Use substitution to solve

Using the unit tile the area of the three diagrams is - square tile = 1 unit², rectangular tile = x unit², and square tile = x² unit².

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

For the square tile with length and breadth = 1 unit, the area is simply the product of the length and breadth, which is 1 unit × 1 unit = 1 unit².

This is the same as the area of the unit tile, so we don't really need to use it to find the area of this tile.

For the rectangular tile with length = x units and breadth = 1 unit, we can use x unit tiles to cover the length, and 1 unit tiles to cover the breadth.

Therefore, the area of the rectangular tile is x unit × 1 unit = x unit².

For the square tile with length and breadth = x units, we can use x unit tiles to cover the length, and x unit tiles to cover the breadth.

Therefore, the area of the square tile is x unit × x unit = x² unit².

In general, for any tile with length = a units and breadth = b units, the area is given by the product of the length and breadth, which is a unit × b unit = ab unit².

Therefore, we can use the unit tile to find the exact area of any tile, as long as we know its dimensions.

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The zeros of the polynomial x4 + 2x3 + 22x2 + 50x - 75 = 0 are x = 3, x = -5, x = ±√(37).

To find all the zeros (real and complex) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0, we can use the Rational Root Theorem and synthetic division.

The Rational Root Theorem states that if p/q is a rational root of a polynomial equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

The factors of the constant term -75 are: ±1, ±3, ±5, ±15, ±25, ±75
The factors of the leading coefficient 1 are: ±1

Therefore, the possible rational roots of the polynomial are: ±1, ±3, ±5, ±15, ±25, ±75

We can use synthetic division to test each of these possible roots until we find one that is a root. Once we find a root, we can use synthetic division again to divide the polynomial by the factor (x - root) to get a smaller polynomial, and then repeat the process until we have found all the roots.

Using synthetic division, we find that 3 is a root of the polynomial. Dividing the polynomial by (x - 3) gives us a smaller polynomial: x^(3)+5x^(2)+37x+25=0

We can repeat the process with this smaller polynomial to find the remaining roots. Using synthetic division again, we find that -5 is a root of the smaller polynomial. Dividing the smaller polynomial by (x + 5) gives us an even smaller polynomial: x^(2)+37=0

This polynomial has no real roots, but it has two complex roots: x = ±√(-37) = ±√(37)i

So, the complete list of zeros (real and complex) of the original polynomial is: 3, -5, ±√(37).

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The equilibrium quantity of items is 25.

The question is asking for the equilibrium quantity of items when the demand and supply functions given are graphed together. The equilibrium quantity can be found by solving for q when the demand and supply functions are equal.

Demand: d(q) = 562.5 - 0.4q2

Supply: s(q) = 0.5q2

Set the demand and supply functions equal to each other and solve for q:

562.5 - 0.4q2 = 0.5q2

0.9q2 = 562.5

q2 = 625

q = 25

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The smallest possible area of a square which can be made from the three pieces is 11833.203125 cm².

To find the smallest possible area of a square, we need to make sure that we use the longest pieces to form the sides of the square. Therefore, we need to divide the 360 cm steel bar into equal parts first, then use the remaining parts to divide the other two steel bars.

Let's call the length of each part x.

The 360 cm steel bar can be divided into n parts of length x, where:

n = 360/x

Similarly, the 300 cm steel bar can be divided into m parts of length x, where:

m = 300/x

And the 210 cm steel bar can be divided into k parts of length x, where:

k = 210/x

To form a square, we need to use all the parts we cut from the steel bars. Therefore, the length of the sides of the square will be nx + mx + kx, which is equal to (n + m + k)x.

The area of the square will be (n + m + k)x²

To find the smallest possible area, we need to minimize (n + m + k)x². Since x can be any positive number, we can focus on minimizing n + m + k.

n + m + k = (360/x) + (300/x) + (210/x)

n + m + k = (870/x)

To minimize (n + m + k), we need to maximize x. However, x cannot be greater than the smallest steel bar, which is 210 cm long.

