A mathematician is analyzing an email message. Out of the first 100 characters, the letter e appears 13 times. The email message is 1,100 characters in length. Based on the information, how many of the characters should be a letter e?

If Ling plans to collect data and plot them in a scatterplot to look for a relationship, the relationship is positive.

In basketball, players who spend more time on the court are likely to have more opportunities to score points. Therefore, as the playing time increases, the number of points scored by a player may also increase. This would result in a positive relationship between playing time and points scored in a basketball game.

A positive relationship in a scatterplot is characterized by a general upward trend in the data points. As one variable increases, the other variable also tends to increase.

In this case, Ling's scatterplot would likely show data points that are positively correlated, meaning that as the playing time increases, the number of points scored by a player would also tend to increase.

It is also possible for there to be no relationship or a negative relationship between playing time and points scored, but given the nature of basketball, a positive relationship is the most likely outcome.

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According to the question the exponential function that goes through the two points is: [tex]y = 4(9^x).[/tex]

Given,

We must first determine the values of a and b before we can build an exponential function with the form y = abx that passes through the points (0,4) and (2,324).

Using the point (0,4), we have:

4 = ab^0

4 = a(1)

a = 4

Using the point (2,324), we have:

324 = 4b^2

b^2 = 81

b = 9 (since b must be positive)

So the exponential function that goes through the two points is:

y = 4(9^x)

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A. Yes, because the rate of change is a constant $70 per year. We can observe that the amount owed increases by $70 for every year that passes.

This is due to the simple interest rate of 7% per year, which means that the boy owes an additional 7% of the original amount ($1000) each year. Thus, the  the amount owed after one year is $1000 + 0.07($1000) = $1070, after two years it is $1000 + 2(0.07($1000)) = $1140, and so on. $70  is the change rate per year.

B. It will be no due to the change rate is $0 per year in 2nd and 3rd years.

This statement is false because, as we calculated in part A, the rate of change is constant at $70 per year. Therefore, the amount owed will increase by $70 even during the second and third years, as long as the boy has not made any payments towards the loan.

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Jacobi measured the diagonals of three TV screens as 5/84 inches, 46 inches and 46. 625 inches.

The lengths of the diagonals in ascending order are 5/84 inches, 46 inches, and 46.625 inches.

The diagonals of the three TV displays are 5/84, 46, and 46.625 inches, respectively.

To determine the ascending order of the diagonals, we first convert the fraction 5/84 to a decimal. We may accomplish this by dividing 5 by 84 with a calculator or long division, yielding:

0.05952381

We can now contrast the three diagonal lengths:

0.05952381 inches

46 inches

46.625 inches

Hence, in increasing order, the diagonals are:

0.05952381 inches

46 inches

46.625 inches

As a result, the diagonal lengths in ascending order are 0.05952381 inches, 46 inches, and 46.625 inches.

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Answer:

Train A= 249 tons

Train B= 87 tons

Step-by-step explanation:

Total weight of Train A and Train B= 336 tons

Weight of Train A > Weight of Train B

Difference of their weights= 162 tons

Let the weight of Train A be t.

Then weight of Train B= t-162

Total weight= 336 tons

         t+t-162= 336

                 2t= 336+162

                 2t= 498

                   t= [tex]\frac{498}{2}[/tex]

                   t= 249

t-162= 249-162

       = 87

∴ the weights of Train A and Train B are 249 tons and 87 tons respectively.

Answer:

Step-by-step explanation:

we can replace 5 with [tex]\sqrt{5}^{2}[/tex]

then.

[tex]\sqrt{5}^{n-1+2}=\sqrt{5}^{n+1}[/tex]

Option P(A) = 5/6 is correct using the probability concept for the scenario n(A')=6.

The scenario is you roll two standard 6-sided dice if n(A′) = 6.
Here, we need to use the concept of probability to solve the problem.

Probability is the branch of mathematics that deals with the measurement of the chance of occurrence of a particular event. It can be defined as the ratio of the number of favorable outcomes to the number of total possible outcomes.

