How do you find the standard deviation of the sum of 4 random variables?

The mass and center of mass of the lamina that occupies the region D and has the given density function p is 57/14 and (14/27, 7/18) respectively.

The center of mass (x―,y―) of a lamina with density function ρ(x,y) is given by

x = M(y)/m, y = M(x)/m

Where, m=∫∫[tex]_{D}[/tex]ρ(x,y)dA

Mx=∫∫[tex]_{D}[/tex] yρ(x,y)dA

My=∫∫[tex]_{D}[/tex] xρ(x,y)dA

Given that, D is bounded by y=x^2 and x=y^2

And ρ(x,y)=19√x

Now, for the point of intersection of y=x^2,x=y^2

we have,

x = (x^2)^2

x = x^4

Subtract x^4 on both side

x - x^4 = 0

x(x^3−1) = 0

x = 0, 1

Now, x=0⇒y=0 and x=1⇒y=1

The points of intersection are (0,0),(1,1)

So, the region D can be written as

D={(x,y): 0≤x≤1, x^2≤y≤x}

So,

m = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}19 \sqrt xdydx[/tex]

m = [tex]19\int_{0}^{1}\sqrt{x}[y]^{x^2}_{x}dx[/tex]

m = [tex]19\int_{0}^{1} \sqrt x(\sqrt{x}-x^2)dx[/tex]

m = [tex]19\int^{1}_{0}(x-x^{5/2})dx[/tex]

m = [tex]19[\frac{x^2}{2}-\frac{x^{7/2}}{7/2}]^1_{0}[/tex]

m = [tex]19[\frac{1}{2}(1^2-0)-\frac{2}{7}(1^{7/2}-0)][/tex]

m = 19(1/2−2/7)

m = 57/14

m = 5714

Now,

Mx = [tex]\int_{0}^{1}\int_{x^2}^{\sqrt x}(19xy)dydx[/tex]

Mx = 19[tex]\int^{1}_{0}x[\frac{y^2}{2}]_{x^2}^{\sqrt x}dx[/tex]

Mx = [tex]\frac{19}{2}\int^{1}_{0}x[(\sqrt{x})^2-(x^2)^2}dx[/tex]

Mx = [tex]\frac{19}{2}\int^{1}_{0}(x^2-x^5)dx[/tex]

Mx = [tex]\frac{19}{2}[\frac{x^3}{3}-\frac{x^6}{6}]_{0}^{1}[/tex]

Mx = [tex]\frac{19}{2}[\frac{1}{3}(1^3-0)-\frac{1}{6}(1^6-0)][/tex]

Mx = 19/2 (1/3−1/6)

Mx = 19/12

And

My = [tex]\int^{1}_{0}\int_{x^2}^{\sqrt x}x(19\sqrt{x})dydx[/tex]

My = [tex]19\int_{0}^{1}x^{3/2}[y]^{\sqrt{x}}_{x^2}dx[/tex]

My = [tex]19\int_{0}^{1}x^{3/2}(\sqrt{x}−x^2)dx[/tex]

My = [tex]19\int_{0}^{1}(x^2-x^{7/2})dx[/tex]

My = [tex]19[\frac{x^3}{3}-\frac{x^{9/2}}{9/2}]_{0}^{1}[/tex]

My = 19[1/3(1^3−0)−2/9(1^{9/2}−0)]

My = 19(1/3−2/9)

My = 19/9

So, x = My/m

x = (19/9)/(57/14)

x = (19/9)×(14/57)

x = 14/27

y = Mx/m

y = (19/12)/(57/14)

y = (19/12)×(14/57)

y = 7/18

Therefore, the solutions are (14/27, 7/18).

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The complete question is given below:

The x-intercept and coordinate of the vertex of the given parabola will be x = 1,7 and (4,-9) respectively.

A pair of numbers that employ the horizontal and vertical distinctions from the two reference axes to represent a point's placement on a coordinate plane. typically expressed by the x-value and y-value pairs (x,y).

As per the given parabola,

y = x² - 8x + 7

At x-intercept, y = 0

x² - 8x + 7 = 0

x² - 7x - x + 7 = 0

x(x - 7) - (x - 7) = 0

(x - 1)(x - 7) = 0

x = 1,7

For the vertex, the slope will be zero,

y' = 2x - 8 + 0 = 0

2x = 8

x = 4

Thus, y = 4² - 8 x 4 + 7

y = -9

Thus, the coordinate of the vertex is (4,-9)

Hence "The given parabola's x-intercept and vertex coordinates are x = 1,7 and (4,-9), respectively.".

