Process _____ is based on comparing the current amount of outcome variation against outcome specifications.

We can see that for c to be greater than 6, sin(C) must be greater than sin(70°). However, since sin(C) cannot be greater than 1, there are no possible values of C that satisfy this condition.

Based on the given information, we have:

[tex]a = 8[/tex] (length of side a)

[tex]b = 2[/tex] (length of side b)

[tex]A = 70^\circ[/tex] (angle A)

To determine whether the given information results in one triangle, two triangles, or no triangle at all, we can apply the triangle inequality theorem.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check if the given sides a, b, and angle A satisfy the triangle inequality theorem:

[tex]a + b > c[/tex]

[tex]8 + 2 > c[/tex]

[tex]10 > c[/tex]

[tex]c > |a - b|[/tex]

[tex]c > |8 - 2|[/tex]

[tex]c > 6[/tex]

where c is the length of the third side.

From the above inequalities, we can see that c must be greater than 6 and less than 10 in order to form a triangle. However, we also know that   [tex]A = 70^\circ[/tex], and the sum of the angles in a triangle is always 180°. So we can use the Law of Sines to find the length of side c:

[tex]sin(A) / a = sin(C) / c[/tex]

[tex]sin(70°) / 8 = sin(C) / c[/tex]

[tex]c = (8 \times sin(C)) / sin(70^\circ)[/tex]

Now we can find the possible values of c using the given information:

[tex]c = (8 * sin(C)) / sin(70^\circ)[/tex]

[tex]c = (8 \times sin(C)) / sin(70^\circ) > 6[/tex] (from triangle inequality)

Therefore,  We can see that for c to be greater than 6, sin(C) must be greater than sin(70°). However, since sin(C) cannot be greater than 1, there are no possible values of C that satisfy this condition.

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Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle. Solve any triangle that result triangle(s). a=8, b=2, A=70°

The equation of the line shown in the given graph is y = x/2 + 4

From the question, we are to write the equation for the line shown in the graph

To write the equation of the line we will use the two points given on the line, (-2, 3) and (2, 5)

Using the formula,

(y - y₁) / (x - x₁) = (y₂ - y₁) / (x₂ - x₁)

Then,

The equation becomes

(y - 3) / (x - (-2)) = (5 - 3) / (2 - (-2))

(y - 3) / (x + 2) = (5 - 3) / (2 + 2)

(y - 3) / (x + 2) = 2 / 4

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

Cross multiplication

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

2y - 6 = x + 2

2y = x + 2 + 6

2y = x + 8

Divide through by 2

y = x/2 + 4

Hence,

The equation is y = x/2 + 4

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Answer:

Perimeter:

[tex]2( \frac{2}{3} + \frac{1}{2} ) = 2( \frac{4}{6} + \frac{3}{6} ) = 2( \frac{7}{6} ) = \frac{7}{3} = 2 \frac{1}{3} [/tex]

The perimeter is 2 1/3 inches.

Area:

[tex] \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} [/tex]

The area is 1/3 of a square inch.

1  1/4

Step-by-step explanation:

The simplified expression is[tex]135v^12y^10u^5[/tex].

A mathematical expression is simplified when it is replaced with a

simpler, typically shorter, equivalent statement. to make it simpler, more

understandable, or easier. arithmetic to eliminate similar elements,

regroup terms in the same variable, etc. to simplify (an equation, fraction, etc.).

To simplify the expression, we'll combine the like terms:

Your expression:

[tex]15v^4y^7u^5 \times 9v^8y^3[/tex]

Multiply the coefficients (15 and 9):

15 × 9 = 135

Add the exponents of the like terms [tex](v^4 and v^8, y^7 and y^3[/tex]):

[tex]v^(4+8) = v^12\\y^(7+3) = y^10[/tex]

Combine the simplified terms:

[tex]135v^12y^10u^5[/tex]

The simplified expression is[tex]135v^12y^10u^5[/tex].

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To exclude the value [tex]$x=0$[/tex] since division by zero is not allowed.

The expression is well-defined for [tex]$x \in (0,1]$[/tex].

Expression is well-defined for [tex]$0\leq x\leq 1$[/tex].

In interval notation, we can write this as [tex]$[0,1]$[/tex].

