The area of a rhombus is 100. Find the length of the two diagonals if one is twice as long as the other.The length of two diagonals of the rhombus are 10 and 20.

Number of the Great Danes that weigh less than 126.6 lbs is: 428 people

The formula for z-score here is:

z = (x' - μ)/σ

Where:

x' is sample mean

μ is population mean

σ is standard deviation

We are given:

x' = 126.6 lbs

μ = 133 lbs

σ = 6.4 lb.

Thus:

z = (126.6 - 133)/6.4

z = -1

We are looking for P(X > 126.6)

Thus, from z-score table, we have:

p-value = 0.1587

Thus:

Number of the Great Danes that weigh less than 126.6 lbs is:

0.1587 * 2700 = 428 people

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The probability that Jim gets free beer is 0.7854.

The terms we need to consider are the square frame, the round board, and the probability p of Jim getting free beer.

To calculate the probability p, we need to find the ratio of the area of the round board to the area of the square frame.

Step 1: Calculate the area of the square frame.
Since the frame is 2x2, its area is A_square = side * side = 2 * 2 = 4 square units.

Step 2: Calculate the area of the round board.
The radius of the round board is 1, so its area is A_round = π * radius² = π * 1² = π square units.

Step 3: Calculate the probability p.
The probability p that Jim gets free beer is the ratio of the round board's area to the square frame's area, which is:
p = A_round / A_square = π / 4

So the probability that Jim gets free beer is π / 4, or approximately 0.7854.

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The probability of making a Type II error (i.e., failing to reject the null hypothesis when the true proportion is actually 0.25) is 0.0104.

What is error?

An error is the difference between a true value and an estimated or approximate representation of that value in applied mathematics. A truncation error occurs when an infinite series is ignored for all but a small number of terms. power 1 P (type II error)

In this case, if the market share in Ontario is actually 0.22, the probability that 90 or fewer Hewlett-Packard computers will be observed out of 427 randomly selected purchases is quite small. The probability of 90 or fewer successes for a binomial distribution with n = 427 and p = 0.22 is approximately 0.219, or 21.9%.

Therefore, the probability of making a Type II error (ie, failing to reject the null hypothesis incorrectly) is approximately 78.1%. This means that if Hewlett-Packard's market share in Ontario is actually 0.22, it is still relatively high. the possibility that the market researcher's test fails to detect this difference and incorrectly concludes that their market share does not exceed 0.198.

Therefore, the probability of a Type II error (ie, failing to reject the null hypothesis when the true proportion is actually 0.25) is 0.0104.

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The average customer spends 0.2 hours, or 12 minutes, in the bank during peak hours.

Little's Law states that the average number of customers in a stable system (i.e., one where the number of arrivals and departures is balanced) is equal to the average arrival rate multiplied by the average time that a customer spends in the system:

L = λW

where L is the average number of customers in the system, λ is the average arrival rate, and W is the average time that a customer spends in the system.

In this case, we are given that the average arrival rate during peak hours is λ = 40 customers per hour, and the average number of customers in the bank is L = 8 customers. We are asked to find the average time that a customer spends in the bank and the probability is unknown.

Plugging in the values, we get:

8 = 40W

Solving for W, we get:

W = 8/40

W = 0.2 hours

Therefore, the average customer spends 0.2 hours, or 12 minutes, in the bank during peak hours.

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Answer:

The answer is 10 and -10

You get this answer because in the middle in a number line is 0 and if it 10 units from 0 then the other side of 0 will be -10 (negative ten) units from 0

Based on the table, it is clear that the larger the size of the

section, the more plants grew. This is supported by the fact

that section2, which had the largest size of 125 square feet,

had the highest number of plants with 38. Section 1 had the

smallest size with only 25 square feet and the lowest number

of plants with only 13. However, itis important to note that the

number of plants can also be affected by other factors such

as the quality of soil, amount of water and sunlight given

to each section, and the type of seeds planted. It would be

beneficial for the scientist to consider these factors in future

experiments to obtain more accurate and reliable results.

For the polygons in the attached figure,

a) Scale factor = 2.5

b) Scale factor = 2.5

c) Scale factor = 1.5

d) Scale factor = 2

We knoa that a scale factor is nothing but the ratio between the scale of a original object and a transformed object.

Here, the polygons are similar.

