In this picture, the white ball _____________ its momentum to the red ball. fill in the blank. please help i have to submit at midnight

The graph representing the inequality 2.5 is less than the product of −0.5 and a number is number line with open circle on point negative 5 and arrow shaded to the left.

Inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠.

Given inequality is,

2.5 is less than the product of −0.5 and a number.

2.5 < -0.5x

We have to isolate x on one side.

Dividing both sides by -0.5,

-5 > x

The inequality sign changes when a negative number is divided.

Or x < -5

So the solutions of the inequality are all the numbers less than -5.

So the graph of the inequality will be a number line where an open circle is at -5, since the point -5 is not included.

Since the solutions are less than -5, arrow is shaded to the left.

Hence the second option is the correct one.

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We have the following response after answering the given question: Hence, the comparable fractions with 12 as the denominator are: 5/12= 60/144 and  1/2 and 5/6 = 10/12

What is the equation?

In a mathematical equation, the equals sign (=), which connects two claims and denotes equality, is utilized. In algebra, an equation is a mathematical statement that proves the equality of two mathematical expressions. For instance, in equation 3x + 5 = 14, the equal sign separates the numbers by a space. Mathematical expressions can be used to describe the relationship between the two sentences on either side of a letter. The logo and the particular piece of software frequently correspond. like, for instance, 2x - 4 = 2.

2 x 2 x 3 is the prime factorization of 12.

2 x 1 is the prime factorization of the number 2.

Two by three is the prime factorization of six.

The LCD is the sum of all the prime factors with the largest powers, giving us 2 × 2 × 3

5/12 = 5/(12 * 1) = 5/12 * 12/12 = 60/144

1/2 = 1/2 * 6/6 = 6/12

5/6 = 5/6 * 2/2 = 10/12

Hence, the comparable fractions with 12 as the denominator are:

5/12 = 60/144

1/2  = 6/12

5/6 = 10/12

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The surface area of the prism is 840 square inches.

What is Prism?

A prism is a three-dimensional geometric shape that has two identical parallel bases and straight sides that connect the bases. The sides of a prism are typically rectangular, but can also be triangular or other polygonal shapes.

To find the surface area of the triangular prism, we need to find the area of the two triangular bases and the three rectangular faces.

Area of the triangular base:

The base is a right triangle with base 10 inches and height 24 inches. Therefore, the area of the base is:

(1/2) x base x height = (1/2) x 10 x 24 = 120 square inches.

Area of the rectangular faces:

The rectangular faces are all congruent, with dimensions of 10 inches by 20 inches. Therefore, the area of each rectangular face is:

length x width = 10 x 20 = 200 square inches.

There are three rectangular faces, so the total area of the rectangular faces is:

3 x 200 = 600 square inches.

Total surface area:

The total surface area of the prism is the sum of the areas of the two triangular bases and the three rectangular faces:

Total surface area = 2 x area of triangular base + 3 x area of rectangular face

= 2 x 120 + 3 x 200

= 240 + 600

= 840 square inches.

Therefore, the surface area of the prism is 840 square inches.

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One possible way to achieve this task is by using the following method:
1. Toss the biased coin twice.
2. If the result is two heads or two tails, discard the result and toss the coin again.
3. If the result is one head and one tail, record the result as a 1 if the first toss was a head and a 0 if the first toss was a tail.

This algorithm works because the probability of getting heads followed by tails or tails followed by heads is equal to 2p(1-p), which is symmetric around p=1/2. Therefore, the probability of outputting 1 is equal to the probability of outputting 0, which is 1/2. This method of generating a random process with a desired probability distribution from another random process is known as rejection sampling.

This method will produce a random process with outputs 1's and 0's with probability 1/2, regardless of the bias of the coin. This is because the probability of getting one head and one tail in two tosses is the same for both biased and unbiased coins, and we are only recording the result when we get one head and one tail. Therefore, this method will produce an unbiased random process from a biased coin.

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Option D, The next step in constructing an angle bisector of angle ABC is to use a straightedge to connect point C to the point where the angle bisector intersects side AB.

To construct an angle bisector of angle ABC, the next step is to use a straightedge to draw a line from vertex C to the point where the angle bisector intersects side AB.

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By using technology, the value of r² is equal to; C. 0.32.

In Mathematics, a coefficient of determination (r² or r-squared) can be defined as a number between zero (0) and one (1) that is typically used for measuring the extent (how well) to which a statistical model predicts an outcome.

