what is the solution to the equality shown?
3m+5>2(m-7)

Answer: -8

Step-by-step explanation:

When the input is 4, according to the input-output table, the output is -8.

Plugging 4 into the function f(x) = 2x - 10 directly also gives the same result:

f(4) = 2(4) - 10 = 8 - 10 = -2

So the output of William's function when the input is 4 is -8.

A  small block is sent through point A with a speed of 6.8 m/s. Its path is without friction until it reaches the section of length L=14 m, where the coefficient of kinetic friction is 0.78. The indicated height are h₁ =5.4 m and h₂=2.8 m. The speeds of the block at (a) Point B is 108m/s and (b) Point C is 7.6m/s

The velocity of an object (usually denoted v) is the amount of its position change over time ; therefore it is a scalar quantity. The average speed of an object over a time interval is the distance traveled by the object divided by the duration of the interval; the instantaneous speed remains zero.

(a) At B the speed is :

   V = [tex]\sqrt{V^{2}+2gh_1 }[/tex]

​⇒ V = [tex]\sqrt{(6.8m/s)+ 2(9.8m/s^2) (5.4)}[/tex]

⇒ V = 108.44 m/s

⇒ V = 108 m/s

(b) Here, matters is the difference in heights (between A and C):

   [tex]V = \sqrt{V_0^{2}+2g(h_1+h_2) }[/tex]

⇒  [tex]V = \sqrt{(6.8)+2 (9.8) (5.4- 2.8) }[/tex]

⇒ [tex]V = \sqrt{6.8 +2(9.8)(2.6)}[/tex]      

⇒ √57.76 = 7.6m/s

(c)  Using the results from part (b), we see that its kinetic energy is just at the beginning of its "rough glide" (horizontally to D) is:

1/2 m(7.6m/s)² = 28.88m (with SI units understood).

We note that this kinetic energy will turn entirely into thermal energy

    28.88m =  μₙ mgd

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The relationship between a dependent and an independent variable expressed as a logarithmic equation is as follows X = Y/2

At times, the relationship between a dependent and an independent variable is expressed as a logarithmic equation. For this question, given a positive constant € and the two equations below, we can find the relationship between X and Y. Equation 1: Log(c) = X/2 Equation 2: Log(c^2) = Y.We can use the basic properties of logarithms to solve this problem. To do so, we need to understand what logarithms are first.Logarithms are the opposite of exponentials. They allow us to find the exponent needed to produce a given value. For instance, log base 2 of 8 is 3 since 2^3 = 8.

A logarithm is represented as log_bx where b is the base and x is the argument or the number being passed to the logarithm.In equation 1, we are given that log(c) = x/2. To isolate x, we need to exponentiate both sides with base e.e^(log(c)) = e^(x/2)This gives us c = e^(x/2).We can rewrite this expression as c^2 = e^x. Now we use equation 2 which states that log(c^2) = y. This is equivalent to 2log(c) = y.Using equation 1, we can substitute log(c) for x/2.2log(c) = y2(x/2) = yx = y/2So the relationship between X and Y is X = Y/2. Therefore, we can conclude that the relationship between a dependent and an independent variable expressed as a logarithmic equation is as follows: X = Y/2.

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Answer:

1 : 4 minutes.

Step-by-step explanation:

Given: 15 minutes / 1 hour.

First convert 1 hour to minutes:

1 hour = 60 minutes

Then write the fraction:

15 : 60

Finally, simplify it:

1 : 4 minutes

The simplified square expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

What is square root ?

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 x 3 = 9.

The square root is denoted by the symbol √, also known as the radical symbol. For instance, the square root of 16 is written as √16 = 4.

The square root can be used to solve various types of equations, including quadratic equations and problems involving areas and volumes. It is also used in various fields such as physics, engineering, and finance.

According to the question:
To simplify the expression [tex]\sqrt{3a^{2}/10b^6}[/tex], we can first separate the numerator and denominator inside the square root:

[tex]\sqrt{3a^2/10b^6} = \sqrt{3a^2}/\sqrt{10b^6}[/tex]

Next, we can simplify the square root of the numerator:

[tex]\sqrt{3a^2} = |3a|,[/tex] where |За| represents the absolute value of За.