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The fraction 3/4 is equivalent to 0.75, or 75%. This means that 3 out of every 4 parts is equal to 75%.

Fraction is a numerical expression that represents a part of a whole. It is represented by a numerator (top number) and a denominator (bottom number). The numerator shows how many parts of the whole are being considered, and the denominator represents the total number of parts that make up the whole. Fractions are used in everyday life, from dividing food and measuring distances to calculating discounts and percentages.

This fraction can also be expressed as a decimal, a percent, or as a mixed number.For example, 3/4 can be written as 0.75, 75%, or 3 1/4.
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As per the concept of ANOVA, the larger the differences among the sample means, the larger the numerator of the F-ratio will be True.

Here we have given that if it is true whether the larger the differences among the sample means the larger the numerator of the F-ratio will be.

In order to find that, we must know the definition of F - ratio.

The term f - ratio is defined as the ratio of the between group variance to the within group variance.

While we consider the given situation, here for both repeated-measures design and independent-measures design then the F-ratio compares the actual mean differences between treatments with the amount of difference that would be expected if there were no treatment effect.

Based on these theory we have identified that the statement is true.

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Answer:

x= -61

Step-by-step explanation:

[tex]87=-2x-35\\(87)+35=(-2x-35)+35\\122=-2x\\\frac{122}{-2}=\frac{-2x}{-2}\\x=-61[/tex]

The equation of the circle in standard form from the given graph is

x² + y² + 2x + 2y - 45 = 0

A circle is a two-dimensional figure with a radius and circumference of 2πr

The standard equation of a circle is (x - h)² + (y - k)² = r²

Where (h, k) is the center of the circle

We have,

The standard equation of a circle is (x - h)² + (y - k)² = r²

The coordinates of the center of the circle from the figure.

= (-1, -2)

This means,

(-1, -2) = (h, k)

The radius of the circle is 5√2.

The distance between (-1, -2) and (-6, 3).

= √(-6 + 1)² + (3 + 2)²

= √(25 + 25)

= √50

= 5√2

Now,

The standard equation of a circle is (x - h)² + (y - k)² = r²

(x + 1)² + (y  + 2)² = 50

x² + 2x + 1 + y² + 2y + 4 = 50

x² + y² + 2x + 2y - 45 = 0

Thus,

The equation of the circle is x² + y² + 2x + 2y - 45 = 0

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Answer:

sqrt{9(x-4)} - 4

3sqrt{x-4} - 4

Step-by-step explanation:

the third option

Answer: C

Step-by-step explanation:

No, the SSA congruence rule is not a valid criterion for proving whether two triangles are congruent with each other.

The SSA Congruence Rule states that two triangles are equal if any two sides and no angle between them are equal to the two sides and the other angle. However, the congruence rule does not say that two triangles are congruent because the sides of the two triangles may not be on the same corresponding side. Both triangles can be different shapes and sizes from each other. Therefore, the SSA Consolidation Rules are not valid. Let's see why SSA doesn't work, see the diagram above. Two triangles ΔABC and ΔDEF have AB = DE, BC = EF, ∠C = ∠F (not included angles). It turns out that two sets of sides and one set of angles are equal (respectively), but the two triangles are not congruent. No two triangles are the same shape and size. Therefore, the SSA Congruence Rule is void.

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The error in the statement will be When reporting correlation, one does not report units because correlation has no units that is option D is correct.

The correct statement would be "The correlation between shoe size and height is 0.87." The correlation coefficient is a measure of the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a strong negative relationship, 0 indicating no relationship, and 1 indicating a strong positive relationship. The correlation coefficient does not have units because it is a standardized measure of the relationship between the variables. If there is a unit given in any correlation statement then it means that the statement is a wrong statement or it has an error.

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Answer:

x can be any number, which we need to find out in an equation

The nature of the roots of the quadratic equation 4[tex]x^{2}[/tex] - 8x + 9 =0 are imaginary.

given equation:

4[tex]x^{2}[/tex] - 8x + 9 =0

now we need to find the nature of the quadratic equation

nature of roots :

Case I: [tex]b^{2}[/tex] – 4ac > 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive, then the roots α and β of the quadratic equation ax2 +bx+ c = 0 are real and unequal.