Therefore, the expression [tex]$\frac{\sqrt{x\cdot 1} \sqrt{1 - x}}{\sqrt{x}}$[/tex] is well-defined for [tex]$ x\in [0,1]$[/tex].

Expression to be well-defined, we need to ensure that we do not perform any illegal operations.

This means that we cannot take the square root of a negative number, and we cannot divide by zero.

Let's consider each of the factors in the expression separately.

The first factor is.

[tex]$\sqrt{x\cdot 1}=\sqrt{x}$[/tex].

To ensure that this is well-defined, we need.

[tex]$x\geq 0$[/tex] since we cannot take the square root of a negative number.

The second factor is

.[tex]$\sqrt{1-x}$[/tex].

To ensure that this is well-defined, we need.

[tex]$1-x\geq 0$[/tex], or equivalently, [tex]$x\leq 1$[/tex], since we cannot take the square root of a negative number.

The denominator is.

[tex]$\sqrt{x}$[/tex].

To ensure that this is well-defined, we need.

[tex]$x\geq 0$[/tex], since we cannot divide by zero.

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A reasonable prediction for the number of times a sixth or seventh grader will be chosen is 24 out of the 40 games.

What is reasonable prediction?

A reasonable prediction is a prediction made with a reasonable degree of accuracy or likelihood, based on available information and knowledge. It is based on facts, past events, and logical assumptions, and is not based on conjecture or guesswork. Reasonable predictions can be made about future events, trends, and outcomes, and can be used to inform decisions, plans, and strategies.

The total number of sixth and seventh graders on the team is:

10 sixth graders + 14 seventh graders = 24 students

The total number of students on the team is:

10 sixth graders + 14 seventh graders + 16 eighth graders = 40 students

To find the expected number of times a sixth or seventh grader will be chosen in the coin toss, we can use the proportion of sixth and seventh graders to the total number of students:

(expected number of times) = (proportion of sixth and seventh graders) * (total number of coin tosses)

(expected number of times) = (24/40) * 40

(expected number of times) = 24

Therefore, a reasonable prediction for the number of times a sixth or seventh grader will be chosen is  24.

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a. The inequality that describes the situation is 8b - 400 ≤ 600

b. The maximum number of 8 pound boxes that can be taken off the plane is 15.

a. Let's denote the number of 8 pound boxes that need to be taken off the plane as "b". We know that a 600 pound crate will be put on the plane, so the weight added by the crate is 600 pounds. Additionally, we know that no more than 400 pounds can be taken off the plane. Therefore, the inequality that describes the situation is

8b - 400 ≤ 600

Simplifying this inequality, we get:

8b ≤ 1000

b ≤ 125

So the number of 8 pound boxes that can be taken off the plane is less than or equal to 125.

b. To find the exact number of boxes that can be taken off the plane, we can divide both sides of the inequality by 8

b ≤ 125

b/8 ≤ 15.625

Since we need to have a whole number of boxes, the maximum number of boxes that can be taken off the plane is 15.

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The given question is incomplete, the complete question is:

An airplane is loaded with cargo and ready to take off before it departs an unknown number of 8 pound boxes, b, Will be taken off the plane and a 600 pound crate will be put on to keep the plane in balance, no more than 400 pounds may be taking off the plane. Right and inequality that best describes this situation.

a. I the inequality that describes the situation.

b. How many 8 pound boxes can be taken off the plane?

The functions 7 are inverse functions while the functions 8 are not inverse functions

To verify that f and g are inverse functions, we need to show that:

Let's apply these conditions to the given functions:

f(x) = 5x + 2; g(x) = (x−2)/5

f(g(x)) = f((x−2)/5) = 5((x−2)/5) + 2 = x − 2 + 2 = x

g(f(x)) = g(5x + 2) = ((5x + 2) − 2)/5 = 5x/5 = x

Since f(g(x)) = x and g(f(x)) = x, we can conclude that f and g are inverse functions.

f(x) = ½x − 7; g(x) = 2x + 14

f(g(x)) = f(2x + 14) = ½(2x + 14) − 7 = x

g(f(x)) = g(½x − 7) = 2(½x − 7) + 14 = x − 7 + 14 = x + 7

Since f(g(x)) = x and g(f(x)) ≠ x, we can conclude that f and g are not inverse functions.

