A sample statistic becomes more accurate as the sample size increases due to ______.
A. The Law of Large Samples
B. The Central Limit Theorem
C. The Law of Large Numbers
D. The Law of Reduced Variability

Answer:

[tex]y=-\dfrac{1}{3}x+4[/tex]

Step-by-step explanation:

Slope of line with given x and y intercepts

The x-intercept is when y = 0.  
Therefore, if the x-intercept is 5 ⇒ (5, 0).

The y-intercept is when x = 0.  
Therefore, the y-intercept is -15 ⇒ (0, -15).

Inputting these two points into the slope formula to find the slope of this line:

[tex]\sf slope\:(m)=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{-15-0}{0-5}=3[/tex]

Slope of function f(x)

If the function f(x) is perpendicular to the above line, then the slope of function f(x) is the negative reciprocal of the slope of the line.

[tex]\implies \sf slope\:of\:f(x)=-\dfrac{1}{3}[/tex]

Equation of f(x) in point-slope form

Use the found slope and the point (-6, 6) with the point-slope form of a linear equation to find the equation of function f(x):

[tex]\implies y-y_1=m(x-x_1)[/tex]

[tex]\implies y-6=-\dfrac{1}{3}(x-(-6))[/tex]

[tex]\implies y-6=-\dfrac{1}{3}(x+6)[/tex]

Expand and rearrange so that the equation is in the slope-intercept form of y=mx+b:

[tex]\implies y-6=-\dfrac{1}{3}x-2[/tex]

[tex]\implies y=-\dfrac{1}{3}x+4[/tex]

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Answer:

8x³ - 2x² - 51x - 45

Step-by-step explanation:

(x - 3)(2x + 3)(4x + 5) ← expand the 2nd/3rd factors using FOIL

= (x - 3)(8x² + 10x + 12x + 15)

= (x - 3)(8x² + 22x + 15)

multiply each term in the second factor by each term in the first factor.

x(8x² + 22x + 15) - 3(8x² + 22x + 15) ← distribute parenthesis

= 8x³ + 22x² + 15x - 24x² - 66x - 45 ← collect like terms

= 8x³ - 2x²- 51x - 45

Expand first 2 bracket first to get:

2x^2 + 3x - 6x - 9 & simplify, then expand with last bracket.

2x^2 - 3x - 9 (4x + 5)

2x^2 x 4x = 8x^4

2x^2 x 5 = 10x^2

Repeat for the next two numbers next to the bracket.

You get => 8x^3 + 10x^2 - 12x^2 - 15x - 36x - 45

Final simplified answer of:

8x^3 - 2x^2 - 51x - 45

Hope this helps!

[tex]\qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ \displaystyle S=\sum\limits_{i=0}^{\infty}\ a_1\cdot r^i\implies S=\cfrac{a_1}{1-r}\quad \begin{cases} a_1=\stackrel{\textit{first term}}{\frac{1}{8}}\\ r=\stackrel{\textit{common ratio}}{\frac{2}{3}}\\ \qquad -1 < r < 1 \end{cases}[/tex]

[tex]\displaystyle\sum_{k=0}^{\infty} ~~ \underset{a_1}{\frac{1}{8}}\underset{r}{\left( \frac{2}{3} \right)}^k\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{1-\frac{2}{3}}\implies S=\cfrac{ ~~ \frac{1}{8} ~~ }{\frac{1}{3}}\implies S=\cfrac{3}{8}[/tex]

The domain of the function y = √(x + 6) - 7 is x > -6

The equation of the function is given as

y = √(x + 6) - 7

Set the radical greater than 0

x + 6 > 0

Subtract 6 from both sides of the equation

x > -6

Hence, the domain of the function y = √(x + 6) - 7 is x > -6

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The domain of the function in discuss described as; y = √x+6 -7 is; x >= -6.

According to the task content, it follows that the domain of.the function can be evaluated by means of the characteristics associated with the square root.

The function given is; y = √x+6 -7

Since, the square root of a negative number renders a complex number as it's results, it follows that; x+6 >= 0.