Therefore, x must be a factor of 210.

Let's try x = 1 cm. In this case, n + m + k = 870 cm, which means we can form a square with sides of length 870 cm/4 = 217.5 cm.

The area of this square is 217.5^2 = 47250.625 cm².

Let's try x = 2 cm. In this case, n + m + k = 435 cm, which means we can form a square with sides of length 435 cm/4 = 108.75 cm.

The area of this square is 108.75² = 11833.203125 cm².

Let's try x = 3 cm. In this case, n + m + k = 290 cm, which means we cannot form a square using all the parts we cut from the steel bars.

Therefore, the smallest possible area of a square which can be made from the three pieces is 11833.203125 cm², and this can be achieved by cutting the steel bars into parts of length 2 cm.

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Answer:

When you multiply two numbers with the same sign (either both positive or both negative), the rule is that the result is always positive.

For example, if you multiply +2 and +3, the result is +6 because both numbers have the same positive sign. Likewise, if you multiply -4 and -5, the result is +20 because both numbers have the same negative sign.

This rule applies to any two numbers with the same sign, regardless of their values or whether they are whole numbers, fractions, or decimals.

The correct answer is option a. 1 - e^-2.

To find the probability of a random variable X with exponential distribution, we use the following formula:P[X > x] = e^(-λx)Where λ is the rate parameter and x is the value we are trying to find the probability of.

In this case, we are given that the mean of the distribution is 1, so we can use this information to find the rate parameter:λ = 1/mean = 1/1 = 1Now, we can plug in the values for λ and x into the formula to find the probability:P[X > 2] = e^(-1*2) = e^-2 = 0.1353Therefore, the correct answer is option a. 1 - e^-2.

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Answer:

m∠E = 57º

Step-by-step explanation:

We can use the isosceles triangle theorem to determine that angle E is congruent to angle D, since side DF is congruent to side EF.

This means that:

m∠E = m∠D

m∠E = (4x + 1)º

Now, we can solve for x using the fact that the interior angles of a triangle sum to 180º.

m∠D + m∠E + m∠F = 180º

↓ substituting the given angles measures (in terms of x)

(4x + 1)º + (4x + 1)º + (5x - 4)º = 180º

↓ grouping like terms

(4x + 4x + 5x)º + (1 + 1 - 4)º = 180º

↓ combining like terms

13xº - 2º = 180º

↓ adding 2º to both sides

13xº = 182º

↓ dividing both sides by 13º

x = 14

With this x value, we can now solve for m∠E using its definition in terms of x.

m∠E = (4x + 1)º

↓ plugging in solved x value

m∠E = (4(14) + 1)º

m∠E = (56 + 1)º

m∠E = 57º

The number of adults is 120, and the total money made from ticket sales is £1120.

The mathematical connection between two or more numbers is called a ratio, and it is represented as the product of the division of two values. Several formats, such as fractions, decimals, or percentages, can be used to express ratios. Mathematicians employ ratios in many different areas, including geometry, probability, and finance. The connection between the lengths of two or more sides of a form is described in geometry using ratios. Ratios, sometimes in the form of odds, are used in probability to indicate the possibility of an event occurring.

Let x be the number of adults.

Given that, ratio of children to adults is 2 : 3.

Thus,

2/3 = 80/x

Cross-multiplying gives:

2x = 240

x = 120

The number of adults is 120.

Each child ticket costs a quarter of the adult ticket, so the cost of a child ticket is:

1/4 * £8 = £2

The total money made from child tickets is:

£2 * 80 = £160

The total money made from adult tickets is:

£8 * 120 = £960

Total money made from ticket sales is:

£160 + £960 = £1120

Hence, the total money made from ticket sales is £1120.

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The simplified result of the polynomial operations is: -14x^5 + 5x^4 + 7x^3 - 4x^2 + x + 17

To complete the polynomial operations and simplify the result, we need to combine like terms. Like terms are terms that have the same variable and the same exponent.