Given that the two standard 6-sided dice are rolled. Now, we need to find the probability of rolling a 7.
Here, the sum of the numbers on the two dice must be 7. We can obtain this sum in the following ways:
(1,6), (2,5), (3,4), (4,3), (5,2), and (6,1)
Thus, the number of ways to roll a 7 is 6 which is n(A') = 6.
Therefore, P(A') = n(A')/n(S) = 6/36 = 1/6.

Now, we are given that n(A′) = 6, which means that the number of ways to roll a 7 is 6.
Thus, the number of ways to not roll a 7 is 36 - 6 = 30.
Therefore, P(A) = n(A)/n(S) = 30/36 = 5/6.

Hence, the correct option is P(A) = 5/6.

Your question is incomplete, but most probably your full question was

For the scenario where you roll two standard 6-sided dice, if n(A′)=6, then:

P(A)=6/36

n(A′)= number of ways to not roll a 7

P(A′)=30/36

n(A)= number of ways to roll a 7

P(A)=5/6​

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Answer:

Yardley is approximately 5.7 miles away from the airport.

Step-by-step explanation:

Let's use the following variables:

x = distance between Yardley and the airport

Using the Pythagorean theorem:

x^2 + 7^2 = 9^2

x^2 + 49 = 81

x^2 = 32

x = sqrt(32)

x ≈ 5.7

Therefore, Yardley is approximately 5.7 miles away from the airport.

Answer:

5.7 miles

Step-by-step explanation:

To find:-

Answer:-

We are given that airport and Wildgrove are 7miles apart and the distance between Yardley and Wildgrove is 9miles . We are interested in finding out tge distance between Yardley and airport.

As you can see from the figure attached, the distance can be calculated using Pythagoras theorem ,

[tex]\rm\implies a^2+b^2 = h^2 \\[/tex]

where ,

Here the longest side is 9 miles and one other side is 7 miles . On substituting the respective values, we have;

[tex]\rm\implies a^2 + (7)^2 = 9^2 \\[/tex]

[tex]\rm\implies a^2 + 49 = 81 \\[/tex]

[tex]\rm\implies a^2 = 81 - 49 \\[/tex]

[tex]\rm\implies a^2 = 32 \\[/tex]

[tex]\rm\implies a =\sqrt{32} \\[/tex]

[tex]\rm\implies a = 5.65 \\[/tex]

[tex]\rm\implies \red{ a = 5.7 } \\[/tex]

Hence the distance between Yardley and airport is 5.7 miles .

We can factor the polynomial as:-32a^2 - 82a + 31 = 32(a - \frac{1}{2})(a - \frac{31}{32})

.To make the polynomial 32a^2 - 82a + C factorable, we need to find an integer value for C such that the quadratic expression can be factored into two binomials.

To do this, we can use the formula for the discriminant: b^2 - 4ac. Since we want the discriminant to be a perfect square, we can set it equal to some integer k^2 and solve for C:

82^2 - 4(32)(C) = k^2

6724 - 128C = k^2

We can try different values of k and solve for C until we find an integer solution. For example, if we let k = 10, we get:

10^2 = 6724 - 128C

C = 31

To verify that this value of C works, we can factor the trinomial using the X formula:

a = 32, b = -82, c = 31

X = (-b ± sqrt(b^2 - 4ac)) / 2a

X = (82 ± sqrt(82^2 - 4(32)(31))) / 2(32)

X = (82 ± 8) / 64

So, the roots of the quadratic are:

a = 32, X1 = 1/2, X2 = 31/32

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The amount of aspirin in the bloodstream will be about 0.838 times what it was before.

The function that represents the amount of aspirin, A, present in the bloodstream x hours after taking a 325 mg pill is:

[tex]A = A_0 \times b^x[/tex]

where A_0 is the initial amount of aspirin in the bloodstream (which is 325 mg), and b is a constant that represents the rate of decay of the aspirin in the bloodstream.

