Let the random variables X, Y have joint density function

3(2−x)y if0
f(x,y) =

(a) Find the marginal density functions fX and fY.

(b) Calculate the probability that X + Y ≤ 1

To solve for x and y in this problem, we need to use the formulas for arithmetic and geometric sequences.

For the arithmetic sequence, we know that the difference between each term is the same. Let's call this difference "d". So we have:

-8 + d = x
x + d = y
y + d = 72

For the geometric sequence, we know that the ratio between each term is the same. Let's call this ratio "r". So we have:

x * r = y
y * r = 72

Now we can use these equations to solve for x and y.

First, we'll use the arithmetic sequence equations to find the value of "d". We can subtract the first equation from the second equation to get:

d = y - x

We can then substitute this into the third equation to get:

y + (y - x) = 72

Simplifying this, we get:

2y - x = 72

Now we can use the geometric sequence equations to find the value of "r". We can divide the second equation by the first equation to get:

r = y/x

We can then substitute this into the first equation to get:

x * (y/x) = y

Simplifying this, we get:

y = x^2

Now we have two equations for "y", so we can substitute one into the other to get an equation in terms of "x" only:

2x^2 - x = 72

Solving this quadratic equation, we get:

x = -8 or x = 9

We can then substitute each of these values back into the equation y = x^2 to get:

y = 64 or y = 81

So the solutions are:

x = -8, y = 64
x = 9, y = 81

Therefore, the first three terms of the sequence are -8, -8+17=9, 9+17=26 and the second, third, and fourth terms are 9, 26, 72.
In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant.

Given the arithmetic sequence: -8, x, y, the difference between consecutive terms is constant, so we can say that x - (-8) = y - x. Simplifying, we get x + 8 = y - x, and then 2x = y - 8 (Equation 1).

Now, considering the geometric sequence: x, y, 72, the ratio between consecutive terms is constant. Therefore, y/x = 72/y. By cross-multiplying, we obtain y^2 = 72x (Equation 2).

To determine x and y, we can solve this system of equations. Using Equation 1, y = 2x + 8. Substitute this expression for y in Equation 2:

(2x + 8)^2 = 72x
4x^2 + 32x + 64 = 72x
4x^2 - 40x + 64 = 0
x^2 - 10x + 16 = 0
(x - 8)(x - 2) = 0

From this quadratic equation, we have two possible values for x: x = 8 or x = 2.

If x = 8, then y = 2x + 8 = 24. This would result in the geometric sequence 8, 24, 72, which has a constant ratio of 3.

If x = 2, then y = 2x + 8 = 12. This would result in the geometric sequence 2, 12, 72, which has a constant ratio of 6.

Both solutions are valid, so we have two possible sets of values for x and y: x = 8, y = 24 or x = 2, y = 12.

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The solutions for x and y are: 1. x = 2, y = 12 and

2. x = 8, y = 24

To determine the values of x and y in the sequence -8, x, y, 72, analyze the information given.

First, consider the arithmetic sequence formed by the first three terms: -8, x, y. In an arithmetic sequence, the common difference between consecutive terms is constant.

Therefore, set up the following equation:

x - (-8) = y - x

Simplifying the equation, we have:

x + 8 = y - x

2x + 8 = y

Next, given that the second, third, and fourth terms form a geometric sequence: x, y, 72. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio.

Express this relationship using the following equation:

y / x = 72 / y

Cross-multiplying, we get:

y² = 72x

Now, we have two equations:

2x + 8 = y (Equation 1)

y² = 72x (Equation 2)

To solve for x and y, we'll substitute Equation 1 into Equation 2:

(2x + 8)² = 72x

Expanding and simplifying:

4x² + 32x + 64 = 72x

Rearranging the terms:

4x² + 32x - 72x + 64 = 0

4x² - 40x + 64 = 0

Dividing the entire equation by 4:

x² - 10x + 16 = 0

Factoring the quadratic equation, we have:

(x - 2)(x - 8) = 0

Setting each factor equal to zero and solving for x, we get:

x - 2 = 0 -> x = 2

x - 8 = 0 -> x = 8

So, x can be either 2 or 8.