Finally, we can simplify the square root of the denominator by factoring out the perfect square[tex]b^2[/tex]:

[tex]\sqrt{10b^6} = \sqrt{10} * \sqrt{b^6} = \sqrt{10} * b^{3}[/tex]

Substituting these values back into the original expression, we get:

[tex]\sqrt{3a^2/10b^6} = |3a| / \(sqrt{10} * b^3[/tex]

Therefore, the simplified expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

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In the given diagram, ΔHIJ ≈ ΔFIG, the value of u in triangle FIG is calculated as: IG = 8 yd

A triangle is a geometric shape that is formed by three line segments that intersect at three non-collinear points. The three line segments are called the sides of the triangle, while the three points where they intersect are called the vertices of the triangle.

A triangle is a three-sided polygon formed by three line segments intersecting at three non-collinear points, and it can be classified based on the length of its sides and the measure of its angles.

∆HIJ similar to ∆FIG

HI/FI = IJ/IG

5/10 = 4/IG

IG = 8 yd

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In the given diagram, ΔHIJ ≈ ΔFIG, the value of u in triangle FIG is calculated as: IG = 8 yd

A triangle is a geometric shape that is formed by three line segments that intersect at three non-collinear points. The three line segments are called the sides of the triangle, while the three points where they intersect are called the vertices of the triangle.

A triangle is a three-sided polygon formed by three line segments intersecting at three non-collinear points, and it can be classified based on the length of its sides and the measure of its angles.

∆HIJ similar to ∆FIG
HI/FI = IJ/IG
5/10 = 4/IG
IG = 8 yd

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The option that is most likely a population as opposed to a sample is registered voters in a county. The correct answer is Option D.

A population is a group of individuals, objects, or events whose properties are being studied. A sample is a smaller subset of the population that is selected to represent the whole population.

The four options given in the question are all sets of individuals who are being studied, but the difference is how they were selected. Respondents to a newspaper survey, the first 5 students completing an assignment, and every third person to arrive at the bank are all examples of samples because they are subsets of a larger group.

However, registered voters in a county are most likely a population because they are the entire group that is being studied, and there is no need for a subset of this group since the entire population is available. Therefore, option d) registered voters in a county is the most likely population as opposed to a sample.

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Step-by-step explanation:

kaitlyn score is   6 points above the mean

                                 z-score =  6 / 12   = .5

kiersten score is 34.8 above the mean   z-score = 34.8/29 = 1.2

rebecca score is .54 above the mean   z -score = .54/ .3 = 1.8

rebecca scored the highest percentile (highes z-score)  of the three....the best

A linear transformation matrix is a square matrix that represents a linear transformation of a vector space. The composition of linear transformation matrices is equivalent to the composition of linear transformations they represent.

In general, the composition of two linear transformations A and B can be represented by the matrix product AB. Therefore, the composition of two linear transformation matrices, say A and B, would be the matrix product AB.

Linear transformations are functions that map vectors from one vector space to another in a linear way. These transformations can be represented by matrices, which can be composed to represent the composition of linear transformations.

The composition of two linear transformations represented by matrices involves multiplying the matrices together. To perform this multiplication, we need to ensure that the number of columns in the first matrix matches the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

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The relation you have described is neither reflexive, nor irreflexive, nor total, nor transitive.

A relation cannot be both reflexive and irreflexive. Hence, these two properties are mutually exclusive. If it is reflexive, then it is not irreflexive. If it is irreflexive, then it cannot be reflexive. Nonetheless, it is possible for a relation to be neither reflexive nor irreflexive.

A reflexive relation is one in which every element is related to itself. A relation is irreflexive if no element is related to itself. A total relation is one in which every element is related to every other element. A transitive relation is one in which, if element A is related to element B and element B is related to element C, then element A is related to element C.

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Suppose a population grows according to a logistic model with initial population 200 and carrying capacity 2,000. If the population grows to 500 after one year, the population will be 852.78 after another three years.

A logistic model, also known as the Verhulst-Pearl model, is a type of function used to describe population growth that is limited. It’s a form of exponential growth that takes into account the carrying capacity of an environment.

Population growth that is limited and slows down as the population approaches its carrying capacity is modeled using the logistic model. It is given by this equation:

[tex]P(t) = K / (1 + Ae^{-rt})[/tex]

where P(t) is the population at time t, K is the carrying capacity, A is the constant of proportionality, and r is the growth rate.