Case II: [tex]b^{2}[/tex]– 4ac = 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is zero, then the roots α and β of the quadratic equation ax2+ bx + c = 0 are real and equal.

Case III: [tex]b^{2}[/tex]– 4ac < 0

When a, b, and c are real numbers, a ≠ 0 and the discriminant is negative, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.

Case IV: [tex]b^{2}[/tex] – 4ac > 0 and perfect square

When a, b, and c are real numbers, a ≠ 0 and the discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational and unequal.

for the above given equation:

a = 4

b = -8

c = 9

=[tex]b^{2}[/tex] - 4ac

= [tex](-8)^{2}[/tex] - 4(4)(9)

= (56) -144

= -88< 0

the roots are imaginary

The nature of the roots of the quadratic equation 4[tex]x^{2}[/tex] - 8x + 9 =0 are imaginary

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Answer:

The answer is 6k⁷ - 10k⁴ + 10k³

Step-by-step explanation:

-2k³ (-3k⁴ + 5k - 5)

6k⁷ - 10k⁴ + 10k³

Thus, The answer is 6k⁷ - 10k⁴ + 10k³

Answer:

Hi! The answer is 120 Db.

Hope it helps!

The mean of number of students who read fictional books is 4.32.

In statistics, the mean refers to the average of a set of values. The mean can be computed in a number of ways, including the simple arithmetic mean (add up the numbers and divide the total by the number of observations).

Given that, the summary of fictional books read by 37 students.

Total number of students =37

Sum =4(2)+10(3)+13(4)+10(7)

= 8+30+52+70

= 160

Now, mean =160/37

= 4.32

Therefore, the mean of number of books read by students is 4.32.

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Answer:

(y2 - y1) / (x2 - x1)

Step-by-step explanation:

The equation that teacher taught to find the slope is

( y2 - y1) / (x2 - x1)

Using the slope-intercept form, the slope is 0 for line y=5.

The slope of a line indicates its steepness. Slope is computed mathematically as "rise over run" (change in y divided by change in x). The slope of a straight line in arithmetic defines how steep it is. It is also known as the gradient. The slope equations. The slope of a line is defined as the "change in y" divided by the "change in x". The slope is defined as the ratio of the vertical change between two places, known as the rise, to the horizontal change between those same two points, known as the run.

Here,

given line,

y=5

y=mx+5

y=0*x+5

m=0

The slope of line y=5 is 0 using the slope-intercept form.

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The value of [tex]-4^{-2}[/tex] without Exponent is  [tex]\frac{1}{16}[/tex]

The exponent of a number indicates how many times a number has been multiplied by itself. For instance, 3^4 indicates that we have multiplied 3 four times. Its full form is 3×3×3×3. Exponent is another name for a number's power. A whole number, fraction, negative number, or decimal are all acceptable.

How many times we must multiply the reciprocal of the base is indicated by a negative exponent. For instance, if a^-n is provided, it can be stretched to 1/a^n. It implies that we must multiply 1/a 'n' times, which is the reciprocal of a. When writing exponentiated fractions, negative exponents are employed.

Given;

(-4)⁻² , That implies we have to multiply reciprocal of -4 "2" times .

⇒[tex]\frac{1}{-4}[/tex]×[tex]\frac{1}{-4}[/tex]=[tex]\frac{1}{16}[/tex]

The value of [tex]-4^{-2}[/tex] without Exponent is  [tex]\frac{1}{16}[/tex]

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Answer:

0.3 or 3/10

Step-by-step explanation:

First guess:

chance it is right: 2/5

chance it is wrong: 3/5

Second guess, if first is right:

chance it is right: 1/4

chance it is wrong: 3/4

therefore we multipy 2/5 and 3/4 and get a simplified answer of 3/10

SAS is a similarity postulate through which we can find if a given two triangles are similar or not

SAS postulate:( side angle side)

It states that if the two sides and one angle of two triangles are equal then the two triangles are said to be similar.

Let the angles ∠ABC, ∠BCA, and AB of triangle ΔABC is equal to angles∠ XYZ and ∠YZX and XY of triangle ΔXYZ then can say that triangle ABC is similar to triangle XYZ, and their remaining sides and angles will be also equal.

SAS is a similar theorem to prove that two triangles are equal.

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The mean of the data 2, 3, 4, 5,0, 1, 3, 3, 4, 3 is 2.8.

Mean is the average of the given numbers and is calculated by dividing the sum of given numbers by the total number of numbers.