If we substitute these values back into Equation 1, we can find the corresponding values of y:

For x = 2:

2(2) + 8 = y

4 + 8 = y

12 = y

For x = 8:

2(8) + 8 = y

16 + 8 = y

24 = y

Therefore, the possible solutions for x and y are:

1. x = 2, y = 12

2. x = 8, y = 24

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Answer:

19

Step-by-step explanation:

Let's call the number we're trying to find "x".

According to the problem:

4x + 25 = 6x - 13

To solve for x, we can start by isolating the x term on one side of the equation. Let's subtract 4x from both sides:

4x + 25 - 4x = 6x - 13 - 4x

25 = 2x - 13

Next, let's add 13 to both sides:

25 + 13 = 2x - 13 + 13

38 = 2x

Finally, we can divide both sides by 2 to solve for x:

38/2 = 2x/2

19 = x

The probability of not picking a yellow marble is 60% (option D)

Probability is the odds that a random event would occur. The chances that a random event would happen has a value that lies between 0 and 1. The more likely it is that the event would happen, the closer the probability value would be to 1.

Probability of not choosing a yellow marble from the bag = number of marbles that are not yellow / total number of marbles

number of marbles that are not yellow = 2 + 4+ 9 = 15

total number of marbles = 2 + 4 + 9 + 10 = 25

Probability of not choosing a yellow marble from the bag = 15/25 = 3/5 = 60%

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The probability P(not yellow) when choosing one marble from the bag is 60%

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

Not Yellow = Red + Green + Purple

This gives

Not Yellow = 2 + 4 + 9

Evaluate

Not Yellow = 15

So, we have the probability notation to be

P(Not Yellow) = Not Yellow/Total

This gives

P(Not Yellow) = 15/(15 + 10)

Evaluate

P(Not Yellow) = 60%

Hence, the value is 60%

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Therefore, if the person works 54 hours, they would make $1,163.04. Rounded to 2 decimal places, the answer is $1,163.00.

The decimal system employs ten decimal digits, a decimal mark, and a minus sign ("-") for negative quantities when writing numbers. The decimal digits are 0 through 9, with the dot (".") serving as the decimal separator in many (mainly English-speaking) nations and the comma (",") in others.

The fractional portion of the number is represented by the place value that follows the decimal. The number 0.56, for instance, is composed of 5 tenths and 6 hundredths.

We can use proportionality to solve this problem. If the person works 25 hours and makes $539, then their hourly rate is:

$539 ÷ 25 hours = $21.56 per hour

They would make if they work 54 hours, we can multiply their hourly rate by the number of hours worked:

$21.56 per hour × 54 hours = $1,163.04

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The P-value is 0.0107 for the sample statistics n-35 and the coefficient of standard deviation is 82.

α = 0. 05

μ = 1620

size (n)= 35

X_bar=1590

σ=82

From the given sample statistics, the test statistics will be calculated as:

t = (X_bar - μ) / (σ / sqrt(n))

t = (1590 - 1620) / (82 / sqrt(35))

t = (-2.5411)

Using the t-distribution table with 34 degrees of freedom, the critical value will be:

t_critical = -1.6909

Here the calculated test statistic is less than the critical value.

P - value = 2*P(-100< t < -1.9720, when df = 34)

P = tcf (-100,-2.4103,34)

P = 0.0107

Therefore we can conclude that the P-value is 0.0107.

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The school which is a better choice is sea side.

We are given that;

The plot

Now,

If you are interested in a smaller class size, Seaside School is a better choice for you because it has a smaller mean and median class size than Bay Side School. This means that on average and in general, Seaside School has fewer students per class than Bay Side School. Also, Seaside School has a smaller maximum class size than Bay Side School (both have a minimum of zero), so you are less likely to encounter a very large class at Seaside School.

Therefore, by algebra the answer will be sea side.

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We can expect a difference of about 6.67 between the two sample means.

To calculate how much difference to expect between two sample means when the population means are equal, we need to compute the standard error of the difference between means (SED).