Suppose a population grows according to a logistic model with initial population 200 and carrying capacity 2,000. If the population grows to 500 after one year, substitute this information into the logistic model: [tex]P(1) = 500[/tex], [tex]K = 2000[/tex], and [tex]P(0) = 200[/tex].

[tex]500 = 2000 / (1 + Ae^{-r(1)})[/tex]

Now, solve for A by dividing both sides by 2000 / (1 + A):

[tex]1 + A = 4A = 3[/tex]

Substitute the value of A back into the logistic model equation:

[tex]P(t) = 2000 / (1 + 3e^{-rt})[/tex]

Solve for r by using the data provided in the problem for the first year (t = 1) and second year (t = 4):

[tex]P(1) = 500 = 2000 / (1 + 3e^{-r(1)})[/tex]

[tex]P(4) = ? = 2000 / (1 + 3e^{-r(4)})[/tex]

Solve the first equation for r:

[tex]500 = 2000 / (1 + 3e^{-r})\\1 + 3e^{-r} = 4e^{-r}\\1 + 3e^r = 4e[/tex]

Solve for e using the quadratic formula to get:

e = 0.4274 and e = 1.1713

Let e = 0.4274:

[tex]1 + 3e^{-r} = 4e^{-r}\\1 + 3(0.4274)^{-r} = 4(0.4274)^{r}\\1 + 0.5746^r = 1.7166^r[/tex]

Take the natural logarithm of both sides:

[tex]ln(1 + 0.5746^r) = ln(1.7166^r) - lnr\\ln(1 + 0.5746^r) = rln(1.7166) - lnr[/tex]

Use a graphing calculator to solve for r:

[tex]ln(1 + 0.5746^r) = rln(1.7166) - lnr[/tex];  -0.1568 < r < 0.7534

Solve for r using the second year’s data:

[tex]2000 / (1 + 3e^{-r(4)}) = P(4)\\2000 / (1 + 3(0.4274)^{-r(4)}) = P(4)\\P(4) = 852.78[/tex]

Thus, the population will be 852.78 after another three years.

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In equation A the missing number is 955, In equation B the missing number is 90 and In equation C the missing number is 805.

A) To find the missing number in the equation 872 + ? + 173 = 2000, we need to subtract 872 and 173 from 2000, which gives us:

2000 - 872 - 173 = 955

Therefore, the missing number is 955.

B) To find the missing number in the equation 9180 ÷ ? = 102, we need to divide 9180 by 102, which gives us:

9180 ÷ 102 = 90

Therefore, the missing number is 90.

C) To find the missing number in the equation ? - 99 = 706, we need to add 99 to 706, which gives us:

706 + 99 = 805

Therefore, the missing number is 805.

To obtain the indicated results with the same numbers and operations, we need to use parentheses to change the order of operations.

A) 3 + (5x7) - 2 = 40

B) (3 + 5) × 7 - 2 = 54

C) 3 + (5 × (7-2)) = 28

Equations are used extensively in various fields of science, engineering, economics, and finance, to name a few. It is formed by placing an equal sign between the two expressions. Equations are used to solve problems and find unknown values.

An equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The variables in an equation represent unknown values that need to be found, while the constants are known values that are already given. Solving an equation involves manipulating the expressions on both sides of the equal sign using mathematical operations to isolate the variable on one side and constants on the other. The final solution obtained is the value of the variable that satisfies the equation..

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Complete Question: -

Find unknown numbers of these operations

A ) 872 +. + 173 = 2000

B ) 9180:. = 102

C ). -99 = 706

With the same numbers and the same operations we can obtain different results, place the parentheses so that the indicated results are obtained.

A ) 3 + 5 x 7-2 = 40

B ) 3 + 5 × 7-2 = 54

C ) 3 + 5 × 7-2 = 28

IT'S FOR TODAY PLEASE ☹, CAN DO IN A LEAF OR WRITE ASI BUT EXPLAIN WELL!!!!!!HELP IF THEY DON'T KNOW NO RESPOND

To understand why b = d = 0 if the function is even, we need to consider the definition of an even function.Therefore If P(x) is an even function, then b = d = 0.

A polynomial is a mathematical expression that consists of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. It can have one or more terms and can be of any degree.

An even function is a function that satisfies the condition f(x) = f(-x) for all x in the domain of the function.

