select the acceptable conclusions to a hypothesis test. the value of the test statistic does not lie in the rejection region. therefore, there is insufficent evidence to suggest that the null hypothesis is false. the value of the test statistic lies in the rejection region. therefore, there is sufficient evidence to suggest that the alternative hypothesis is true. the value of the test statistic lies in the rejection region. therefore, there is sufficient evidence to suggest that the null hypothesis is not true. the value of the test statistic does not lie in the rejection region. therefore, there is sufficient evidence to suggest that the null hypothesis is true. the value of the test statistic does not lie in the rejection region. therefore, we accept the null hypothesis.

The given statement is false becasue If the p-value of the coefficient of an X variable in a regression is 0.001, then it suggests that there is significant evidence that the coefficient is different from 0 in the population regression line.

A p-value is a value that represents the probability of obtaining the observed results of a statistical hypothesis test. The p-value is used as an alternative to rejection point to give the smallest level of significance at which the null hypothesis would be rejected. It can be used to test hypotheses in statistics.

If the p-value is less than the significance level (α), then the null hypothesis is rejected. However, if the p-value is greater than the significance level (α), then the null hypothesis is not rejected. Therefore, we reject the null hypothesis if the p-value is less than or equal to the level of significance.

So, if the p-value of the coefficient of an X variable in a regression is 0.001, it suggests that the coefficient is statistically significant and different from 0 in the population regression line. Hence, the statement "it is reasonable to say that the coefficient of the X variable is 0 in the population regression line" is false.

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Answer:

This graph shows a quadratic curve with roots 0 and -4

The polynomial is therefore

x² -x(α+β) + αβ

x² -x(0-4) + 0(-4)

x² +4x +0

x² + 4x

But it's an n-shaped curve, meaning a is negative

Multiply through by -1

-x² - 4x

y = -x² - 4x

So when the input value x = -1

y = -(-1)² + 4(-1)

y = -(1) - 4

y = -1 - 4

y = -3

Check the graph to confirm

Answer: Area = 47.5m(2)

Step-by-step explanation:

Area = Pie X r(2) / 2

Area = 3.14 X 5.5(2)/2

Area = 47.4925

Ty made more money in tips on night 5 than usual because $49 is more than the standard $22 tip level.

A mathematical statement that demonstrates the equality of two expressions is known as an equation. Mathematical operations like addition, subtraction, multiplication, and division may be used, and it frequently involves one or more variables that are represented by letters. Equations are employed in a variety of mathematical and scientific contexts, from the solution of straightforward algebraic problems to the modeling of intricate physical systems. Finding the values of the variables that equalize both sides of the equation is necessary to solve an equation.

given

On night 5, he received the following in tips:

Total tips: ($23 + $29 + $25 + $16) multiplied by the sum of the tips from the first four nights equals $22 * 5.

Ty made more money in tips on night 5 than usual because $49 is more than the standard $22 tip level.

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Answer:

Step-by-step explanation:

The number of toy soldiers that Leo has is 48.

The first step is to determine the multiples of 4 that lie between 27 and 54. A multiple of a number is the product of an integer and that number.

Multiplies of 4 that lie between 27 and 54 = 28, 32, 36, 40, 44, 48, 52

Now divide each of the multiples by 7 :

28 / 7 = 4

32 / 7 = 4 remainder 4

36 / 7 = 5 remainder 1

40 / 7 = 5 remainder 5

44 / 7 = 6 remainder 2

48 / 7 = 6 remainder 6

52 / 7 = 7 remainder 3

It only 48 that meets this criteria. This means that the number is 48.

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An offshore oil well is 5 kilometers off the coast. The refinery is 6 kilometers down the coast. Laying pipe in the ocean is twice as expensive as on land. Kilometers down the coast should the pipe be laid in order to minimize the cost, the pipe should be laid approximately 8.62 km down the coast to minimize the cost.