The formula for SED in the independent-measures t-test is:

SED = sqrt((s1^2/n1) + (s2^2/n2))

where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

a) For the first situation, we have:

s1^2 = SS1/(n1-1) = 500/(6-1) = 100

s2^2 = SS2/(n2-1) = 524/(12-1) = 49.45

Plugging these values into the formula, we get:

SED = sqrt((100/6) + (49.45/12)) = 5.76

Therefore, we can expect a difference of about 5.76 between the two sample means.

b) For the second situation, we have:

s1^2 = SS1/(n1-1) = 600/(6-1) = 120

s2^2 = SS2/(n2-1) = 696/(12-1) = 69.6

Plugging these values into the formula, we get:

SED = sqrt((120/6) + (69.6/12)) = 6.67

Therefore, we can expect a difference of about 6.67 between the two sample means.

When the samples have larger variability (bigger SS values), the standard error for the sample mean difference will increase. This is because larger variability means that the scores are more spread out around their respective means, which increases the amount of variability in the difference between the two sample means. In contrast, when the variability is smaller, the scores are more tightly clustered around their means, and the standard error for the sample mean difference will be smaller.

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The segment length and the conversion of radian and degree are given below.

We have,

In order to solve for segment length in relation to circles, chords, secants, and tangents, we need to first define some terms:

Circle: A set of all points in a plane that are equidistant from a given point called the center of the circle.

Chord: A line segment joining two points on a circle.

Secant: A line that intersects a circle in two points.

Tangent: A line intersecting a circle at exactly one point, called the point of tangency.

Segment: A part of a circle bounded by a chord, a secant, or a tangent and the arc of the circle that lies between them.

Now, let's consider the following cases:

Chord-chord intersection:

If two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. That is:

AB × BC = DE × EF

where AB and BC are the lengths of the segments of one chord, and DE and EF are the lengths of the segments of the other chord.

Secant-secant intersection:

If two secants intersect outside a circle, the product of the length of one secant and its external segment is equal to the product of the length of the other secant and its external segment. That is:

AB × AC = DE × DF

where AB and AC are the length of one secant and its external segment, and DE and DF are the length of the other secant and its external segment.

Secant-tangent intersection:

If a secant and a tangent intersect outside a circle, the product of the length of the secant and its external segment is equal to the square of the length of the tangent. That is:

AB × AC = AD^2

where AB and AC are the length of the secant and its external segment, and AD is the length of the tangent.

Tangent-tangent intersection:

If two tangents intersect outside a circle, the lengths of the two segments of one tangent are equal to the lengths of the two segments of the other tangent. That is:

AB = CD

BC = DE

where AB and BC are the lengths of the two segments of one tangent, and CD and DE are the lengths of the two segments of the other tangent.

Using these formulas, we can solve for segment length in various situations involving circles, chords, secants, and tangents.

To convert the degree measure to radian measure, we use the fact that 360 degrees is equal to 2π radians.

Therefore, we can use the following conversion formula:

radian measure = (degree measure × π) / 180

For example:

Convert 45 degrees to radians:

radian measure = (45 degrees × π) / 180

radian measure = (45/180)π

radian measure = π/4

So 45 degrees is equal to π/4 radians.

Convert 120 degrees to radians:

radian measure = (120 degrees × π) / 180

radian measure = (2/3)π

So 120 degrees is equal to (2/3)π radians.

Convert 270 degrees to radians:

radian measure = (270 degrees × π) / 180

radian measure = (3/2)π

So 270 degrees is equal to (3/2)π radians.

Note that radians are a more natural unit for measuring angles in many mathematical contexts, as they relate directly to the arc length of a circle.

To convert the radian measure to degree measure, we use the fact that 180 degrees equal π radians.

Therefore, we can use the following conversion formula:

degree measure = (radian measure × 180) / π

For example:

Convert π/3 radians to degrees:

degree measure = (π/3 radians × 180) / π

degree measure = 60 degrees

So π/3 radians is equal to 60 degrees.