We know that the corresponding sides of similar figure are in proportion.

a) 10/6 = 2.5

15/6 = 2.5

18/7.2 = 2.5

So, the scale factor would be 2.5

b)

25/10 = 2.5

15/6 = 2.5

S0, the scale factor = 2.5

c)

12/8 = 1.5

6/4 = 1.5

9/6 = 1.5

So, the scale factor = 1.5

d)

12/6 = 2

16/8 = 2

20/10 = 2

so, the scale factor is 2

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Yes, I can help you with these Laplace transform problems.

1. To find the Laplace transform of f(t)=e-2t sin (5t), we can use the formula:

L{e-at sin(bt)} = b / (s+a)2 + b2

Applying this formula, we get:

L{e-2t sin (5t)} = 5 / (s+2)2 + 52

Therefore, the Laplace transform of f(t)=e-2t sin (5t) is:

L{f(t)} = 5 / (s+2)2 + 25

2. To find the Laplace transform of f(t)=tsin (3t), we can use integration by parts, followed by applying the Laplace transform:

L{f(t)} = L{t} L{sin (3t)} - L{dt/ds} L{sin (3t)}

Using the Laplace transform of t and sin(3t), we get:

L{t} = 1 / s2

L{sin(3t)} = 3 / (s2 + 32)

Differentiating sin(3t) with respect to t gives:

d/dt sin(3t) = 3 cos(3t)

Taking the Laplace transform of both sides gives:

L{d/dt sin(3t)} = s L{cos(3t)} - cos(0)

Since L{cos(3t)} = s / (s2 + 32), we can simplify to:

L{d/dt sin(3t)} = 3s / (s2 + 32)

Therefore, the Laplace transform of f(t)=tsin (3t) is:

L{f(t)} = (2s3) / (s2 + 32)2

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To verify if the set {-696, -36, -19, 3, 7, 12, 99} is a complete system of residues modulo 7, we need to check if all residue classes modulo 7 are represented by elements in the set.

The residue classes modulo 7 are {0, 1, 2, 3, 4, 5, 6}. To check if the given set is a complete system of residues modulo 7, we need to check if each residue class is represented by at least one element in the set.
- 0: None of the elements in the set is divisible by 7, so none of them leave a residue of 0 when divided by 7.
- 1: 12 and -696 leave a residue of 1 when divided by 7.
- 2: -36 leaves a residue of 2 when divided by 7.
- 3: 99 and -19 leave a residue of 3 when divided by 7.
- 4: None of the elements in the set leave a residue of 4 when divided by 7.
- 5: 7 leaves a residue of 5 when divided by 7.
- 6: 3 leaves a residue of 6 when divided by 7.
Since every residue class modulo 7 is represented by at least one element in the set, we can conclude that the set {-696, -36, -19, 3, 7, 12, 99} is a complete system of residues modulo 7.

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The difference between the two possibilities is based on theory and mathematics. The experimental probability is based  on the results of several tests or experiments, but the theortical result is  calculated by comparing the positive results with all the results.

Theoretical probability of an event  occurring based on theory and reasoning. It is determined by dividing  number of favourable results by total  result. On the other hand, the experimental depend on the results of  various trials or tests.

The difference between theoretical probability and testing probability is that theory is based on knowledge and mathematics. Theoretical probability is what it should be. The test will appear as a result. For example, if I flip a coin, 50 times, the theoretical number of heads of the coin is 25. Coin flip probability = 0.5

Number of flips = 50

Theoretical number of heads = 0.5 × 50

= 25

If I actually flip a coin 50 times, 25 heads may or may not come up. If we have 21 heads, the test probability is 21 out of 50 heads, or 0.42. So the theoretical probability of getting heads in this example = 0.5

The experimental probability of landing heads = 0.42. Hence, both probabilities are not the same.

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The reference angle for -400 is 5.729 degrees.

We have,

To find the reference angle for -400, we need to find the acute angle formed by the terminal side of the angle and the x-axis.

We start by drawing the angle in the standard position, which means placing the initial side of the angle along the positive x-axis and rotating the terminal side in the clockwise direction.

Since -400 is in the fourth quadrant, the terminal side of the angle would lie 400 units clockwise from the negative x-axis.

To find the reference angle, we need to find the acute angle formed by the terminal side and the x-axis.

This is simply the angle formed by the terminal side and a perpendicular line dropped from the endpoint of the terminal side to the x-axis.

In this case, the perpendicular line would drop 40 units to the x-axis, forming a right triangle with legs of 40 and 400 units.