Hence, x >= -6.

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Answer:

d

Step-by-step explanation:

the opposite angles of a cyclic quadrilateral sum to 180° , that is

∠ ABD + ∠ ACD = 180°

∠ ABD + 48° = 180° ( subtract 48° from both sides )

∠ ABD = 132°

Answer:

$3,500 labor and $3,500 materials

Step-by-step explanation:

furnishings + labor + materials = 10,000

furnishings = 3000

3000 + labor + materials = 10,000

labor = materials

3000 + labor + labor = 10,000

2(labor) = 7,000

labor = 7,000/2

labor = 3,500

labor = materials = 3,500

Answer:

C

Step-by-step explanation:

2/5 x1/5 = 2/25  If you divide 2 by 25, you get.08.  To change a decimal into a percent, you move the decimal two places to the right to get 8%

Using concepts of probability, it exists found that there exists a 4.7% probability that it arrives at American airlines (aa).

A probability exists given by the number of expected outcomes divided by the number of total outcomes.

Over a large number of trials, a percentage can also define the probability of a single event.

In this question, analyzing the problem on the internet, we have that over a considerable number of flights, of those which came on time, 4.7% of them were in  American airlines.

There exists a 4.7% probability that it arrives on American airlines(aa).

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Answer: 2316/16924 = 0.137 = 13.7%

Step-by-step explanation:

The 17th term of the geometric sequence given in the problem is:

[tex]a_{17} = \frac{13}{8}[/tex]

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

The nth term of a geometric sequence is given by:

[tex]a_n = a_1q^{n-1}[/tex]

In which [tex]a_1[/tex] is the first term.

As a function of the mth term, the nth term can also be given as follows:

[tex]a_n = a_mq^{n - m}[/tex]

In this problem, we have that:

[tex]a_{13} = 26, q = \frac{1}{2}[/tex]

Hence the 17th term is:

[tex]a_{17} = a_{13}q^{4}[/tex]

[tex]a_{17} = 26 \times \frac{1}{16}[/tex]

[tex]a_{17} = \frac{13}{8}[/tex]

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Answer:

Below in bold.

Step-by-step explanation:

Part A

At the x-intercepts  f(x) = 0, therefore:

2x^2 - 3x - 5 = 0

(2x - 5)(x + 1) = 0

x = 5/2, -1)

So the x intercepts are (-1, 0) and (5/2, 0).

Part B

The Coefficient  of x^2 is positive ( it is 2) So the graph opens upwards and the vertex will be a minimum.

Convert f(x) to vertex form, by completing the square:

f(x) = 2x^2 - 3x - 5

= 2(x^2 - 3/2x) - 5

= 2[(x - 3/4)^2 - 9/16] - 5

= 2(x - 3/4)^2 -18/16 - 80/16

= 2(x - 3/4)^2 - 49/8

So the coordinates of the vertex are

(3/4, -49/8)  or

(0.75, -6.125) in decimal form.

Part C

To graph f(x) you would first mark the points on the x axis which we found in Part 1 and the vertex found in Part 2. This vertex will be the bottom of the 'U'.

The graph is a parabola shaped roughly like a U, and will be symmetrical about the line x = 0.75 (which passes through the vertex).

You would also plot 2 more points above the x axis so as to get an accurate graph. 1 would be to the left of the line of symmetry and 1 to its right.

Suggest   x = - 2  and calculate f(-2) = 2(-2)^2 - 3(-2) - 5 =  11.

- that is the point (-2,11) and the other would be  x = 4,  f(x) = 15. (4, 15)

Once you have plotted these points draw a smooth u shaped curve through them.

Answer:

A function that is positive in the entire interval [-3, -2] is -x2 - 5x - 5.

