In a Gallup poll conducted nationwide in July 2020, it was found that 63% of a sample of female
adults supported a ban on public smoking.
a) Describe the population parameter of interest in this study.

Answer:

let video game be x

let DVDs be y

2x+3y=148.85

3x +5y=230.75

make x the subject of the formula

x=148.85-3y/2

3(148.85-3y/2)+5y =230.75

446.55-3y/2=230.75

461.5=446.55-3y

461.5-446.55 =-3y

-3y = 14.95

y=-4.987

x=

Yοu wοuld need apprοximately $274,394.92 in yοur accοunt at the beginning tο be able tο withdraw $20,000 each year fοr 30 years at an 8% interest rate.

Tο calculate the amοunt needed in the accοunt at the beginning tο withdraw $20,000 each year fοr 30 years, we can use the fοrmula fοr the present value οf an annuity: [tex]PV = PMT * ((1 - (1 / (1 + r)^n)) / r)[/tex]

Where:

PV = present value (amοunt needed in the accοunt at the beginning)

PMT = payment per periοd (annual withdrawal amοunt = $20,000)

r = interest rate per periοd (8% per year = 0.08)

n = tοtal number οf periοds (30 years)

Plugging in the given values, we get:

[tex]PV = 20,000 * ((1 - (1 / (1 + 0.08)^{30})) / 0.08)[/tex]

PV ≈ $274,394.92

Therefοre, yοu wοuld need apprοximately $274,394.92 in yοur accοunt at the beginning tο be able tο withdraw $20,000 each year fοr 30 years at an 8% interest rate.

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Assuming Sanford paid no sales tax, he received $10.15 in change.

What are basic arithmetic operations?

Basic arithmetic calculations are the fundamental operations used in mathematics to perform numerical computations. These operations include addition, subtraction, multiplication, and division of numbers. In addition, arithmetic calculations may also involve other concepts, such as fractions, decimals, percentages, and ratios.

The total cost of the two shirts is 2 x $24.95 = $49.90.

The total cost of the two shirts and the pair of pants is $49.90 + $39.95 = $89.85.

Sanford paid with a $100 bill, so his change would be:

$100 - $89.85 = $10.15

Therefore, Sanford received $10.15 in change.

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Since opposite angles are equal, this means that GHJK is a parallelogram, because opposite sides are parallel.

Additionally, each pair of opposite sides of a parallelogram are congruent (of equal length), and the opposite angles of a parallelogram are also congruent (of equal measure). These properties make parallelograms useful in various fields such as engineering, architecture, and geometry.

There are different types of parallelograms, including rectangles, rhombuses, and squares, which have additional properties such as perpendicular sides, equal sides, or right angles. A rectangle is a type of parallelogram with four right angles, while a rhombus has four sides of equal length, and a square has both equal sides and right angles.

To prove that GHJK is a parallelogram, we need to show that opposite sides are parallel.

Since GK is parallel to HJ, we have:

∠LGK + ∠KHL = 180° (because GK // HJ)

But we are given that:

∠LGK = ∠LKL (given)

Therefore:

∠LKL + ∠KHL = 180°

This means that ∠KHL is a supplementary angle to ∠LKL.

Similarly, we can show that ∠KJH is a supplementary angle to ∠JKL:

∠JKL + ∠KJL = 180° (because GK // HJ)

But we are given that:

∠LGK = ∠LKL

Therefore:

∠LGK + ∠KJL = 180°

This means that ∠KJL is a supplementary angle to ∠LGK.

Now, we can use the fact that the opposite angles of a parallelogram are equal to show that GHJK is a parallelogram:

∠LKL = ∠JKL (alternate interior angles)

∠KHL = ∠KJH (alternate interior angles)

Since opposite angles are equal, this means that GHJK is a parallelogram, because opposite sides are parallel.

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The answer of this question is in every 5 hour 6 m water height increase in tank. So we have to learn about proportional.

Proportional is the right amount, size , degree or quantity compared with something else. We can simply say that it is comparison between one object or thing with another object or thing.

Their are two types of proportional properties:

a. Directly Proportional, In this when amount , size or degree of any object increases than the amount ,size, or degree of another object are also increases with same ratio.

b. Indirectly Proportional, In this when amount , size or degree of any object increases than the amount , size or degree of another object is decreases in same ratio.

Let Solve this question, in this the first property used.