Based on the given data, the correlation can be determined by using an online graphing calculator as shown in the image attached below. Since the value of r is equal to -0.56, the coefficient of determination (r²) can be calculated as follows;

r² = (-0.56)²

r² = 0.3136 ≈ 0.32

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Complete Question:

A guidance counselor wants to determine if there is a relationship between a student's number of absences, x, and

their grade point average (GPA), y. The given data list the number of absences and GPAs for 15 randomly selected students.

Number of Absences

15

1

0

6

9

12

3

3

1

2

7

0

4

9

10

GPA

2.1

4.3

4.5

3.2

4.0

1.7

3.8

2.9

3.6

3.4

2.6

3.1

2.8

2.8

4.1

Using technology, what is the value of r²?

O-0.56

O-0.32

O 0.32

O 0.56

If $8000 is invested at 8% compounded continuously, the amount 8 years is $12,904.95.

To find out how long it will take $3,000 to grow to $20,000 if it is invested at 4% compounded monthly, we can use the formula A=P(1+r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Plugging in the given values, we get:

20,000=3,000(1+0.04/12)^(12t)

Dividing both sides by 3,000, we get:

6.66667=(1+0.04/12)^(12t)

Taking the natural logarithm of both sides, we get:

ln(6.66667)=12t*ln(1+0.04/12)

Solving for t, we get:

t=ln(6.66667)/(12*ln(1+0.04/12))

t=41.2 years

So it will take approximately 41.2 years for $3,000 to grow to $20,000 if it is invested at 4% compounded monthly.

To find out the amount of $8,000 invested at 8% compounded continuously for 8 years, we can use the formula A=Pe^(rt), where A is the final amount, P is the principal amount, r is the annual interest rate, and t is the number of years. Plugging in the given values, we get:


A=8,000e^⁰ ⁶⁴

A=12,904.95

So the amount of $8,000 invested at 8% compounded continuously for 8 years is $12,904.95.

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The present value is $7,307.31.

To calculate the present value, we can use the formula: PV = A / (1 + r/n)^(n*t) where PV is the present value, A is the future value, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

In this case, we have A = $35,000, r = 15% = 0.15, n = 4 (since it is compounded quarterly), and t = 10.

Plugging these values into the formula, we get:

PV = $35,000 / (1 + 0.15/4)^(4*10)

PV = $35,000 / (1.0375)^40

PV = $35,000 / 4.79

PV = $7,307.31

Therefore, the present value is $7,307.31.

I ignored the "Ineedarim?" part of the question as it is irrelevant to the calculation.

$7,307.31.

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By taking the longer route, the amount of money Kelsey will save is $7.38.

To find out how much money Kelsey saves by taking the longer route, we need to calculate the total cost of each route and compare them.

For the shorter route, the total cost is the cost of gas plus the cost of tolls.
- The cost of gas is the distance traveled divided by the miles per gallon of the car, multiplied by the price of gas: (214 mi / 34.0 mpg)($3.29/gal) = $20.48
- The cost of tolls is $9.75
- So the total cost of the shorter route is $20.48 + $9.75 = $30.23

For the longer route, the total cost is just the cost of gas, since there are no tolls:
- The cost of gas is (236 mi / 34.0 mpg)($3.29/gal) = $22.85
- So the total cost of the longer route is $22.85

Now we can compare the two costs to find out how much money Kelsey saves by taking the longer route:
$30.23 (shorter route) - $22.85 (longer route) = $7.38

Therefore, Kelsey saves $7.38 by taking the longer route.

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The answer of the given question based on the histogram of Investment Account Values the correct option is c) 63%.

Investment refers to act of allocating resources, like  money or time, with  expectation of generating income or profit in  future. In other words, it involves sacrificing something of a value in  present, in hope of obtaining something of the greater value in  future

Investments can be taken  in many forms, including stocks, bonds, mutual funds, real estate, and commodities, among others. The choice of investment depends on variety of factors, including investor's risk tolerance, financial goals, and the investment time horizon.

Based on the histogram of Investment Account Values provided, it appears that approximately 63% of investments have a value of $150 or less. This can be determined by adding up the percentage of investments in each bar up to and including the $150 bar.

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So the area of one tile is 50 square inches. Therefore, the length of the triangle's base is 20 inches.