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Answer:

To dilate a triangle using a scale factor of 3.5, we multiply the distance between each vertex and the center of dilation (in this case, the origin) by 3.5.

Using this information, we can find the coordinates of each vertex of triangle A'B'C':

A' = (43.5, 33.5) = (14, 10.5)

B' = (33.5, -23.5) = (10.5, -7)

C' = (-33.5, 13.5) = (-10.5, 3.5)

Therefore, the vertex of point C' is (-10.5, 3.5).

So the correct answer is C′(−10.5, 3.5).

The odds ratio of 5.91 suggests that vaccination significantly increases the likelihood of remaining healthy and not developing COVID-19 symptoms compared to no vaccination.

Calculate the odds of staying healthy for each vaccination status based on the data provided:

Unvaccinated:

Number of individuals who stayed healthy = 44

Number of individuals who developed COVID-19 = 25

Odds of staying healthy = Number of individuals who stayed healthy / Number of individuals who developed COVID-19 = 44/25 = 1.76

Vaccinated:

Number of individuals who stayed healthy = 52

Number of individuals who developed COVID-19 = 5

Odds of staying healthy = Number of individuals who stayed healthy / Number of individuals who developed COVID-19 = 52/5 = 10.4

Next, let's calculate the odds ratio from the odds of staying healthy for each vaccination status:

Odds ratio = Odds of staying healthy for vaccinated / Odds of staying healthy for unvaccinated

= 10.4 / 1.76

= 5.91

Now, let's interpret the odds ratio and answer the research question. An odds ratio of 5.91 indicates that the odds of staying healthy among vaccinated individuals is 5.91 times higher compared to unvaccinated individuals.

This suggests that vaccination is associated with a significantly higher likelihood of staying healthy and not developing COVID-19 symptoms. Therefore, the data collected by the researcher suggests that the vaccine is effective in preventing the development of COVID-19 symptoms.

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Stop using brainly. This whole website has became a big money grab over the last few months and they will remove all your posts and answers unless you pay them money. I've had it happen several times now.

Martha travels 8 miles per hour, which is the same as saying that she covers 8 miles in one hour.

We can express Martha's weekend bicycling information as a rate by dividing the distance traveled by the time taken. In this case, her rate would be:

[tex]\frac{12 miles}{1.5 hours} = $8 miles per hour[/tex]

This tells us that Martha travels 8 miles for every hour that she spends bicycling.

We can express Martha's weekend bicycling information as a unit rate by dividing the distance traveled by the time taken and expressing the result in terms of one unit of time (usually one hour). In this case, her unit rate would be:

[tex]$\frac{12 miles}{1.5 hour}. \frac{1 hour}{1} = \frac{8 miles}{1 hour}[/tex]

This tells us that Martha travels 8 miles per hour, which is the same as saying that she covers 8 miles in one hour.

Expressing information as a rate or a unit rate allows us to compare different situations more easily. In this case, we could compare Martha's weekend bicycling rate with her weekday bicycling rate, or with the rates of other bicyclists, to see how they differ.

Therefore, Martha travels 8 miles per hour, which is the same as saying that she covers 8 miles in one hour.

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Specific function is not provided. To calculate, we need to find predict, and determine population.

1. Growth rate: This is the rate at which the population increases or decreases over time.

2. Measures: In this context, "measures" refers to the way the population is represented in the function, which is in millions.

3. Function: This is a mathematical relationship that describes how the population of Japan changes with respect to time (years since 2000).

To answer your question with a given function, follow these steps:

Step 1: Find the population in 2007.
Plug in the value of the number of years since 2000 (7) into the function and calculate the population in millions.

Step 2: Predict the population in 2020.
Plug in the value of the number of years since 2000 (20) into the function and calculate the population in millions.

Step 3: Determine when the population reaches 2 billion.
Set the function equal to 2000 (since 2 billion people = 2000 million) and solve for the number of years since 2000. Convert this result to the actual year by adding the number of years to 2000.

Once you have the specific function, you can follow these steps to find the answers to your questions.

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The growth rate continues, the population of Japan will reach 2 billion people in approximately the year

2226 (adding 226.3 to 2000).

To find the population in 2007, we simply need to substitute 7 for t in the function:

[tex]P(7) = 127.7(1.002)^7 ≈ 127.7(1.015) ≈ 129.6 million[/tex]

Therefore, the population of Japan in 2007 was approximately 129.6 million.