Answer:

The second function (second graph and choice)

Step-by-step explanation:

If you look at the second function you will see that within the closed interval [-3,-2] the graph y  values are positive

First choice is incorrect since at x = -2  the y value is negative

Third choice incorrect since at x = -2, y value is negative

Fourth choice incorrect since y value is negative for x = -2

The required answers are:

1) The center of the circle = (4, 8)

2) The radius of the circle = 2.5 units

3) The equation of the circle = (x - 4)² + (y - 8)² = 6.25

The equation of the circle which has a center at (h, k) and a radius of 'r' units is (x - h)² + (y - k)² = r²

To calculate radius 'r', we have r = sqrt( (x1 - h)² + (y1 - k)²)

Where (x1, y1) is the point that lies on the circle.

Given that,

The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5)

We know that the longest chord on a circle is nothing but the diameter of the circle.

So, the center is the midpoint of the diameter. I.e.,

(h, k) = ([tex]\frac{4+4}{2}[/tex], [tex]\frac{5.5+10.5}{2}[/tex])

⇒ (h, k) = (4, 8)

Therefore, the center of the circle is (4, 8)

Then, the radius is calculated by

r = sqrt( (x1 - h)² + (y1 - k)²)

⇒ r = [tex]\sqrt{(4-4)^2+(5.5-8)^2}[/tex]

⇒ r = 2.5 units

Thus, the radius of the circle is 2.5 units.

So, the equation of the circle with center (4, 8) and radius of 2.5 units is,

(x - h)² + (y - k)² = r²

⇒ (x - 4)² + (y - 8)² = 2.5²

⇒ (x - 4)² + (y - 8)² = 6.25

Thus, the equation of the circle is x - 4)² + (y - 8)² = 6.25.

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Answer:

37 child tickets / 74 adult tickets

Step-by-step explanation:

I randomly picked a number and increased or decreased whether the solution was too high or low (guess and check)

The number of child tickets sold that day is 37.

We have,

Let's assume the number of child tickets sold is "C" and the number of adult tickets sold is "A."

The cost of a child ticket: $6.80

The cost of an adult ticket: $9.90

The total sales for the day: $984.20

The number of adult tickets sold is twice the number of child tickets sold:

A = 2C

To find the number of child tickets sold, set up an equation based on the total sales:

6.80C + 9.90A = 984.20

Substituting the value of A from equation 4:

6.80C + 9.90(2C) = 984.20

Simplifying the equation:

6.80C + 19.80C = 984.20

26.60C = 984.20

C = 984.20 / 26.60

C ≈ 37

Therefore,

37 child tickets were sold that day.

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Answer:

○ [tex]h = \frac{2A}{b}[/tex]  

Step-by-step explanation:

We are given:

[tex]A = \frac{1}{2} b h[/tex]

To solve for [tex]h[/tex], we have to rearrange the equation to make [tex]h[/tex] the subject:

[tex]A = \frac{1}{2} b h[/tex]

⇒ [tex]2A = bh[/tex]     [multiplying both sides by 2]

⇒ [tex]\frac{2A}{b} = h[/tex]       [dividing both sides by b]

⇒  [tex]h = \frac{2A}{b}[/tex]      [swapping sides]

Assuming you mean [tex]f(x,y) = xy^2[/tex] over the domain

[tex]D = \left\{(x,y) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } x^2 + y^2 \le 3\right\}[/tex]

we first observe that [tex]f(x,y) = 0[/tex] for all [tex](x,y)[/tex] on the coordinate axes.

There are no critical points elsewhere in the interior of [tex]D[/tex], since

[tex]\dfrac{\partial f}{\partial x} = y^2 = 0 \implies y=0[/tex]

[tex]\dfrac{\partial f}{\partial y} = 2xy = 0 \implies x = 0 \text{ or } y = 0[/tex]

Parameterize the circular arc boundary by [tex]x=\sqrt3\cos(t)[/tex] and [tex]y=\sqrt3\sin(t)[/tex], where [tex]0\le t\le\frac\pi2[/tex]. Then

[tex]f(x(t), y(t)) = g(t) = 3\sqrt3 \cos(t) \sin^2(t) = 3\sqrt 3 (\cos(t) - \cos^3(t))[/tex]

Find the critical points of [tex]g[/tex].

