is added as an attachment

With the given information and the image attached below, we can state that the two triangles are congruent by SAS,

Then go ahead to use the CPCTC to show that any of their corresponding parts are congruent as well.

The proof is given as:

Statement Reason

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According to the given statement, 37.9 cups of tomatoes in each jar.

A system of linear equations is a set of equations with two or more variables. A linear equation is one that can be expressed as ax + b = c, where x is a variable and a, b, and c are constants.

A group of two or more linear equations that share the same variables is known as a system of linear equations. As an illustration, consider the set of linear equations below:

2x + 3y = 10

4x + 6y = 20

x and y are the same variables, and the equations are linear equations. The solution to this system of linear equations is x = 2, y = 2.

This question can be solved using the system of linear equations.

Solving the system of equations gives:

T = 7.4 cups of tomatoes in each jar

Step 1: Calculate the total amount of tomatoes used.

Total tomatoes = 8 jars x cups of tomatoes per jar

Total tomatoes = 8 x (41.6 - 2.1)

Total tomatoes = 303.2 cups

Step 2: Divide the total amount of tomatoes by the total number of jars.

Cups of tomatoes per jar = Total tomatoes ÷ 8

Cups of tomatoes per jar = 303.2 ÷ 8

Cups of tomatoes per jar = 37.9 cups

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A function built from pieces of different functions over different intervals.

Finding a Piecewise Function’s Equation: Find the equation for both y=mx+b lines. If there is an O, the equation is > or; if there is a •, the equation is or. In the equation, the line represents X.

A piecewise function is one that is constructed from parts of distinct functions at different intervals. For example, we may write a piecewise function f(x) that returns -9 when -9 x -5, 6 when -5 x -1, and -7 when -1.

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Answer: 30%

Step-by-step explanation:

because e9 people are chosen and 9 divided by 3 is 3 and 3 x 10 =30

The number of large box is 12 and small box is 8.

What is cube?

The term "cube" refers to a solid three-dimensional shape in mathematics or geometry that has six square faces, eight vertices, and twelve edges. It is also asserted to be a conventional hexahedron. You must be familiar with the three-by-three Rubik's cube, which is the most prevalent example in daily life and can help to increase brain capacity. Similar to this, you'll run into a lot of real-world examples, like 6 sided dice, etc. Solid geometry is the study of three-dimensional objects with surface areas and volumes. Cube, cylinder, cone, and sphere are some of the other solid shapes. Here, we'll talk about its definition, attributes, and relevance to mathematics.

Given :

The volume of small box is 9 cubic feet

The volume of the large box is 12 cubic feet

The volume of all box is 216 cubic feet

There are 20 boxes of paper shipped.

Total Volume covered by small box will be  9x

Total Volume covered by large box will be 24x

Total volume shipped = 216

9x + 24x  = 216

33x = 216

x = 6

So The number of large box is 12 and small box is 8.

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The solution to the differential equation with the initial condition given is y(x)= (x² + 1)⁴.

Step 1: The given differential equation is dy/dx = 2xy/(x² + 1). Rearranging the equation, we get dy = 2xy dx/(x² + 1).

Step 2: Integrating both sides, we get ∫dy = ∫2xy dx/(x² + 1). On the left side, we have ∫dy = y + C, where C is the constant of integration. On the right side, we have ∫2xy dx/(x² + 1) = x² y + C.

Step 3: Substituting the initial condition y(3) = 6 in the solution, we get 6 = 3² y + C. Solving for C, we get C = 6 - 9y.

Step 4: Substituting C = 6 - 9y in the solution, we get y(x) = (x² + 1)⁴

The solution to the differential equation with the initial condition given is y(x)= (x² + 1)⁴.

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A direct variation equation that relates x and y is y = (5/2)x.

What is the direct variation equation?

Direct variation is a linear function defined by an equation of the form y = kx when x is not equal to zero. Inverse variation is a nonlinear function defined by an equation of the form xy = k when x is not equal to zero and k is a nonzero real number constant.

When two variables are directly proportional, they vary directly with each other.

The relationship between the variables can be represented by a direct variation equation of the form:

y = kx

where k is the constant of proportionality.

Given that y = 5 and x = 2, we can use these values to find the value of k:

y = kx

5 = k(2)

k = 5/2

So the direct variation equation that relates x and y is:

y = (5/2)x

This equation shows that for every value of x, the value of y is 5/2 times that value.

For example, if x = 3, then y = (5/2) * 3 = 7.5, and if x = 4, then y = (5/2) * 4 = 10.

Hence, A direct variation equation that relates x and y is y = (5/2)x.

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The equation y = 5x for x ≤ 2 and y = 10 + 2.5(x-2) for x ≥ 2.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

In other words, an equation is a set of variables that are constrained through a situation or case.

Given that,

First, the two-hour rate of charge is $5 per hour

y = 5x for x ≤ 2

Over the 2 hours, the rate of charge is $2.5 per hour

y = 10 + 2.5(x-2) for x ≥ 2.

Hence "The equation y = 5x for x ≤ 2 and y = 10 + 2.5(x-2) for x ≥ 2".

