Which rational function models the time, y, in hours, that it takes the train to travel between the two cities at an average speed of x miles per hour?

y= x/6.4
y= x/160
y= 6.4/x
y= 160/x​

Answer:

x=46 4/5, x=46.8

Step-by-step explanation:

To find x-intercept/zero, substitute y=0

-222=5x-3(-7*0-4)

-222=5x-3(-7*-4) Solve

x= -234/5

x=46 4/5, x=46.8

Answer:

Step-by-step explanation:

Considering the central limit theorem, the distribution is normal since the number of samples is large. Also, the population standard

deviation is known. We would determine the z score.

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.90 = 0.1

α/2 = 0.1/2 = 0.05

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.05 = 0.95

The z score corresponding to the area on the z table is 2.05. Thus, confidence level of 90% is 1.645

Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Confidence interval = mean ± z × σ/√n

Where

σ = population standard Deviation

Confidence interval = x ± z × σ/√n

x = 22.8 hours

σ = 6.4 hours

n = 175

i) Confidence interval = 22.8 ± 1.645 × 6.4/√175

= 22.8 ± 0.80

The lower end of the confidence interval is

22.8 - 0.80 = 22

The upper end of the confidence interval is

22.8 + 0.80 = 23.6

ii) error bound is the same as the margin of error

Error bound = 0.8

Answer:

The boat is approaching the dock at rate of 2.14 ft/s

Step-by-step explanation:

The situation given in the question can be modeled as a triangle, please refer to the attached diagram.

A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow that means x = 5 ft.

The length of rope from bow to pulley is 13 feet that means y = 13 ft.

We know that Pythagorean theorem is given by

[tex]x^{2} + y^{2} = z^{2}[/tex]

Differentiating the above equation with respect to time yields,

[tex]2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}[/tex]

[tex]x\frac{dx}{dt} + y\frac{dy}{dt} = z\frac{dz}{dt}[/tex]

dx/dt = 0  since dock height doesn't change

[tex]y\frac{dy}{dt} = z\frac{dz}{dt}[/tex]

[tex]\frac{dy}{dt} = \frac{z}{y} \frac{dz}{dt}[/tex]

The rope is being pulled in at a rate of 2 feet per second that is dz/dt = 2 ft/s

First we need to find z

z² = (5)² + (13)²

z² = 194

z = √194

z = 13.93 ft

So,

[tex]\frac{dy}{dt} = \frac{z}{y} \frac{dz}{dt}[/tex]

[tex]\frac{dy}{dt} = \frac{13.93}{13}(2)[/tex]

[tex]\frac{dy}{dt} = 2.14[/tex] [tex]ft/s[/tex]

Therefore, the boat is approaching the dock at rate of 2.14 ft/s

So, the boat approached the dock with a speed of 2.1337 m/sec.

Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle.

So, the formula is,

[tex]a^2+b^2=c^2[/tex]

Differentiating the above equation,

[tex]2a\frac{da}{dt} +2b\frac{db}{dt} =2c\frac{dc}{dt} ...(1)[/tex]

It is given that,

[tex]a=5m\\\frac{da}{dt}=0\\ c=13\\\frac{dc}{dt}=2 m/s[/tex]

[tex]b=\sqrt{13^2-5^2} \\b=12 m/s[/tex]

Substituting the above values in equation (1) we get,

[tex]2\times5\times0+2\times12\frac{db}{dt} =2\times13\times2\\\frac{db}{dt} =\frac{26}{12}\\ \frac{db}{dt} =2.1337 m/s[/tex]

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The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.

Place value is the basis of our entire number system. This is the system in which the position of a digit in a number determines its value.

The number 42,316 is different from 61,432 because the digits are in different positions.

here, we have,

given that,

6,582,390,417

so, the values of each of the digits in 6,582,390,417 in word form is:

6: six billon

5: five hundred

8: eighty two million

3: three hundred thousand

and ninety four hundred

seventeen.

Hence, The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.

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Answer:

it can be written in eqn form as

x-8=>17

or,x=>25

Answer:

x-8 ≥17

If you need to find x, Add 8 on both sides

x ≥25

Hope this helped

Answer:

2

Step-by-step explanation:

Multiply all of the fractions together.

Answer:

[tex]25.00\geq 3.50p[/tex]

Step-by-step explanation:

since you have $25.00, you can only spend up to that much money, so it will have to be less than or equal to , and since p = slices of pizza, you multiply that by 3.50 to know how much you can buy.

hope this helps :)

Answer:

$3.5p<=$25.00

Step-by-step explanation:

If you only have $25.00 to spend on pizza it can not exceed that limit, so your total amount needs to be more than or equal to money the pizza slices cost

Answer:

320

Step-by-step explanation:

2/5 = .4

800 * .4 =320

Answer:

x = 92

Step-by-step explanation:

Supplementary angles add to 180 degrees,

x+y = 180

We know y =88

x+88 = 180

Subtract 88 from each side

x+88-88=180-88

x =92

Answer:

RANDOM

Step-by-step explanation:

9514 1404 393

Answer:

  1/5000

Step-by-step explanation:

The scale is ...

  map distance / ground distance = (0.095 m)(475 m) = 0.0002 = 1/5000

To find the map scale, convert the actual distance from meters to centimeters and then set up a ratio with the map distance. After simplifying, the scale of the map is determined to be 1:5000, meaning 1 centimeter on the map equals 5000 centimeters in reality.