To predict the population in 2020, we substitute 20 for t in the function:

[tex]P(20) = 127.7(1.002)^20 ≈ 127.7(1.044) ≈ 133.2 million[/tex]

Therefore, the predicted population of Japan in 2020 is approximately 133.2 million.

To find the year in which the population of Japan reaches 2 billion people, we need to solve for t in the equation:

[tex]2,000 = 127.7(1.002)^t[/tex]

Taking the natural logarithm of both sides and solving for t:

ln(2,000/127.7) = t ln(1.002)

t ≈ 226.3

Therefore, if the growth rate continues, the population of Japan will reach 2 billion people in approximately the year

2226 (adding 226.3 to 2000). However, it is important to note that this is a theoretical calculation and does not take

into account any changes in the growth rate or other factors that may affect population growth in Japan.

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There are 504 boys at the school out of the total 720 students, and 168 of them are from poor homes.

We must first ascertain the number of female students in order to calculate the number of guys enrolled in the school. Since there are 30% more female students overall, we can compute percentage as follows:

216 girls is equal to 0.3 x 720 males.

The number of boys can be calculated as follows:

Total students - Number of girls equals Number of boys

Boys' number = 720 - 216

Amount of boys: 504

There are 504 guys enrolled in the school as a result. Furthermore, we are informed that 168 of these boys come from poor homes, which may be helpful for additional computations or research.

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As a result, if a bicycle covers 50 miles in 25 minutes, their speed is 120 miles per hour.

Speed is defined in physics as the distance covered in a unit of time. What determines the velocity vector's magnitude is a scalar quantity. An item is said to be travelling faster or slower depending on its speed. It has zero speed if it isn't moving at all.

Therefore, a bicycle travelling 50 miles in 25 minutes is moving at a speed of 120 miles per hour.

Miles might mean two different things. As a noun, it refers to a linear measurement unit that is 1,760 yard. It also has the adverbial meanings of greatly or far.

When the distance is 50 miles, the relationship between speed, s, and time, t, can be written as follows:

s = d/t

where d is the distance traveled, and t is the time it takes.

The distance is 50 miles in this instance, and the travel time is 25 minutes, or 25/60 hours.

When we enter these values into the formula above, we obtain:

120 miles per hour is equal to s = 50/(25/60).

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Answer: C. experimental study.

Experimental studies are used to establish cause-and-effect relationships between variables by manipulating one variable (independent variable) and observing the effect on another variable (dependent variable) while controlling for other potential factors. Correlational studies examine the relationship between two variables but do not establish causality, descriptive studies describe a phenomenon without manipulating variables, and meta-analysis is a statistical method that combines the results of multiple studies to provide an overall summary.

Step-by-step explanation:

The equivalent expression for c is c = P + b - 6.

To solve for c in the equation P=6-b+c, we need to isolate the variable c on one side of the equation.

We can start by adding -6 and b to both sides of the equation to eliminate the 6 and b terms from the right-hand side:

P - 6 + b = -b + c + 6 - 6

Simplifying the left-hand side:

P - 6 + b = P - (6 - b)

Substituting this into the right-hand side:

-c + P - (6 - b) = -c + b + P - 6

Finally, to isolate c on one side of the equation, we can add c to both sides:

-c + P - (6 - b) + c = b + P - 6

Simplifying the left-hand side:

P - 6 + b = b + P - 6

Simplifying the right-hand side:

P - 6 + b = P - 6 + b

We can see that both sides of the equation are equal, so the solution is:

c = P - 6 + b

Therefore, c = P + b - 6.

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The number of students surveyed who plays an instrument is 114 students.

The ratio of students who play instruments to those who do not is 7/20, which means that out of every 7+20=27 students, 7 of them play an instrument and 20 of them do not.

If we know that 440 students were surveyed, we can set up a proportion to find the number of students who play an instrument:

7/27 = x/440

To solve for x, we can cross-multiply:

7 * 440 = 27 * x

3080 = 27x

x = 114.07

Since we can't have a fraction of a student, we should round up to the nearest whole number, which means that approximately 114 students surveyed play an instrument.

Therefore, the required answer is 114.

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A(n) "spanned" volume is similar to a primary partition, but in fact, is not a primary partition.