Let the distance down the coast where the pipe is laid be x. Therefore, the distance from the refinery to the point where the pipe meets the shore will be (x + 5) km. The total distance of the pipe can be found using the Pythagorean Theorem.[tex]D = \sqrt{((x + 5)^2 + 6^2) } = \sqrt{(x^2+ 10x + 61)}[/tex] km

Let C(x) be the cost of laying the pipe down the coast at a distance x. Then

Now, to minimize the cost, we have to find the value of x that minimizes C(x). The first derivative of C(x) is:[tex]C'(x) = 2 + 1.5 [x^2 + 10x + 61]^{-1/2} [2x + 10][/tex] After simplifying,[tex]C'(x) = [2(x^2 + 10x + 61) + 1.5(2x + 10)] [x^2 + 10x + 61]^{-1/2}= (3x^2 + 23x + 82) [x^2 + 10x + 61]^{-1/2}= 0[/tex] (at a minimum point)

Now we can solve for x using the above equation.[tex](3x^2 + 23x + 82) = 0[/tex] ⇒ [tex]3(x^2 + 7.67x + 27.33) = 0[/tex] Using the quadratic formula; x = {-b ± √(b² - 4ac)}/2 a We get, x = {-23 ± √(23² - 4(3)(82))}/2(3)x = {-23 ± √(529 - 984)}/6x = {-23 ± √(-455)}/6x = -3.03 or -8.62 Since x must be positive, x = -3.03 is not possible. Hence, the distance down the coast where the pipe should be laid in order to minimize the cost is :x = -8.62Therefore, the pipe should be laid approximately 8.62 km down the coast to minimize the cost.

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The cookers at healthy Bakery could make 323 bagels in 17 hours, and their rate is 19 bagels according to hour.

To find out how many bagels the cookers could make in 17 hours, we will use the unitary method, which involves finding the rate at which the cookers can make bagels and additionally multiplying that price through the wide variety of hours labored.

Let the rate at which the cookers can make bagels be r bagels in line with hour. We also can set up the subsequent share

190 bagels/ 10 hours = r bagels 1 hour

Simplifying this proportion, we get

r = 190 bagels/ 10 hours

r = 19 bagels/ 1 hour

So the cookers can make 19 bagels in keeping with hour.

To find out how many bagels they could make in 17 hours, we can multiply the rate via the number of hours

19 bagels/ hour × 17 hours = 323 bagels

Therefore, the cookers at healthy Bakery could make 323 bagels in 17 hours, and their rate is 19 bagels according to hour.

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We need to use mathematical logic to prove that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

     Let's take an example of two sets A and B.

      Let's say, A={a, b} and B={1, 2, 3}

      To find the cardinality of the cross-product of A and B, we need to make all possible ordered pairs with A and B.

             {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

 Counting the number of ordered pairs, we get six.

  So, the cardinality of the cross-product of two sets A and B equals the cardinality of each set multiplied by the other. Here,

           |A| = 2 and |B| = 3,

          so |A x B| = 2 x 3

                           = 6.

          Therefore, it is proved that the cardinality of the cross product of two sets is the cardinality of each individual set multiplied by each other.

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The following statements correctly describe how the graph of the geometric sequence: 640, 160, 40, 10, ... should appear:

The given sequence is 640, 160, 40, 10, ... which is a geometric sequence.

Here, the first term is 640 and the common ratio is ¼

The terms of a geometric sequence can be written as an = a₁(r)⁽ⁿ⁻¹⁾

Here, a₁ = 640, and r = ¼.

Hence, the nth term of the given sequence is given by the formula:

an = 640(1/4)⁽ⁿ⁻¹⁾

The graph of the given sequence will appear as shown below:

The given sequence is a decreasing sequence, which means the terms of the sequence keep decreasing as the value of n increases.

Therefore, the graph will show exponential decay.

The domain of the sequence will be the set of natural numbers, which is {1, 2, 3, ...}, since we cannot find any term before the first term.

Therefore, the first term is the initial term and we can count the other terms of the sequence in natural numbers.

Hence, the domain will be the set of natural numbers.

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Answer:

the graph will show exponential decay.

the domain will be the set of natural numbers.

Step-by-step explanation:

The answer above is correct.

As per unitary method, the number of months would take her to break - even on the points purchase is 128

To break even on the points purchase, Jessie needs to recoup the cost of the point through the savings in her monthly mortgage payments. In this case, the cost of the point is 1% of the loan amount, which is $1,350.

To calculate the break-even point, we can use the following formula:

Break-even point = Cost of point / Monthly savings

In this case, the monthly savings is the difference between the original mortgage payment ($766.51) and the new mortgage payment with the point ($755.96), which is $10.55 per month.