To determine the scale of the map we'll convert the given distance from George's house to the school into the same units and then create a ratio. Since the map shows the distance as 9.5 centimeters and the actual distance is 475 meters, we first need to convert meters to centimeters because the scale needs to have the same units for both measurements.

1 meter = 100 centimeters, so 475 meters is equivalent to 475 x 100 = 47500 centimeters.

Now, we can set up the scale of the map as a ratio:
9.5 centimeters (map distance) / 47500 centimeters (actual distance).

We can simplify this ratio to find the scale of the map by dividing both the numerator and the denominator by 9.5:

9.5 cm / 47500 cm = 1 cm / (47500 / 9.5) = 1 cm / 5000 cm

This means that every centimeter on the map represents 5000 centimeters in real life. Therefore, the map scale is 1:5000.

Answer:

The confidence interval for the mean is given by the following formula:  

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

The margin of error is given by:

[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:

Of confidence intervals with this margin of error, 99% will contain the population proportion

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Solution to the problem

The confidence interval for the mean is given by the following formula:  

[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

The margin of error is given by:

[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]

For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:

Of confidence intervals with this margin of error, 99% will contain the population proportion

Final answer:

To compare the populations of the two bird species, we calculate the average rate of change for Bird A by dividing the change in population by time over 18 months, and for Bird B by evaluating the given exponential function at the endpoints of the time interval.

Explanation:

To compare the population changes of Bird A and Bird B and interpret the average rates of change over the interval [0, 18], we first need to calculate the average rate of change for Bird A. Given Bird A's population at different times, we can calculate the average rate of change by dividing the change in population by the change in time, over the interval [0, 18].

For Bird A, the population increases from 8.3 to 9.1 thousand over 18 months. The average rate of change for Bird A is thus (9.1 - 8.3) / (18 - 0) = 0.8 / 18 = 0.0444 thousand per month.

For Bird B, the population change is given by a function y= 3.6(1.06)ˣ, where y is the population in thousands and x is the time in months. To find the average rate of change over [0, 18], we evaluate the function at the endpoints of the interval: y(0) = 3.6 and y(18) = 3.6(1.06)¹⁸. After calculating y(18), we'd use the same average rate of change formula.

Interpreting the results, if Bird A's average rate of change is less than that of Bird B, it means Bird B's population is growing faster on average than Bird A's population over the 18 months.

9514 1404 393

Answer:

  cut the pizza ±1.855 inches from the centerline

Step-by-step explanation:

Here's an interesting approach that actually gives amazing accuracy.

If the pizza were a 14" square, the cuts would be 14/6 = 2.33 inches from the centerline. If we draw a diagram of the pizza as a circle of radius 7 centered at the origin, we propose to create a running total of the areas of trapezoidal slices in the first quadrant parallel to the y-axis out to a distance of x=2.3. We want to find where the accumulated area is close to 1/12 of the area of the circle. In order to get sufficient resolution as to where the cut should be, we choose to use a total of 20 slices to cover that area. (n=20 for the Riemann sum)

Of course, the equation of the circle is x^2 +y^2 = 49. Then the value of y of interest is ...

  f(x) = √(49 -x^2)

where the value of x is some multiple of 2.3/20, the slice width for our Riemann sum. If we number the slices 0 to 19, the accumulated area up to slice k is ...

  [tex]\displaystyle A_k=\dfrac{2.3}{20}\sum_{n=0}^{k}\dfrac{f(x_n)+f(x_{n+1})}{2}[/tex]

As you can tell, this is using the trapezoidal method of computing the Riemann sum. We considered left, right, and midpoint integration methods but settled on this as having the kind of accuracy we wanted.

We want an accumulated area of 1/3 of our quarter circle, or (1/12)(π)(7^2) = 49π/12, so we want to find the zero of the difference Ak -49π/12.

As it turns out, the desired area is bracketed by k=16 and k=17. By linearly interpolating between the area values for these numbers of slices, we find that x=1.8546 is the location we need to cut the pizza. The problem is symmetrical, so the other cut is at x = -1.8546 inches from center.

_____

When evaluating functions multiple times, it is convenient to use a graphing calculator or spreadsheet. With a spreadsheet, you could list function values in one column, the trapezoidal method area in another column, and the accumulated area in yet another column. As here, the interpolation required for a final answer is also easily handled in a spreadsheet.

__

Additional comment

The arc α that encloses a sector equal to 1/3 the area of the pizza will satisfy the equation α -sin(α) -2π/3 = 0. That is about 2.6053256746 radians. The cut distance we're trying to find is the 7cos(α/2) ≈ 1.85452459222 inches, which differs by about 0.005% from the value we found.

Answer:

  2x -3y = 12

Step-by-step explanation:

For some horizontal change Δx and some vertical change Δy between the two points, an equation of the line through points (x1, y1) and (x2, y2) can be written as ...