[tex]\frac{1}{4} = \frac{3}{10} , we\ get\ 5 = 6[/tex]

[tex]\frac{1}{2} = \frac{3}{5}\ we\ get\ 5 = 6[/tex]

[tex]\frac{3}{4} =\frac{9}{10} \ we\ get\ 5 =6[/tex]

So, In every 5 hour 6 m water height increase in tank.

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The values of angles A , B, C are 140 , 40 , 140 respectively if other angle

is 40 degrees.

What are parallel lines ?

Parallel lines can be defined as lines which are equidistant from each other and which never intersect .

from the given figure ,

the angle b and 40 degrees are vertically opposite angles ,

vertically opposite angles are always equal

so,

b = 40

and sum of angles A and 40 is 180 degrees

so,

A + 40 = 180

A = 180 - 40

A = 140

And here similarly ,

angles A and C are vertically opposite angles ,

so,

C = 140 degrees.

Therefore, The values of angles A , B, C are 140 , 40 , 140 respectively  if other angle is 40 degrees.

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a) The total interest that Angelica must pay on the loan during the period that payments are not being made is $2,998.58.

b) Angelica's monthly payment on the loan is $204.07.

The total interest when there were no repayments is determined using an online finance calculator.

The same calculator is used to determine the monthly payments after her graduation.

The total interest and the monthly payments are computed based on monthly compounding interest.

The amount of the non-subsidized student loan = $13,000

Annual compound interest rate = 5.2%

Loan period = 12 years

Non-repayment period = 4 years

N (# of periods) = 48 months (4 years x 12)

I/Y (Interest per year) = 5.2%

PV (Present Value) = $13,000

PMT (Periodic Payment) = $-0

Results:

FV = $-15,998.58

Total Interest = $2,998.58

N (# of periods) = 96 months (8 years x 12)

I/Y (Interest per year) = 5.2%

PV (Present Value) = $15,998.58

FV (Future Value) = $0

Results:

Monthly Payments (PMT) = $-204.07

Sum of all periodic payments = $-19,590.49

Total Interest = $3,591.91

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Answer:

175

Step-by-step explanation:

To find the median of a set of numbers, we must first arrange them numerically.

Arranging the given set in ascending order, we get:

125, 150, 150, 150, 175, 175, 200, 225, 250

The median is the middle number when the set is ordered this way. If the set has an odd number of values, the median is the middle value. If the set has an even number of values, the median is the average of the two middle values.

In this case, the set has nine values, so it is odd. The middle value is the fifth number, which is 175. Therefore, the median of the set is 175.

Answer:

The value of "c" is 4.

Here's the solution:

2^-4 • 2² = a = c

2^-4 = 1/2^4 = 1/16

2² = 4

a = c = 1/16 • 4 = 1/4

Therefore, c = 1/4 = 0.25

The relative frequency οf value οf 10 is 0.3.

The relative frequency is calculated as the prοduct οf the number οf times a given value οf the data appears in the set οf all οutcοmes divided by the tοtal number οf οutcοmes.

Here the given data set is 10, 14, 12, 16, 13, 14, 16, 10, 12, and 10.

Fοr value οf 10, the number οf times 10 οccured is 3 and tοtal number οf data set is 10.

Relative frequency,

=> Number οf times οf value οf 10/ Tοtal number οf data set

=> 3/10

=> 0.3.

Hence the relative frequency οf value οf 10 is 0.3.

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Answer:

Step-by-step explanation:

The peaches cost $2.50 per pound.

Conversion rate is the percentage of website visitors or potential customers who take a desired action, such as making a purchase, signing up for a newsletter, or filling out a form. In other words, it is the rate at which website visitors are converted into customers or subscribers.

To find the cost per pound of peaches, we need to divide the total cost by the weight in pounds.

Cost of 3.1 pounds of peaches = $7.75

Cost of 1 pound of peaches = $7.75 / 3.1

Cost of 1 pound of peaches = $2.50 (rounded to the nearest cent)

Therefore, the peaches cost $2.50 per pound.

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The ratio of the sides of the small box and the medium box is 1:3

The ratio of the area of the bases of the small box and the medium box is 1:9.

The ratio of the volumes of the small box and the medium box is 1:27.

The volume of a medium box is 5832 cubic inches.

What is a cube?

The solid shape of a cube has six square faces. Because every square face has the same side length, each face is the same size. A cube has 8 vertices and 12 edges. Each vertex is the intersection of three cube edges.

Given that a cube-shaped small packaging box has a volume of 216 cubic inches.