We are told that after 3 hours, there is about 199.6 mg of aspirin remaining in the patient's bloodstream. Using the function above, we can set up an equation based on this information:

[tex]199.6 = 325 \times b^3[/tex]

To solve for b, we can divide both sides by 325 and then take the cube root of both sides:

[tex]b^3[/tex]= 199.6 / 325

b = [tex](199.6 / 325)^{(1/3)[/tex]

b ≈ 0.838

So the value of b is approximately 0.838. In the context of the problem, this means that the amount of aspirin in the bloodstream decreases by a factor of approximately 0.838 every hour. In other words, if we wait one hour,  If we wait two hours, it will be about (0.838)^2 times what it was before, and so on. This exponential decay model can be useful in predicting how long it will take for the aspirin to be completely eliminated from the bloodstream.

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Check the picture below.

let's notice the polygon has 6 sides, so

[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ S = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ n=6 \end{cases}\implies S=180(6-2)\implies S=720 \\\\[-0.35em] ~\dotfill\\\\ 2b+110+116+112+110~~ = ~~720 \\\\\\ 2b+448=720\implies 2b=272\implies b=\cfrac{272}{2}\implies b=136[/tex]

The value of x is 25 using Pythagoras theorem.

A pοint where twο lines meet prοduces an angle.

The breadth οf the "οpening" between these twο rays is referred tο as a "angle". It is depicted by the figure.

Radians, a unit οf circularity οr rοtatiοn, and degrees are twο cοmmοn units used tο describe angles.

By cοnnecting twο rays at their ends, οne can make an angle in geοmetry. The sides οr limbs οf the angle are what are meant by these rays.

The limbs and the vertex are the twο main parts οf an angle.

The cοmmοn terminal οf the twο beams is the shared vertex.

Pythagοras theοrem,

perpendicular²= hypοtenuse² + base²

Find the hypotenuse of lower triangle

c² = 15² + 9²

c = [tex]\sqrt{15^2 + 8^2}[/tex]

c = 17

Now, c is the base altitude of upper triangle

So

x² = [tex]\sqrt{20^2 + 15^2}[/tex]

x = 25

Thus, the value of x is 25 using Pythagoras theorem.

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Answer:

Median = 42
LQ = 33
UQ = 52

Step-by-step explanation:

median is given by the term that divides the groups in two equally quantities. In this case is (n+1)/2 = (13+1)/2 = 14/2 = 7

the 7th term is: 42
(notice this means 6 values are below 42 and 6 values are above 42, the definition of a median)

the first (lower) quartile is given then by the (n+1)/4 value

(13+1)/4=3.5, this is the half between 3th and 4th terms.

since the term is the same (3th value is 33 and 4th value is 33)

LQ=33
(25% of the values are below 33)

for the upper quartile the value represents the top 75%, this is given by

3(13+1)/4 = 10.5

this is the half between 10th and 11th terms

(51+53)/2=52

(25% of the values are above 52)

A sample of customers from barnsboro national bank shows an average account balance of 315 with a standard deviation of 87. a sample of customers from wellington savings and loan shows an average account balance of 8350 with a standard deviation of 1800. Option A: Barnsboro Bank has more variation.So, the correct option is A) Barnsboro Bank has more variation.

The given data is:

A sample of customers from Barnsboro National Bank shows an average account balance of 315 with a standard deviation of 87.

A sample of customers from Wellington Savings and Loan shows an average account balance of 8350 with a standard deviation of 1800.

To compare the variation in the account balances of both banks, we use the coefficient of variation (CV).

CV = Standard deviation / Mean

CV of Barnsboro National Bank = 87/315 ≈ 0.2762

CV of Wellington Savings and Loan = 1800/8350 ≈ 0.2150

The CV of Barnsboro National Bank is greater than the CV of Wellington Savings and Loan.