Convert 2π/5 radians to degrees:

degree measure = (2π/5 radians × 180) / π

degree measure = (360/5) degrees

degree measure = 72 degrees

So 2π/5 radians is equal to 72 degrees.

Convert 3π/4 radians to degrees:

degree measure = (3π/4 radians × 180) / π

degree measure = (540/4) degrees

degree measure = 135 degrees

So 3π/4 radians is equal to 135 degrees.

Note that degree measure is commonly used in everyday life and in many technical fields, whereas radian measure is often used in advanced mathematics, physics, and engineering.

Thus,

The segment length and the conversion of radian and degree are given above.

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(a) The statement is false because we have found a case where n - m² is even.

(b) The statement holds true.

(c) The statement is false because we have found a case where s^2 is rational despite s being irrational.

(d) It is not possible to find two odd integers n and m such that n²m² - 1 is odd. Thus, the statement is false.

(a) The statement "for any integers n and m, if both n and m are odd, then n - m² is even" is incorrect. Let's consider a counterexample:

Take n = 3 and m = 1. Both n and m are odd.

n - m² = 3 - 1² = 3 - 1 = 2, which is an even number.

Therefore, the statement is false because we have found a case where n - m² is even.

(b) The statement "for any prime number p, p-2 is not prime" is generally true. Let's consider the cases:

If p is an odd prime greater than 2, then p-2 is an even number, and the only even prime number is 2. Therefore, p-2 cannot be prime in this case.

If p = 2, then p-2 = 0, which is not considered a prime number.

In both cases, p-2 is not a prime number. Therefore, the statement holds true.

(c) The statement "for any real number s, if s is irrational, then s^2 is irrational" is incorrect. Let's consider a counterexample:

Take s = √2. √2 is an irrational number.

s^2 = (√2)^2 = 2, which is a rational number.

Therefore, the statement is false because we have found a case where s^2 is rational despite s being irrational.

(d) The statement "There are two odd integers n and m such that n²m² - 1 is odd" is true. Let's consider the following example:

Take n = 1 and m = 1. Both n and m are odd.

n²m² - 1 = 1² * 1² - 1 = 1 * 1 - 1 = 0, which is an even number.

However, if we take n = 3 and m = 1, both n and m are still odd.

n²m² - 1 = 3² * 1² - 1 = 9 * 1 - 1 = 9 - 1 = 8, which is an even number.

Therefore, it is not possible to find two odd integers n and m such that n²m² - 1 is odd. Thus, the statement is false.

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The measure of <BAE is 130 degree.

We have,

m <C = 65

As, opposite sides are parallel then ABCD is a parallelogram.

Then, opposite angles of a parallelogram are congruent

∠ BAD = ∠ BCD = 65°

∠ ADE = ∠ BAD = 65° ( alternate angles )

Since, DE = AE then Δ ADE is isosceles Triangle.

So, ∠ DAE = ∠ ADE = 65°

We can write,

∠ BAE = ∠ BAD + ∠ DAE = 65° + 65° = 130°

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The probability that less than 61% of sampled teenagers own smartphones is 0.934 or 93.4%.

To find the probability that less than 61% of sampled teenagers own smartphones, we need to use the standard normal distribution. We first need to standardize the value of 61% using the formula:

z = (x - μ) / σ

where x is the value we want to standardize (in this case, 61%), μ is the mean (0.55), and σ is the standard deviation (0.0397).

Plugging in the values, we get:

z = (0.61 - 0.55) / 0.0397 = 1.511

We can then look up the probability of getting a z-score less than 1.511 in a standard normal distribution table or calculator. The probability is approximately 0.934.

Therefore, the probability that less than 61% of sampled teenagers own smartphones is 0.934 or 93.4%.

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Complete question:

e) Find the probability that less than 61% of sampled teenagers own smartphones.

(a) Find the mean :The mean μ p is 0.55

(b) Find the standard deviation

The standard deviation σp is 0.0397

There was a significant reduction in peoples over estimation of the line length, t = XXX with df = XXX, p < 0.01 two-tailed, with M = 10%, n = 25, s = 15%, Cohen's d = XXX , M = 10%, 95% CI [XXX, XXX].

