A ball is dropped from a height of 10 ft. Assuming that on each bounce, the ball rebounds to one-fifth of its previous height, find the total distance traveled by the ball.

The zeros and their multiplicities of the function are 1/([tex]\sqrt{3}[/tex],-1/[tex]\sqrt{3}[/tex],-3,3 and 1,1,1,1.

The value which satisfy the polynomial. that is on substituting the value in the polynomial will give 0.

To find the zeros of the function, we need to solve the equation f(x) = 0. We can do this by factoring the polynomial using the difference of squares formula:

[tex]\begin{aligned} f(x)&=9 x^{4}-82 x^{2}+9 \ &= (3x^2 - 1)(3x^2 - 9) \ &= 3(x-\frac{1}{\sqrt{3}})(x+\frac{1}{\sqrt{3}})(x-3)(x+3) \end{aligned}[/tex]

From this factorization, we can see that the zeros of the function are:

x = 1/([tex]\sqrt{3}[/tex],-1/[tex]\sqrt{3}[/tex],-3,3

Therefore, the zero of f is (1/3, -1/3, 3, -3). To determine the multiplicities of each zero, we can look at the degree of each factor.

For the factors (x - 3) and (x + 3), we have a degree of 1, which means that each of these zeros has multiplicity 1.

For the factor (x - 1/√3), we have a degree of 1 as well, which means that this zero also has multiplicity 1.

For the factor (x + 1/√3), we have a degree of 1, which means that this zero also has multiplicity 1.

Therefore, the zeros and their multiplicities of the function are 1/([tex]\sqrt{3}[/tex],-1/[tex]\sqrt{3}[/tex],-3,3 and 1,1,1,1.

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3.31 is an estimate for the radius of the circle.

The circle fοrm is a clοsed twο-dimensiοnal shape because every pοint in the plane that makes up a circle is evenly separated frοm the "centre" οf the fοrm.

Each line tracing the circle cοntributes tο the fοrmatiοn οf the line οf reflectiοn symmetry. In additiοn, it rοtates arοund the center in a symmetrical manner frοm every perspective.

he radius of the circle, we need to use the formula for the area of a sector, which is:

Area of sector = (θ/360) x π x r

the radius of the circle "x" for now, and set up an equation using the given information

51.3 = (120/360) x π x

Simplifying this equation, we get:

= (51.3 x 360)/(120 x π)

x² ≈ 10.95

x ≈ √10.95

x ≈ 3.31

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Answer:

47.6 inches

Step-by-step explanation:

diameter is just double the radius

23.8×2= 47.6 inches

Answer:

1.     m∠1 = 60° because angle ∠1 and the 60° angle are vertical angles

2.    ∠1 and ∠2 are adjacent.

3.    m∠2 = 180° - 60°

Step-by-step explanation:

m∠2 is not equal to 60° because it is the complementary angle of the one labeled 60°. Therefore, m∠2 = 30°. m∠2 can be determined by the information given. m∠1 ≠ m∠2 because m∠1 = 60° and m∠2 = 30°

Answer:

[tex]14in^2[/tex]

Step-by-step explanation:

Volume of a rectangluar prism = Length x Width x Height

Volume, Length, and Width is given now we can solve for Height, y

[tex]V = l * w * h[/tex]

[tex]3in^3 = 1in * 1in* h[/tex]

[tex]h = 3in[/tex]

Surface area of a rectangular prisim is A = 2(wl +hl +hw)

A = [tex]2(1in * 1in + 3in * 1in + 3in * 1in)\\2(1 in^2+ 3 in^2+ 3in^2)\\2(7in^2)\\14in^2[/tex]

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Answer:

3y-16

y-5=x

4x-y+4

4(y-5)-y+4

4y-20-y+4

3y-16  

Therefore , the solution of the given problem of equation comes out to be  y = 0.5x²- 3

The foundation of a model of linear regression is the equation y=mx+b. The inclination is B, and the y-intercept is m. Although it is true that y and y are separate parts, the the above sentence is frequently referred to as a "mathematics problem with two variables". Y=mx+b is the formula for a singular system of equations, where m denotes the gradient and b the y-intercept. The equation is Y=mx+b, where m denotes slopes and b denotes the y-intercept.