Using the Pythagorean theorem, we can find the hypotenuse of this right triangle, which is the distance from the origin to the endpoint of the terminal side:

h = √(40² + 400²) = 404

The sine of the reference angle is the ratio of the opposite leg to the hypotenuse:

sin Ф = opposite/hypotenuse = 40/404 = 0.099

Taking the inverse sine of this value, we can find the reference angle:

Ф = [tex]Sin^{-1}[/tex](0.099) = 5.729 degrees

Therefore,

The reference angle for -400 is 5.729 degrees.

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The value of X in the diagram provided is

We can use the trigonometric function of sine to find the angle θ, where θ is the angle between the opposite side and the hypotenuse.

sin(θ) = opposite / hypotenuse

sin(θ) = 12 / 13

To find θ, we can take the inverse sine of both sides:

θ = sin⁻¹(12/13)

θ = sin⁻¹(0.9231)

θ = 67.38°

Note that we use calculator to find the θ

Therefore, the angle in the right-angled triangle with opposite side 12, adjacent side 5, and hypotenuse 13 is approximately 67.38 degrees.

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The midpoint of the second class ([40-49]) is calculated by adding the lower and upper limits of the class and dividing by 2. So, (40+49)/2 = 44.5.

The relative frequency of the second class is calculated by dividing the frequency of the second class by the total frequency. So, 8/38 = 0.211.

The class width is the difference between the upper and lower limits of the class. So, [59.5-69.5] has a class width of 10.

The fourth actual class is [60-69], which has a midpoint of (60+69)/2 = 64.5.

Therefore, the answer is a. 44.5, 0.211, 10 and [59.5-69.5].

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Myra should choose the container with a volume of 343 in.³

This container is the closest to her requirement of 288 in.³ and will have less leftover sand compared to the other options.

We have,

Myra needs a container with a volume of at least 288 in.³ of sand.

Out of the three options, the only one that meets this requirement is the container with a volume of 336 in.³

This container has more than enough sand for Myra to build her sandcastle.

However, Myra will be deducted points for having too much sand left over in her container.

So, she should choose the smallest container that meets her requirement of 288 in.³.

The container with a volume of 336 in.³ is too large and will likely result in too much leftover sand.

Therefore,

Myra should choose the container with a volume of 343 in.³

This container is the closest to her requirement of 288 in.³ and will have less leftover sand compared to the other options.

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The probability that the answer is correct given that the

a) Let's use Bayes' theorem to calculate the probability that the answer is correct given that the person you asked said "East". Let H be the event that the person is honest, L be the event that the person is a liar, E be the event that Julia resides in the East and W be the event that Julia resides in the West. Then we have:

P(E|H) = 2/3 (the probability that an honest person gives the correct answer)

P(E|L) = 1/2 (the probability that a liar gives the correct answer)

P(H) = 1/4 (the probability that the person is honest)

P(L) = 3/4 (the probability that the person is a liar)

By the law of total probability, we have:

P(E) = P(E|H)P(H) + P(E|L)P(L) = (2/3)(1/4) + (1/2)(3/4) = 5/12

Then, using Bayes' theorem, we have:

P(H|E) = P(E|H)P(H)/P(E) = (2/3)(1/4)/(5/12) = 2/5

So the probability that the answer is correct given that the person said "East" is 2/5.

b) The probability that the same person gives the same answer twice in a row is:

P(E∩E) = P(E)P(E|H)P(H) + P(E)P(E|L)P(L) = (5/12)(2/3)(1/4) + (5/12)(1/2)(3/4) = 5/24

Using Bayes' theorem again, we have:

P(H|EE) = P(EE|H)P(H)/P(EE) = (2/3)^2(1/4)/(5/24) = 8/15

So the probability that the answer is correct given that the person said "East" twice in a row is 8/15.

c) The probability that the same person gives the same answer three times in a row is:

P(E∩E∩E) = P(E)P(E|H)^2P(H) + P(E)P(E|L)^2P(L) = (5/12)(2/3)^2(1/4) + (5/12)(1/2)^2(3/4) = 5/32

Using Bayes' theorem again, we have:

P(H|EEE) = P(EEE|H)P(H)/P(EEE) = (2/3)^3(1/4)/(5/32) = 4/5

So the probability that the answer is correct given that the person said "East" three times in a row is 4/5.

d) The probability that the same person gives the same answer four times in a row is:

P(E∩E∩E∩E) = P(E)P(E|H)^3P(H) + P(E)P(E|L)^3P(L) = (5/12)(2/3)^3(1/4) + (5/12)(1/2)^3(3/4) = 5/48

Using Bayes' theorem again, we have:

P(H|EEEE) = P(EEEE|H)P(H)/P(EEEE) = (2/3)^4(1/4)/(5/48) = 16/25

So the probability that the answer is correct given that the

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The given statement, "You can infer statistical significance from a 95% CI. "A 95% CI gives you information about the precision of the association." and "A study with a small sample will have a wider 95% CI." are true and "A 95% CI gives you information about the precision of the association, but not the strength of the association." is false.