First, let's rewrite the expression to make it easier to see the like terms:
(7x^3 + 3x^4 - x^5 + 12) + (-13x^5 + 2x^4 - 4x^2 + x + 5)

Next, let's combine the like terms:
7x^3 + 3x^4 + 2x^4 - x^5 - 13x^5 - 4x^2 + x + 12 + 5

Simplify the expression by adding or subtracting the coefficients of the like terms:
5x^4 + 7x^3 - 14x^5 - 4x^2 + x + 17

The simplified result of the polynomial operations is:
-14x^5 + 5x^4 + 7x^3 - 4x^2 + x + 17

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Answer:

infinity

Step-by-step explanation:

There are "infinity" rational numbers between 0 and 5​.

What are rational numbers :

A rational number is one that has the form p/q, where p & q are both integers and q is not zero.

How to find rational numbers :

The denominators must be equal to get the rational numbers between two rational numbers with differing denominators.

Finding the LCM of the denominators or multiplying the denominators of one to both the numerator and denominator of the other are two options for equating the denominators.

The polar coordinates corresponding to the rectangular coordinates (0, 5) are (5, π/2).

To find the distance from the origin to the point (r), we can use the Pythagorean theorem. The distance from the origin to a point (x, y) is given by the formula √(x² + y²).

In this case, since the x-coordinate is 0, we only need to find the distance from the origin to the y-coordinate. Therefore, r = √(0² + 5²) = 5.

To find the angle (θ) that the line from the origin to the point makes with the positive x-axis, we can use trigonometry.

However, this is undefined since we cannot divide by zero. Therefore, we need to use a special case. Since the x-coordinate is 0, the point lies on the y-axis.

Therefore, the angle θ is either π/2 or 3π/2. However, we are given the constraint 0 ≤ θ < 2π. Therefore, the angle θ = π/2 since it satisfies the constraint.

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Complete Question:

Convert the following rectangular coordinates into polar coordinates. Always choose 0 ≤ θ < 2 π . (a) ( 0 , 5 )

In respοnse tο the query, we can state that While persistence might be equatiοn crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.

A. Flexibility fοsters inventiοn and creativity.

In a math equatiοn, twο assertiοns are cοnnected by the equals sign (=), which denοtes equivalence. A mathematical assertiοn used in algebraic equatiοns establishes the equivalence οf twο mathematical statements. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign creates a space between the values 3x + 5 and 14.

Tο cοmprehend the relatiοnship between the twο sentences written οn οppοsing sides οf a letter, utilise a mathematical fοrmula. The lοgο and the specific prοgramme typically cοrrespοnd. An illustratiοn wοuld be 2x - 4 = 2.  

Peοple and οrganizatiοns whο are flexible are better equipped tο adjust tο shifting cοnditiοns, explοre new avenues, and adapt. Its adaptability encοurages experimentatiοn and taking chances, which can result in fresh, creative ideas.

Fοr reaching οbjectives and cοmpleting wοrk quickly, realistic expectatiοns and well-οrganized preparatiοn are crucial, but they may nοt always fοster creativity and inventiοn. While persistence might be crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.

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To find the equation of the line that passes through (-3,2) and is parallel to the line defined by 5x+2y=-10, we need to follow these steps:

Step 1: Find the slope of the given line. Since the equation is in the form of Ax + By = C, we can rearrange it to the slope-intercept form, y = mx + b, where m is the slope.
5x + 2y = -10
2y = -5x - 10
y = (-5/2)x - 5
So, the slope of the given line is -5/2.

Step 2: Since the two lines are parallel, they have the same slope. So, the slope of the new line is also -5/2.

Step 3: Use the point-slope form of a line to find the equation of the new line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.
y - 2 = (-5/2)(x - (-3))
y - 2 = (-5/2)x - 15/2
y = (-5/2)x - 15/2 + 2
y = (-5/2)x - 11/2
So, the equation of the new line is y = (-5/2)x - 11/2.

Therefore, the equation of the line that passes through (-3,2) and is parallel to the line defined by 5x+2y=-10 is
y = (-5/2)x - 11/2.