Assume the length of sides is a.

The volume of a cube is side³.

The volume of the small box is a³.

Thus,

a³ = 216

a³ = 6³

a = 6

The length of the small box is 6.

The area of the base of small box is 6×6  = 36 square inches.

The dimensions of a medium box are tripled in length, width, and height.

Thus the length of the medium box is 3×6 = 18 inches.

The area of the base of the medium box is 18×18 = 324 square inches.

The volume of cube medium box is 18×18×18 = 5832 cubic inches.

The ratio of the sides of small box and medium box is

6:18 = 1:3

The ratio of the area of bases of small box and the medium box is

36:324 = 1:9

The ratio of the volumes of small box and the medium box is

216:5834 = 1:27

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The value of XS in the perpendicular lines is 16 units.

The line VU is perpendicular to the line  XS. Perpendicular lines are defined as two lines that meet or intersect each other at right angles.

Therefore, perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point.

Hence,

b + 5 = 3b - 1

b - 3b = -1 - 5

-2b = -6

divide both sides by -2

b = -6 / -2

b = 3

Therefore, let's find XS as follows:

XS = b + 5 + 3b - 1

XS = 4b + 4

XS = 4(3) + 4

XS = 12 + 4

XS = 16 units

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The first 5 elements of set A are: {-1, -2, 1, -3, 2}.

What is the set of elements?

The objects in a set are called the elements (or members ) of the set; the elements are said to belong to the set (or to be in the set), and the set is said to contain the elements. Usually, the elements of a set are other mathematical objects, such as numbers, variables, or geometric points.

Set A is defined as:

A = {x | x = k(-1)ⁿ where k ∈ Z* and n ∈ N}

To find the first 5 elements of A, we can substitute different values of k and n:

For k = 1 and n = 1, x = 1(-1)¹ = -1

For k = 2 and n = 1, x = 2(-1)¹ = -2

For k = 1 and n = 2, x = 1(-1)² = 1

For k = 3 and n = 1, x = 3(-1)¹ = -3

For k = 2 and n = 2, x = 2(-1)² = 2

Therefore, the first 5 elements of set A are: {-1, -2, 1, -3, 2}.

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From the given scatter plot,  point L is an outlier. The association is negative linear association.  The line is not the best fit accurate. Graph shows a negative linear association. As the number of weeks increases, the number of tickets sold decreases

What is scatter plot?

A scatter plot is a type of data visualization that displays the relationship between two numerical variables. It is a graph with points plotted on it that represent the values of two different variables.

1. In a scatter plot, an outlier is a data point that lies far away from the other data points in the plot, and is not in line with the overall pattern or trend that the other data points display.

2. The scatter plot for this data is likely to show a negative linear association between the number of people in a group and the zoo ticket price per person. As the number of people in a group increases, the zoo ticket price per person decreases.

The rate of change of Zoo Ticket Price with respect to the the number of persons:

[tex]m=14.50-1500/2-1=-0.5[/tex]

3. No because the line should touch every point.  When we draw a line passing through the origin, it passes through the data points (2,3), (4,3), and (4,5), but it does not pass through the other data points (6,5), (6,6), and (8,7).

This indicates that the line of best fit may not accurately represent the overall trend of the data.

4. The given data points are (1,5), (2,4), (3,3), (4,2), (5,1), and (6,0). When we plot these points on a graph with x-axis representing variable x and y-axis representing variable y, we get a downward sloping straight line passing through all the points.

This pattern suggests a negative linear association between the variables x and y. As the value of x increases, the value of y decreases at a constant rate.

Therefore, the graph shows a negative linear association between x and y.

5. The answer is option (a) "As the number of weeks increases, the number of tickets sold decreases".

As per the given data points and the line passing through (1,100) and (6,35), we can observe that the number of tickets sold decreases as the number of weeks a movie has been in theaters increases. Therefore, the statement "As the number of weeks increases, the number of tickets sold decreases" best describes the relationship between the number of weeks a movie has been in theaters and the number of tickets sold.

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The exponential growth of bacteria gives the growth percentage to be 4.05%.

What is an exponential function?

The formula for an exponential function is [tex]f(x) = a^x[/tex], where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

Let N(t) be the number of bacteria in the population at time t.

According to the continuous exponential growth model, we have -

[tex]N(t) = N0 \times e^{(rt)}[/tex]

where N0 is the initial number of bacteria, r is the hourly growth rate parameter, and t is the time elapsed in hours.

