i need help with these

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).

To learn more about mass and center of mass of the density function link is here

brainly.com/question/15089285

#SPJ4

The complete question is given below:

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

Answer : Finding the preimage (s) of a value a by a function f is equivalent to solving equation f(x)=a f ( x ) = a . Finding the preimage (s) of a value a by a function f , which has a known curve, is equivalent to find the abscissae of the intersection(s) of the curve with the ordinate line y=a

In the following list of properties, f: A → B indicates than f is function (or map) from A to B:

If f : A → B then

f [A] = ∅

f -1[B] = ∅.

If f : A → B and X, Y ⊆ B then f-1[X ∩ Y] = f-1[X] ∩ f-1[Y].

If f: A → B and X, Y ⊆ B then f-1[X ∪ Y] = f-1[X] ∪ f-1[Y]

If f: A → B and X, Y, ⊆ A then

f [A ∪ B] = f|A| ∪ f |B|

f [A ∩ B] ⊆ f|A| ∩ f |B|In the following list of properties, f: A → B indicates than f is function (or map) from A to B:

If f : A → B then

f [A] = ∅

f -1[B] = ∅.

If f : A → B and X, Y ⊆ B then f-1[X ∩ Y] = f-1[X] ∩ f-1[Y].

If f: A → B and X, Y ⊆ B then f-1[X ∪ Y] = f-1[X] ∪ f-1[Y]

If f: A → B and X, Y, ⊆ A then

f [A ∪ B] = f|A| ∪ f |B|

f [A ∩ B] ⊆ f|A| ∩ f |B|

The image of 5 is √5.

The preimage of 5 is 25; for ℕ(the set of natural numbers) it is the set of all perfect squares in ℕ [2].

Step-by-step explanation:

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)

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.

Learn more about Logarithm at:

https://brainly.com/question/27389970

#SPJ4

Answer:

(- 6,4) or x = - 6,y = 4

Step-by-step explanation:

[tex]y=\frac{1}{3}x+6[/tex]

(0,6) and (- 6,4)

(0,5) and (- 6,4)

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]

To refer more about Exponents, visit:

https://brainly.com/question/13669161

#SPJ1

Domains having continuous boundaries, Lipschitz boundaries and C1 boundaries are three types of domain.

The collection of all potential inputs for a function is its domain. For instance, the domain of f(x)=x^2 and g(x)=1/x are all real integers with the exception of x=0. Additionally, we can define special functions with more constrained domains.

Consider the function y = f(x), which has the independent variable x and the dependent variable y. A value for x is said to be in the domain of a function f if it successfully allows the production of a single value y using another value for x.

There are three types of domain in mathematics. Domains having continuous boundaries, Lipschitz boundaries, C1 boundaries, and others are frequently thought of as types of domains. An exterior or external domain is the interior of a bounded domain's complement, whereas a bounded domain is a domain that is a bounded set.

To learn more about domain visit the link:

https://brainly.com/question/26098895

#SPJ4

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).

To know more about distance click-
https://brainly.com/question/23848540
#SPJ4

The height of the rose of Esther's rose is 10 cm while Mikes's rose is 17 cm.

Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.

As per the given,

Height of Esther's rose = x cm

Height of Mike's rose = 2x - 3

The sum of their heights decreased by 5 is 22.

x + 2x - 3 - 5 = 22

3x - 8 = 22

x = 10

Thus, Esther's rose = 10 cm and Mike's rose = 2 x 10 - 3 = 17 cm

Hence "Mike's rose is 17 cm tall, compared to Esther's 10 cm high rose".

For more about the equation,

https://brainly.com/question/10413253

#SPJ1

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.

To learn more about an arithmetic mean visit:

https://brainly.com/question/15196910.

#SPJ1

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.

To know more about the mean visit: brainly.com/question/29895356

#SPJ4

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

The solutions to the equation are x = ±1/2 and x = √-2 and the factored form is (4x² - 1)(x² + 2) = 0

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

4x4 + 7x2 – 2 = 0

Rewrite the equation properly

So, we have the following representation

4x⁴ + 7x² – 2 = 0

Expand the equation

This gives

4x⁴ + 8x² - x² – 2 = 0

Factorize the expanded equation

So, we have

4x²(x² + 2) - 1(x² + 2) = 0

Factor out x² + 2

So, we have the following representation

(4x² - 1)(x² + 2) = 0

When the equation is solved, we have

x = ±1/2 and x = √-2

Read more about polynomial equations at

https://brainly.com/question/17517586

#SPJ1

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.

Learn more about Correlation at:

brainly.com/question/4219149

#SPJ4

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

Learn more about complex numbers at:

brainly.com/question/12241782

#SPJ4

Answer:

Hi! The answer is 120 Db.

Hope it helps!

Answer:

Answer below

Step-by-step explanation:

The meaning of congruence in Maths is when two figures are similar to each other based on their shape and size.

Answer:exactly equal shape and size

Step-by-step explanation:

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.

learn more about congruent here

brainly.com/question/12413243

#SPJ4

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.

To learn more about SSA Congruence rule, refer:

https://brainly.com/question/20687931

#SPJ4

The median of 1 to 10 is 5.5.

The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.

Here we have to find the median of 1-10.

Here we can see that there are 10 numbers given

so, n= 10

Where the number of terms is even.

We know that the formula to find the value of the median whose n value is given,

Median= {(n/2 +1 )th term +(n/2)th term} / 2

We can now substitute the value of n in the above formula, we get

Median= {(10/2+1) +(10/2)} / 2

We can now simplify the above step, we get

Median= 11/2 = 5.5

Therefore, the median of 1 to 10 is 5.5.

To know more about the median visit: brainly.com/question/28060453

#SPJ4

The roots of the quadratic equation 3x²-14x-5 = 0 are 5, -1/3

The meaning of quadratic equation is any equation containing one term in which the unknown is squared and no term in which it is raised to a higher power.

Given that, a quadratic equation, 3x²-14x-5 = 0

On factorizing, we get,

3x²-14x-5 = 0

3x(x-5)+1(x-5) = 0

(3x+1)(x-5) = 0

(x-5) = 0

x = 5

And,

(3x+1) = 0

x = -1/3

Hence, The roots of the quadratic equation 3x²-14x-5 = 0 are 5, -1/3

For more references on quadratic equation, click;

https://brainly.com/question/17177510

#SPJ4

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.

Learn more about Growth rate at:

https://brainly.com/question/28869954

#SPJ4

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

Learn more about circle here:

https://brainly.com/question/11833983

#SPJ1

The step that is the same when bisecting segments and angles is "Draw a ray from the vertex to another point."

When bisecting a segment, you must draw a line that divides the segment into two equal parts. One way to do this is to draw a ray with one endpoint at the midpoint of the segment and the other endpoint at any point beyond the segment. This ray will bisect the segment. When bisecting an angle, you must draw a line that divides the angle into two equal angles. One way to do this is to draw a ray from the vertex of the angle to any point on the angle. This ray will bisect the angle.

Learn more about Segments here:

https://brainly.com/question/28554913

#SPJ4

Answer: Mark the intersection points of the arcs.

Step-by-step explanation: Draw a ray with one endpoint. is incorrect

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]

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.

To know more about time visit: brainly.com/question/28050940

#SPJ4

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.

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.

To know more about slope,

https://brainly.com/question/29184253

#SPJ4

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