The statement You can infer statistical significance from a 95% CI is true, as it is a measure of the precision of the association between two variables.

A 95% CI will be wider for a study with a smaller sample size, but this does not necessarily indicate a weaker association. In other words, the width of a 95% CI does not indicate the strength of the association, and so the statement that A 95% CI gives you information about the precision of the association, but not the strength of the association is false.

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Full Question ;

Identify the true and false statements about 95% confidence intervals.

- You can infer statistical significance from a 95% CI.

- A 95% CI gives you information about the precision of the association.

- A study with a small sample will have a wider 95% CI.

-A 95% CI gives you information about the strength of the association.

The list of the temperature from coldest to warmest temperature includes:

The temperatures (Fahrenheit) of the 4 cities from coldest to warmest includes -9 degrees, -7 degrees, -6 degrees, 5 degrees and 12 degrees.

The coldest temperature is -9 degrees, followed by -7 degrees, then -6 degrees. The positive temperature is  5 degrees and 12 degrees.

We must note these temperatures are in Fahrenheit which is not the standard unit of measurement used in all countries, so, we must specify the unit when reporting temperatures.

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To determine the magnitude of the force F exerted at the lever of the assembly, we'll need to consider the moment at point O, the distances involved, and the angle at which the force is applied.

The moment at point O is given as 150 kN·m. Let's consider the lever arm distance, which is the horizontal distance between point O and the line of action of the force F. This can be found by looking at the given measurements: 4m (distance from O to P) + 3m (distance from P to the line of action of force F) = 7m.

Now, we'll take the angle of the force into account. The force F is applied at a 60° angle. To find the horizontal component of the force, we can use the cosine of the angle:
Horizontal component of F = F * cos(60°)

The moment at point O is the product of the horizontal component of force F and the lever arm distance (7m):
Moment = (F * cos(60°)) * 7m

Given that the moment at point O is 150 kN·m, we can now solve for the magnitude of the force F:
150 kN·m = (F * cos(60°)) * 7m

To solve for F:
F = (150 kN·m) / (cos(60°) * 7m)

Calculating the value, we find the magnitude of the force F to be approximately 43.3 kN.

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Answer:

At Marco Market, a chewy toy costs $2 and a cat collar costs $6. Meanwhile, at Sonia Superstore, a chewy toy for dogs costs $4 and a cat collar costs $4. To represent the quantities of each item that can be purchased at each store, an equation can be used.

Step-by-step explanation:

At Marco Market, a chewy toy costs $2 and a cat collar costs $6. Meanwhile, at Sonia Superstore, a chewy toy for dogs costs $4 and a cat collar costs $4. To represent the quantities of each item that can be purchased at each store, an equation can be used.

Angle A is measured as 39°.

Inverse trigonometric functions have been what they sound like.

The opposite direction functions of trigonometry are somewhat the inverse functions of the basic trigonometric functions. The basic trigonometric function sin = x can be replaced with sin-1 x =. In this case, x is able to be expressed as a whole number, a decimal number, a fraction, as well as an exponent.

Now, we have AB (Hypotenuse)= 54 m BC (opposite side)= 38 m in triangle ABC.

To find the angle A's measurement

By employing inverse trigonometric keys.

We are aware of the following:

The sin inverse formula is as follows:

[tex]\theta = Sin^-^1(\frac{opposite side}{hypontenuse} )[/tex]

[tex]\theta= Sin^-^1(\frac{54}{38} )[/tex]

[tex]\theta= Sin^-^1(\frac{27}{19} )[/tex]  ≈1.570796326794897−0.888179846706129

θ = 39°

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Answer:

[tex]\large \boxed{\boxed{\textsf{$(x+2)(x-4)$}}}[/tex]

[tex]\boxed{\boxed{\large \textsf{$x=-2, x=4$}}}[/tex]

This is a quadratic expression, in the form:

[tex]\boxed{\large \textsf{$ ax^2+bx+c$, where$\ a \neq 0$}}[/tex]

To factorise this expression, we will need to have 4 terms. Currently there are only 3. To do this, we need to find 2 integers, that add together to form the middle term, -2, and multiply together to form the constant term, -8.