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Using inequality, the daughter's age to her mother's is 12x < 40.

Inequality is a mathematical statement that two algebraic expressions are unequal.

Inequalities are depicted as:

The age of the mother = 40

The age of the daughter = 12

The number of times the mother's age is to her daughters = 3.33 times (40/12).

Let the number of times the mother's age is more than her daughter's age = x

40 > 12x

or 12x < 40

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The statement that √s^2 is equals to the value of s is false, as s² is an even function, hence √s^2 can be equal either to the value of s or to -s.

The s² function is even, hence, for example:

sqrt[(-3)²] = sqrt(9) = 3.

Which proves by contradiction that the statement is false.

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Based on the box-and-whisker plot, the numbers that are considered possible outliers include the following: B. 31, 87, 90.

In Mathematics, an outlier is also referred to as anomalous data and it can be defined as a numerical value that is either unusually too small or large (big) in comparison with the overall pattern of the numerical values contained in a data set.

In Mathematics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

By critically observing the box-and-whisker plots or box plot, we can logically deduce that 31, 87, and 90 are possible outliers.

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[tex]10^{32} \times 10^{36}[/tex] can be written as [tex]10^{68}[/tex] using a single exponent.

What is an exponents?

In mathematics, exponents are a way to indicate the repeated multiplication of a number or phrase.

Exponents are numbers that are superscripted above other numbers. In other words, it denotes that a certain level of power has been conferred upon the base. Index and power are other names for the exponent. If m is a positive number and n is its exponent, the expression Mn means that m has been multiplied by itself n times.

Exponents are required for a more comprehensible representation of numerical quantities. Repeated multiplication is simple to write down when using exponents. If both n and x are positive integers, the expression xn means that x has been multiplied by itself n times.

When multiplying two numbers with the same base, we can add their exponents. Therefore:

[tex]10^{32} \times 10^{36 }= 10^{(32+36) }= 10^{68}[/tex]

Hence, [tex]10^{32} \times 10^{36}[/tex]can be written as [tex]10^{68}[/tex] using a single exponent.

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The price of the item 10 years from today will be approximately $37,607.56

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

To find the current price of the item, we need to substitute t = 0 in the given equation.

So we get:

[tex]p(0) = 25000(1.034)^0 = 25000(1) = 25000[/tex]

Therefore, the current price of the item is $25,000.

To find the price 10 years from today, we need to substitute t = 10 in the given equation. So we get:

[tex]p(10) = 25000(1.034)^{10} = 37607.56[/tex]

Therefore, the price of the item 10 years from today will be approximately $37,607.56.

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Answer:

Step-by-step explanation:

A. The sand will not fit into the tank because the volume of the sand is higher than that of the container

B. the difference in the volume of the first and second tank is 70in³

C. The measurement of the tank that will hold exactly 375 is 5in × 7in × 10.7 in

A prism is a solid shape that is bound on all its sides by plane faces.

The volume of a prism is expressed as;

volume = base × height

The volume of the tank = 5×7×9

= 315 in³

The volume of the sand is 375 .

Therefore the volume of the sand is greater than that of the tank, this means the sand will not fit into the tank.

B. The height of the second tank = 9+2 = 11

The volume of the second tank = 5×7 × 11 = 385

therefore the difference in the volume of the first and second tank = 385-315

= 70in³

C. If the tank has thesame base, then the height will be

375 = 5× 7 × h

375 = 35h

h = 375/35

h = 10.7 in

therefore the measurement of the tank that will hold exactly 375 is 5in × 7in × 10.7 in

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Answer:15.75 min

Step-by-step explanation:

7/8)60= 52.5

3/10)52.5= 15.75

Answer: We will use long division to find the quotient of (x³ - 3x² + 5x + 3) ÷ (x + 1).

    x² - 4x + 9  

___________________

x + 1 | x³ - 3x² + 5x + 3

- (x³ + x²)

--------------

-4x² + 5x

-(-4x² - 4x)

------------

9x + 3

-(9x + 9)

-------

6

Therefore, the quotient of (x³ - 3x² + 5x + 3) ÷ (x + 1) is x² - 4x + 9 with a remainder of 6.