Substituting the values, we get:

Break-even point = $1,350 / $10.55

Break-even point = 128 months

Therefore, it would take Jessie 128 months, or just over 10 years, to break even on the points purchase.

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Each subsequent jump will be exactly one unit longer in length than the one before it after the initial jump, which can be one unit in length. Each jump has the option of going either left or right.

The translation of the question is

A robot jumps on a number line starting at zero. The rules of the game are the following if you look at your right hand, the length of each jump is expressed with a positive number and, if facing your left hand, with a negative number.

Also: if you jump straight ahead (that is, in the direction you are facing), the number number of jumps is expressed as a positive number, and if you jump backwards (i.e., on the opposite direction to the one you are looking at), the number of hops is considered with a negative number. Observe the following jumps of the robot, its final position in each case and answer in your notebook the questions

When closely examined, it is simple to conclude that:

If you have consistently jumped in the appropriate direction, you will arrive at point p = 1 + 2 + 3 + 4 +... + n after n jumps.

You would be at point p - 2k in any of these n leaps if you leapt left in the kth hop (k=n) rather than right.

Moreover, after n leaps, you can be anywhere between n * (n + 1) / 2 and -(n * (n + 1) / 2) with the same parity as n * (n + 1) / 2 by carefully selecting which jumps to go left and which to go right.

The trick is to imitate jumping while keeping the aforementioned facts in mind. Always jump to the right, and if at some time you arrive at a location that has the same parity as X and is at or beyond X, you'll know the answer.

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Table of values:

x y

-2 5

-1 3/2

0 1

1 1/2

2 1

We may solve for y by substituting the supplied  values of x into the equation [tex]y=1/2x3-2x+1[/tex] and completing Table of values .

For instance,[tex]y=1/2(-2)3-2(-2)+1[/tex] Equals 5 when[tex]x=-2.[/tex] The equivalent values of y for x=-1, 0, 1, and 2 are similarly found.

The ordered pairs (x,y) that fulfil the equation are displayed in the resultant table. These points allow us to graph the equation and see how the curve looks. In this instance, the equation describes a cubic function with two turning points at (-1,3/2) and (1,1/2), as well as a horizontal intercept at (0,1).

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(a) The solution to the differential equation isx(t) = 10(1 - e^(-0.005t))

To solve the differential equation dx/dt = 0.005(10 - x), we can use separation of variables:

dx / (10 - x) = 0.005 dt

Integrating both sides:

-ln|10 - x| = 0.005t + C

where C is the constant of integration. Solving for x:

|10 - x| = e^(-0.005t - C)

Since x cannot be negative, we can drop the absolute value sign and solve for C using the initial condition that x(0) = 0:

C = -ln(10)

Therefore, the solution to the differential equation is:

x(t) = 10 - e^(-0.005t - ln(10))

Simplifying:

x(t) = 10(1 - e^(-0.005t))

(b) New bills will make up 90% of the currency in circulation after approximately 461 days.

We want to find the value of t such that x(t) = 0.9(10) = 9. Plugging this into our solution from part (a):

9 = 10(1 - e^(-0.005t))

Dividing both sides by 10 and taking the natural logarithm:

ln(0.1) = -0.005t

Solving for t:

t = 200 ln(10) = 460.51

Therefore, new bills will make up 90% of the currency in circulation after approximately 461 days.

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The expression that shows the measurement of the length of the land is L = 3x - 10a.

Let L be the length of the rectangular plot.

We know that the perimeter of a rectangle is given by the sum of the lengths of all its sides. In this case, the perimeter is 10x - 28a, so we can write:

Perimeter = 2L + 2W

10x - 28a = 2L + 2(2x - 4a)

10x - 28a = 2L + 4x - 8a

2L = 6x - 20a

L = 3x - 10a

In other words, the length of the land is equal to three times the width minus ten times the value of 'a'.

To explain this equation in a little more detail, we can say that the length of the rectangular plot is dependent on both the width of the land and the value of 'a'.

The expression tells us that as the value of 'a' increases, the length of the land decreases, and as the width of the land increases, the length of the land also increases. This equation is useful because it allows us to calculate the length of the land without having to measure it directly, as long as we know the value of 'a' and the width of the land.

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The line of best fit for the data is approximately y = -1.07x + 92.87: A) True.

In order to determine a linear equation for the line of best fit that models the data points contained in the table, we would have to use a graphing calculator (scatter plot).