8a. A. was
8b. B. t = XXX with df = XXX
8c. A. < 0.01 two-tailed
8d. C. M = 10%, n = 25, s = 15%, Cohen's d = XXX , M = 10%, 95% CI [XXX, XXX].

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In this case, the scientist is studying if students' beliefs about illegal drug use change as they go through college. The null hypothesis (H0) for McNemar's test in this context would state that there is no significant change in students' beliefs about illegal drug use between the time they enter college and four years later. In other words, the proportion of students who change their beliefs about illegal drug use is not significantly different from the proportion who do not change their beliefs.

Null hypothesis (H0): There is no significant difference in students' beliefs about illegal drug use before and after going through college.

Let 0-1 represent students who participate in sports and let 2-9 represent students who do not participate in sports.

Let 0-9 represent students selected for the sample, with digits 0-8 representing students who participate in sports and digit 9 representing a student who does not participate in sports.

So, an appropriate assignment of digits for this simulation would be: Let 0-1 represent students who participate in sports and let 2-9 represent students who do not participate in sports.

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There are four possible values for the random variable X: 0, 1, 2, and 3

To find the probability distribution of X, which represents the number of girls in a family with three children, we first need to determine the possible values for the random variable X.

Part 1: Find the number of possible values for the random variable X.
There can be 0, 1, 2, or 3 girls in the family. Therefore, there are 4 possible values for the random variable X.

The random variable X represents the number of girls in a family with three children. To determine the possible values for X, we consider the number of girls that can exist in the family. In this case, there can be zero, one, two, or three girls.

When no girls are present, X takes the value 0. If there is one girl, X takes the value 1. If there are two girls, X takes the value 2. Finally, if there are three girls, X takes the value 3.

Therefore, there are a total of four possible values for the random variable X, which correspond to the different combinations of the number of girls in the family.

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While it is not very likely that there will be exactly 14 births on any given day, it is still possible, and the probability of it happening is about 8.3%.

Let's start by calculating the mean or average number of births per day. To do this, we divide the total number of births in a year (4126) by the number of days in a year. Since there are 365 days in a year, the mean number of births per day is:

4126 / 365 = 11.3

This means that on average, there are about 11 to 12 births per day in this hospital.

In this case, the average rate of occurrence is 11.3 births per day. Using the Poisson distribution formula, we can calculate the probability of having 14 births in a day as follows:

P(X=14) = (e⁻¹¹°³) x (11.3¹⁴) / 14!

where e is the mathematical constant approximately equal to 2.71828, X is the random variable representing the number of births in a day, and ! represents the factorial function.

Using a calculator or a software tool, we get:

P(X=14) = 0.083

This means that the probability of having exactly 14 births in a day is about 8.3%.

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The requried relation is, the higher the grade level in school, the more hours spent on homework. Option A is correct.

From the given data, we can see that the hours spent on homework per day increase as the grade level in school increases. This suggests that there is a positive correlation between grade level and homework hours.

In general, higher grade levels in school involve more advanced coursework and greater academic expectations, which can require more time and effort outside of class to complete homework and study. Therefore, it is reasonable to expect that students in higher grade levels will spend more hours on homework per day compared to students in lower grade levels.

Thus, the requried relation is, the higher the grade level in school, the more hours spent on homework. Option A is correct.

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Jonty is correct as the original cost of paint is £196 which is less than £200 as said by him.

The area of part of cuboid to be painted will be the sum of all the unpainted areas. So, the area remaining to be painted will be = (2 × 3 × 2.5) + (2 × 3 × 12) + (12 × 2.5)

Remaining area = 15 + 72 + 30

Remaining area = 117 m²

Let us assume the original cost of paint be x. So,

x + 10%x = 26.95

110x = 26.95 × 100

110x = 2695

x = £24.5

Now, number of required tins = total unpainted area/area covered by one tin

Number of required tins = 117/15

Number of required tins = 7.8 tins

Taking it as 8 tins.

Previous cost of tins = 8 × 24.5

Previous cost = £196

Since the original cost is less than £200, Jonty is stating truth.