Step-by-step explanation:

Answer: f(g(2)) = 1   and g(f(1)) = 2

Step-by-step explanation:

The equation of the parabola is f(x) = x² - 4x + 4

The equation of the line is g(x) = x + 1

To find f(g(2)), you must first find g(2)

g(2) = (2) + 1

g(2) = 3

Now find f(g(2)) by using 3 for g(2)

f(3) = (3)² - 4(3) + 4

f(3) = 9 - 12 + 4

f(3) = 1

f(g(2)) = 1

To find g(f(1)), you must first find f(1)

f(1) = (1)² - 4(1) + 4

f(1) = 1 - 4 + 4

f(1) = 1

Now find g(f(1)) by using 1 for f(1)

g(1) = (1) + 1

g(1) = 2

g(f(1)) = 2

Hope this helps!

The trinomial in standard form is x^2 + 10x + 25.

To express (x+5)^(2) as a trinomial in standard form, we need to expand the expression using the distributive property.

First, we will distribute the first term, x, to each term inside the parentheses:

(x+5)(x+5) = x(x) + x(5) + 5(x) + 5(5)

Next, we will simplify the terms:

= x^2 + 5x + 5x + 25

Finally, we will combine like terms to get the trinomial in standard form:

= x^2 + 10x + 25

Therefore, the trinomial in standard form is x^2 + 10x + 25.

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Since 100% of the children are served both meals, this means that all 25 children are served both meals. Therefore, the answer is 25 children.

To find the number of children who are served both meals, we can use the following formula:

Number of children served both meals = (percentage of children served both meals / 100) x total number of children

Plugging in the given values, we get:

Number of children served both meals = (100 / 100) x 25

Number of children served both meals = 1 x 25

Number of children served both meals = 25

Therefore, the number of children who are served both meals is 25. .

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The given function, n(x)=x^(2)+10x+24, can be written in vertex form by completing the square. Vertex form is given by y=a(x-h)^2+k.

The vertex is at (h,k). To find h and k, first find the average of the x-values of the two roots:

h = ( -b +- sqrt(b^2 - 4ac) ) / 2a
 = ( -10 +- sqrt( 10^2 - 4(1)(24) ) ) / 2(1)
 = ( -10 +- sqrt(100 - 96) ) / 2
 = ( -10 +- sqrt(4) ) / 2
 = ( -10 +- 2 ) / 2
 = -6

Substituting h into the equation y=a(x-h)^2+k, we have:

k = y - a(x-h)^2
 = n(x) - a(x+6)^2
 = x^2 + 10x + 24 - a(x+6)^2
 = 24 - a(x+6)^2

We know that when x=-6, k=24, so

24 = a( -6+6 )^2
24 = 36a
a = 2/3

Therefore, the equation in vertex form is y = 2/3(x+6)^2 + 24.

The vertex is (h,k) = (-6, 24).

The x-intercepts (s) are the roots of the equation, so they can be found by setting the equation equal to 0 and solving for x.

0 = x^2 + 10x + 24
0 = (x+6)(x+4)

Therefore, the x-intercepts are x=-6 and x=-4.

The y-intercept is when x=0, so it is y = 24.

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The streetlight is approximately 16.41 feet tall, and the length of its shadow is approximately 27.35 feet.

What is the proportion?