If P(x) is an even function, then we have P(x) = P(-x) for all x. Substituting -x for x in the expression for P(x), we get:

P(-x) = a(-x)⁴ + b(-x)³ + c(-x)² + d(-x) + e

= a(x⁴) - b(x³) + c(x²) - d(x) + e

Since P(x) = P(-x), we can equate the two expressions for P(x) and P(-x) to get:

a(x⁴) + b(x⁴) + c(x²) + d(x) + e = a(x⁴) - b(x³) + c(x²) - d(x) + e

Simplifying this equation, we get:

2b(x³) + 2d(x) = 0

Since this equation holds for all values of x, we can set x = 0 to get:

2d(0) = 0

which implies that d = 0. Similarly, setting x = 1, we get:

2b(1³) + 2d(1) = 0

2b = 0

b = 0

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Answer: The first restaurant had a greater median time

the range for the first restaurant is 35 minutes and the range for the second restaurant is 39 minutes

The second restaurant had more wait times from 30-39 minutes

Answer: A = 55* or 55 degrees

Step-by-step explanation:

The interior angles of a triangle always have a sum of 180 degrees.

75 + 50 + x = 180

Solve for x

FInal answer: 55 degrees

hope i explained it :)

The figures created are a square and a right triangle, and the perimeter of the entire figure is (13x/3) + x × sqrt(10).

When Namandu divides the top side of the square into three equal parts, he creates two segments of length x/3 each. By connecting the first point of division with the vertex of the parallel side, he creates a right triangle with legs of length x/3 and x, and hypotenuse of length y.

Using the Pythagorean theorem, we can solve for y:

y^2 = (x/3)^2 + x^2

y^2 = x^2/9 + x^2

y^2 = (10x^2)/9

y = x×sqrt(10)/3

Now we can find the perimeter of each figure that is created

Perimeter of the original square = 4x

Perimeter of the right triangle = x + x/3 + y = x + x/3 + xsqrt(10)/3

Perimeter of the entire figure = 4x + x + x/3 + xsqrt(10)/3 = (13x/3) + x×sqrt(10)

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1.1) If Thabo and Amon decide to share the R400 in a ratio of 3.2, the amount Thabo will receive is R240.

1.2) If Thabo used 1/5 (a fraction) of his money towards a secondhand video game, the cost of the secondhand video game is R48.

1.3.1) If Thabo's transport fare to school has increased from R800 to R1,150, R1,150 written in words, is One Thousand, One Hundred and Fifty rupees.

1.3.2) If Thabo's transport fare to school has increased from R800 to R1,150, the percentage increase in the transport fare is 43.75%.

The ratio refers to the relative size of one quantity compared to another.

Ratios are formally written using the ratio standard form (:).  Ratios can also be expressed as decimals, percentages, and fractions.

The total amount received from the cleaning job = R400

The number of hours Thabo worked = 3 hours

The number of hours Amon worked = 2 hours

1.1) Sharing ratio = 3:2

The sum of ratios = 5

Thabo's share of R400 = R240 (R400 x 3/5)

1.2) The cost of the video game = R48 (R240 × ¹/₅)

Old transport fare = R800

Current transport fare = R1,150

Increase in transport fare = R350 (R1,150 - R800)

Percentage increase in transport fare = 43.75% (R350/R800 x 100)

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The statement that is correct is (D) We can use the binomial distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.

The binomial distribution can be used to calculate the probability of a certain number of successes in a given number of trials, where each trial has a fixed probability of success.

The probability of a flight being delayed is 0.24, and the probability of a flight not being delayed is 0.76. Therefore, the probability of exactly 10 flights out of 100 being delayed can be calculated using the binomial distribution with n = 100, k = 10, and p = 0.24.