In this scenario, the latitude would be plotted on the x-axis of the scatter plot while the average temperature would be plotted on the y-axis of the scatter plot.

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the latitude and average temperature, a linear equation for the line of best fit is given by:

y = -1.07x + 92.87

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Answer:

1/2 or .5

Step-by-step explanation:

If y is inversely proportional to the square of x, we can express this relationship using the formula:

y = k/x^2

where k is a constant of proportionality. We are told that y = 8 for a particular value of x, so we can substitute these values into the equation:

8 = k/x^2

To find the value of k, we can solve for it:

k = 8x^2

Now we are asked to find the new value of y when x increases by 300%. This means that the new value of x will be 4 times the original value (since an increase of 300% means an increase by a factor of 3, and we need to add the original value to get the new value). So we can substitute 4x for x in our equation:

y = k/(4x)^2 = k/16x^2

We already know the value of k, so we can substitute it in and simplify:

y = (8x^2)/(16x^2) = 1/2

Therefore, the new value of y is 1/2 when x increases by 300%.

X = 2 is the solution of  the given equation for "x".

What does a math equation mean?

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. For instance, 3x + 5 = 14 is an equation where 3x + 5 and 14 are two expressions separated by the 'equal' sign.

                           A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. By solving for x, we discover that x equals 7, which is the value for the variable.

8ˣ = (25)

 8ˣ =  (5)²

  X = 2

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After collecting the data, Maria found that the number of points per game for a particular basketball player is normally

with a mean of 14 points and a standard deviation of 2 points. Based on the empirical rule, what is the probability that the player scored less than 8 points in a randomly selected game?

To solve this problem, we'll need to use the formula for a standard normal distribution:z = (x - μ) / σwhere z is the standard score, x is the raw score, μ is the population mean, and σ is the population standard deviation. We'll need to calculate the standard score for x = 8, which gives us:[tex]z = (8 - 14) / 2z = -3.0[/tex]

Now, using the standard normal distribution table or calculator, we can find the probability that a randomly selected game results in a score less than 8 points. The area to the left of z = -3.0 is approximately 0.0013, which means that the probability of a randomly selected game resulting in a score less than 8 points is 0.0013 or 0.13%.Therefore, the probability that in a randomly selected game, the player scored less than 8 points is 0.13%.

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X = -80538738812075974, Y = 80435758145817515,

and Z = 12602123297335631 are the values of  x,y,z of whole numbers.

What in mathematics is a whole number?

Complete numbers the sum of all natural numbers plus zero. not a decimal or fraction. {0, 2, 3, 4, 5 6, 7, 8, 9, 10, 11 …} Integer. a negative number, zero, or a counting number.

                            Natural numbers with a starting value of 1 and higher are considered whole numbers. Let's examine a few instances of entire numbers. The alphabet "W" stands for the collection of whole numbers. W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,.…}

x3+y3+z3=k, with k being all the numbers from one to 100

X = -80538738812075974,

Y = 80435758145817515,

and Z = 12602123297335631

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Answer:

120 degrees

Step-by-step explanation:

The sum of the interior angles of a pentagon is given by the formula, where S is the sum of the interior angles, and n is the number of sides of the polygon:

S = (n - 2) × 180 degrees

For a pentagon, n = 5, so we have:

S = (5 - 2) × 180 = 540 degrees

If four of the angles are equal, we can let x be the measure of each of those angles. Then, we can write an equation for the sum of the angles:

4x + 60 = 540

Solving for x, we get:

4x = 540 - 60 = 480

x = 480/4 = 120

Therefore, each of the four equal angles in the pentagon has a measure of 120 degrees.

A) x = 5 is not a linear function since it is a vertical line and does not have a slope. B) The table description does not provide enough information to determine if it is a linear function. C) x + 7 = 4y is a linear function in slope-intercept form (y = (1/4)x + 7/4).

A linear function is a mathematical function that can be represented by a straight line with a constant slope. The equation of a linear function can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis). Option A (x = 5) is not a linear function, as it is a vertical line with an undefined slope. Option B is a linear function, as the table describes points that can be plotted to form a straight line. Option C is also a linear function, but it is in a different form (x + 7 = 4y). This equation can be rearranged to y = (1/4)x + 7/4, which is in the standard form of a linear function.