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The complete question is attached in figure.

The probability that the randomly selected point falls in the green sector is 1/6.

Option B is the correct answer.

We have,

The area of the green sector can be found by using the formula for the area of a sector:

A = (θ/360)πr²,

Where θ is the central angle and r is the radius.

In this case,

θ = 60° and r = 6 inches,

So the area of the green sector is:

A = (60/360)π(6)²

A = π(6)²/6

A = 6π

So,

The total area of the spinner is π(6)² = 36π.

So the probability of the randomly selected point falling in the green sector is:

P = (Area of green sector)/(Total area of spinner)

P = (6π)/(36π)

P = 1/6

Therefore,

The probability that the randomly selected point falls in the green sector is 1/6.

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The required solution for the given right angle triangle is given below.

Blank #1: We can use the Pythagorean theorem to find AB:

[tex]AB = \sqrt{AC^2 + CB^2} \\= \sqrt{12^2 + 9^2}\\ =15[/tex]

Therefore, AB = 15.

Blank #2: We can use the inverse tangent function to find the angle A:

tan(A) = opposite / adjacent = CB / AC = 9 / 12

[tex]A = tan^{-1}(9/12) = 36.86^o[/tex]

Therefore, angle A ≈ 36 degrees.

Blank #3: We can use the inverse cosine function to find the angle C:

cos(C) = adjacent / hypotenuse = CB / AB = 9 / 15

C = arccos(9/15) ≈ 53.14

Therefore, angle C ≈ 53.14 degrees.

Blank #4: We can use the fact that the sum of angles in a triangle is 180 degrees to find angle B:

B = 180 - A - C ≈

B= 90

Therefore, angle B ≈ 90

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To approximate the area of the region defined by[tex]y = √(3x)[/tex] using upper and lower sums, we first divide the interval [0,1] into n subintervals of equal width [tex]Δx = 1/n[/tex]. We then compute the upper and lower sums using the formulae above, and take their average to obtain the approximate area.

To approximate the area of the region defined by the function [tex]y = √(3x)[/tex]using upper and lower sums, we first need to divide the interval of integration [0,1] into subintervals of equal width. Let n be the number of subintervals, then the width of each subinterval is[tex]Δx = 1/n[/tex].

The upper sum is the sum of the areas of rectangles whose heights are taken from the upper endpoints of each subinterval. Specifically, for each i from 1 to n, we compute the height of the rectangle as f(xi), where xi is the upper endpoint of the i-th subinterval.

Upper sum =[tex]Δx [f(x1) + f(x2) + ... + f(xn)], where x1 = 0, x2 = Δx, x3 = 2Δx, ..., xn = (n-1)Δx.[/tex]Similarly, the lower sum is the sum of the areas of rectangles whose heights are taken from the lower endpoints of each subinterval.

Lower sum = [tex]Δx [f(x0) + f(x1) + ... + f(xn-1)][/tex], where[tex]x0 = 0, x1 = Δx, x2 = 2Δx, ..., xn-1 = (n-1)Δx.[/tex] To find the approximate area of the region using upper and lower sums, we simply compute the upper and lower sums using the given number of subintervals, and take their average: Approximate area = (Upper sum + Lower sum)/2.

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If cruz purchased a large pizza for $12.75. It serves 5 people, the cost per serving of the pizza is $2.55. So, correct option is A.

To find the cost per serving of the pizza, we need to divide the total cost of the pizza by the number of servings. In this case, the pizza costs $12.75 and serves 5 people.

Therefore, the cost per serving can be calculated as:

Cost per serving = Total cost of pizza / Number of servings

Cost per serving = $12.75 / 5

Cost per serving = $2.55

So, the cost per serving of the pizza is $2.55.

When working with fractions or dividing quantities, we need to pay attention to the units involved. In this case, the units of the cost and the servings must match for the division to be meaningful.

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The approximate area of composite figure is 80.01cm2, the correct option is A.