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Throughout his innings, Kohli scored 432 runs in total.. Rohit scored 422 runs and Rahul scored 427 runs. In total, they scored 442 runs.

Kohli, Rohit and Rahul are three of the greatest batsmen in Indian cricket. In a match, they scored 442 runs in total.Rohit scored the most runs (422 total),. Kohli scored 10 runs less than Rohit, making his total score 432 runs. Rahul was the third highest scorer, with 5 runs more than Rohit, making his total score 427 runs. This shows the great batting prowess of the three batsmen and how they were able to score a combined 442 runs between them. This goes to show the great talent and skill that the three players possess and how they are able to make such a great contribution to their team's total score.

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Answer: We are given that GH = 4x - 3 and IJ = 3x + 14.

To find x, we can set GH equal to IJ and solve for x:

4x - 3 = 3x + 14

x = 17

Therefore, x = 17.

To find the length of GH, we can substitute x = 17 into the expression for GH:

GH = 4x - 3

GH = 4(17) - 3

GH = 68 - 3

GH = 65

Therefore, the length of GH is 65.

Step-by-step explanation:

Answer: o find the maximum value of the objective function F=5x+2y and the values of x and y for which it occurs, we need to graph the constraints and determine the feasible region. Then, we can evaluate the objective function at the vertices of the feasible region to find the maximum value.

First, we will graph the constraints x+2y<6 and 2x+y<6. To do this, we can rewrite the inequalities as equations and graph the corresponding lines:

x+2y=6 (solid line)

2x+y=6 (dashed line)

Next, we need to determine which side of each line satisfies the inequality. To do this, we can choose a point on one side of each line and substitute its coordinates into the inequality. If the inequality is true, then that side of the line satisfies the inequality. If not, then the other side satisfies the inequality.

For example, let's choose the point (0,0) as a test point for the inequality x+2y<6:

0+2(0)<6

This is true, so the side of the line on which (0,0) lies satisfies the inequality. Similarly, we can choose the point (0,0) as a test point for the inequality 2x+y<6:

2(0)+0<6

This is also true, so the side of the line on which (0,0) lies satisfies the inequality.

Now, we can shade in the feasible region, which is the region of the graph that satisfies all of the constraints. It is the region bounded by the two lines and the axes, and it is shown in the figure below:

lua

   |              .

   |            .

   |          .

 3 |        .

   |      .

   |    .

 2 |  .

   |.

   ----------------

     0  1  2  3  4  5  6

The vertices of the feasible region are (0,3), (2,2), and (3,0). To find the maximum value of the objective function, we evaluate it at each vertex:

F(0,3) = 5(0) + 2(3) = 6

F(2,2) = 5(2) + 2(2) = 14

F(3,0) = 5(3) + 2(0) = 15

Therefore, the maximum value of the objective function is 15, and it occurs when x=3 and y=0.

enjoy (:

The equation of the line shown is y = 1x + 9.

An equation is a statement of equality between two expressions. It is typically expressed in mathematical notation, with the left side of the equation equal to the right side. Equations are used to describe how two values are related, and can be used to solve problems or predict outcomes. Equations can be used in any field of study, from physics to economics.

The equation of the line shown can be found by using the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is m = (10 - 9)/(10 - 0) = 1.

The y-intercept is b = 9, as this is the point at which the line crosses the y-axis. Therefore, the equation of the line shown is y = 1x + 9.

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Complete questions as follows-
this is happening do proper format and calculation
Find the equation of the line shown.
y
10
S87654321
9
0 1 2 3 4 5 6 7 8 9 10
X

The distance AC is approximately 6.47 km, the bearing of A from C is approximately 313.8°, and the area of triangle ABC is approximately 14.8 km².

To solve this problem, we can use the law of cosines to find the length of AC and the law of sines to find the angle at A.