Area is a measure of the amount of space inside a two-dimensional (2D) shape or figure. It is usually measured in square units, such as square meters (m²) or square centimeters (cm²). The area of a shape is calculated by multiplying the length of one side or dimension of the shape by the length of another side or dimension. The resulting value represents the total number of square units that are contained within the shape.Area is an important concept in mathematics and is used in various fields, such as geometry, physics, and engineering. It helps in measuring and comparing the sizes of shapes and figures, and in determining how much material is needed to cover or fill a certain area.

Here,

The area of a parallelogram can be calculated as the product of its base and height. In the case of the given tile, the base and height are 10 inches and 5 inches respectively. Therefore, the area of one tile is:

Area of tile = Base x Height

Area of tile = 10 in x 5 in

Area of tile = 50 square inches

If the two tiles have the same area, then we can assume that the area of the triangle tile is also equal to 50 square inches (the area of one parallelogram tile that we calculated in the previous question).

The formula for the area of a triangle is:

Area of triangle = (1/2) x Base x Height

where the base is one of the sides of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

In this case, we know that the area of the triangle is 50 square inches and the height is 5 inches, so we can solve for the base as follows:

Area of triangle = (1/2) x Base x Height

50 = (1/2) x Base x 5

Base = (2 x 50) / 5

Base = 20

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Complete question:

An artist is using tiles in the shape of a parallelogram to make a mosaic. The tiles are in the shape of parallelogram with height as 5 inch and length as 10 inch. What is the area of one tile? if the two tiles have the same area, what is the length of the triangles base?

Answer:

look down

Step-by-step explanation:

let length be" x"

accroding to question

breadth= 2x/3

we know,

Area of rectangle= l × b

384=x × 2x/3

calculate

Answer: the circles diameter is 14 meters

Step-by-step explanation:                                {Formula for circumference

2TUR = 43.96                                                      of a circle : L 27UR}

2x3 . 14r = 43.96

r=7

d=2r=2x7=14                                                         {d=2r}  

Answer:

The circumference of a circle can be calculated using the formula:

C = 2πr

where C is the circumference, π is the mathematical constant pi (approximately 3.14), and r is the radius of the circle.

To find the diameter of the circle, we can use the formula:

d = 2r

where d is the diameter and r is the radius.

Given that the circumference of the circle is 43.96 meters, we can use the formula for circumference to solve for the radius:

C = 2πr

43.96 = 2 × 3.14 × r

r = 43.96 / (2 × 3.14)

r = 6.998 meters

Now we can use the formula for diameter to find the diameter:

d = 2r

d = 2 × 6.998

d = 13.996 meters

Therefore, the diameter of the circle is approximately 13.996 meters.

Step-by-step explanation:

mag delete nito napaka bob0 nya promise

The expanded form of (10a - b)(10a + b) is 100a^2 - b^2.

To expand the given expression, we need to distribute the first term inside the parentheses to the second term inside the parentheses. This can be done by multiplying each term inside the first parentheses by each term inside the second parentheses. Here are the steps:

1. 10a * 10a = 100a^2
2. 10a * b = 10ab
3. -b * 10a = -10ab
4. -b * b = -b^2

Now, we can combine like terms and simplify the expression:

100a^2 + 10ab - 10ab - b^2 = 100a^2 - b^2

So, the final answer is:

100a^2 - b^2

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To complete the sentence, we need to choose the expression that correctly represents the number of books owned by the university library. The expression given in parentheses is "nine million three hundred thirty-nine thousand five hundred sixty" or "9,339,560".

To write this number in words, we can break it down into its different place values. The digit in the millions place is 9, so we can say "nine million."The digit in the hundred thousands place is 3, so we can say "three hundred thousand." The digit in the ten thousands place is 3, so we can say "thirty thousand."The digit in the thousands place is 9, so we can say "nine thousand." The digit in the hundreds place is 5, so we can say "five hundred.

"The digit in the tens place is 6, so we can say "sixty." The digit in the ones place is 0, so we can say "zero."Putting it all together, we can say that the university library owns more than "nine million, three hundred thirty-nine thousand, five hundred sixty" books.

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Answer:

Refer to pic..........

The amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces

The weight of Kingston at birth = 9 pounds 2 ounces

The weight of Karmichael at birth = 7 pounds 11 ounces.

Next, the weights are converted to ounces such as:

1 pound = 16 ounces

For 9 pounds = 16×9 = 144 ounces+ 2 ounces = 146 ounces

For 7 pounds = 16×7 = 112 ounce + 11 ounces = 123 ounce

The difference between the birth weight of Kingston and Karmichael = 146-123 = 23 ounce

Convert 23 ounce to pounds = 23/16 = 1 pound 4 ounces

Therefore, amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces.