[tex]\large \textsf{the 2 integers $\Rightarrow$ 2 and -4}\\ \textsf{$-4+2 = -2$\ (coefficient of middle term)}\\ \textsf{$-4 \times 2=-8$\ (constant term)}[/tex]

Now we can split the middle term into 2 terms, using the integers we just found:

[tex]\large \textsf{$x^2+2x-4x-8$}[/tex]

Now we can factorise this.

[tex]\large \textsf{Group each pair of terms together, and take out a common factor.}\\ \\ \large \textsf{$x(x+2)-4(x+2)$}\\ \\ \large \textsf{Now take out the common factor from the expression: (x+2)} \\ \\ \large \textsf{$(x+2)(x-4)$}[/tex]

This leaves us with our fully factorised expression:

[tex]\large \boxed{\boxed{\textsf{$(x+2)(x-4)$}}}[/tex]

To solve the quadratic expression, we can make it equal to zero:

[tex]\large \textsf{$x^2-2x-8=0$}[/tex]

Primarily, to solve this, we can used the factorised form from above, and apply the zero-product property.

The zero-product property states that:[tex]\large \textsf{If $a\times b=0$, then $a=0$ or $b=0$ (or both $a=0$ AND b=0)}[/tex]

[tex]\large \textsf{$\therefore$ if $(x+2)(x-4)=0$, then $(x+2)=0$, and/or $(x-4)=0$ }[/tex]

[tex]\large \textsf{$\implies \boxed{\boxed{x=-2, x=4}}$ }[/tex]

Similarly, we can also use the quadratic formula to solve this equation:

[tex]\boxed{\Large \textsf{$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$} \large \textsf{, for $ax^2+bx+c=0$}}[/tex]

[tex]\large \textsf{$\Rightarrow a=1, b=-2, c=-8$}\\ \\ \Large \textsf{$x=\frac{-(-2)\pm \sqrt{(-2)^2-4(1)(-8)}}{2(1)}$}\\ \\ \boxed{\boxed{\large \textsf{$\therefore x=4, x=-2$}}}[/tex]

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If point A is located at (-5, 8) and point C is located at (8, -2), the x-coordinate of point B is 29.375.

First, we need to find the coordinates of point B. We can use the ratio of AB to BC to find the distance from A to B and from B to C:

Let the distance from A to B be 3x, and the distance from B to C be 2x.

Then we can use the distance formula to find the coordinates of point B:

x-coordinate of B = x-coordinate of A + (distance from A to B)

x-coordinate of B = -5 + 3x

To find x, we need to solve for x given that B lies on the line segment AC.

The slope of the line segment AC is:

m = (y2 - y1) / (x2 - x1) = (-2 - 8) / (8 - (-5)) = -10 / 13

The equation of the line segment AC is:

y - y1 = m(x - x1)

Substituting in the values we know:

y - 8 = (-10 / 13)(x + 5)

Simplifying:

y = (-10 / 13)x - 250 / 13 + 104 / 13

y = (-10 / 13)x - 146 / 13

Since point B lies on the line segment AC, we can substitute the x-coordinate of B and solve for y:

y = (-10 / 13)(-5 + 3x) - 146 / 13

Now we have a system of equations:

y = (-10 / 13)(-5 + 3x) - 146 / 13

y = (2 / 3)x + 38 / 3

We can solve for x by setting the two expressions for y equal to each other:

(-10 / 13)(-5 + 3x) - 146 / 13 = (2 / 3)x + 38 / 3

Multiplying both sides by 39 to get rid of the fractions:

-30(5 - 3x) - 146 = 26x + 494

Expanding and simplifying:

-150 + 90x - 146 = 26x + 494

Subtracting 26x and adding 296 to both sides:

64x = 840

Dividing by 64:

x = 13.125

Therefore, the x-coordinate of point B is:

x-coordinate of B = -5 + 3x

x-coordinate of B = -5 + 3(13.125)

x-coordinate of B = 29.375 (rounded to the nearest tenth)

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Answer:

[tex]3^{21}[/tex]

Step-by-step explanation:

using the rules of exponents

[tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]

[tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]

given

(3² × [tex]3^{5}[/tex] )³

= ([tex]3^{(2+5)}[/tex] )³

= ([tex]3^{7}[/tex] )³

= [tex]3^{7(3)}[/tex]

= [tex]3^{21}[/tex]

The calculated equation of the line is y = -2x

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

The linear graph

The points on the graph are

(-4,8) (0,0)

It passes through the origin, the slope is calculated as

slope = y/x

This gives

y/x = 8/-4

Evaluate

y/x = -2

This gives

y = -2x

Hence, the equation is y = -2x

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The number 36 should be placed in the blank space. So, Multiplying by 1. 36 is the same as increasing by 36 percent from a decimal number.