First, we can use the given information to draw a diagram and label the angles and sides:

       C

      / \

     /   \

 5 /     \ 7

   /       \

  /         \

A /___θ_____\ B

From the given information, we know that:

AB = 7 km

BC = 5 km

∠ABC = 100° (since ∠ABD = 180° - 30° - 70° = 80° and ∠DBC = 180° - 280° = 100°)

Using the law of cosines, we can find the length of AC:

AC^2 = AB^2 + BC^2 - 2ABBCcos(∠ABC)

AC^2 = 7^2 + 5^2 - 2(7)(5)cos(100°)

AC^2 ≈ 41.84

AC ≈ 6.47 km

Next, we can use the law of sines to find the angle at A:

sin(∠CAB)/AC = sin(∠ABC)/AB

sin(∠CAB)/6.47 = sin(100°)/7

sin(∠CAB) ≈ 0.276

∠CAB ≈ 16.2°

To find the bearing of A from C, we can use the fact that the bearing is the angle measured clockwise from north. Since ∠CAB is acute and the bearing of B from A is 30° east of north, the bearing of A from C is:

360° - (30° + ∠CAB) ≈ 313.8°

Finally, to find the area of triangle ABC, we can use the formula:

Area = (1/2)ABBCsin(∠ABC)

Substituting the given values, we get:

Area = (1/2)(7)(5)sin(100°)

Area ≈ 14.8 km²

Therefore, the distance AC is approximately 6.47 km, the bearing of A from C is approximately 313.8°, and the area of triangle ABC is approximately 14.8 km².

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The answer is 4A+2B+5. This can be achieved by combining the like terms in the expression 4a-1+2B+6. First we add 4 and -1 to get 3 for the A-coefficient, then we add 2 and 6 to get 8 for the B-coefficient, and finally we combine the coefficients to get 4A+2B+5.

To combine the like terms in the expression 4a-1+2B+6, we need to simplify it. We can do this by combining the A-coefficients and the B-coefficients.

The A-coefficient is 4, so we add 4 and -1 to get 3. This means the expression is now 4A+2B+6.

The B-coefficient is 2, so we add 2 and 6 to get 8. This means the expression is now 4A+2B+8.

Finally, we combine the A-coefficient and the B-coefficient to get 4A+2B+5.

Therefore, the answer is 4A+2B+5.

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By simplification the quadratic function [tex]y = x^2 + x - 8[/tex] in vertex form is[tex]y = (x + 1/2)^2 - 33/4[/tex], and its vertex is (-1/2, -33/4).

What is the vertex form of a quadratic equation?

The standard form of a quadratic function is:

[tex]y = ax^2 + bx + c[/tex]

where a, b, and c are constants and x is the variable.

The vertex form of a quadratic function is:

[tex]y = a(x - h)^2 + k[/tex]

where a, h, and k are constants and (h, k) is the vertex of the parabola.

To write the quadratic function [tex]y = x^2 + x - 8[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

[tex]y = a(x - h)^2 + k[/tex]

where (h, k) is the vertex of the parabola.

So, first we need to factor out the coefficient of the x^2 term:

[tex]y = 1(x^2 + x) - 8[/tex]

Now, we need to complete the square for the expression inside the parentheses:

[tex]y = 1(x^2 + x + 1/4 - 1/4) - 8[/tex]

[tex]y = 1[(x + 1/2)^2 - 1/4] - 8[/tex]

Finally, we can simplify and write the function in vertex form:

[tex]y = (x + 1/2)^2 - 33/4[/tex]

Therefore, the quadratic function [tex]y = x^2 + x - 8[/tex] in vertex form is[tex]y = (x + 1/2)^2 - 33/4[/tex], and its vertex is (-1/2, -33/4).