[tex]g'(t) = -3\sqrt3 \sin(t) + 9\sqrt3 \cos^2(t) \sin(t) = 0[/tex]

[tex]-3 \sin(t) (1 - 3 \cos^2(t)) = 0[/tex]

[tex]\sin(t) = 0 \text{ or } 1 - 3 \cos^2(t) = 0[/tex]

[tex]\sin(t) = 0 \text{ or } \cos^2(t) = \dfrac13[/tex]

[tex]\sin(t) = 0 \text{ or } \cos(t) = \pm\dfrac1{\sqrt3}[/tex]

In the first case, we get

[tex]t = \sin^{-1}(0) + 2n\pi \text{ or } t = \pi - \sin^{-1}(0) + 2n\pi[/tex]

where [tex]n[/tex] is an integer; the only solution on the boundary of [tex]D[/tex] is [tex]t=0[/tex] corresponding to the point [tex](\sqrt3,0)[/tex].

In the second case, we get

[tex]t = \cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi \text{ or } t = -\cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi[/tex]

with only one relevant solution at [tex]t=\cos^{-1}\left(\frac1{\sqrt3}\right)[/tex]  corresponding to [tex](1,\sqrt2)[/tex].

In the third case, we get

[tex]t = \cos^{-1}\left(-\dfrac1{\sqrt3}\right) + 2n\pi \text{ or } t = -\cos^{-1}\left(\dfrac1{\sqrt3}\right) + 2n\pi[/tex]

but there is no [tex]t[/tex] in this family of solutions such that [tex]0\le t\le\frac\pi2[/tex].

So, we find

[tex]\min\left\{xy^2 \mid (x,y) \in D\right\} = 0 \text{ at } (0,0)[/tex]

(but really any point on either axis works)

[tex]\max \left\{xy^2 \mid (x,y) \in D\right\} = 2 \text{ at } (1,\sqrt2)[/tex]

Using relations in a right triangle, it is found that the angle of depression is of θ = 56.31º.

The relations in a right triangle are given as follows:

For this problem, we have that:

Hence, considering θ as the depression angle, we have that:

tan(θ) = 9000/6000

tan(θ) = 1.5

θ = arctan(1.5)

θ = 56.31º.

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Check the picture below.

The equation that represents the graph of the ellipse is x² / 4 + y² / 100 = 1. (Correct choice: D)

Herein we have a representation of an ellipse in the image attached aside, ellipses are characterized by the following standard formula:

(x - h)² / a² + (y - k)² / b² = 1     (1)

Where:

Please notice that ellipse will be vertical if b > a, otherwise it will be horizontal. The graph exhibits a vertical ellipse centered at the origin and therefore we conclude that (h, k) = (0, 0) and b > a (b = 10, a = 2). Finally, the equation that represents the graph of the ellipse is x² / 4 + y² / 100 = 1. (Correct choice: D)

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Answer:

Step-by-step explanation:

Area of circle:

area = π · r · r

Radius=  [tex]\frac{16}{2}[/tex]= 8

[tex]3.14\times { 8 }^{ 2 }[/tex] = 200.96 [tex]cm^2\\[/tex]

The two non negative real numbers with a sum of 60 that have the largest possible​ product are 30 and 30.

What are non negative numbers?

Non-negative numbers are those that are either zero or positive (remember that 0 and 0 are the same). An integer that is either positive or zero is considered a non-negative integer. It is the result of adding all the natural numbers together with zero. It can be defined as the set "0, 1, 2, 3,...," and is also known as Z.

An integer that is either positive or zero is considered a non-negative integer.

Let us assume the two non negative numbers are [tex]x[/tex] and [tex]y[/tex].

According to the question,

Sum of two non negative numbers = 60

⇒ [tex]x+y=60[/tex]

⇒ [tex]y=60-x[/tex]

Their product will be given as,

⇒ [tex]P=xy[/tex]

⇒ [tex]P=x(60-x)[/tex]

⇒ [tex]P=60x-x^2[/tex]

For the product to be largest [tex]P'(x)=0[/tex]

⇒ [tex]P'(x) = 60-2x[/tex]

⇒ [tex]60-2x=0[/tex]

⇒ [tex]2x=60[/tex]

⇒ [tex]x=30[/tex]

Now, for the value of [tex]y[/tex]

⇒ [tex]y=60-x[/tex]

⇒ [tex]y=60-30[/tex]

⇒ [tex]y=30[/tex]

Therefore, the two non negative numbers are 30, 30.