We know that a sample of 1800 bacteria reached the size of 2204 bacteria in five hours, so we can use this information to find the hourly growth rate parameter.

First, we can set up a ratio of the final size to the initial size -

[tex]\frac{2204}{1800} = e^{5r}[/tex]

Simplifying, we get -

[tex]e^{5r} = 1.2244[/tex]

Taking the natural logarithm of both sides, we get -

5r = ln(1.2244)

Solving for r, we get -

r = ln(1.2244) / 5

On further simplifying, we get -

r = 0.20245 / 5

r ≈ 0.04049

To express the growth rate as a percentage, we can multiply by 100 -

r percentage = 100 × r ≈ 4.049

Rounding to the nearest hundredth, we get -

r percentage ≈ 4.05%

Therefore, the hourly growth rate parameter is approximately 4.05%.

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Answer:

the answer is 19/20

Step-by-step explanation:

the explanation is 3/4 x 5/5=15/20+ 4/20=1/5 x 4/4

The length would decrease at first to AB - BC then it would increase to the maximum of B + BC

From the diagram we have that two triangles are formed when we have AB = 3 cm and when we have BC = 1.5cm

This would happen when at point AC we have C moving in a counterclockwise direction around A

As AC goes towards A, then we would have AC decrease at first to AB - BC. Then it would increase. The maximum length would be AB + BC

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Answer:

It decreases

Step-by-step explanation:

The dimensions of the rectangle are, length = 73 cm and width = 17 cm.

Finding the values of the variables employed in a system of equations entails solving the set of equations. By keeping the equations balanced on both sides, we compute the values of the unknown variables. Finding the value of the variable that makes the condition of all the provided equations true is the primary goal when solving an equation system. A given system of equations may have a variety of solutions.

Let us suppose the width of the rectangle = w.

Given that, length is 5 more than four times its width.

Thus,

l = 5 + 4w

Now, the perimeter of the rectangle is given as:

P = 2(l + w)

Substitute the value of P = 180:

180 = 2l + 2w

The system of equations is:

l = 5 + 4w

180 = 2l + 2w

Substitute the value of l from equation 1 into equation 2:

180 = 2(5 + 4w) + 2w

180 = 10 + 8w + 2w

180 - 10 = 10w

170 = 10w

w = 17

Substitute the value of w = 17 in the first equation:

l = 5 + 4w

l = 5 + 4(17)

l = 73

Hence, the dimensions of the rectangle are, length = 73 cm and width = 17 cm.

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Answer: 565.49ft^3/ 565.49ft cubed

Hopefully it is correct!

Answer:

[tex]x = \frac{40\sqrt{3}}{3}\\\\y = \frac{20\sqrt{3}}{3}[/tex]

===================================================

Explanation:

The short leg is y because it is opposite the 30 degree angle.

The connection between the long leg and short leg is through this formula

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}[/tex]

which applies to 30-60-90 triangles only.

Let's use that template to solve for y.

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\20 = y\sqrt{3}\\\\y = \frac{20}{\sqrt{3}}\\\\y = \frac{20\sqrt{3}}{\sqrt{3}\sqrt{3}}\\\\y = \frac{20\sqrt{3}}{\sqrt{3*3}}\\\\y = \frac{20\sqrt{3}}{\sqrt{9}}\\\\y = \frac{20\sqrt{3}}{3}\\\\[/tex]

Then to find the hypotenuse x, we double the short leg.

[tex]x = 2y\\\\x = 2*\frac{20\sqrt{3}}{3}\\\\x = \frac{2*20\sqrt{3}}{3}\\\\x = \frac{40\sqrt{3}}{3}\\\\[/tex]

This formula works for 30-60-90 triangles only.

-----------

Side notes:

[tex]\frac{20\sqrt{3}}{3} \approx 11.54701\\\\\frac{40\sqrt{3}}{3} \approx 23.09401\\\\[/tex]

Answer:

x = 23.09 (2 d.p.)

y = 11.55 (2 d.p.)

Step-by-step explanation:

The interior angles of a triangle sum to 180°.  Therefore, from inspection of the given diagram, the interior angles of the given right triangle are:

This means the triangle is a special right triangle and that the 30-60-90 theorem can be applied to quickly find the measures of the missing sides.

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.