The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of the sum of all the terms and the total number of terms. Hence, we can say;

Mean = Sum of the Given Data/Total number of Data

Mean= 2 +3 +4 +5 +0 +1 +3 +3 +4 +3 / 7 = 28/10 = 2.8

Hence, The mean of the data 2, 3, 4, 5,0, 1, 3, 3, 4, 3 is 2.8.

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If there is a y-intercept, we can find it by plugging a zero for x and evaluating the function. If this produces an undefined value, then there is no y-intercept. To find the x-intercept, set y to zero and solve for x.

Now, According to the question:

Logarithmic functions will always have an x-intercept, but they may not have a y-intercept. Let's look at a simple example:

                        y = [tex]log_1_0x[/tex]

A y-intercept would be located at the y value we get by plugging in a zero for x. The problem is that the function does not have zero as a part of its domain. It is not defined there, so there is no y-intercept. The x-intercept is the x value that causes y to be zero. For a basic logarithmic function like this, that is always x equals 1.

                            0 = [tex]log_1_0x[/tex]

                            [tex]10^0[/tex] = x

                            1 = x

Now we can shift, stretch, and reflect this model to change these results, but the basic idea is still the same. For example, let's consider this model:

                          y = [tex]log_1_0[/tex] (x + 100) + 2

As always, we look for a y-intercept by plugging in a zero for x. In this case, we do get an answer. This is because of the 100 that is added to the argument of the logarithm.

                          y = [tex]log_1_0[/tex] (0 + 100) + 2

                          y = 2 + 2 = 4

As always, to find the x-intercept, we set y to zero and solve for x

                              y = [tex]log_1_0[/tex] (x + 100) + 2

                             -2 =  [tex]log_1_0[/tex] (x + 100)

                             [tex]10^-^2[/tex]  = x + 100             [By taking antilog on both sides ]

                             0.01 = x + 100

                               x = -99.99

So, no matter what the logarithmic function is, we can find the intercepts in this way. If there is a y-intercept, we can find it by plugging a zero for x and evaluating the function. If this produces an undefined value, then there is no y-intercept. To find the x-intercept, set y to zero and solve for x.

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The growth rate of of is smaller than that of of as approaches ∞. Even though the function of equals is growing as increases, it isn't growing as fast as of equals squared. And so the limit of of over of is zero.

Now, According to the question:

Let's know:

What is the growth rate meaning?

The growth rate of a value (GDP, turnover, wages, etc.) measures its change from one period to another (month, quarter, year). It is very generally expressed as a percentage.

The growth rate of of is smaller than that of of as approaches ∞. Even though the function of equals is growing as increases, it isn't growing as fast as of equals squared. And so the limit of of over of is zero.

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Answer:

x+2-x+3

x-x+2+3

2+3

5

2x+x-2-2x-1

2x-2x+x-2-1

x-2-1

x-3

In total, Menaha traveled 97km 1000m (97.1km).

Distance is a numerical measurement of how far apart two objects, points, or places are in space. Distance can be measured in linear units such as meters, kilometers, feet, miles, etc. It can also be measured in angular units such as degrees or radians.

Distance can also refer to the space between two points in time, such as the time between two events. Distance can be used to measure physical distance, time, or even emotional distance.
To calculate this, the two distances must be added together.
The train distance is =86km 520m (86.52km)

and the car distance is =11km 480m (11.48km).
When added together, =86km 520m+11km 480m = 97.52km.
However, since the distances are measured in km and m,
it is necessary to convert the measurements into a single unit of measurement.
To do this, the measurements must be converted into metres.
The train distance is 86,520 metres

And the car distance is 11,480 metres.
When added together,

the total distance is 97,000 metres (97km).

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Galileo's discovery was that the period of swing of a pendulum is independent of its amplitude--the arc of the swing--the isochronism of the pendulum.

Christiaan Huygens, a Dutch mathematician, astronomer, physicist, and horologist, created the pendulum clock in 1656 and received a patent for it in 1657. With the help of this technology, clocks now lose less than 15 seconds every day instead of nearly 15 minutes.

Huygens was motivated by Galileo Galilei's pendulum research, which he began in about 1602. Isochronism, or the fact that a pendulum's period of swing is about the same for swings of all sizes, is the main characteristic that makes pendulums ideal timekeepers, was discovered by Galileo.