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Answer: C.) x + 7 = 4y and  D

Step-by-step explanation: i hope this helps

For point, p divides the directed line segment xy so that the ratio of xp to py is 3 to 5 correct option is d. Point R is three-fifths of the way from point P to point Y along PY.

The directed line segments divide a line into ratios, and each ratio has different properties. The properties of ratios of points on a line are dependent on the way the line is divided. For instance, suppose point p divides the directed line segment xy so that the ratio of xp to py is 3 to 5. We can describe point r that divides the directed line segment yx so that the ratio of yr to rx is 5 to 3 as follows:
We can solve this problem using the concept of directed line segments and the properties of ratios of points on a line.

Since P divides the directed line segment XY in the ratio 3:5, we can write:

XP = (3/8)XY and PY = (5/8)XY

Now, let's consider the directed line segment YX. We want to find a point R on this segment such that YR:RX = 5:3.

We can express YR and RX in terms of XY using the fact that YX = -XY:

YR = (5/8)YX and RX = (3/8)YX

Substituting -XY for YX, we get:

YR = (-5/8)XY and RX = (-3/8)XY

To find the location of point R, we need to find the distance from Y to R along the directed line segment YX. We can do this by adding the distances from Y to P and from P to R:

YR = YP + PR

Using the ratios we derived earlier, we can express YP and PR in terms of XY:

YP = PY = (5/8)XY

PR = R - P = (3/8)XY - (3/8)XY = 0

Therefore, YR = (5/8)XY + 0 = (5/8)XY

This means that point R is located at a distance of 5/8 of the length of YX from Y. So, the correct answer is (d) Point R is three-fifths of the way from point P to point Y along PY.

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Answer:

2125 ft/min

33,000 ft

y = -2125x + 33,000

Step-by-step explanation:

A. -2125 feet per minute. You get this number when you divide 17,000 by 8 (rise over run). You could also use the formula y2-y/x2-x1 with the points (0, 33,000) and (8, 17,000).

B. 33,000 feet is the height of the plane before it starts descending, so it must be the starting value.

C. Plug in the values you got for A and B into the slope formula y = mx + b

y = -2125x + 33,000

The length of BC is approximately 3.5 cm

A triangle is described as a closed, 2-dimensional shape with 3 sides, 3 angles, and 3 vertices.

We construct triangle ABC, through the following steps:

The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

Mathematically shown as:

a/sin(A) = b/sin(B) = c/sin(C)

5/sin(95) = BC/sin(34)

BC = (5*sin(34))/sin(95)

BC ≈ 3.5cm

In conclusion, the length of BC is approximately 3.5 cm.

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In Δ ABC and Δ PQR, if AB = PR, BC = RQ and AC = PQ, then Δ ABC is congruent to Δ PRQ, which means option D is the right answer.

The congruency theorem is used to determine the relation between two similar looking figures in two dimensional space. The word congruent itself means being in harmony. There are different rules which are used to determine the congruency between the triangles.

These rules are given as follows:

In the given question, it is given that sides AB = PR, BC = RQ and AC = PQ, this implies that the congruency can be setup using SSS rule, which if followed will suggest that Δ ABC will be congruent to Δ PRQ.

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Answer:

[tex]\bf a^2* (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

Step-by-step explanation:

Factorize:

 First take out the common term a².

[tex]a^5 - 0.064a^2= a^2*(a^3 - 0.064)\\\\[/tex]

Now, factorize using the identity a³ - b³

a³ - b³ = (a - b) (a²  + ab + b²)

[tex]a^2 * (a^2 - 0.064) = a^2 * (a^3 - 0.4^3)[/tex]

                          [tex]= a^2 * (a - 0.4) * (a^2 + a*0.4 + 0.4^2)\\\\=a^2 * (a - 0.4)(a^2 + 0.4a + 0.16)[/tex]

The volume of the solid whose base is the region inside the circle with radius 3 if the cross sections taken parallel to one of the diameters are equilateral triangles is 81/2*\sqrt3 by using the slicing method.

To find the volume of the solid whose base is the region inside the circle with radius 3, we need to integrate the area of the cross sections taken parallel to one of the diameters, which are equilateral triangles.

Let's consider a cross section of the solid taken at a distance x from the center of the circle.

Since the cross section is an equilateral triangle, all its sides have the same length.