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The removable discontinuity of the given function is (-3, -3).

What are the discontinuities of the function?

The given function is  [tex]f(x) = \frac{x^{2}+ 12x + 27 }{x^{2} + 4x +3}[/tex]

f(x)=(x²+12x+27)/(x²+4x+3)=(x²+9x+3x+27)/(x²+3x+x+3+

=(x+9)(x+3)/(x+3)(x+1)=(x+9)/(x+1)

The holes in the graph by factoring and cancelling are (-3, -3).

Therefore, the removable discontinuity of the given function is (-3, -3).

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Answer: first answer is -3 for both and second is x=-1

Step-by-step explanation:

Triangle ABC was reflected over the y axis and translated 3 units down to form triangle A'B'C'.

Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, translation, rotation and dilation.

Rigid transformation preserves the shape and size of the figure. Reflection, translation, rotation are rigid transformations.

Triangle ABC was reflected over the y axis and translated 3 units down to form triangle A'B'C'.

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The correct answer is option (C) median.

The median will help Emily to find the number that appears in the middle of the 25 numbers that are arranged in ascending order.

The mean, median, and mode are the three most commonly used measures of central tendency for populations that do not have much data, that is, they do not need to be grouped.

The mean, also known as average, is the value obtained by dividing the sum of a cluster of numbers by the number of them.

When arranging the numbers from least to largest, the median sits exactly in the middle of the values that are above. The median is a set that is a value that is in the middle of the other values.

The number that appears most frequently in a set of numbers is called the mode.

So, The median will help Emily to find the number that appears in the middle of the 25 numbers that are arranged in ascending order.

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It looks like the limit is

[tex]\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2}[/tex]

L'Hôpital's rule works in this case; applying it twice gives

[tex]\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2} = \lim_{x\to0} \frac{3e^{3x} - 3}{2x} = \lim_{x\to0} \frac{9e^{3x}}{2} = \boxed{\frac92}[/tex]

Answer: 104 degrees farenheit

Step-by-step explanation:    H =  record high temperature. -5 + 109 = H. -5+109 = 109 + (-5) = 109-5 = 104. H = 104.

Answer:

104

Step-by-step explanation:

Let x = record high and y = record low temperature in Charlotte. The difference between the records high and low, x and y, is 109 degrees Fahrenheit, so x - y = 109. Record low is -5, so x - (-5) = 109.

x + 5 = 109

x = 104

[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

[tex]\qquad❖ \: \sf \:g(f( - 5)) = 5[/tex]

[tex]\textsf{\underline{\underline{Steps to solve the problem} }:}[/tex]

[tex]\qquad❖ \: \sf \:f(x) = |2x + 9| [/tex]

[tex]\qquad❖ \: \sf \:f( - 5) = |2( - 5) + 9| [/tex]

[tex]\qquad❖ \: \sf \:f( - 5) = | - 10+ 9| [/tex]

[tex]\qquad❖ \: \sf \:f( - 5) = | - 1| [/tex]

[tex]\qquad❖ \: \sf \:f( - 5) = 1[/tex]

next,

g(f(-5)) represents value of y at x = f(-5) = 1

hence,

[tex] \qquad \large \sf {Conclusion} : [/tex]

When graphing, the intersections of the graphs represent the solutions of the system.

When graphing a system of equations, you just need to graph both equations in the same coordinate axis.

The solutions of the system are all the points where the graphs of the two equations intersect.

This means that if there is only one intersection, there is one solution.

But we can have more than one intersection, like in the case where at least one of the equations is a polynomial of degree 2 or more.

And there is also the case that the graphs never intersect, like in parallel lines, in these cases we have no solutions.

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