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The number of hours she can afford to rent the paddle boat is  x ≤ 6.8. Option D

we have the equation in the inequality as

1.25x + 6.50 ≤ 15

we would have to solve for x

hence we would take the like terms first

1.25x ≤ 15 - 6.50

then we would have

1.25x ≤ 8.5

Next we have to divide through by 1.25

1.25x / 1.25 ≤ 8.5 / 1.25

x ≤ 6.8

Hence the solution to the inequality is x ≤ 6.8. Elise can afford to rent the paddle boat for x ≤ 6.8

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Elise has $15 to spend on a paddle boat ride she uses this inequality to determine x the number of hours she can afford to rent the paddle boat

1.25x + 6.50 ≤ 15

What is the solution to the inequality?

x ≥ 17.2

x ≥ 6.8

x ≤ 17.2

x ≤ 6.8

​

Answer:

x ≤ 6.8

Step-by-step explanation:

Answer:

The formula for the area of a circle is A = πr^2, where r is the radius of the circle.

We know that the diameter of the circle is 10 ft, so the radius is half of that:

r = 10 ft ÷ 2 = 5 ft

Now we can plug in this value for r into the formula:

A = π(5 ft)^2

A = π(25 ft^2)

A ≈ 78.54 ft^2

Rounding this to the nearest whole number, we get:

A ≈ 79 ft^2

Therefore, the area of the circle to the nearest whole number is 79 square feet.

Answer:

79ft^2

Step-by-step explanation:

diameter=10ft

radius=5ft

5^2=25ft

25×π=25π

25π= 78.539....

to nearest whole number =79ft^2

The measured angle at position 1 is 63 degree. The angle which is  between the two tangents drawn from a outer side point to a circle is supplementary to  angle subtended

An angle is a geometric figure formed by two rays, or line segments that share a common endpoint, called the vertex of the angle. Angles are measured in degrees, radians, or other units, and are typically denoted using the symbol  θ

In geometry, the tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency. The tangent line is perpendicular to the radius of the circle that passes through the point of tangency.

In the given question ,

The angle which is  between the two tangents drawn from a outer side point to a circle is supplementary to  angle subtended by the line segment joining the points of contact at the center.

From the above theorem  we can write

angle at position 1 x= 360 - ((360-243)+90+90)

we get x=63

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Answer:

41

Step-by-step explanation:

a^2+b^2=c^2

So...

40^2+9^2=1681

√1681=41

Option (c) is true i.e. y is not a function of x, since the x-values do not correspond to exactly (precisely) one y-value.

A function in mathematics from a set X to a set Y allocates exactly (precisely) one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

The items that make up a set are referred to as its elements (or members), and the set is said to contain the elements when they are claimed to belong to it.

The mapping shows a relationship between x and y in this mapping is y is not a function of x, since the x-values do not correspond to exactly one y-value.

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= 5x = 5(12) = 60 sandwiches, the number of ham sandwiches. Tina makes 60 ham sandwiches.

First, we need to find the total number of each type of sandwich that Tina makes:

3/8 of the sandwiches are tuna, so the number of tuna sandwiches

= 3/8 x 480

= 180 sandwiches

35% of the sandwiches are beef, so the number of beef sandwiches

= 35% x 480

= 0.35 x 480

= 168 sandwiches

The remaining sandwiches must be ham and cheese sandwiches. Let's call the number of ham sandwiches "h" and the number of cheese sandwiches "c". We know that h:c = 5:6, which means that h = 5x and c = 6x for some common factor x. The total number of sandwiches is 480, so:

h + c = 480 - 180 - 168

h + c = 132

Substituting h = 5x and c = 6x, we get:

5x + 6x = 132                              

11x = 132

x = 12

Therefore, the number of ham sandwiches = 5x = 5(12) = 60 sandwiches. Tina makes 60 ham sandwiches.

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The correct option is b. Sample_size for (0.01) = 4*Sample_size for (0.04).

When answering questions on Brainly, you should always be factually accurate, professional, and friendly. Additionally, you should be concise, provide a step-by-step explanation, and use the following terms in your answer: "estimate," "sample," and "confidence."For the given problem, we need to determine the sample size of the retailer's customers to estimate the proportion of Hispanic customers in a particular state with 0.04 accuracy and 99% confidence since there is no planning value for p*.