Here,

=> x  y

  -4  5

   -2  -1

   0  -3

    2  -1

To find : the equation which passes through the point

=> y = 0.5x²- 3

We can see,

=> y = 0.5(-4)² - 3

=> y = 8 -3

=> y =5

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The present ages of the child and woman are 4 and 16, respectively.

A quadratic equation is a polynomial equation of second degree in one variable. It has the structure:

ax² + bx + c = 0

where the variable x and the constants a, b, and c. There may be one, two real solutions to a quadratic equation or two complex solutions. They are helpful in finding the roots of polynomials and other parabolic shape-related problems, such as projectile motion. Quadratic equations can be solved using a variety of strategies, including factoring, completing the square, and the quadratic formula.

Let us suppose the current age of the child = x.

The current age of the woman is 4x.

5 years ago, the age of the child = x - 5.

The age of the woman was 4x - 5.

Thus,

(x - 5)(4x - 5) = 175

4x² - 25x + 20 = 0

4x² - 25x + 20 = 0

(4x - 5)(x - 4) = 0

x = 5/4 or x = 4.

We can discard the first solution, as it implies a negative age for the woman.

Therefore, the present ages of the child and woman are 4 and 16, respectively.

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A sequence An is defined recursively by the equation aₙ = 0.5( aₙ₋₁ + aₙ₋₂ ) for n ≥ 3

First five terms of this sequence are 14, 14, 14, 14, 14 respectively.

A sequence is an ordered list of numbers. Like 1,2,3,.... The three dots implies to continue forward by following the pattern. Each number in the sequence is called a term. We have a sequence 'aₙ' and it is defined as aₙ = 0.5( aₙ₋₁ + aₙ₋₂ ) for n≥ 3 and

First term of sequence, a₁ = 14

second term of sequence, a₂ = 14

We have to determine value of first five terms of sequence.

Third term of sequence, n = 3

a₃ = 0.5( a₂ + a₁ )

=> a₃ = 0.5( 14 + 14)

=> a₃ = 0.5 × 28 = 14

Forth term of sequence, n = 4

a₄ = 0.5( a₃ + a₂ )

=> a₄ = 0.5( 14 + 14 ) = 14

Fifth term of sequence, n= 5

a₅ = 0.5( a₄ + a₃ )

=> a₅ = 0.5( 14+ 14) = 14

Hence, required terms are obtained and sequence is 14,14,14,14,..... ( constant sequence).

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Complete question:

A sequence, An is defined recursively by an = 0.5(an-1 + an-2) for n ≥ 3, where a1 = 14 and a2 = 14. Find the first five terms of the sequence.

Answer:

The remainder when 5x3 + 2x2 - 7 is divided by x + 9 is -692.

Explanation:

We can use long division to divide 5x^3 + 2x^2 - 7 by x + 9:

-5x^2 + 43x - 385

x + 9 | 5x^3 + 2x^2 + 0x - 7

5x^3 + 45x^2

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

-43x^2 + 0x

-43x^2 - 387x

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

387x - 7

Therefore, the remainder when 5x^3 + 2x^2 - 7 is divided by x + 9 is 387x - 7.

Hope this helps, sorry if this is wrong! :]

Answer:

a) x = 20

b) x = 9

Step-by-step explanation:

  a)     [tex]\boxed{\bf Mean =\dfrac{sum \ of \ all \ the \ data}{Number \ of \ data}}[/tex]

          [tex]\dfrac{16 +x + 3 + 14 +57}{5}=22\\[/tex]

                                 x + 90 = 22*5

                                 x + 90 = 110

                                         x = 110 - 90

                                        x = 20

b)

         [tex]\bf \dfrac{4 + x + 5 +6 + 1}{5} = 5[/tex]

                          x + 16   = 5 * 5

                            x + 16 = 25

                                   x = 25 - 16

                                   x = 9

Answer:

Mario's heart beats 112 times per minute.