Therefore, the formula for the ratio of the sides is b : b√3 : 2b where:

From inspection of the given triangle, the side opposite the 60° angle is 20.  Therefore:

[tex]\implies b\sqrt{3}=20[/tex]

[tex]\implies b=\dfrac{20}{\sqrt{3}}[/tex]

[tex]\implies b=\dfrac{20\sqrt{3}}{3}[/tex]

Substitute the found value of b into the expressions for the other two sides to find the values of x and y.

x is the side opposite the right angle. Therefore:

[tex]\implies x=2b[/tex]

[tex]\implies x=2\cdot \dfrac{20\sqrt{3}}{3}[/tex]

[tex]\implies x=\dfrac{40\sqrt{3}}{3}[/tex]

[tex]\implies x=23.09\; \sf units\;(2\;d.p.)[/tex]

y is the side opposite the 30° angle. Therefore:

[tex]\implies y=b[/tex]

[tex]\implies y=\dfrac{20\sqrt{3}}{3}[/tex]

[tex]\implies y=11.55\; \sf units\;(2\;d.p.)[/tex]

Answer:

A'(4,-3) B'(6,11) C'(-7,-2)

Step-by-step explanation:

X-axis is (x,y) --> (x,-y)

A(4,3) --> (4,-3)

B(6,-11) --> (6,11)

C(-7,2) --> (-7,-2)

♡♡♡

Answer:

a.   (-2, 4) only

==================================================

Explanation:

Let's check point A.

We have x = -2 and y = 4 pair up together.

Plug these coordinates into the first inequality.

x + 4y > 10

-2 + 4*4 > 10

-2 + 16 > 10

14 > 10

We get a true statement, so (-2,4) is in the shaded solution region of the inequality x+4y > 10.

Repeat for the other inequality.

3x - 2y < 12

3*(-2) - 2*4 < 12

-6 - 8 < 12

-14 < 12

This is also true.

The point (-2,4) is also in the shaded solution region for 3x-2y < 12

Since it's in both regions simultaneously, it satisfies the system of inequalities.

Therefore, (-2,4) is a solution to the system.

See the diagram below. Point A is in the shaded region.

-------------------------------

Now we need to check point B.

We have x = 3 and y = 1. Let's plug those into the 1st inequality.

x + 4y > 10

3 + 4*1 > 10

3 + 4 > 10

7 > 10

That final statement is false, so (3,1) is not in the shaded region for x+4y > 10. We don't need to check the other region since this point needs to satisfy both inequalities to be considered a solution overall.

See the diagram below. Point B is outside the shaded region.

Answer:

The quote "Elie sees the death of the boy as the death of the God within" illustrates that Elie has learned from Moishe the Beadle that one's spirituality and relationship with God is a personal experience that resides within oneself. Moishe's teachings suggest that God is not necessarily an external force that determines the fate of individuals or the world, but rather a presence within oneself that must be nurtured and maintained. Elie's interpretation of the boy's death as the death of God within himself shows that he has internalized this lesson and is searching for meaning and understanding within his own experiences. This reflects Moishe's teachings that the "real answers lie in oneself," rather than in external sources of authority or dogma.

Step-by-step explanation:

Answer:

The equation is tan(25.4°) = x/1,600, so

x = 1,600 tan(25.4°) = about 759.74 feet

The vector components and direction of the river and motorboat are;

(a) 12·i

(b) 12·i + 12·√3·j

(c) 24·i + 12·√3·j

(d) 31.7 mi/h

A vector is a quantity or measurement that indicates a magnitude and a direction of the measurement.

The direction the river flows = East

The speed of the river = 12 mi/h

The direction the boater heads = 60° from the shore

Speed of the motor boar = 24 mi/h

The unit vector in the east direction = i

Unit vector in the north direction = j

(a) The velocity of the river, which is 12 mi/h east expressed in vector form is therefore;

(b) The velocity of the motorboat relative to the river, in vector component form is therefore;

Component of the velocity in the east direction = 24 × cos(60°)·i = 12·i

Component in the north direction = 24 × sin(60°)·j = 12·√3·j

The component form of the velocity of the boat is therefore;

(c) The true velocity of the boat is the vector sum of the velocity of the river and the velocity of the boat, therefore;

True velocity of the boat = 12·i + 12·i + 12·√3·j = 24·i + 12·√3·j

(d) The true speed is the magnitude of the velocity of the boat which can be found ads follows;

The (true) direction of the motorboat, θ, can be obtained from the arctangent of the ratio of the velocity component in the northern direction to the velocity in the eastern direction as follows;

The velocity component of the motorboat in the northern direction = 12·√3

The component of the velocity in the eastern direction = 24

Therefore;

tan(θ) = 12·√3/(24)

θ = arctan(12*√3/(24)) ≈ 40.9°, which is about East 40.9° North

The direction from the north, therefore is found as follows;

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Answer: Recall that E ∩ F ∩ G denotes the intersection of the three sets E, F, and G, and the superscript "c" denotes the complement of this set (i.e., everything outside of this intersection).