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Answer:

[tex]\large\boxed{\textsf{x = 7}}[/tex]

Step-by-step explanation:

[tex]\textsf{For this problem, we are asked to find the value of x.}[/tex]

[tex]\textsf{We should simply isolate the x so that it's only on one side.}[/tex]

[tex]\large\underline{\textsf{How?}}[/tex]

[tex]\textsf{Simply use the Distributive Property for the right side of the equation.}[/tex]

[tex]\textsf{Simplify the equation to where x is by itself.}[/tex]

[tex]\large\underline{\textsf{What is the Distributive Property?}}[/tex]

[tex]\textsf{The Distributive Property is a Property that allow us to distribute expressions further.}[/tex]

[tex]\textsf{Commonly, the form is a(b+c); Where b and c are multiplied by a.}[/tex]

[tex]\large\underline{\textsf{Use the Distributive Property;}}[/tex]

[tex]\mathtt{5x-2=3(x+4)}[/tex]

[tex]\mathtt{5x-2=(3 \times x)+(3 \times 4)}[/tex]

[tex]\mathtt{5x-2=3x+12}[/tex]

[tex]\large\underline{\textsf{Add 2 to Both Sides of the Equation;}}[/tex]

[tex]\mathtt{5x-2 \ \underline{+ \ 2}=3x+12 \ \underline{+ \ 2}}[/tex]

[tex]\mathtt{5x=3x+14}[/tex]

[tex]\large\underline{\textsf{Subtract 3x from Both Sides of the Equation;}}[/tex]

[tex]\mathtt{5x-3x=3x-3x+14}[/tex]

[tex]\mathtt{2x=14}[/tex]

[tex]\large\underline{\textsf{Divide the Whole Equation by 2;}}[/tex]

[tex]\mathtt{\frac{2x}{2} = \frac{14}{2} }[/tex]

[tex]\large\boxed{\textsf{x = 7}}[/tex]

Answer:

[tex] \sf \: x = 7[/tex]

Step-by-step explanation:

Now we have to,

→ Find the required value of x.

The equation is,

→ 5x - 2 = 3(x + 4)

Then the value of x will be,

→ 5x - 2 = 3(x + 4)

→ 5x - 2 = 3(x) + 3(4)

→ 5x - 2 = 3x + 12

→ 5x - 3x = 12 + 2

→ 2x = 14

→ x = 14 ÷ 2

→ [ x = 7 ]

Hence, the value of x is 7.

Sgr A has a diameter of about 27.3 million miles; that's roughly the same distance from Mercury to The Sun.

Distance is a measurement of how far apart two objects or points are, either numerically or rarely qualitatively. Distance can relate to a physical length in physics or to an estimate based on other factors in everyday language (e.g. "two counties over"). The term is frequently used metaphorically to refer to a measurement of the difference between two similar objects (such as statistical distance between probability distributions or edit distance between strings of text) or a degree of separation, as spatial cognition is a rich source of conceptual metaphors in human thought (as exemplified by distance between people in a social network). The concept of a metric space is used in mathematics to formalize the majority of these ideas of distance, both literal and figurative.

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The triangles are similar by the AA similarity criterion since they have two pairs of congruent angles (41° and 109°, and 30°). Therefore, the answer is B. AA.

A triangle is a polygon that has three sides, three vertices, and three angles. It is the simplest polygon in Euclidean geometry and one of the most studied shapes in mathematics. The sum of the three angles in a triangle is always 180 degrees, and the length of one side of a triangle is always less than the sum of the other two sides. There are many different types of triangles, including equilateral triangles (where all sides are equal), isosceles triangles (where two sides are equal), and scalene triangles (where no sides are equal). Triangles are used in many areas of mathematics and science, including trigonometry, geometry, and physics.

Here,

To determine whether the two triangles are similar using AA similarity, we need to compare the angles of both triangles.

Triangle 1 has angles 41°, 30°, and 109°.

Triangle 2 has angles 30°, 41°, and 109°.

We see that the two triangles have two pairs of congruent angles, namely 41° and 41°, and 109° and 109°. Therefore, the two triangles are similar by the AA similarity criterion.

Note that the order of the angles does not matter when comparing triangles for similarity.

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the principal in Account B is 4.8 times the principal in Account A.

How to solve?

Let the principal in Account A be P and the principal in Account B be Q. Also, let n be the number of months.

From the given information, we have:

Interest earned in Account A = $6.00

Interest rate in Account A = 4%

Number of months = n

Using the formula for simple interest, we have:

Interest earned in Account A = (P × 4%× n) / 12

Substituting the given values, we get:

6.00 = (P × 4% × n) / 12

P× n = 150

Similarly, we have:

Interest earned in Account B = $30.00

Interest rate in Account B = 5%

Number of months = n

Using the formula for simple interest, we have:

Interest earned in Account B = (Q× 5% × n) / 12

Substituting the given values, we get:

30.00 = (Q× 5% ×n) / 12

Q × n = 720

To compare the principals, we can divide the equation for Account B by the equation for Account A:

(Q × n) / (P× n) = 720 / 150

Simplifying, we get:

Q / P = 4.8

Therefore, the principal in Account B is 4.8 times the principal in Account A.