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Answer:

mean 36.2 median is 33

Step-by-step explanation:

For median you put the numbers in least to greatest and then find the middle which in this case is 33 and 34 so 33 is in between those two numbers so that would be the median. For the mean you add all the number together which is 362 and then divide by the number of numbers that there is which is 10

According to Descartes' Rule of Signs, the possible number of positive real zeros for a polynomial function is determined by the number of sign changes in the coefficients of the terms in the function.

Similarly, the possible number of negative real zeros is determined by the number of sign changes in the coefficients of the terms in the function when x is replaced with -x.

For the polynomial function P(x)=6x^(4)-x^(3)+5x^(2)-x+9, there are 2 sign changes in the coefficients of the terms: from 6 to -1, and from 5 to -1. Therefore, the possible number of positive real zeros is 2 or 0.

For the polynomial function P(-x)=6(-x)^(4)-(-x)^(3)+5(-x)^(2)-(-x)+9, there are 0 sign changes in the coefficients of the terms. Therefore, the possible number of negative real zeros is 0.

In conclusion, the possible number of positive real zeros for the polynomial function P(x)=6x^(4)-x^(3)+5x^(2)-x+9 is 2 or 0, and the possible number of negative real zeros is 0.

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Ann average speed for the entire walk was 1 mile per hour.

Time (hours) Distance from Home (miles)

0                                0

1                                 2

2                                4

3                                6

4                                5

5                                4

6                                3

7                                 2

8                                 1

9                                0

Ann walked for three hours, covering a distance of 6 miles, the furthest she had walked from her house.

2. When did Ann make her lunch break?

At 4 hours, Ann took a break for lunch.

3. What time did Ann arrive at her house?

After the nine-hour mark, Ann went back home.

4. What was Ann's average walking speed for the first three hours?

The total distance Ann walked during the first three hours of her walk (6 miles) divided by the time required to complete that distance can be used to determine her average pace (3 hours). She moved at an average speed of 2 miles per hour for the first three hours.

During the entire trek, Ann travelled a total of 9 miles (since she returned to her starting point). It took 9 hours to travel the entire trip. She walked at a mile an hour on average for the entire distance.

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The equation of the line with gradient 2 and that passes through the point (1,1) is y= 2x -1.

A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,

m = Δy/Δx where, m is the slope

Given:

Gradient = 2

and, line passes through the point (1,1).

We know using Slope intercept form y= mx+ b

and, y= 2x+ b

Now, line passes through the point (1,1).

1 = 2(1) + b

b= -1

So, the equation of line is y= 2x -1.

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By answering the above question, we may state that The rectangle's equation measurements are 9/4 miles by 9/20 miles.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

A is equal to (9/20) sq miles wide at x miles.

[tex]A = (9/20) (width x length) = x(x/5)\\9x/20 = x^2/5\\45x = 20x^2\sx^2 - 45x/20 = 0[/tex]

[tex]x=(-b \sqrt(b2 - 4ac)) / 2a[/tex]

Here, an equals 1, b equals -45/20, and c equals 0. By entering these values, we obtain:

[tex]x = (-(-45/20) +\sqrt((-45/20)^2 - 4(1)(0))) / 2(1) x = (45/20 ± sqrt(2025/400)) / 2\sx = (45/20 ± 45/40) / 2\sx = (9/4) or (1/4)[/tex]

We may disregard the negative answer since a rectangle's length cannot be negative. As a result, the rectangle has a length of 9/4 miles and a width of (1/5) x (9/4) = 9/20 miles.

The rectangle's measurements are 9/4 miles by 9/20 miles.

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The simplified result is (-x + 4)/(x^(2) - 16).

To add the given expressions, we need to combine the like terms in the numerator and keep the denominator the same since they are already common denominators.

The combined expression will look like this:

(7x + 10 - 8x - 6)/(x^(2) - 16)

Now we need to simplify the numerator by combining the like terms:

(-x + 4)/(x^(2) - 16)

This is the simplified expression. We cannot simplify it any further since there are no common factors between the numerator and denominator.

Therefore, the answer to the given question is (-x + 4)/(x^(2) - 16).

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Using basic arithmetic operation, the solution to the equation 7/6 + 2/3y = 4/3 is y = 3/4.

What is the basic arithmetic operation?

The four basic mathematical operations are Addition, subtraction, multiplication, and division.