First, let's find E ∩ F ∩ G:

E ∩ F ∩ G = {24, 29}

Now, let's find the complement of this set:

(E ∩ F ∩ G)c = S - {24, 29}

This means that we take the set S and remove the elements 24 and 29.

So,

(E ∩ F ∩ G)c = {4, 9, 14, 19}

Step-by-step explanation:

The statements that are true are: Option c) Kelsey spent more on the pencils than she spent on the markers.

Option e) Kelsey spent $24 in all.

The problem provides us with some information about the cost and quantity of packages of pencils and markers, as well as the quantities of pencils and markers that Kelsey bought. We need to use this information to determine which statements are true.

We know that a package of pencils has 10 pencils and costs $2, and a package of markers has 8 markers and costs $4.

Kelsey bought 40 pencils and 24 markers. We can use this information to determine the number of packages of each item that Kelsey bought:

Pencils: Kelsey bought 40 pencils, and each package has 10 pencils, so she must have bought 4 packages of pencils.

Markers: Kelsey bought 24 markers, and each package has 8 markers, so she must have bought 3 packages of markers.

Now we can use this information to evaluate the statements:

a) Kelsey bought 4 packages and 3 packages of markers - This statement is false. Kelsey actually bought 4 packages of pencils and 3 packages of markers.

b) Kelsey spent $4 in pencils - This statement is false. Kelsey bought 4 packages of pencils, and each package costs $2, so she spent a total of $8 on pencils.

c) Kelsey spent more on the pencils than she spent on the markers - This statement is true. Kelsey bought 4 packages of pencils, which cost a total of $8, and 3 packages of markers, which cost a total of $12. Therefore, she spent more on markers than on pencils.

d) Kelsey spent $20 dollars in all - This statement is false. Kelsey spent $8 on pencils and $12 on markers, so she spent a total of $20.

e) Kelsey spent $24 in all - This statement is true. Kelsey spent $8 on pencils (4 packages x $2 per package) and $12 on markers (3 packages x $4 per package), so she spent a total of $20.

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Answer:

Step-by-step explanation:

The value of the integral is 6.

Integration is a mathematical operation that involves finding the integral of a function. The integral of a function is a measure of the area under the curve of the function in a given interval. The process of integration is the opposite of differentiation, and it involves finding a function whose derivative is the given function.

Integration is an important tool in many areas of mathematics, science, and engineering. It can be used to calculate quantities such as velocity, acceleration, displacement, and area, among others. It also has applications in fields such as physics, economics, and statistics.

There are different methods of integration, including substitution, integration by parts, partial fractions, and trigonometric substitution. The choice of method depends on the form of the function being integrated and the desired outcome.

We can use integration by parts to evaluate this integral:

Let u = x and dv = g'(x) dx, then du/dx = 1 and v = g(x).

Using the formula for integration by parts:

[tex]\int\limits^5_1[/tex] xg'(x) dx = [xg(x)]₁⁵ - [tex]\int\limits^5_1[/tex] g(x) dx

We can use the given information to find the value of each term in this expression:

[xg(x)]₁⁵ = 5g(5) - 1g(1) = 5(1) - 1(5) = 0

[tex]\int\limits^5_1[/tex] g(x) dx = -6 (given)

Substituting these values into the integration by parts formula, we get:

[tex]\int\limits^5_1[/tex] xg'(x) dx = 0 - (-6) = 6

Therefore, the value of the integral is 6.

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The relationship between the variables in the table is a direct variation.

A function that models the relationship is y = 3x.

In Mathematics, a direct variation is also referred to as direct proportion and it can be modeled by using the following mathematical expression or function:

y = kx

Where:

Under direct variation, the value of x represent an independent variable while the value of y represents the dependent variable. Therefore, the constant of proportionality (variation) can be calculated as follows:

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 9/3 = 27/9 = 39/13 = 60/20

Constant of proportionality (k) = 3.

Therefore, the required function is given by;

y = kx

y = 3x

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