Now, we can use the formula for sample size to estimate the number of customers needed for the retailer, which is given by:n= z^2 p q/E^2 Where, z = the critical value from the standard normal distribution at the given confidence level p = the estimate of the proportion q = 1 - p E = the maximum likely difference between the sample and population proportion p = Proportion of Hispanic customers Let's use the values of z and E for the given problem, which isz=2.58 (at 99% confidence level, two-tailed)E=0.04 We need to find the sample size for the retailer's customers, so we can substitute the given values in the sample size formula and solve forn as shown:n = z^2pq/E^2 = (2.58)^2(0.5)(0.5)/(0.04)^2= 664.53 ≈ 665 So, the sample size of 665 customers would be needed to estimate the proportion of Hispanic customers in a particular state with 0.04 accuracy and 99% confidence.Now, we are asked to find the true statement if we decide to go for 0.01 accuracy instead of 0.04 accuracy.

The formula to calculate sample size for 0.01 accuracy can be written asn = (z^2pq)/(0.01)^2 Now, we need to find the relationship between sample size for 0.01 accuracy and 0.04 accuracy, which is given as follows :n1/n2 = (z1/z2)^2 (E1/E2)^2= (2.58/2.33)^2 (0.04/0.01)^2≈ 3.41 × 16n1/n2 = 54.56 Since sample size for 0.01 accuracy is larger than the sample size for 0.04 accuracy, we can say that option b. Sample_size for (0.01) = 4*Sample_size for (0.04) would be true if we decide to go for 0.01 accuracy instead of 0.04 accuracy. Hence, the correct option is b. Sample_size for (0.01) = 4*Sample_size for (0.04).

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Answer: D. Population: all the first-year students at the university

Sample: a group of 185 incoming first-year students

Conclusion: The reason for the majority of the students to choose the university was affordability.

Answer:

D. Population: all the first-year students at the university

Sample: a group of 185 incoming first-year students

Conclusion: The reason for the majority of the students to choose the university was affordability..

Answer:

[tex]\dfrac{dy}{dx}=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]

Step-by-step explanation:

Note: I will provide links with descriptions/steps to each of the rules used in this differential equation at the bottom of this answer.

Given

[tex]-26x^2+4x^{39} y+y^9=-21[/tex]

Find [tex]\dfrac{dy}{dx}[/tex]

Differentiate the left side of the equation.

[tex]\dfrac{d}{dx}\left(-26x^2+4x^{39}y+y^9\right)= \dfrac{d}{dx}\left(-21\right)[/tex]

Focus the left side of the equation.                                                         Apply the Sum Rule for derivatives.

[tex]\dfrac{d}{dx} \left[-26x^2\right]+\dfrac{d}{dx} \left[4x^{39}y\right]+\dfrac{d}{dx} \left[y^9\right][/tex]

Lets evaluate each derivative.

1.   Evaluate [tex]\frac{d}{dx} \left[-26x^2\right][/tex]

Differentiate using the Power Rule for derivatives.

[tex]\dfrac{d}{dx} \left[-26x^2\right]=2*26x^{2-1}[/tex]

Simplify.

[tex]2*-26x^{2-1}\\-52x^{2-1}\\-52x[/tex]

2.   Evaluate [tex]\frac{d}{dx} \left[4x^{39}y \right][/tex]

Differentiate using the Product Rule for derivatives.

[tex]4\left(x^{39}\dfrac{d}{dx}\left[y\right]+y\dfrac{d}{dx}\left[x^{39}\right]\right)[/tex]

Rewrite  [tex]\dfrac{d}{dx}\left[y\right][/tex] as [tex]y'[/tex]

[tex]4\left(x^{39}y'+y\dfrac{d}{dx}\left[x^{39}\right]\right)[/tex]

Differentiate using the Power Rule for derivatives.

[tex]4\left(x^{39}y'+y\left(39x^{39-1} \right)\right)[/tex]

Simplify.