A spanned volume is similar to a primary partition, but in fact, is not a primary partition. In a spanned volume, you can combine free space from multiple disks to create a larger logical storage unit, while a primary partition is a segment of a hard disk drive that the operating system recognizes as a separate logical storage unit. A volume can span multiple partitions depending on how it is configured, or it can encompass a single partition. Different operating systems have different ways of configuring and managing volumes.

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The dimension for the largest area is 181 sq ft.

The rectangle has width. [tex]$w$[/tex] and height [tex]$h$[/tex],and let the radius of the semicircle be [tex]$r$[/tex]. We want to maximize the area [tex]$A$[/tex] of the window, which is given by:

[tex]A= 21 \pi r 2 +wh[/tex]

We also have the constraint that the total length of framing material is 12 feet:

[tex]2r+2w+h=12[/tex]

Solving this equation for [tex]$h$[/tex] we get:

[tex]h=12-2r-2w[/tex]

Substituting this expression for [tex]$h$[/tex] into the equation for [tex]$A$[/tex], we get:

[tex]A= 21 \pi r ^2+w(12-2r-2w)[/tex]

Expanding this expression, we get:

[tex]A= 21\pi r^2 +12w-2rw-2w^2[/tex]

To find the values of [tex]$r$[/tex], [tex]$w$[/tex], and [tex]$h$[/tex] that maximize[tex]$A$[/tex], we take the partial derivatives of [tex]$A$[/tex] with respect to each variable, set them equal to zero, and solve for the variables.

We get:

[tex]\partial r/\partial A=\pi r-2w=0 \Rightarrow r= 2w/\pi[/tex]

[tex]\partial w/\partial A=12-2r-4w=0 \Rightarroww= (6-r)/2= (6-2w/\pi)/2=3- w/\pi[/tex]

Substituting the expression for [tex]$w$[/tex] into the equation for [tex]$r$[/tex], we get:

[tex]r= 2/\pi (3-w/\pi )= 6/\pi - 2/\pi^2w[/tex]

Substituting these expressions for [tex]$r$[/tex] and [tex]$w$[/tex] into the equation for [tex]$h$[/tex], we get:

[tex]h=12-2r-2w=12- (4/\pi)w[/tex]

So the dimensions that maximize the area are:

[tex]w= \pi/2,r= 3\pi /4 ,h= (48-8\pi) / \pi[/tex]

The area of the window is:

[tex]A= (1/2)\pir^2 +wh= (1/2)\pi ( 3\pi/4)^2 + (\pi/2)\cdot (48-8\pi)/\pi = 9\pi/8+24-4\pi\approx 18.1 sq ft[/tex]

​

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Therefore, the dimensions of the largest area rectangular window with a semicircle top that can be made with 12' of framing material are approximately 1.55' * 3.38'. The maximum area is: Area = 4.82 square feet.

To find the dimensions for the largest area of a rectangular window with a semicircle top using 12' of framing material, we need to use optimization techniques.

Let's denote the width of the rectangular window as "w" and the height of the rectangular portion as "h". We also know that the semicircle top will have a radius equal to the width of the rectangular portion, so its diameter will be 2w.

The perimeter of the window is given by:

Perimeter = 2w + h +[tex]\pi w[/tex]

Since we have 12' of framing material available, we can write:

2w + h + [tex]\pi w[/tex] = 12

We can rearrange this equation to solve for h:

h = 12 - 2w - [tex]\pi w[/tex]

The area of the window is given by:

Area = w * h + 1/2 * [tex]\pi w^2[/tex]

Substituting the expression for h from the perimeter equation, we get:

Area = w(12 - 2w - [tex]\pi w[/tex]) + 1/2 * [tex]\pi w^2[/tex]

Expanding and simplifying, we get:

Area = 12w - [tex]2\pi w^2[/tex] - [tex]w^2[/tex]

To find the dimensions that maximize the area, we need to take the derivative of the area equation with respect to w and set it equal to zero:

d(Area)/dw = 12 - [tex]4\pi w[/tex] - 2w = 0

Solving for w, we get:

w = 12/(4\pi+2) ≈ 1.55'

Substituting this value back into the perimeter equation, we can find the height:

h = 12 - 2w - \piw ≈ 3.38'

Therefore, the dimensions of the largest area rectangular window with a semicircle top that can be made with 12' of framing material are approximately 1.55' * 3.38'. The maximum area is:

Area ≈ 4.82 square feet.