To find out, we can use the following equation:

Heart rate = number of beats / time

Since we're given that Mario's heart beats 28 times in 1/4 minute, we can set up the equation like this:

Heart rate = 28 / 1/4

To divide by a fraction, we can multiply by its reciprocal:

Heart rate = 28 x 4/1

Heart rate = 112

Therefore, Mario's heart rate is 112 beats per minute.

The volume of the top half of the cone (in terms of π) is 4π whose height is 6 inches and radius is 4 inches.

The volume of the top half of the cone (in terms of π) is 4π whose height is 6 inches and radius is 4 inches.

It is a physical quantity that is typically expressed in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).

When the cone is sliced in half by a horizontal plane, the resulting cross-section is a circle with radius 2 inches. This circle has half the diameter of the original base of the cone, which means that it has half the area. We can use this fact to find the volume of the top half of the cone.

The original cone has height 6 inches and radius 4 inches, so its volume is given by:

V = (1/3)π(4²)(6) = (1/3)π(96) = 32π

When the cone is sliced in half, the top half has volume equal to half the volume of the original cone that lies above the horizontal plane. The height of this top half can be found using similar triangles: the radius of the top half is half the radius of the original cone, so the height of the top half is half the height of the original cone. Therefore, the height of the top half is 3 inches.

The radius of the top half is given as 2 inches, which means that its volume is:

V = (1/3)π(2²)(3) = (1/3)π(12) = 4π

Therefore, the volume of the top half of the cone (in terms of π) is 4π.

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Answer:

A)  [tex]15x^2+22x+8[/tex]

B)  Degree 2, Quartic Expression

C)  The dimensions of the rectangle are polynomials. When multiplied together, the area of the rectangle is also a polynomial.

Step-by-step explanation:

A)  The formula for area of a rectangle is Area = L * W. The length and width are represented by (3x+2) and (5x+4). So we can say that

[tex]Area = (5x+4)(3x+2)[/tex]

Use FOIL to multiply the the polynomials. First, Outside, Inside, Last

[tex]Area = (5x)(3x) + (5x)(2) + (3x)(4) + (2)(4)\\Area = 15x^2 + 10x + 12x + 8\\Area = 15x^2 + 22x + 8[/tex]

The expression that represents the area of the rectangle is

[tex]15x^2 + 22x + 8[/tex].

B)  The degree of the expression is 2, because two is the highest power of x. The classification of the expression is quadratic because the graph of the expression is a parabola.

Degrees vs. Classification

Degree 0:  Zero Polynomial or Constant

Degree 1:  Linear (line)

Degree 2:  Quadratic (parabola)

Degree 3: Cubic

Degree 4: Quartic

...

C)  Closure property for polynomials applies to addition, subtraction, and multiplication. It means that the result of multiplying two polynomials will also be a polynomial. Part A demonstrates polynomial closure under multiplication because the dimensions of the rectangle are polynomials and so is the area.

Answer:

To evaluate this expression, we need to follow the order of operations, which is commonly remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction):

1. Start with any operations inside parentheses. There are no parentheses in this expression.

2. Evaluate any exponents. There are no exponents in this expression.

3. Evaluate multiplication and division, from left to right. In this expression, we have:

7 + (35/3) + 5 - (7 + (35/3) / (3 - 5))

= 7 + (35/3) + 5 - (7 + (11.67) / (-2))

= 7 + (35/3) + 5 - (7 - 5.835)

= 7 + (35/3) + 5 + 5.835 - 7

= 15.835 + (35/3) - 7

4. Finally, evaluate addition and subtraction, from left to right:

15.835 + (35/3) - 7

= 15.835 + 11.67

= 27.505

Therefore, the value of the expression is 27.505.