To solve this equation for y, we need to isolate the variable y on one side of the equation. We can do this by performing algebraic operations to simplify the equation. Here are the steps:

Subtract 7/6 from both sides:

7/6 + 2/3y - 7/6 = 4/3 - 7/6

2/3y = 1/2

Multiply both sides by the reciprocal of 2/3:

(3/2)(2/3y) = (3/2)(1/2)

y = 3/4

Hence, the solution to the equation 7/6 + 2/3y = 4/3 is y = 3/4.

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There are 432 possible outcomes when tossing a coin and rolling three number cubes.

Yes, it is possible to calculate the total number of outcomes when tossing a coin and rolling three number cubes using the basic counting method.

According to the basic counting principle, if there are m ways to do one task and n ways to complete another, then there are m * n methods to complete both tasks sequentially.

There are two possible outcomes from this coin toss (heads or tails), and each of the three number cubes can be rolled six different ways (1 through 6). As a result, according to the basic counting principle, there are:

2 * 6 * 6 * 6 = 432

Hence, when rolling three number cubes and tossing a coin, there are 432 possible results.

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a) Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. So the option E is correct.

b) There is insufficient information to draw the conclusion that the variables are connected even though the value is greater than the 0.05 threshold for significance.

c) The difference between these percentage are fairly similar which suggests that sex and risk of lower-extremity Injury in  interscholastic soccer are not related.

a) The null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer is:

Null hypothesis: Sex and lower-extremity injury risk in interscholastic soccer are unrelated factors. Alternative hypothesis: sex and the likelihood of sustaining a lower-extremity injury in collegiate soccer are associated variables.

So the option E is correct.

b) The conclusion is that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries. This conclusion is based on the fact that the chi-square statistic (1.63) and the corresponding p-value (0.202) are both greater than the 1.63 X standard for significance. This indicates that there is not enough evidence to show that the variables are related.

c) Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

Girls = 79/997 × 100 = 7.92%

Boys = 156/1664 × 100 = 9.38%

The difference between the two percentages is consistent with the conclusion that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries because the difference between the two percentages is small.

This indicates that the risk of lower-extremity injuries is relatively similar for both boys and girls, which is consistent with the conclusion that there is no significant relationship between gender and the risk of lower-extremity injuries.

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The complete question is:

Researchers studied a random sample of high school students who participated in interscholastic athletics to learn about the risk of lower-extremity injuries (anywhere between hip and toe) for interscholastic athletes. Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

(a) Write null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer.

A. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. Alternative hypothesis: Sex and risk of a lower-extremity Injury in interscholastic soccer are not related variables.

B. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not explanatory variables.

C. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are not explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are explanatory variables.

D. Null hypothesis: There is a weak relationship between sex and risk of a lower-extremity injury in interscholastic soccer. Alternative hypothesis: There is a strong relationship between sex and risk of a lower-extremity injury in interscholastic soccer

E. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables.

(b) For these data, the value of the chi-square statistic is 1. 63, and the p-value for the chi-square test is 0. 202. Based on these results, state a conclusion about the two variables in this situation and explain how you came to this conclusion. The p-value is greater than the 1. 63 X standard for significance, so there is not sufficient evidence to be able to conclude that the variables are related.

(c) For each sex separately, calculate the percent of participants who had a lower-extremity injury. (Round your answers to one decimal place. ) girls 7.9% boys 9.4%. Explain how the difference between these percentages is consistent with the conclusion you stated in part (b). These percentages are fairly similar , which suggests that sex and risk of a lower-extremity injury in interscholastic soccer are not related

The completely factored form of the given polynomial is 4(v-6)(v+2).

To factor the given polynomial, 4v^(2)-16v-48, we first need to factor out the constant, which is 4. This will give us:

4(v^(2)-4v-12)

Now, we can use the factoring method to factor the remaining polynomial, v^(2)-4v-12. We need to find two numbers that multiply to -12 and add to -4. These numbers are -6 and 2. So, we can write the polynomial as:

4(v-6)(v+2)

Therefore, the completely factored form of the given polynomial is 4(v-6)(v+2).

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Answer:

1440

Step-by-step explanation:

hope this helps.......

Answer:

See below for proof.