Let  this length be y. Since the triangle is equilateral, its height can be found using the Pythagorean theorem as follows:

[tex]height = \sqrt{(y^2 - (y/2)^2)} = \sqrt{(3/4y^2)}= \sqrt{3/2y}[/tex]

Therefore, the area of the cross section at a distance x from the center of the circle is:

[tex]A(x) = (1/2)y\sqrt{3/2y} = \sqrt{3/4y^2}[/tex]

Now, we need to integrate this area over the range of x from -3 to 3 (since the circle has radius 3):

[tex]V = \int\ [-3,3]\sqrt{3/4*y^2} dx[/tex]

To find the limits of integration for y, we need to consider the equation of the circle:

[tex]x^2 + y^2= 3^2[/tex]

Solving for y, we get:

[tex]y =\pm\sqrt{(3^2 - x^2)}=\pm\sqrt{(9^2 - x^2)}[/tex]

Since we want the cross sections to be equilateral triangles, we know that y is equal to the height of an equilateral triangle with side length equal to the diameter of the circle, which is 2*3 = 6. Therefore, we can write:

[tex]y = 3*\sqrt{3}[/tex]

Substituting this into the integral, we get:

[tex]V = \int\ [-3,3] \sqrt{3/4*(3\sqrt3)^2} dx[/tex]

[tex]= \int\ [-3,3] 27/4*\sqrt{3} dx[/tex]

Integrating, we get:

[tex]V = [27/4\sqrt{3x}]*[-3,3][/tex]

[tex]= 81/2*\sqrt{3}[/tex]

Therefore, the volume of the solid is [tex]81/2*\sqrt3[/tex]cubic units

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The correct statement explain for the given ratio of boys to girls of 3:4 is - this fraction of girls in the class is found to be 4/7.

Irrespective whatever how a ratio is expressed, it is crucial to reduce it to the fewest whole numbers, just like with any fraction. To accomplish this, divide the integers by their largest common factor after discovering it.

Ratios can also be expressed as a fraction because they are straightforward division problems. Some folks prefer to use merely words to express ratios.

Class contains a ratio of boys to girls of 3: 4.

So, Boys / Girl = 3 / 4

Total students = boys  + girls.

Total students = 3 + 4 = 7

So,

Girl / Total = 4/7

Thus, the correct statement explain for the given ratio of boys to girls of 3:4 is - this fraction of girls in the class is found to be 4/7.

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The complete question is-

A class has a ratio of boys to girls 3:4.

Select correct option:

a) The fraction of boys in the class is 3/4

b) The fraction of girls in the class is 4/7

c) The number of boys in the class is 6

d) The number of pupils in the class is 12

Answer:The prices of the adult movie tickets and the child movie tickets are $15 and $5 respectively.

Given that, the Jones family spent $45 on movie tickets for 2 adults and 3 children.

Step-by-step explanation:What is a linear system of equations?

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

Let cost of adult tickets be x and the cost of children tickets be c.

The Jones family spent $45 on movie tickets for 2 adults and 3 children.

2a+3c=45 ------(I)

The Smith family spent $40 for 2 adults and 2 children.

2a+2c=40

a+c=20 ------(II)

From equation (II), we have a=20-c

Substitute a=20-c in equation (I), we get

2(20-c)+3c=45

⇒ 40-2c+3c=45

⇒ c=$5

Put c=5 in equation (II), we get

a+5=20

⇒ a=$15

We can plot the lines x = 260, y = 320, and x + y = 380 to graph these limitations, and we can highlight the area that satisfies each requirement.

We refer to x and y as being limited by g(x,y)=c, which is also referred to as the constraint equation. Constrained maximum or constrained minimum points, respectively, are positions (x,y) that meet the constraint equation g(x,y)=c and are maxima or minima of f(x,y).

Let's indicate the quantity of black walnut and mahogany boards sold, respectively, using the variables x and y.

x ≤ 260 (constraint on the number of mahogany boards)

x + y ≤ 380 (constraint on the total number of boards sold)

y ≤ 320 (constraint on the number of black walnut boards)

We also understand that the quantity of sold boards cannot be zero, hence we have:

x ≥ 0

y ≥ 0

Last but not least, our goal is to maximise profit, as indicated by the expression:

P = 20x + 6y

When all the restrictions are combined, the following inequality system results:

x ≤ 260

y ≤ 320

x + y ≤ 380

x ≥ 0

y ≥ 0

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