Step-by-step explanation:

a) The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2.

b) P(W1 = 1)P(W2 > 1) + P(W1 = 1)P(W2 > 2) + P(W1 = 2)P(W2 > 3) + ...= p1(1 - p2) + p1(1 - p2)p2 + p1p2(1 - p2) + ...= p1(1 - p2)(1 +p2 + p22 + ...) = p1(1 - p2)/(1 - p2)2= p1/(1 - p2)2

c) The probability that their minimum value is k is P(min(W1, W2) = k) = P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1(1 - p2)k-1p2 + p2(1 - p1)k-1p1

d) The probability that their maximum value is k is P(max(W1, W2) = k) = P(W1 = k, W2 = k) + P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1p2k-1 + p1(1 - p2)k-1 + p2(1 - p1)k-1

a) P(W1 = W2)The probability that W1 = W2 is 0. If W1 and W2 have different values, then W1 is equal to either 1 or 2. The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2. The probability of that happening is P(W1 = 1, W2 = 2) = P(W1 = 2, W2 = 1) = p1p2.b) P(W1 < W2)The probability of W1 being less than W2 is P(W1 = 1, W2 = 2) + P(W1 = 1, W2 = 3) + P(W1 = 2, W2 = 3) + ... This may be written as P(W1 = 1)P(W2 > 1) + P(W1 = 1)P(W2 > 2) + P(W1 = 2)P(W2 > 3) +...= p1(1 - p2) + p1(1 - p2)p2 + p1p2(1 - p2) + ...= p1(1 - p2)(1 + p2 + p22 + ...) = p1(1 - p2)/(1 - p2)2= p1/(1 - p2)2

d) Distribution of min(W1, W2)If W1 and W2 are independent geometric random variables with parameters p1 and p2, respectively, then the probability that their minimum value is k is P(min(W1, W2) = k) = P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1(1 - p2)k-1p2 + p2(1 - p1)k-1p1e) Distribution of max(W1, W2)If W1 and W2 are independent geometric random variables with parameters p1 and p2, respectively, then the probability that their maximum value is k is P(max(W1, W2) = k) = P(W1 = k, W2 = k) + P(W1 = k, W2 > k) + P(W1 > k, W2 = k) = p1p2k-1 + p1(1 - p2)k-1 + p2(1 - p1)k-1

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The equation p = 8.50h allows us to calculate Eric's pay for any number of hours worked.

An equation is a mathematical statement that shows the equality of two expressions or quantities. It typically contains variables, constants, and mathematical operators such as addition, subtraction, multiplication, and division. Equations are often used to solve problems and find unknown values.

Here,

Eric's pay is $8.50 per hour, so we can write:

p = 8.50h

where p is his pay and h is the number of hours worked.

Let's take an example, if Eric works for 20 hours, we can substitute h = 20 into the equation to find his pay:

p = 8.50(20)

p = 170

So Eric's pay for working 20 hours is $170.

Similarly, if Eric works for 40 hours, we can substitute h = 40 into the equation:

p = 8.50(40)

p = 340

So Eric's pay for working 40 hours is $340.

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Commission is usually calculated as a percentage of the selling price of a product or service. Therefore, the correct answer is option B, where the commission is calculated as (percent commission) x (selling price). For example, if the commission rate is 5% and the selling price is $100, the commission earned would be $5, calculated as follows:

Commission = (5%)( $100) = $5

This means that the salesperson or agent would earn $5 in commission for selling the product or service at a price of $100. The commission rate can vary depending on the type of product or service and the agreement between the parties involved.

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The formula would be: =BINOM.DIST(3,9,0.25,TRUE) + BINOM.DIST(2,9,0.25,TRUE) + BINOM.DIST(1,9,0.25,TRUE) + BINOM.DIST(0,9,0.25,TRUE). Probability of this outcome is 0.372.

The probability of drawing at most three spade cards when drawing nine cards with replacement from a standard 52-card deck is 0.628. This probability was calculated using the binomial distribution formula in Microsoft Excel. The formula takes into account the number of trials (nine), the probability of success (drawing a spade card), and the number of successes desired (at most three). The result shows that there is a relatively high likelihood of drawing at least four spade cards, as the probability of this outcome is 0.372.