Galileo came up with the concept for a pendulum clock in 1637, and his son started building one in 1649, but neither of them survived to see it completed. The precision of clocks was greatly improved by the advent of the pendulum, the first harmonic oscillator used for timekeeping, going from around 15 minutes per day to 15 seconds per day.

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Galileo made the discovery that a pendulum's period of swing is independent of its amplitude, or the swing's arc, or the isochronism of the pendulum.

The pendulum clock was developed in 1656 by Dutch mathematician, astronomer, physicist, and horologist Christiaan Huygens, who also got a patent for it in 1657. With the use of this technology, clocks no longer lose roughly 15 minutes of time each day, but rather lose less than 15 seconds.

Galileo Galilei's pendulum research, which he started about 1602, inspired Huygens. Galileo discovered isochronism, or the fact that a pendulum's period of swing is roughly the same for swings of all sizes, which is the main quality that makes pendulums great timekeepers.

The idea for a pendulum clock was developed by Galileo in 1637, and his son began construction on one in 1649, but neither of them lived to see it finished. With the invention of the pendulum, the first harmonic oscillator used for timekeeping, the accuracy of clocks was substantially increased, decreasing from roughly 15 minutes per day to 15 seconds per day.

Two congruent figures are similar can be proved by using theorem of similarity given by : SSS, SAS, ASA, AA, and RHS.

Congruent figures are also representing similar figure can be proved in the following ways:

Therefore, two congruent figures are similar proved by theorem of similarity given by : SSS, SAS, ASA, AA, and RHS.

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We check the validity that i² is equal to -1

The first thing we will do is define numerical sets and complex numbers.

Numerical sets are groupings of numerical values that have a particularity in common, they can be integers, decimals, fractions, among others.

Among the numerical sets there is one that we call complex numbers, which include values that are not real, such as "i" a letter that denotes that it is an imaginary number.

The definition of a letter value that identifies a type of complex number is the "i" which is the result of the square root of -1, then we have:

√(-1) = i

√(-1) x √(-1) = i²

[√(-1)]² = i²

(-1) = i²

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Answer:

Area of the figure =[tex]100.26[/tex]

Step-by-step explanation:

Firstly we need to find the area of the rectangle

[tex]Area \\ of \\rectangle= lb[/tex]

[tex]12X6\\= 72[/tex]

[tex]Area\\of \\the \\semicircles= \frac{\pi }{2} Xr^{2}[/tex][tex]X2[/tex]

= [tex]\frac{3.14}{2} X(3)^{2} X2[/tex]

=[tex]28.26[/tex]

Area of the figure = Area of rectangle + Area of the 2 semi circles

Area of the figure = [tex]72+28.26[/tex]

[tex]100.26[/tex]

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The stone face is approximately 57.8512 feet in height.

Height is a mathematical term that refers to the vertical distance between an object's top and base.

Sometimes, it has the designation "altitude."

The measurement of an item along the y-axis in coordinate geometry is referred to as height in geometry.

So, let h be the stone's face's height.

One sight is 750 feet away from the mountain's base, and there is a 33° elevation difference between it and the bottom of the face.

The distance between the mountain's base and the base of the stone face

= 750 * tan33°

= 750 * 0.64940759319

= 487.055694898

A different location, which is 36° in elevation and 750 feet from the mountain's base, can be seen from the summit of the face.

The separation between the mountain's base and the top of the stone face = (h + 487.055694898) ft

Now, using trigonometry:

h + 487.055694898/750 = tan36°

h + 487.055694898 = 0.726542525 * 750

h = 544.906896004 - 487.055694898 = 57.8512011059 = 57.8512

Therefore, the stone face is approximately 57.8512 feet in height.

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Correct question:
To measure a stone face carved on the side of a​ mountain, two sightings 750 feet from the base of the mountain are taken. if the angle of elevation to the bottom of the face is 33° and the angle of elevation to the top is 3636°​, what is the height of the stone​ face?

Step-by-step explanation:

What is the formula for mean median and mode?

Mean: is the average of values

Median: center most value when values are arranged in numerical order

Mode: most frequent occurring value

_______________________________________

Example

Given values: 0 , 3 , 2 , 10 , 11 , 31 , 2

mean: (0 + 3 + 2 + 10 + 11 + 31 + 2)/7  = 8.429

median: 0 , 2 , 2 , 3 , 10 , 11 , 31

mode: 2 (because 2 shows up twice and the others numbers only once)