Step-by-step explanation:

Given equations for x and y:

  [tex]x=v_0 \cdot t \cdot \cos \theta[/tex]

  [tex]y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

Rearrange the equation for x to isolate t by dividing both sides of the equation by v₀ · cos θ:

[tex]\implies t=\dfrac{x}{v_0 \cdot \cos \theta}[/tex]

Square the equation for x:

[tex]\implies x^2=(v_0 \cdot t \cdot \cos \theta)^2[/tex]

[tex]\implies x^2={v_0}^2 \cdot t^2 \cdot \cos^2 \theta[/tex]

Rearrange to isolate t² by dividing both sides of the equation by v₀² · cos²θ:

[tex]\implies t^2= \dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}[/tex]

Now we have created expressions for t and t² in terms of x.

Substitute these into the equation for y:

[tex]\implies y=-16 \cdot t^2 + v_0 \cdot t \cdot \sin \theta[/tex]

[tex]\implies y =-16 \cdot \left(\dfrac{x^2}{{v_0}^2 \cdot \cos^2 \theta}\right)+ v_0 \cdot \left(\dfrac{x}{v_0 \cdot \cos \theta}\right) \cdot \sin \theta[/tex]

Simplify:

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{x^2}{\cos^2 \theta}\right)+\left(\dfrac{x}{\cos \theta}\right) \cdot \sin \theta[/tex]

[tex]\implies y =-\dfrac{16}{{v_0}^2} \cdot \left(\dfrac{1}{\cos^2 \theta}\right) \cdot x^2+x \cdot \left(\dfrac{\sin \theta}{\cos \theta}\right)[/tex]

[tex]\textsf{Use\;the\;identities\;\;$\sec^2\theta=\dfrac{1}{\cos^2\theta}$\;\;and\;\;$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$}:[/tex]

[tex]\implies y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta[/tex]

[tex]\textsf{Hence\;proving\;that\;\;$y=-\dfrac{16}{{v_0}^2} \cdot \sec^2 \theta \cdot x^2+x \cdot \tan \theta$}.[/tex]

The equation can be written as,

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex].

An object or particle that is projected in a gravitational field, such as from the surface of the Earth, and moves along a curved path while only being affected by gravity is said to be in projectile motion.

To begin with, we know that the horizontal displacement of the projectile is given by [tex]x = v_ot cos(\theta)[/tex], where vo is the initial velocity of the projectile and θ is the angle it makes with the positive x-axis. Therefore, we can rearrange this equation to solve for time t:

[tex]t = \dfrac{x} { (v_o cos(\theta))}[/tex]

Next, we can use the equation for vertical displacement under constant acceleration due to gravity: [tex]y = -16t^2+ v_ot sin(\theta)[/tex]. Substituting the expression for t derived above, we obtain:

[tex]y = -16[\dfrac{x} { (vo cos(\theta)}]^2 + v_o([\dfrac{x} { (v_o cos(\theta)}] sin(\theta)[/tex]

Simplifying this expression, we get:

[tex]y = \dfrac{-16x^2} { (v_o^2 cos^2(\theta)) + x tan(\theta)}[/tex]

Now, we need to eliminate the angle θ in terms of x and y. We can do this by using the fact that tan(θ) = y / x, which gives:

[tex]\theta= tan^{-1}(\dfrac{y} { x})[/tex]

Substituting this expression for θ into our previous equation, we obtain:

[tex]y =\dfrac{ -16x^2 }{(v_o^2 cos^2(tan^{-1}(\dfrac{y} { x}))} + x\ tan(tan^{-1}(\dfrac{y} { x})[/tex]

Since tan(tan⁻¹(z)) = z, we can simplify this expression further:

[tex]y = \dfrac{-16x^2} { (v_o^2 (1 + (\dfrac{y} { x})^2))} + y[/tex]

Finally, by rearranging terms, we get the desired equation:

[tex]y = \dfrac{-15x^2} { (v_o^2) + (\dfrac{1} { v_o^2})tan^{-1}(\dfrac{y} { x})}[/tex]

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The true statement is all irrational and rational numbers have decimal expansions. (second option)

A rational number is a number that can be expressed as a fraction of two integers . Rational numbers have terminating decimals and they have decimal expansions. Examples of rational numbers are 1.5, 2 and 4.25

A irrational number is a number that cannot be expressed as a fraction of two integers . Irrational numbers have repeating decimals. Examples of irrational numbers are 22/7, 1/3.

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The slope of this situation is equal to 5.48.

In Mathematics, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

Based on the information, an equation that models the situation is given by this mathematical expression;

y = mx + c

y = 5.48x + 3.50

In conclusion, we can reasonably infer and logically deduce that the slope is 5.48 and the y-intercept is equal to 3.50.

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