A proportion is a statement that two ratios are equal. In other words, a proportion is an equation that shows that two fractions or two ratios are equivalent. A proportion can be written in the form of:

a/b = c/d

We can use the properties of similar triangles to solve this problem. Let's call the height of the streetlight h, and the length of the shadow cast by the streetlight x. We can set up the following proportion:

(height of person) / (length of person's shadow) = (height of streetlight) / (length of streetlight's shadow)

or

6 / 10 = h / x

Simplifying this proportion, we get:

x = (10h) / 6

We also know that the person is standing 18 feet from the streetlight, and that the length of the person's shadow is 10 feet. Using the Pythagorean theorem, we can set up the following equation:

6^2 + 10^2 = (18 + x)^2

Simplifying and substituting x, we get:

36 + 100 = (18 + (10h/6))^2

136 = (18 + (10h/6))^2

Taking the square root of both sides, we get:

√136 = 18 + (10h/6)

Simplifying, we get:

√136 - 18 = (10h/6)

Multiplying both sides by 6, we get:

6(√136 - 18) = 10h

Simplifying, we get:

h ≈ 16.41 feet

Now, we can substitute this value of h into the expression for x that we derived earlier:

x = (10h) / 6 ≈ 27.35 feet

Therefore, the streetlight is approximately 16.41 feet tall, and the length of its shadow is approximately 27.35 feet.

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Step-by-step explanation:

Using perfect square trinomial rules,

[tex](a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} [/tex]

Here, a is sin x

b is cos x

So

[tex]( \sin(x) + \cos(x) ) {}^{2} = \sin {}^{2} (x) + 2 \sin(x) \cos(x) + \cos {}^{2} (x) [/tex]

We know

[tex] \sin {}^{2} (x) + \cos {}^{2} (x) = 1[/tex]

So we know have

[tex]1 + 2 \sin(x) \cos(x) [/tex]

[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]

So our final answer is

[tex]1 + \sin(2x) [/tex]

Limitations in the design and techniques used depending on the specifics of the dataset.

Problem Description: This problem seeks to use an experimental design, specifically Completely Randomized Design (CRD), to solve a problem. By using a dataset of the user's choice, the goal is to analyse this data using the R statistical software and draw meaningful conclusions from the results. The significance of this problem lies in the ability to understand the nature of the dataset and use this to solve the problem.

Method: CRD is a type of design that is used to determine how various factors, such as treatments, affect the response of interest. In this problem, the user will be using the R statistical software to perform the CRD, by selecting a dataset of their choice that is suited to the method. This dataset may either be a built-in dataset, manually entered data, or data imported into R from an online source. The data must be relevant to the problem and the method being used.

Objectives: The main objective of this problem is to use the CRD method to solve the problem. Through this analysis, it is also intended to gain an understanding of the data and the relationships between the variables. This may be done through testing hypotheses related to multiple comparison tests and the statistical model.

Statistical Analysis: The statistical model used for the experimental design is a linear model with the response variable being a linear combination of explanatory variables. These explanatory variables can include treatments, levels, and covariates, all of which will be determined by the dataset. The related hypotheses that will be tested are that the means of the response variables are equal among all the different treatments, levels, and covariates. The hypotheses related to the multiple comparison tests will be determined by the specifics of the dataset.

Results: After entering the R code and the corresponding dataset, the output from the R console will be generated. This output includes the linear model, coefficients, summary statistics, and residual plots that can be used to analyse the results.

Analysis of Results: Using the R output, the hypotheses related to the model and the multiple comparison tests can be tested at a certain level of significance. In addition, the residual plots can be used to determine whether the assumptions of the model have been met.

Conclusion: Through the use of CRD and the R statistical software, this problem was solved and the objectives were met. The dataset chosen allowed for an analysis of the data and a deeper understanding of the relationships between the variables. However, there may be limitations in the design and techniques used depending on the specifics of the dataset.

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The number of cases which Judge Jerry hears every hour is 2 2/19

An element of a whole or, more broadly, any number of equal pieces, is represented by a fraction.

When used in conversational English, a fraction indicates the number of components of a particular size, as in one-half, eight-fifths, and three-quarters.

How to calculate:

GIven that he tries 5 cases in 2 3/8 hours

The unit rate of cases per hour is calculated by dividing the number of cases by the number of hours.

5/(2 3/8) = 5/(19/8) = 5 * 8/19 = 40/19 = 2 2/19

Therefore, the number of cases which Judge Jerry hears every hour is 2 2/19

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