The decimal system has a base of ten . These numbers are generally represented by the dot "." between the digits called "decimal point". We can express an integer as a decimal by putting a decimal point after the digit in one's place and writing 0 onwards. The term "percent" is a number or ratio that represents a fraction of 100. Steps to convert decimal to Percent :

We have to fill up a blank space with a number. We have a decimal number 1.36. From above discussion, we need to convert this decimal value into a percent : 0.36 × 100 = 36% So, multiplying by 1.36 is the same as increasing by 36%. Hence, required value is 36.

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Since the graph was obtained by transforming the graph of the square root function, an equation for the function the graph represent is: [tex]g(x) = -\sqrt{9(x-1)} +2[/tex]

In Mathematics and Geometry, a square root function is a type of function that typically has this form f(x) = √x, which basically represent the parent square root function i.e f(x) = √x.

In Mathematics and Geometry, a horizontal translation to the right is modeled by this mathematical equation g(x) = f(x - N) while a vertical translation to the positive y-direction (downward) is modeled by this mathematical equation g(x) = f(x) + N.

Where:

Therefore, the required square root function can be obtained by applying a set of transformations to the parent square root function as follows;

f(x) = √x

g(x) = -√9(x - 2) + 2

[tex]g(x) = -\sqrt{9(x-1)} +2[/tex]

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The solution of the inequality is q < -8.

We have,

38 < 30 - q

Now, solving the inequality

Subtract 30 from both of inequality as

38 - 30 < 30 - q - 30

8 < -q

Now, to make the variable q is positive then the sign of inequality change.

-8 > q

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The quadratic function is f(x)= x²+1-6

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

If f(x) = y, a quadratic equation is written as;

y = ax²+bx+c

if alpha and beta are the roots of a quadratic equation when y = 0, the the equation of the quadratic is formed by the formula;

x²-(alpha+beta) x + alpha × beta

alpha = -3

beta = 2

alpha × beta = -3×2 = -6

alpha +beta = -3+2 = -1

Therefore the quadratic function = x²-(-1) +(-6)

f(x)= x²+1-6

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Answer:

  -0.1, 1.3

Step-by-step explanation:

You want the solutions to the quadratic equation 5x² -2x -1 = 4x.

The equation can be put in standard form by subtracting 4x:

  5x² -6x -1 = 0

  5(x² -6/5x +(6/10)²) -1 -5(6/10)² = 0 . . . . . complete the square

  5(x -0.6)² = -2.8 . . . . . . . . . . . . . subtract 2.8

  x = 0.6 ± √0.56 = -0.1 or 1.3 . . . . . . . divide by 5 and take square root

Solutions to the equation are x = -0.1 and x = 1.3.

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Additional comment

The square is completed by making the trinomial in parentheses have the form x² -2ax +a², where 'a' is half the coefficient of the x-term. When we add a² inside parentheses, we need to subtract an equivalent quantity outside parentheses.

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Therefore, the value of the given double integral is approximately 1634.449.

The double integral:

∫ from y=0 to y=9 [ ∫ from x=In y to x=9 of [tex](7y/x) e^y dx ][/tex] dy

Using integration by parts, we can evaluate the inner integral as:

∫ from x=In y to x=9 of [tex](7y/x) e^y dx = [7y/e^x][/tex] evaluated from x=In y to x=9

= [tex]7y(e^{-9} - e^{(-lny)}) = 7y(1/y - 1/e^9) = 7 - 7e^{(9-y)[/tex]

Substituting this back into the original double integral and evaluating the integral with respect to x, we get:

∫ from y=0 to y=9 [tex][ 7y - 7y e^{(9-y)} ] dy[/tex]

Using integration by parts again, we can evaluate this integral as:

[tex][ 7y^2/2 + 7y e^{(9-y)} - 49 e^{(9-y) ][/tex] evaluated from y=0 to y=9

= [tex]3309/2 - 343 e^{-9[/tex]

So, the value of the given double integral is approximately 1634.449.

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