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Using the Empirical Rule, it is discovered that the proportion of people in a population with IQs ranging from 80 to 120 is 0.95 = 95%.

It specifies the following for a normally distributed random variable:

68% of the measurements are within one standard deviation of the mean.

95% of the measurements are within 2 standard deviations of the mean.

99.7% of the measurements are within 3 standard deviations of the mean.

Given a mean of 100 and a standard deviation of 20, we can calculate:

80 = 100 - 2 x 20.

120 = 100 + 2 x 20.

Because these are both the most extreme values within two standard deviations of the mean, then the proportion of people in a population with IQ scores between 80 and 120 is 0.95 = 95%.

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The first lefty is the fifth person chosen is7.45%

There are some lefties among the 5 people is, 50.16%

The first lefty is the fifth person chosen is

[tex](.87)^4(.13)=.07447=7.45 \%[/tex]

There are some lefties among the 5 people is,

[tex]$$1-(.87)^5=.50157=50.16 \%$$[/tex]

The first lefty is the second or third person.

[tex]$(.87)(.13)+(.87)^2(.13)=.211497=21.15 \%_0$[/tex]

There are exactly 3 lefties in the group.

[tex]$(5, .13,3)=.0166=1.66 \%$[/tex]

There are at least 3 lefties in the group,

[tex]1 \text { - binomadf }(5, .13,2)=.0179=1.79 \%[/tex]

There are no more than 3 lefties in the group

[tex]\text { binomcdf }(5, .13,3)=.9987=99.87 \%[/tex]

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Both questions can be answered by using the function f(x).The function f(x) represents the total output of a farm in tons in the week of 2015, and the value of x is known. This means that by using the value of x, the total output of the farm in 2015 and the output of the farm in the week of 2015 can both be calculated.

The function f(x) is a mathematical representation of the total output of a farm in tons for the week of 2015. The value of x is known, which means that by using this value, the total output of the farm in 2015 as well as the output of the farm in the week of 2015 can both be calculated. Therefore, both of the questions can be answered by using the function f(x). Knowing the total output of the farm in 2015 can help to better understand the production of the farm over the course of the year, while the output of the farm in the week of 2015 can be used to gain insights into the weekly production of the farm. With this information, decisions can be made to improve the production and efficiency of the farm.

The function f(x) can be used to answer both questions regarding the total output of a farm in 2015 and the output of the farm in the week of 2015.

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X and Z are not independent random variable as P(X = x and Z = z) and  P(X = x) * P(Z = z) are not equal for most values of x and z by using their joint probability

To prove that two random variables are independent,  to show that their joint probability distribution is equal to the product of their individual probability distributions. Start with part (a) and then move on to part (b).

(a) Let X be the first digit and Y be the second digit of the chosen number from the set {10, 11, 12, ..., 99}.

Step 1: Calculate the probabilities of X and Y individually.

To find the probability of X taking any particular value,

The probability of X taking any value x (where x is a digit from 1 to 9) is given by:

P(X = x) = (number of numbers starting with digit x) / (total numbers)

= 9 / 90

= 1 / 10

Similarly, for Y, since there are 10 digits (0 to 9) and  choose the second digit uniformly at random, the probability of Y taking any value y is:

P(Y = y) = (number of numbers with digit y as the second digit) / (total numbers)

= 10 / 90

= 1 / 9

Step 2: Calculate the joint probability P(X, Y).

The joint probability is the probability that X takes a particular value x and Y takes a particular value y simultaneously.

P(X = x and Y = y) = (number of numbers with digit x as the first digit and digit y as the second digit) / (total numbers)

= 1 / 90

Step 3: Check if X and Y are independent.

Two random variables X and Y are independent if and only if their joint probability equals the product of their individual probabilities.

P(X = x and Y = y) = P(X = x) * P(Y = y)

1 / 90 = (1 / 10) * (1 / 9)

Since the equation holds for all values of x and y,  conclude that X and Y are independent random variables.

(b) Let X be the first digit of the chosen number and Z be the sum of the two digits.

Step 1: Calculate the probabilities of X and Z individually.

Calculated the probability distribution for X in part (a), which is P(X = x) = 1 / 10 for x = 1 to 9.

To  calculate the probability distribution for Z, the sum of the two digits:

For Z = 0: There is only one number, 10, with a sum of digits equal to 0.

P(Z = 0) = 1 / 90

For Z = 1: There are two numbers, 10 and 01, with a sum of digits equal to 1.

P(Z = 1) = 2 / 90

= 1 / 45

For Z = 2: There are three numbers, 11, 20, and 02, with a sum of digits equal to 2.

P(Z = 2) = 3 / 90

= 1 / 30

For Z = 3: There are four numbers, 12, 21, 03, and 30, with a sum of digits equal to 3.

P(Z = 3) = 4 / 90

= 2 / 45

For Z = 4: There are five numbers, 13, 31, 04, 40, and 22, with a sum of digits equal to 4.

P(Z = 4) = 5 / 90

= 1 / 18

For Z = 5: There are four numbers, 14, 41, 23, and 32, with a sum of digits equal to 5.