We are given that;

Measurements= 7cm, 10cm and 6cm

Now,

Area of triangle= 1/2 x 7 x 6

=21cm2

Area of semicircle= 3.14*7/2

=10.99cm2

Area of rectangle= 10*7

=70cm2

Area of figure= 21 + 70 - 10.99

=80.01cm2

Therefore, by area the answer will be 80.01cm2.

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The minimum sample size required to estimate an unknown population mean is 148. So, the correct option is D) 148.

To find the minimum sample size required to estimate an unknown population mean with 90% confidence, within 3.4 lb of the population mean, and a population standard deviation of 25 lb, follow these steps:

1. Identify the given values:
  - Confidence level = 90%
  - Margin of error (E) = 3.4 lb
  - Population standard deviation (σ) = 25 lb

2. Find the corresponding z-score for the 90% confidence level. Using a standard normal distribution table or calculator, the z-score is 1.645.

3. Use the formula to find the sample size (n):

  n = (z * σ / E)^2
  n = (1.645 * 25 / 3.4)^2

4. Calculate the sample size:

  n ≈ 147.267

Since we cannot have a fraction of a person, round up to the nearest whole number to ensure the required confidence level.

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1. The discriminant is positive, it has two real solutions

2. The discriminant is zero, it has a real solution

3. The discriminant is negative, it has no real solution

The discriminant of a graph is expressed as the part of the quadratic formula that is found under the square root symbol: b²-4ac.

It describes and gives information on whether there are two solutions, one solution, or no solutions.

It is important to note the following about discriminants;

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The missing figures in the diagrams are:      88km                    616km²,   37.7km                      113.14km respectively

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the center. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.

                Radis          diameter            circumference            area

S/N           r                        2r                         2πr                           πr²

Applying the above formulae in each of the questions we have as follows:

1             3                  2*3=6             2*3.14*3=18.84        3.14*3*3=28.26       2        3.5                   7                 2*3.14*3.5=21.98    3.14*3.5*3.5=38.47      

3       7.5                 15               2*3.14*7.5=47.1ft      3.14*7.5*7.5=176.25

4       14km           28km         2*3.14*14=87.92km    3.14*14*14=615.44km²

5        5mi           10mi        2*3.14*5=31.4mi               3.14*5*5= 78.5mi²

6       2.5cm         5cm         2*3.14*2.5=15.7cm          3.14*2.5*2.5=19.63cm²

7      

14                     14*2 28             2*22/7*14                    22/7*14*14  

                                                     88km                    616km²

6                             12                2*22/7*6                     22/7*6*6

                                                    37.7km                      113.14km

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The solution of the formula for the variable r is given as follows:

r = (I - t)/P.

The formula in this problem is defined as follows:

I = Pr + t.

To solve the formula for the variable r, we first must isolate the term with the variable r, as follows:

Pr = I - t.

Then we isolate the variable r applying the division operation, which is the inverse operation to the multiplication, giving the solution as follows:

r = (I - t)/P.

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The probability of exactly one success in the binomial experiment would be 20. 4 %.

The probability that there is one success in a binomial probability which has a chance of success of 5 % can be found by the formula :

P ( X = 1) = (5 choose 1) x ( 0.05 ) x  (0.95 ) ⁴

= ( 0.05 ) x  ( 0. 95 ) ⁴

= 0.05 x 0.8145

= 0.040725

Multiplying both gives:

P(X = 1) = 5 x 0.040725

= 0.203625

In conclusion, the probability of one success is 0.203625 or 20. 4 %.

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The cost of Legos, given that this is based on vowels and consonants would be $ 19.

The vowels and consonants can be arranged such that:

Doll - 1 vowel (o), 3 consonants ( d , l , l ) - $ 17

Kite - 2 vowels ( i, e ) , 2 consonants ( k, t) - $ 14

Skates - 2 vowels ( a, e ), 4 consonants ( s, k , t , s) - $24

Using this, we can solve for vowels and consonants such that cost per vowel is $2, and the cost per consonant is $5.

The cost of Legos is based on 2 vowels (e, o) and 3 consonants (L, g, s)

= ( 2 x 2 ) + ( 5 x 3 )

= $ 19

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