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The exact volume of the cylinder is 192π cubic inches

The volume of a cylinder can be calculated using the formula:

V = πr²h

where V is the volume, r is the radius, and h is the height.

Given the radius r = 8 inches and the height h = 3 inches, we can substitute these values into the formula to get:

V = π(8²)(3)

V = π(64)(3)

V = 192π

So the exact volume of the cylinder is 192π cubic inches. Since this is an exact answer, we can leave it in terms of π, or we can use a calculator to get a numerical approximation of V

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The pass rate in the afternoon subgroup is greater than 1, we cannot draw a reliable conclusion without further investigation or clarification of the data.

To compare the probability that a student will pass the challenge in the morning with the probability that a student will pass the challenge in the afternoon, we need to calculate the proportion of students who passed in each subgroup.

The proportion of students who passed the challenge in the morning subgroup is:

Pass rate in the morning subgroup = Number of students who passed in the morning subgroup / Total number of students in the morning subgroup

Pass rate in the morning subgroup = 40 / 60

Pass rate in the morning subgroup = 0.67

The proportion of students who passed the challenge in the afternoon subgroup is:

Pass rate in the afternoon subgroup = Number of students who passed in the afternoon subgroup / Total number of students in the afternoon subgroup

Pass rate in the afternoon subgroup = 50 / 40

Pass rate in the afternoon subgroup = 1.25

that the pass rate in the afternoon subgroup is greater than 1, which is not possible as probabilities must be between 0 and 1. This implies that there might be a data error.

Assuming the data is correct, we can conclude that the probability of passing the challenge in the afternoon subgroup is higher than the probability of passing the challenge in the morning subgroup. However, since the pass rate in the afternoon subgroup is greater than 1, we cannot draw a reliable conclusion without further investigation or clarification of the data.

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Step-by-step explanation:

2.1 Factoring 3x2-2x-23

The first term is, 3x2 its coefficient is 3 .

The middle term is, -2x its coefficient is -2 .

The last term, "the constant", is -23

Step-1 : Multiply the coefficient of the first term by the constant 3 • -23 = -69

Step-2 : Find two factors of -69 whose sum equals the coefficient of the middle term, which is -2 .

-69 + 1 = -68

-23 + 3 = -20

-3 + 23 = 20

-1 + 69 = 68

Answer:

4

Step-by-step explanation:

(is the equation below x - 7 or x +7, I am working with x + 7)

9+ (2x - 6) is equivalent to x + 7

9 + (2x - 6) = x + 7

Open the bracket

9 + 2x - 6 = x + 7

2x + 3 = x +7

Subract 3 from both sides

2x = x + 4

Subtact x from both sides

X = 4

Confirm if the equation in the question is x +7 or x - 7

If the volume of a right cone is 3,525, then the height of the cone is approximately 5 units.

We can use the formula for the volume of a cone to solve for the height of the cone:

V = (1/3)πr²2h

where V is the volume of the cone, r is the radius (half of the diameter), and h is the height of the cone.

Given that the diameter of the cone is 30 units, the radius is 15 units. Substituting this and the volume V = 3525, we get:

3525 = (1/3)π(15)²2h

3525 = 225πh/3

h = 3(3525)/(225π)

h ≈ 5

Therefore, the height of the cone is approximately 5 units.

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The value of x must be x = 18, assuming that K is a point on the segment JL.

Here we have a segment JL, such that K is a point between J and L.

Here we know that the lengths are:

JK = x + 7

KL = 3x + 25

JL = 7x - 22

We know that the sum of the two first ones should be equal to the total segment:

JK + KL = JL

Then we can write the equation:

x+ 7 + 3x + 25 = 7x - 22

Solving this for x.

4x + 32 = 7x - 22

32 + 22 = 7x - 4x

54 = 3x

54/3 = x = 18

that is the value of x.