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The correlation coefficient, rounded to three decimal places, is approximately 0.992.

A statistical measure known as the correlation coefficient expresses the strength and direction of the linear link between two variables. It has a range of -1 to +1, with a negative value signifying a negative (inverse) relationship, a positive value a positive relationship, and a zero a no relationship. A perfect positive relationship is shown by a correlation value of 1, whereas a perfect negative (inverse) relationship is indicated by a correlation coefficient of -1.

The regression coefficient is given by the formula:

r = [nΣ(xy) - ΣxΣy] / √([nΣx² - (Σx)²][nΣy² - (Σy)²])

Now, for the given data we have:

n = 10

Σx = 31

Σy = 675

Σx² = 124.25

Σy² = 60743

Σ(xy) = 3475.5

Substituting the value we have:

r = [10(3475.5) - (31)(675)] / √([10(124.25) - (31)²][10(60743) - (675)²])

r ≈ 0.992

Hence, the correlation coefficient, rounded to three decimal places, is approximately 0.992.

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If sandbox is 4 ft long, 3 ft wide, and 1 ft deep, then the number of "sand-bags" which are needed to fill the sandbox is 20 bags.

The "Volume" of a cuboid is defined as a measure of the amount of three-dimensional space it occupies. The volume of a cuboid can be calculated using the formula : Volume = Length × Width × Height,

To find out how many bags of sand are needed to fill the sandbox, we first need to calculate the volume of the sandbox.

The sandbox is 4 feet long, 3 feet wide, and 1 foot deep.

So, its volume can be calculated as : Volume = Length × Width × Depth,

Substituting the values,

We get,

⇒ Volume = 4 ft × 3 ft × 1 ft = 12 cubic feet

So, Volume of the sandbox is 12 cubic feet.

Next, we divide the "total-volume" of sandbox by "volume of each bag" of sand (0.6 cubic feet) to determine the number of bags needed,

⇒ Number of bags = (Total volume)/(Volume per bag),

⇒ Number of bags = 12/0.6 = 20 bags,

Therefore, 20 bags of sand are needed to fill the sandbox.

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Answer:

X; Y

Step-by-step explanation:

Let I be the midpoint of UV

Coordinates of midpoint: x = (5-7)÷2 = -1;   y = (8-6)÷2 = 1

(W) has center I(-1,1) and radius = IU = [tex]\sqrt{(5-(-1))^2+(8-1)^2}=\sqrt{85}[/tex]

Equation of (W): [tex](x+1)^2+(y-1)^2=85[/tex]

We see that X(8,3); Y(6,-5) satisfy the equation of (W)

Answer:
No

Step-by-step explanation:

3(-20-3)-5>23

simplify:

Answer:

35° 21

Step-by-step explanation:

In an isosceles triangle, the angles opposite to the equal sides of an isosceles triangle are equal in measurement1. Therefore, if ∠A = 110° and ABC is an isosceles triangle, then ∠B = ∠C = (180° - ∠A)/2 = (180° - 110°)/2 = 35° 21.

I hope this helps!

Hence,the area of circle [tex]=144\pi[/tex][tex]cm^{2}[/tex]

In a plane all points  are at a specific distance from a specific point, the center, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant.

The patch of size on a surface is determined by its area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, therefore  the area of a plane area or .plane region  refers to the area of a form  of plane lamina .

we know that ,

area of circle =[tex]\pi r^{2}[/tex]

according to question,

r= 12 cm.

So,the area of circle

[tex]=\pi *12^{2} \\=\pi *144[/tex]

or the area of circle [tex]=144\pi[/tex][tex]cm^{2}[/tex]

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An equation that could be solved using this application of the quadratic formula include the following: D. 2x² + 12x + 4 = 13.

In Mathematics and Geometry, a quadratic equation can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the standard form of a quadratic equation is represented by the following equation;

ax² + bx + c = 0

Mathematically, the quadratic formula is modeled or represented by this mathematical equation:

[tex]x = \frac{-b\; \pm \;\sqrt{b^2 - 4ac}}{2a}[/tex]

For the given quadratic equation 2x² + 12x + 4 = 13, we have:

2x² + 12x + 4 = 13

2x² + 12x + 4 - 13 = 0

2x² + 12x - 9 = 0

By substituting, we have;

[tex]x = \frac{-(12)\; \pm \;\sqrt{(12)^2 - 4(2)(-9)}}{2(2)}[/tex]

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Histogram would be the most appropriate graphical representation for the collected data,  because it shows each individual data point.