P(Z = 5) = 4 / 90

= 2 / 45

For Z = 6: There are three numbers, 15, 51, and 24, with a sum of digits equal to 6.

P(Z = 6) = 3 / 90

= 1 / 30

For Z = 7: There are two numbers, 25 and 52, with a sum of digits equal to 7.

P(Z = 7) = 2 / 90

= 1 / 45

For Z = 8: There are two numbers, 26 and 62, with a sum of digits equal to 8.

P(Z = 8) = 2 / 90

= 1 / 45

For Z = 9: There is only one number, 36, with a sum of digits equal to 9.

P(Z = 9) = 1 / 90

Step 2: Calculate the joint probability P(X, Z).

The joint probability is the probability that X takes a particular value x and Z takes a particular value z simultaneously.

Formula:

P(X = x and Z = z) = (number of numbers with digit x as the first digit and the sum of digits equal to z) / (total numbers)

Check the case where X = 1 and Z = 2 as an example:

Numbers with digit 1 as the first digit and a sum of digits equal to 2: 12 and 21 (two numbers)

P(X = 1 and Z = 2) = 2 / 90

= 1 / 45

Step 3: Check if X and Z are independent.

To determine if X and Z are independent, we need to compare their joint probability to the product of their individual probabilities for all values of x and z.

However, if compare the joint probabilities P(X = x and Z = z) to P(X = x) * P(Z = z), find that they are not equal for most values of x and z.

Therefore, conclude that X and Z are not independent random variables.

as P(X = x and Z = z) and  P(X = x) * P(Z = z) are not equal for most values of x and z.

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The greatest number of different line segment that can be drawn so that each line passes through two of these points is 36. This is because every pair of points can be connected by a line, and with ten points, there are 45 pairs.

The greatest number of different lines that can be drawn so that each line passes through two of these points is 36. This is because every pair of points can be connected by a line, and with ten points, there are 45 pairs. Since each line can only be drawn once, the total number of different lines that can be drawn is 45 divided by two, which is equal to 36. In addition, the number of lines that can be drawn from each point to the rest of the points is 9, and since there are 10 points, the total number of lines that can be drawn is 90. However, this number is redundant since each line can only be drawn once, so the greatest number of lines that can be drawn is 36.

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Answer:

42 is the answer

Step-by-step explanation:

for distributive property of multiplication you could use...

2 x 3 x 7

as 3 x 7 is 21

The term, [tex]\frac{1}{49}[/tex] is simplified form of the expression [tex]7^{-\frac{5}{6}} \ \cdot\ 7^{-\frac{7}{6}}[/tex].

The top and bottom of a fraction are said to be in its simplest form if they only share the number one. For the top & bottom to remain whole numbers, you cannot divide them further.

Additionally, lowest terms, or simple form, may be used.

The simplest form of a fraction, for instance, is 4/5. Apart from 1, there isn't a number that divides both 4 and 5. Not in its simplest form is the fraction 3/6. Numerator & denominator can both be divided by 3.

Given to simplify the expression

[tex]7^{-\dfrac{5}{6}} \ \cdot\ 7^{-\dfrac{7}{6}}[/tex]

[tex]7^{(-\dfrac{5}{6}-\dfrac{7}{6})}[/tex]

[tex]7^{(\dfrac{-5-7}{6})}[/tex]

[tex]7^{(\dfrac{-12}{6})}[/tex]

[tex]7^{(-2)}[/tex]

[tex]\dfrac{1}{7} \times \dfrac{1}{7}[/tex]

[tex]\dfrac{1}{49}[/tex]

Thus, [tex]\frac{1}{49}[/tex] is simplified form of the expression [tex]7^{-\frac{5}{6}} \ \cdot\ 7^{-\frac{7}{6}}[/tex].

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Zaire has a total of 7 holes * 4 points/hole = 28 points from scores of 4 from last week.To reach 60 points, he needs to score 4 on 60 - 28 = 32 more holes today. So, Zaire needs to score a 4 on 32 holes today.

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The rule demonstrating the logical equality of the two claims. -((w V p) ^(-91q1w)) DeMorgan's law has the form -(w V p) v-(-91qAw).

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The equation that shows the number of pounds of celery(y) the cook will use each day is 8y+17= 31 and y = 1 ¼

A linear equation is an algebraic statement where each term is either a constant or a variable raised to the first power. In other words, none of the exponents can be greater than 1.

For each day , the cook will use 2 1/8 pounds of carrot and y pounds of celery

therefore total = 17/8 +y

for 5 days the total carrot and celery used =( 17/8 + y) ×5

for 5 days 19 3/8 pounds of carrot and celery is used . Therefore;

155/8 = 5(17/8+y)

for each day, divide both sides by 5

155/8×1/5 = 17/8+y

31/8 = 17/8+y

collect like terms

y = 31/8-17/8

multiply through by 8

8y = 31-17

8y+17 = 31

therefore the equation that can be used to find the pound of celery used per day is 8y+17= 31

and y = 14/8 = 7/4 = 1 ¼

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[tex]f(x)=5x^2-3x+3\implies \left. \cfrac{df}{dx}=10x-3 \right|_{x=1}\implies 10(1)-3\implies 7[/tex]