Since the interest rate in Account B is higher than the interest rate in Account A, we can conclude that the principal in Account B is greater than the principal in Account A.

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Answer:

24.4 degrees

Step-by-step explanation:

This is a right triangle so you can use trig to solve. If you take the arctan of 5/11 you get 24.4(rounded to the nearest tenth)

Answer:

25

Step-by-step explanation:

2x-14=36

2x=36+14

2x=50

x=25

Answer:

Step-by-step explanation:

The area of a circle is given by the formula:

A = πr^2

where A is the area and r is the radius.

Substituting the given values, we get:

A = π(2.1 m)^2 = 13.85 m^2

So the total cost of the glass with a price of $7.10 per square meter is:

Cost = Price per square meter x Total area

Cost = $7.10/m^2 x 13.85 m^2 = $98.24

Therefore, the cost of the glass is $98.24.

Answer:

For WXYZ to be a parallelogram, both pairs of opposite sides must be parallel.

So, we have 7m + 4 = 9m and 5y + 13 - (2n - 1) = 6y - 8 - (n + 6).

Solving the first equation, we get m = 2.

Solving the second equation, we get y - n = 4.

Thus, for WXYZ to be a parallelogram, m = 2 and y - n = 4, so option B is correct.

The probability of getting an even number and a yellow disc is 1/6 1/3 = 1/18. This is due to the fact that there is only one method to obtain an even number and a yellow disc.

Probabilistic theory is a branch of mathematics that calculates the chance of an event or a claim being true. A risk is a number between 0 and 1, where 1 represents certainty and a probability of about 0 shows how likely an event appears to be to occur. Probability is a mathematical term for the likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1, or as percentages ranging from 0% to 100%. The proportion of occurrences among equally likely choices that result in a certain event in compared to all possible outcomes.

the answers to the questions:

a) The chance of getting a yellow disc is 1/6 + 1/6 = 1/3. This is because there are two methods to acquire a yellow disc: receive an even number and then choose the yellow disc, or get an odd number and then choose the yellow disc.

b) The chance of getting a blue disc is 1/6 + 1/6 = 1/3. This is because there are two methods to receive a blue disc: get an even number and then choose the blue disc, or get an odd number and then choose the blue disc.

c) The chance of getting an odd number with a red disc is (1/6 + 1/6) 1/3 = 1/18. This is because there are two ways to acquire an odd number, and there is only one way to get a red disc for each of those two methods.

d) The chance of getting an even number and a yellow disc is 1/6 1/3 = 1/18. This is due to the fact that there is only one method to obtain an even number and a yellow disc.

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a) the lengths of each side:

AC = 3, BC = 1, CE = 3, CD = √2.

b) < C ≅ < C, corresponding angles of congruent triangles.

c) ΔACE ≅ ΔBCD by Side-Angle-Side (SAS) criterion,

d) ∠CBD  ≅  ∠A, vertical angles.

What is the similarity of a triangle?

If the two sides of a triangle are in the same proportion of the two sides of another triangle, and the angle inscribed by the two sides in both triangles are equal, then two triangles are said to be similar.

a)

AC = √[(2-2)² + (4-1)²] = √9 = 3

BC = √[(2-2)² + (4-3)²] = 1

CE = √[(5-2)² + (1-1)²] = 3

CD = √[(3-2)² + (3-4)²] = √2

b) < C ≅ < C because they are corresponding angles of congruent triangles ΔACE and ΔBCD.

c) ΔACE ≅ ΔBCD by Side-Angle-Side (SAS) criterion, since AC = BC, CE = CD and < C ≅ < C.

d) < CBD < A because the point B lies on AC, and therefore < A and < CBD are vertical angles, and by definition, vertical angles are congruent.

Hence, a) the lengths of each side:

AC = 3, BC = 1, CE = 3, CD = √2.

b) < C ≅ < C, corresponding angles of congruent triangles.

c) ΔACE ≅ ΔBCD by Side-Angle-Side (SAS) criterion,

d) ∠CBD  ≅  ∠A, vertical angles.

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