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Answer:

8.41

Step-by-step explanation:

8.14+3.45=11,59

20-11,59=8.41

Here is an example table for tracking the flowers stocked in the flower shop, based on the information provided in Question 4, Part I:

A flower is the reproductive structure found in flowering plants.

Flower Type Color Initial Quantity

Roses          Red 50

Roses          Pink 30

Tulips         Yellow 25

Tulips          Red 20

Lilies         White 40

Lilies          Pink 15

In this table, each row represents a specific flower type and color, and the initial quantity of that flower type and color that the shop stocks. This table can be used to track inventory levels and monitor when certain types and colors of flowers need to be restocked.

Additional columns can be added as needed to track other information, such as supplier information, purchase dates, or prices.

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value of variable x is 65 degree.

A straight line is a geometric object that extends infinitely in both directions and has a constant slope or gradient. It is the shortest distance between two points, and it can be described by an equation in the form of y = mx + c, where m is the slope or gradient of the line and c is the y-intercept.

The term "supplementary angles" refers to two angles whose sum is equal to 180 degrees. In other words, if angle A and angle B are supplementary angles, then:

∠A +∠B = 180 degrees

To find value of x

A straight line's angle total is 180°.

2x+50°=180°

2x=130°

x=65

Hence, value of variable x is 65 degree.

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a)  approx 0.562$$

When playing backgammon, the doubling cube is an important feature that allows players to increase the stakes of the game. Imagine playing with a "tripling cube" instead, with faces of 3, 9, 27, 81, 243, and 729. Using the analysis given for the doubling cube, we can compute the probability of winning above which a triple should be accepted.To determine the probability of winning above which a triple should be accepted, we need to use the formula derived from the analysis of the doubling cube:$$P_w = \frac{q^2}{1-2q^2}$$where Pw is the probability of winning, and q is the probability of losing. We can substitute 1-q for p, the probability of winning, to get:$$P_w = \frac{(1-p)^2}{1-2(1-p)^2}$$Now, we need to modify this formula to account for the tripling cube. If we triple the current stakes, then we have effectively tripled the value of the doubling cube. In other words, if the current stakes are 1, then the value of the tripling cube is 3. If the current stakes are 2, then the value of the tripling cube is 9. More generally, if the current stakes are n, then the value of the tripling cube is 3^n. Using this information, we can modify the formula as follows:$$P_w = \frac{q^{3^n}}{1-2q^{3^n}}$$. This is the formula we need to use to compute the probability of winning above which a triple should be accepted. We can solve for q using the quadratic formula:$$q = \frac{1\pm\sqrt{1-4(1-2P_w)(-P_w^{3^n})}}{2}$$The value of q we want is the smaller one, because we want to compute the probability of losing. Once we have q, we can substitute it into the formula for Pw to get the probability of winning above which a triple should be accepted. Example Suppose we have a backgammon game with a current stake of 4, and we are considering accepting a triple from our opponent. Using the formula above, we can compute the probability of winning above which a triple should be accepted as follows:$$n = \log_3{3^2} = 2$$$$P_w = \frac{q^{3^2}}{1-2q^{3^2}}$$$$q = \frac{1-\sqrt{1-4(1-2P_w)(-P_w^{3^2})}}{2}$$$$q = \frac{1-\sqrt{1-8P_w^9}}{2}$$$$q = \frac{1-\sqrt{1-8(0.5)^9}}{2}$$$$q \approx 0.515$$$$P_w = \frac{q^{3^2}}{1-2q^{3^2}}$$$$P_w = \frac{(0.515)^9}{1-2(0.515)^9}$$$$P_w \approx 0.562$$. Therefore, if we have a probability of winning greater than 0.562, then we should accept the triple.

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Answer:

2)

3x + 13 = 16

3x + 13 + (-13) = 16 + (-13)

3x = 3

3x / 3 = 3 / 3

x = 1

3)

2x / 2 = 20 / 2

x = 10

Activity 3:

1) x = 13, m‹O = 137°

2) x = 5, PG = 18

Step-by-step explanation: