A survey of 7th and 8th grade student who were asked whether or not they were in favor of or against school uniforms. This two-way table shows the results. How many 7th grade students are against school uniforms?​

Answer:

63.3%

Step-by-step explanation:

30 divided by 100 and then multiplied by 19 gives u the answer

Ricardo used w/100 Ccf of water.

Given that, Ricardo household used w cubic feet of water in a month of summers,

we need to express the number of ccf of water they used algebraically,

Ccf = A CCF is a unit of measurement of volume.

The CCF is used to measure gas consumption, used in the billing of natural gas and water delivered to households.

1 CCF = 100 cubic feet

CCF means Centum Cubic Feet.

1 cubic feet = 0.01 Ccf

Therefore,

w cubic feet = 0.01 × w Ccf

= w / 100

Hence, Ricardo used w/100 Ccf of water.

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The area of the small triangle is 4 sq.cm.. The area of the medium triangle is 12 sq.cm. The area of the large triangle is 24 sq. cm.

With three sides, three angles, and three vertices, a triangle is a closed, two-dimensional object. A polygon also includes a triangle.

Given data:

Dimensions-

area of triangle = 1/2 *base * height

The area of the small triangle = 1/2*2*4 = 4 sq.cm.

The area of the medium triangle = 1/2*4*6 = 12 sq.cm.

The area of the large triangle = 1/2*6*8 = 24 sq. cm

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Complete question:

Dimensions-

small triangle: base = 2 cm, height = 4cm

medium triangle: base = 4 cm , height = 6 cm

Larger triangle: base = 6 cm ,height = 8 cm

The area of the small triangle is_______

The area of the medium triangle is______

The area of the large triangle is______

The relative frequency of spinning a 1 is 0.32 or 32%.

The given table shows the results of spinning a spinner 50 times. The outcomes of the spins are listed in the first column, and the frequencies are listed in the second column. To find the relative frequency of spinning a 1, we need to divide the frequency of spinning a 1 by the total number of trials (50).

According to the table, the frequency of spinning a 1 is 16. Therefore, the relative frequency of spinning a 1 can be calculated as follows:

Relative frequency of spinning a 1 = (frequency of spinning a 1) / (total number of trials)

Relative frequency of spinning a 1 = 16 / 50

Relative frequency of spinning a 1 = 0.32 or 32%

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The Gabrielsons ran a total of 30 kilometers in the family relay race.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

It seems that there are five family members who participated in the relay race, and the distance run by each of them in kilometers is listed as follows:

11, 4, 8, 2, 5

To find the total distance run by the family, we simply add up the distances:

11 + 4 + 8 + 2 + 5 = 30

Therefore, the Gabrielsons ran a total of 30 kilometers in the family relay race.

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Answer:

Step-by-step explanation:

the 32 pack because it costs just slightly more

Using the above information, we can calculate the missing figures in the table and this is shown on the table attached.

For the "Shorts" row: It's given that 42 people were wearing shorts.

Therefore, the "Close-Toed Shoes" column in the "Shorts" row is also 42, and the "Total" column is the sum of open-toed and closed-toed shoes, which is 84.

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See text below

30 people surveyed were wearing open-toed shoes.

1/3 of the people with open-toed shoes were wearing

pants.

• 42 people were wearing shorts.

Twice as many people were wearing closed-toed shoes

than open-toed shoes.

Complete the table using the information given.

                   Open-Toed Shoes              Close-Toed Shoes         Total

Shorts                     -------                           ------                               --------

Pants                         ------                           ----------                            ------

Total                          ------                           ----------                            ---------

Terrell hit 0.8 of his home runs during the first half of the baseball season, which is the decimal equivalent of the fraction (4)/(5).

A fraction is a way of representing a part of a whole, where a whole is divided into equal parts and the fraction represents the number of those equal parts that are being considered.

A decimal number is a number expressed in the base-ten positional system, where each digit can take one of ten possible values and the value of each digit is determined by its position relative to the decimal point.

To write a fraction as a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). In this case, the fraction is (4)/(5), which means Terrell hit 4 out of 5 home runs during the first half of the season

(4)/(5) = 0.8

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The given question is incomplete, the complete question is:

Terrell hit a total of 10 home runs during the baseball season. He hit (4)/(5) of the home runs during the first half of the season. Write the fraction of home runs hit during the first half of the season as a decimal.

9. The volume of the triangular pyramid is given by:

V = (1/3)Bh

Here, B is the base area.

The base is shaped as a right triangle thus, using the Pythagoras Theorem we have:

h = sqrt(26.7² - 11.7²) = 23.274 km

The area if the base is:

B = (1/2)bh = (1/2)(11.7 km)(23.4 km) = 136.89 sq. km

Now, the volume is:

V = (1/3)(136.89)(15) = 2053.35 cubic km

Hence, the volume of the triangular pyramid is 2053.35 cubic km.

9. The volume of the triangular pyramid is 2053.35 cubic km. 11. The area of the shaded portion is 348.19 cubic in 13. The slant height of the cone is 8.53 meters.

A fundamental conclusion in geometry relating to the lengths of a right triangle's sides is known as Pythagoras' theorem. According to the theorem, the square of the length of the hypotenuse, the side that faces the right angle, in any right triangle, equals the sum of the squares of the lengths of the other two sides, known as the legs.

9. The volume of the triangular pyramid is given by:

V = (1/3)Bh

Here, B is the base area.

The base is shaped as a right triangle thus, using the Pythagoras Theorem we have:

h = √(26.7² - 11.7²) = 23.274 km

The area if the base is:

B = (1/2)bh = (1/2)(11.7 km)(23.4 km) = 136.89 sq. km

Now, the volume is:

V = (1/3)(136.89)(15) = 2053.35 cubic km

Hence, the volume of the triangular pyramid is 2053.35 cubic km.

11. The volume of a cone is given by:

V = (1/3)πr²h

The dimension of the bigger cone is radius is 9 in, and height 15 in:

V1 = (1/3)π(9 in)²(15 in) = 381.7 cubic in

The dimension of the smaller cone is radius is 4 in, and height 10 in:

V2 = (1/3)π(4 in)²(10 in) = 33.51 cubic in

Now, the area of the shaded portion is:

V1 - V2 = 381.7 - 33.51  = 348.19

13. The volume of a cone is given by:

V = (1/3)πr²h

Substituting the values we have:

542.87 = (1/3)π(6 m)²h

h = 542.87 / [(1/3)π(6 m)²] = 6.05 m

Now, using the Pythagoras Theorem for the slant height we have:

s² = r² + h²

s² = (6 m)² + (6.05 m)²

s² = 72.9

s = √(72.9) = 8.53 m

The slant height of the cone is 8.53 meters.

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Scenic Cinemas should offer discounted weekday matinee showings for seniors and continue to focus on weekend showings for their broader audience.

Based on the survey results, it would be wise for Scenic Cinemas to tailor their offerings to accommodate both preferences.

One option would be to offer discounted weekday matinee showings targeted toward seniors. This could incentivize them to visit on weekdays while also offering them a more affordable option. At the same time, Scenic Cinemas could continue to focus on weekend showings to cater to their broader audience.

Another approach would be to offer more diverse programming during the weekdays, such as classic films or independent movies that may appeal more to seniors. This could create a niche for Scenic Cinemas and attract a more loyal customer base.

Ultimately, it's important for Scenic Cinemas to balance the needs of their different audience segments to maximize revenue and customer satisfaction. By offering targeted promotions and programming, they can ensure that both seniors and other moviegoers feel valued and have a reason to visit the theatre.

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a) The probability of a given woman in this age group having a cholesterol level less than 200mg/dl is 6.68%.

b) The probability of a given woman in this age group having a cholesterol level more than 200mg/dl is 93.32%.

c) The probability of a given woman in this age group having a cholesterol level between 190mg/dl and 210mg/dl is 15.87%.

d) If a woman in this age group has a cholesterol level higher than 230mg/dl, it is considered high and puts her at risk of heart disease

To calculate the probability of a given woman in this age group having a cholesterol level less than 200mg/dl, we need to find the z-score first. The z-score is the number of standard deviations that a given value is from the mean. The formula to calculate the z-score is:

z = (x - μ) / σ

where x is the given value, μ is the mean, and σ is the standard deviation.

For a cholesterol level of 200mg/dl, the z-score is:

z = (200 - 230) / 20 = -1.5

We can then use a z-table or calculator to find the probability of a z-score being less than -1.5, which is 0.0668 or approximately 6.68%.

Next, to find the probability of a given woman in this age group having a cholesterol level more than 200mg/dl, we can use the same process but subtract the probability of a z-score being less than -1.5 from 1 because the total probability is always 1.

So, the probability of a given woman in this age group having a cholesterol level more than 200mg/dl is:

1 - 0.0668 = 0.9332 or approximately 93.32%.

Finally, to find the probability of a given woman in this age group having a cholesterol level between 190mg/dl and 210mg/dl, we need to find the z-scores for both values.

For a cholesterol level of 190mg/dl, the z-score is:

z = (190 - 230) / 20 = -2

For a cholesterol level of 210mg/dl, the z-score is:

z = (210 - 230) / 20 = -1

We can then use the z-table or calculator to find the probability of a z-score being between -2 and -1, which is 0.1587 or approximately 15.87%.

Finally, a brief report on the guidance that you would give a woman having high cholesterol levels in this age group is:

It is essential to make lifestyle changes such as eating a healthy diet, exercising regularly, quitting smoking, and managing stress to lower cholesterol levels.

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Therefore , the solution of the given problem of coordinates comes out to be  (1, -4) are the coordinates of the vertex of the function  F(x) = x² - 2x - 3.

When used in connection with particular other algebraic components on this place, such as Euclidean space, a parameter can reliably detect placement using a number of qualities or coordinates. Coordinates, which appear as collections of numbers when traversing in reflected space, can be utilised to identify particular places or things. Using the two y & x measurements, one can find something over both sides.

Here,

The formula x = -b/2a can be used to determine the vertex of a quadratic function with the form

F(x) = ax² + bx + c. A and B are equal in the given function

=>  F(x) = x² - 2x - 3.

With these values entered into the formula, we obtain:

=> x = -b/2a x = -(-2)/(2*1)

=> x = 2/2 x = 1

Consequently, the vertex's x-coordinate is 1.

Now, we can enter the value of x into the following function, F(x), to determine the vertex's y-coordinate:

=> F(1) = 1² - 2(1) - 3

=> F(1) = 1 - 2 - 3

=> F(1) = -4

Therefore, the vertex's y-coordinate is -4.

As a result, (1, -4) are the coordinates of the vertex of the function

=> F(x) = x² - 2x - 3.

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Answer:

Step-by-step explanation:

Graph #          Matching equation

1                             |-3x|

2                            |-x|

3                           -|2x|

You can tell which one matches by finding the slope and whether the V points up or down

The cardholder will pay an additional $1,471.66 in interest by making monthly payments of $200 until the balance is paid off instead of paying off the current balance in full.

First, we need to calculate the total interest that will accrue on the current balance of $4,528.34. We can do this using the formula

Interest = Balance x (APR/12)

where APR is the annual percentage rate and is divided by 12 to get the monthly interest rate. Plugging in the values, we get:

credit card Interest = $4,528.34 x (22.99%/12) = $87.80

So the total interest that will accrue on the current balance is $87.80.

Next, we need to calculate how long it will take to pay off the balance by making monthly payments of $200. We can use a credit card repayment calculator to do this, but we'll use a simplified formula here

Months = -log(1 - (Balance x (APR/12))/Payment) / log(1 + (APR/12))

where Payment is the monthly payment amount. Plugging in the values, we get

Months = -log(1 - ($4,528.34 x (22.99%/12))/$200) / log(1 + (22.99%/12)) = 29.6 months

So it will take about 30 months (or 2.5 years) to pay off the balance by making monthly payments of $200.

Finally, we can calculate how much more the cardholder will pay in total by subtracting the current balance from the total amount paid over 30 months

Total amount paid = $200 x 30 = $6,000

Total interest paid = $6,000 - $4,528.34 = $1,471.66

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The equations for which 8 is a solution are: x - 4 = 4, 2x = 16, x/2 = 16, and x/8 = 1.

An equation is a mathematical statement that shows that two expressions are equal to each other. It typically consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The goal is usually to solve for the value of the variable that makes the equation true.

In the given question,

The equations in which 8 is a solution are:

x - 4 = 8 (which simplifies to x = 12)

2x = 16 (which simplifies to x = 8)

x/2 = 16 (which simplifies to x = 32)

x/8 = 1 (which simplifies to x = 8)

Therefore, the correct answers are:

x - 4 = 8

2x = 16

x/2 = 16

x/8 = 1.

The equations for which 8 is a solution are: x - 4 = 4, 2x = 16, x/2 = 16, and x/8 = 1.

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x² - 16x + 64 = 0 is the equation whose solution is 8.

How to check for which equations is 8 a solution?

To check if 8 is a solution for each equation, we substitute x = 8 into each equation and see if the equation is true or not.

x + 6 = 8 + 6 = 14

and

2x + 2 = 2(8) + 2 = 18, which is not equal to the value of (x+6).

Therefore, 8 is not a solution to this equation.

4x - 4 = 4(8) - 4 = 28,

and

10 - 2x = 10 - 2(8) = -6, which is not equal to 28.

Therefore, 8 is not a solution to this equation.

2x - 4 = 2×8-4 = 12

and

3x-24 = 3×8-24 = 0 which is not equal to 12. Therefore, 8 is not a solution to this equation.

Now,

x² - 16x + 64 = 0 (which can be factored as (x-8)² = 0)

x = 8 (which is always true)

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Correct question is "For which equations is 8 a solution? Select the correct answers from the following,

1) x + 6 = 2 x + 2

2) 10 - 2x = 4 x - 4

3) 2 x-4 = 3 x - 24

4) x² - 16x + 64 = 0

The correct answer is: Fast Chicken, because it has a smaller median.

What is median?

Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude. To find the median, you need to arrange the values in order from smallest to largest and then find the middle value.

Based on the information provided, the drive-thru restaurant with less wait time is Fast Chicken, as it has a smaller median wait time of 12.5 minutes compared to the median wait time of 12 minutes for Super Fast Food. The mean wait time is not given, and even if it were, the median is a better measure of central tendency to compare in this scenario, as the box plots suggest some potential outliers.

Therefore, the correct answer is: Fast Chicken, because it has a smaller median.

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The exponential function to represent the number of new participants of the challenge as a function of the day number, is f(x) = 12 × 4ˣ.

Let f(x) be the number of new participants on day x.

On day 0 (today), Aliyah, Kim, and Reese are the only participants. They each send selfies to 4 friends, so there will be 3 x 4 = 12 new participants on day 1.

From day 1 onwards, each new participant will send selfies to 4 friends. Therefore, the number of new participants will quadruple each day. This gives us an exponential growth function:

f(x) = 12 × 4ˣ

where f(x) represents the number of new participants on day x.

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Answer:

5 units is the distance

Step-by-step explanation:

7,2 is point c

7,7 is point d

7 - 7 = 0

7 - 2 = 5

5 is the answer

The area of circle is with radius 3xy³ is 9πx²y⁶.

Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula, A = πr2.

Radius of a circle is the distance from the center of the circle to any point on it's circumference. It is usually denoted by 'R' or 'r'.This quantity has importance in almost all circle-related formulas. The area and circumference of a circle are also measured in terms of radius. Circumference of circle = 2π (Radius)

area of circle=πr²

=π×(3xy³)²

=9πx²y⁶

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A control group on the dependent variable, P-value is higher than the usual cutoff of 0.05 for rejecting the null hypothesis that there is no difference between the groups, this is frequently regarded as evidence that the difference between the groups is not statistically significant.

In an analysis testing difference between an experimental and a control group on the dependent variable, a p-value of 0.07 means that there is a 7% chance of obtaining a difference as large or larger than the observed difference between the two groups, assuming that there is no true difference between the groups in the population.

Interpreted as evidence that the difference between the groups is not statistically significant, since the p-value is greater than the conventional threshold of 0.05 for rejecting the null hypothesis of no difference between the groups.

Statistical significance does not necessarily imply practical significance, and there may still be a meaningful difference between the groups that is not detected by the statistical test.

Additionally, the interpretation of a p-value depends on the study design, sample size, and other factors, and should be considered in conjunction with other information about the study.

True difference between the groups in the population, a p-value of 0.07 indicates that there is a 7% chance of finding a difference as large or larger than the observed difference between the two groups in an analysis testing the difference between an experimental and a control group on the dependent variable.

P-value is higher than the usual cutoff of 0.05 for rejecting the null hypothesis that there is no difference between the groups, this is frequently regarded as evidence that the difference between the groups is not statistically significant.

It's crucial to remember that statistical significance does not always imply practical relevance.

There could still be a significant difference between the groups that is not shown by statistical analysis.

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The value of "a" is approximately 1.83 given that the absolute value of the difference of the two roots of the quadratic equation "ax squared plus 5x minus 3 equals 0" is the square root of 61 divided by 3, and "a" is positive.

We are given that the absolute value of the difference between the two roots of the quadratic equation "ax squared plus 5x minus 3 equals 0" is the square root of 61 divided by 3, and "a" is positive. We need to find the value of "a".

Let the two roots of the equation be r1 and r2, where r1 is not equal to r2. Then, we have:

|r1 - r2| = √(61) / 3

The sum of the roots of the quadratic equation is given by r1 + r2 = -5 / a, and the product of the roots is given by r1 × r2 = -3 / a.

We can express the difference between the roots in terms of the sum and product of the roots as follows:

r1 - r2 = √((r1 + r2)² - 4r1r2)

Substituting the expressions we obtained earlier, we have:

r1 - r2 = √(((-5 / a)²) + (4 × (3 / a)))

Simplifying, we get:

r1 - r2 = √((25 / a²) + (12 / a))

Taking the absolute value of both sides, we get:

|r1 - r2| = √((25 / a²) + (12 / a))

Comparing this with the given expression |r1 - r2| = √(61) / 3, we get:

√((25 / a²) + (12 / a)) = √(61) / 3

Squaring both sides and simplifying, we get:

25 / a² + 12 / a - 61 / 9 = 0

Multiplying both sides by 9a², we get:

225 + 108a - 61a² = 0

Solving this quadratic equation for "a", we get:

a = (108 + √(108² + 4 × 61 × 225)) / (2 × 61)

Since "a" must be positive, we take the positive root:

a = (108 + √(108² + 4 × 61 × 225)) / (2 × 61) ≈ 1.83

Therefore, the value of "a" is approximately 1.83.

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The question is -

Given that the absolute value of the difference of the two roots of the quadratic equation "ax squared plus 5x minus 3 equals 0" is the square root of 61 divided by 3, and "a" is positive, what is the value of "a"?

Answer:

[tex]\frac{24\sqrt{5} }{5}[/tex]

Step-by-step explanation:

Let's start by assigning one of the unknown legs with the variable x.

We know that the other leg is 2 times the length of x, so we can write:

2x

We also know that the length of the hypotenuse is 12.

From here, we can use the Pythagorean Theorem.

Recall that the Pythagorean Theorem is:

[tex]a^2+b^2=c^2[/tex]

where a is the length of one leg, b is the length of the other leg, and c is the length of the hypotenuse.

Let's substitute the values. We have:

[tex]x^2+(2x^2)=12^2=\\x^2+4x^2=144=\\5x^2=144=\\x^2=\frac{144}{5}=\\x=\frac{12}{\sqrt{5} }[/tex]

Let's rationalize the denominator by multiplying the numerator and denominator by [tex]\sqrt{5}[/tex], like so:

[tex]\frac{12}{\sqrt{5} } =\\\frac{12\sqrt{5} }{5}[/tex]

Therefore, [tex]x=\frac{12\sqrt{5} }{5}[/tex]

Let's solve for 2x:

[tex]2x=\\2(\frac{12\sqrt{5} }{5})=\\ \frac{24\sqrt{5} }{5}[/tex]

So, the length of the longest leg is [tex]\frac{24\sqrt{5} }{5}[/tex]

The equation of the line 2x + 3y + 1 = 0 after it has been reflected across y = -1 and rotated about O(2,3) by 180 degrees is -2x + y = 15.

To reflect the line 2x + 3y + 1 = 0 across the line y = -1, we need to subtract twice the y-coordinate of each point on the line from the equation of the line.

This is because the reflection of a point across a line is obtained by reflecting it across the line's perpendicular bisector, which is also the line with the same x-coordinates and twice the distance from the point to the line.

Subtracting twice the y-coordinate, we get:

2x + 3(-2y - 1) + 1 = 0

Simplifying the equation, we get:

2x - 6y - 5 = 0

Now, to rotate the line about the point O(2,3) by 180 degrees, we can use the rotation formula:

x' = 2a - x

y' = 2b - y

where (x,y) are the coordinates of a point on the original line, and (x',y') are the coordinates of the same point after rotation about the point (a,b).

Substituting the coordinates of two points on the line, we get:

(0,-1) → x' = 2(2) - 0 = 4, y' = 2(3) - (-1) = 7

(-1/2,0) → x' = 2(2) - (-1/2) = 4 1/2, y' = 2(3) - 0 = 6

Now, we can use the two points (4,7) and (4 1/2,6) to find the equation of the rotated line using point-slope form:

slope = (6-7) / (4 1/2 - 4) = -2

y - 7 = -2(x - 4)

y = -2x + 15

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Answer:

D: 27(12x² - 5x - 2).

Step-by-step explanation:

The formula for the surface area of a cylinder is: A = 2πr² + 2πrh

Given that the radius of the cylinder is 3x - 2 cm and the height is x + 3 cm, we can substitute these values in the formula and simplify:

A = 2π(3x - 2)² + 2π(3x - 2)(x + 3)

A = 2π(9x² - 12x + 4) + 2π(3x² + 7x - 6)

A = 18πx² - 24πx + 8π + 6πx² + 14πx - 12π

A = 24πx² - 10πx - 4π

A = 2π(12x² - 5x - 2)

Therefore, the answer is option D: 27(12x² - 5x - 2).

Answer:

D

Step-by-step explanation:

The sample space are:

{AR, AS, AT, AE, RA, RS, RT, RE, SA, SR, ST, SE, TA, TR, TS, TE, EA, ER, ES, ET}

The favorable outcomes care:

{RS, RT, SR, ST, TR, TS}

The probability is  0.3 or 30%

Part A:

The sample space represents all possible outcomes that can occur when two cards are randomly selected without replacement from the given pile of cards.

The sample space can be represented as follows:

{AR, AS, AT, AE, RA, RS, RT, RE, SA, SR, ST, SE, TA, TR, TS, TE, EA, ER, ES, ET}

Part B:

The favorable outcomes are those outcomes in which both cards are consonants. In this case, the consonants are R, S, and T.

The favorable outcomes can be represented as follows:

{RS, RT, SR, ST, TR, TS}

Part C:

To calculate the probability of selecting 2 cards that are consonants, we need to find the ratio of favorable outcomes to the sample space.

The number of favorable outcomes is 6, and the size of the sample space is 20.

Therefore, the probability of selecting 2 cards that are consonants is:

P(consonants) = favorable outcomes / sample space

P(consonants) = 6 / 20

P(consonants) = 0.3 or 30%

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The percentage of the large pizza they ate is 100%

To calculate the percentage of pizza they both ate we have to add and then covert the fractions to percentage.

Given from the question

Frank ate 20% of the large pizza and Joey ate 4/5 portion of the large pizza

Converting 4/5 into percentage

For converting into percentage we have to multiply 100 to the fraction

= 4/5 x 100

= 80%

Now, let us proceed by adding the collected two percentages to find out the total percentage of pizza consumed by Frank and Joey

20% + 80%

= 100%

The percentage of the large pizza they ate is 100%.

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Step-by-step explanation:

Ok do what you need to do is label the line opposite the square angle as H for the hypotenuse. Then label the line opposite the circular angle as O for the opposite. And finally, the last line remaining should be labelled as A for Adjacent. Does this help?

The solution to the IVP given by y''+y=t, y(0)=1, y'(0)=-2 is y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

To solve the Initial Value Problem, we can use the method of undetermined coefficients, which involves assuming a particular form for the solution to the non-homogeneous equation y'' + y = t, and then finding the coefficients of the terms in that form by substituting it back into the equation.

First, we find the general solution to the homogeneous equation y'' + y = 0

The characteristic equation is r² + 1 = 0, which has solutions r = ±i. Therefore, the general solution to the homogeneous equation is

y_h(t) = c₁ cos(t) + c₂ sin(t),

where c₁ and c₂ are constants determined by the initial conditions.

Next, we assume a particular form for the non-homogeneous solution, based on the form of the right-hand side t. Since t is a linear function, we assume that the particular solution has the form

y_p(t) = a t + b.

Substituting this into the differential equation, we get

y''_p + y_p = t

2a + (at+b) = t.

Equating coefficients, we get

a = 1/2, b = 0.

Therefore, the particular solution is

y_p(t) = (1/2) t.

The general solution to the non-homogeneous equation is then the sum of the homogeneous and particular solutions

y(t) = y_h(t) + y_p(t)

= c₁ cos(t) + c₂ sin(t) + (1/2) t.

To determine the constants c₁ and c₂, we use the initial conditions:

y(0) = c₁ cos(0) + c₂ sin(0) + (1/2) (0) = c₁ = 1,

y'(0) = -c₁ sin(0) + c₂ cos(0) + (1/2) (1) = c₂ - (1/2) = -2,

so c₂ = -3/2.

Therefore, the solution to the IVP is

y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

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Sue works 5 out of 7 days a week, which implies that she has two days off. We need to discover how numerous conceivable plans there are for her to work on Tuesday or Friday or both.

There are two cases to consider:

1. Sue works on Tuesday as it were, Friday as it were, or both Tuesday and Friday.

2. Sue does not work on Tuesday or Friday.

For the primary case, there are three conceivable outcomes:

1. Sue works on Tuesday as it were and has Friday off.

2. Sue works on Friday as it were and has Tuesday off.

3. Sue works on both Tuesdays and Fridays.

For the moment case, there are two conceivable outcomes:

1. Sue works on one of the other 5 days of the week and has both Tuesday and Friday off.

2. Sue has Tuesday and Friday off.

In this manner, there are added up to 3 + 2 = 5 conceivable plans for Sue to work on Tuesday or Friday or both. 

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The equivalent sum to the given equation is -9c - 20.

An algebraic expression is consists of variables, numbers with various mathematical operations.

Equivalent sums refers to addition or subtraction from the other number to maintain the same total value.

= 9c-12-15c-8-3c

To find the equivalent sum, first we can simplify this expression by first combining like terms:

= 9c - 15c - 3c - 12 - 8

= (9c - 15c - 3c) - (12 + 8)   (grouping the like terms)

Solving the expression for terms c and  for constant terms,

= -9c - 20

Therefore, the equivalent sum is -9c - 20.

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