A histogram is a graphical representation of a distribution of continuous data that is used to visualize the underlying frequency distribution of a set of data. Here are the general steps to plot a histogram:

1. Choose the appropriate number of bins: The first step in creating a histogram is to choose the number of bins or intervals that the data will be divided into. This decision is based on the number of data points and the range of the data.

2. Determine the range of the data: Determine the range of the data, which is the difference between the largest and smallest values in the data set.

3. Calculate the width of each bin: Divide the range of the data by the number of bins to determine the width of each bin.

4. Plot the horizontal axis: The horizontal axis of the histogram represents the range of the data and is divided into the chosen number of bins.

5. Plot the vertical axis: The vertical axis of the histogram represents the frequency or density of the data.

6. Count the number of observations: Count the number of observations that fall into each bin.

7. Plot the bars: Plot the bars of the histogram such that the height of each bar corresponds to the number of observations in each bin.

8. Label the axes and title: Finally, label the horizontal and vertical axes appropriately and give the histogram an appropriate title.

Histogram would be the most appropriate graphical representation for the collected data of milligrams of Vitamin C from 100 different gummy vitamins sold in the world.

A histogram is a graph that displays the distribution of a continuous data set using bars. It is useful for showing the frequency distribution of a large data set and helps to identify patterns or outliers in the data.

In this case, the milligrams of Vitamin C is a continuous variable, and the histogram would display the frequency of different milligram ranges. It would show the concentration range of the vitamin C in the gummy vitamins sold worldwide and provide an overall picture of the distribution of the data.

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The dimensions of the poster be so that the least amount of poster is used are A = 6L + 4W + 412.

Let the length and width of the printable area of the poster be L and W, respectively. Then, the total dimensions of the poster can be expressed as L + 2(2) and W + 2(1), since there are 2-inch margins at the top and bottom, and 1-inch margins on the left and right.

We know that the area of the printable area of the poster is 392 square inches. Therefore, we can write the equation: LW = 392

We want to minimize the total area of the poster, which is given by:

A = (L + 2(2))(W + 2(1)) = (L + 4)(W + 2)

Expanding this expression, we get:

A = LW + 2L + 4W + 8

Substituting the equation for LW, we get:

A = 392 + 2L + 4W + 8

Simplifying, we get:

A = 2L + 4W + 400

To minimize this expression, we can take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 2 = 0 => L = 0[/tex]

[tex]

∂A/∂W = 4 = 0 => W = -100[/tex]

These values do not make sense in the context of the problem. Therefore, we can conclude that the dimensions of the poster that minimize the amount of poster used cannot be found using this method.

Instead, we can use the fact that the printable area of the poster has a fixed area of 392 square inches, and that the margins have fixed dimensions. We can express the area of the poster as:

A = (L + 4)(W + 2) = LW + 4L + 2W + 8

Substituting the equation for LW, we get:

A = 392 + 4L + 2W + 8

Simplifying, we get:

A = 4L + 2W + 400

To minimize this expression, we can again take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 4 = 0 => L = 0[/tex]

[tex]∂A/∂W = 2 = 0 => W = -200[/tex]

These values do not make sense in the context of the problem. Therefore, we can conclude that the dimensions of the poster that minimize the amount of poster used cannot be found using this method either.

We can try a different approach. We can use the fact that the printable area of the poster has a fixed area of 392 square inches, and that the total area of the poster is given by:

A = (L + 4)(W + 2) + 2(L + 4) + 2(W + 2)

Expanding this expression, we get:

A = LW + 6L + 4W + 20

Substituting the equation for LW, we get:

A = 392 + 6L + 4W + 20

Simplifying, we get: A = 6L + 4W + 412

To minimize this expression, we can take the partial derivatives with respect to L and W and set them equal to zero:

[tex]∂A/∂L = 6 = 0 => L = -2/3[/tex]

[tex]∂A/∂W = 4 = 0 => W = -3/2[/tex]

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