[tex]4\left(x^{39}y'+39yx^{38} \right)[/tex]

3.   Evaluate [tex]\frac{d}{dx} \left[y^9\right][/tex]

Differentiate using the Chain Rule for derivatives.

[tex]9y^8\dfrac{d}{dx} \left[y\right][/tex]

Rewrite  [tex]\dfrac{d}{dx}\left[y\right][/tex] as [tex]y'[/tex]

[tex]9y^8y'[/tex]

Add up all of the derivatives then simplify.

[tex]-52x+4\left(x^{39}y'+39yx^{38} \right)+9y^8y'[/tex]

[tex]-52x+4\left(x^{39}y'\right)+4\left(39yx^{38} \right)+9y^8y'[/tex]

[tex]-52x+4x^{39}y'+156yx^{38}+9y^8y'[/tex]

[tex]4x^{39}y'+156x^{38}y+9y^8y'-52x[/tex]

Focus the right side of the equation.

[tex]\dfrac{d}{dx}\left(-21\right)[/tex]

Since [tex]-21[/tex] is constant with respect to [tex]x[/tex] , the derivative of [tex]-21[/tex] with respect to [tex]x[/tex] is 0 .

Now we have

[tex]4x^{39}y'+156x^{38}y+9y^8y'-52x=0[/tex]

Finally lets solve for [tex]y'[/tex].

Subtract [tex]156x^{38}y[/tex] from both sides of the equation.

[tex]4x^{39}y'+9y^8y'-52x=-156x^{38}y[/tex]

Add [tex]52x[/tex] to both sides of the equation.

[tex]4x^{39}y'+9y^8y'=-156x^{38}y+52x[/tex]

Factor [tex]y'[/tex] out of [tex]4x^{39}y'[/tex]

[tex]y'\left(4x^{39}\right) +9y^8y'=-156x^{38}y+52x[/tex]

Factor [tex]y'[/tex] out of [tex]9y^8y'[/tex]

[tex]y'\left(4x^{39}\right)+y'\left(9y^8\right)=-156x^{38}y+52x[/tex]

Factor [tex]y'[/tex] out of [tex]y'\left(4x^{39}\right)+y'\left(9y^8\right)[/tex]

[tex]y'\left(4x^{39}+9y^8\right)+=-156x^{38}y+52x[/tex]

Divide each term by [tex]4x^{39}+9y^8[/tex]

[tex]\dfrac{y'\left(4x^{39}+9y^8\right)}{4x^{39}+9y^8} =\dfrac{-156x^{38}y}{4x^{39}+9y^8} +\dfrac{52x}{4x^{39}+9y^8}[/tex]

Cancel the common factor of [tex]4x^{39}+9y^8[/tex] on the left side of the equation.

[tex]y'=\dfrac{-156x^{38}y}{4x^{39}+9y^8} +\dfrac{52x}{4x^{39}+9y^8}[/tex]

Combine the numerators over the common denominator.

[tex]y'=\dfrac{-156x^{38}y+52x}{4x^{39}+9y^8}[/tex]

Factor [tex]52x[/tex] out of [tex]-156x^{38}y+52x[/tex] and simplify.

[tex]y'=\dfrac{52x\left(-3x^{37}y\right)+52x}{4x^{39}+9y^8}[/tex]

[tex]y'=\dfrac{52x\left(-3x^{37}y\right)+52x(1)}{4x^{39}+9y^8}[/tex]

[tex]y'=\dfrac{52x\left(-3x^{37}y+1\right) }{4x^{39}+9y^8}[/tex]

[tex]y'=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]

Rewrite [tex]y'[/tex] as [tex]\dfrac{dy}{dx}[/tex].

[tex]\dfrac{dy}{dx}=-\dfrac{52x\left(3x^{37}y-1\right) }{4x^{39}+9y^8}[/tex]

Learn more about the Power Rule for derivatives here

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Learn more about the Sum Rule for derivatives here

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Learn more about the Chain Rule for derivatives here

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