  Δy·x -Δx·y = Δy·(x1) -Δx·(y1)

Here, we have ...

  Δy = y2 -y1 = 2 -(-2) = 4

  Δx = x2 -x1 = 9 -3 = 6

So, our equation can be ...

  4x -6y = 4·3 -6·(-2) = 24

Factoring out a common factor of 2 makes the equation be ...

  2x -3y = 12 . . . . . . equation of the line in standard form

Solving for y gives the equation in slope-intercept form:

  y = 2/3x -4

_____

More conventional solution

Plotting the points and drawing the line, you see that the y-intercept is -4. You also see that there is a "rise" of 2 grid squares for each "run" of 3 grid squares. Thus the slope of the line is 2/3. With this information, you can write the equation directly in slope-intercept form:

  y = mx + b . . . . . . line with slope m and y-intercept b

  y = 2/3x -4 . . . . . . the line through the given points

Final answer:

The line passing through the points (3, -2) and (9, 2) has a slope of 2/3, and its equation is y = (2/3)x - 4, which can be graphed by plotting the given points and ensuring the slope is represented correctly.

Explanation:

To graph the line that passes through the points (3, -2) and (9, 2), we first find the slope of the line. The slope formula is (y2 - y1) / (x2 - x1). Plugging in our points, we get (2 - (-2)) / (9 - 3) which simplifies to 4 / 6, further reduced to 2 / 3. Therefore, the slope of the line is 2 / 3.

Next, we use one of the points and the slope to write the equation in point-slope form, y - y1 = m(x - x1). Using the point (3, -2), the equation becomes y + 2 = (2/3)(x - 3). After distributing the slope and moving -2 to the other side, we get the equation y = (2/3)x - 4.

Finally, we can graph the line by plotting the two given points and drawing a straight line through them, ensuring that the rise over the run matches the slope of 2 / 3. The equation of the line y = (2/3)x - 4 can be verified using various x-values to see if the resulting y-values fall on the line plotted.

Answer:

Any number that is not a fractional value positive and negative but technically  all numbers are irrational

Step-by-step explanation:

Any number that is not a fractional value positive and negative but technically all numbers are irrational.

We have to find the numbers that are positive negative and 0 that are not irrational

Fractional values are represented using fixed-point arithmetic and are useful for DSP applications.

For a fractional division, we first scale the denominator to the range 0.5 ≤ d < 1.0.

Then we use a table lookup to provide an estimate of x0 to d−1.

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Answer:

The dimensions should be length=width=height=74 cm.

Step-by-step explanation:

We have an optimization with restriction problem.

We have to maximize the volume, subject to restriction in the sum of the length, width and height.

Let x be the length and width, that are equal, and z be the height.

The restriction can be expressed as:

[tex]x+x+z\leq222\\\\2x+z\leq222[/tex]

We can express z in function of x as:

[tex]2x+z=222\\\\z=222-2x[/tex]

The volume, the function to be optimized, can be expressed as:

[tex]V=x^2z=x^2(222-2x)=222x^2-2x^3[/tex]

To optimize, we derive and equal to zero.

[tex]\dfrac{dV}{dx}=\dfrac{d}{dx}[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74[/tex]

We have the optimum length. We can now calculate the height z:

[tex]z=222-2(74)=222-148=74[/tex]

Answer:

= 10√3

Step-by-step explanation:

[tex]2 \sqrt{27} - \sqrt{48} + 4 \sqrt{12} \\ = (2 \times \sqrt{9 \times 3}) - (\sqrt{16 \times 3}) + (4 \times \sqrt{4 \times 3} )\\ = (2 \times 3 \sqrt{3}) - 4 \sqrt{3} + (4 \times 2 \sqrt{3} ) \\ = 6 \sqrt{3} - 4 \sqrt{3} + 8 \sqrt{3} \\ = 10 \sqrt{3} [/tex]

Answer:

10√3

Step-by-step explanation:

2√27 − √48 + 4√12

2√(3²×3) − √(4²×3) + 4√(2²×3)

6√3 − 4√3 + 8√3

√3(6 - 4 + 8)

10√3

Answer:

Step-by-step explanation:

See attached file for answer pls

Answer:

[tex]P(X>34) = 0.9889[/tex]

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 50

Standard Deviation, σ = 7

We are given that the distribution of random variable X is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

P(X greater than 34)

[tex]P( X > 34) = P( z > \displaystyle\frac{34 - 50}{7}) = P(z > -2.2857)[/tex]

[tex]= 1 - P(z \leq -2.2857)[/tex]

Calculation the value from standard normal z table, we have,  

[tex]P(X>34) = 1 - 0.0111= 0.9889= 98.89\%[/tex]

The attached image shows the normal curve.

Answer:

A)   2704 hands

B)   16 hands

C)   144 hands

D)   384 hands

E)   208 hands

F)   400 hands

Step-by-step explanation:

See the attached file for explanation

There are 2704 possible hands, 16 hands consist of a pair of aces, 144 hands have all face cards, 384 hands have 1 king, 208 hands consist of two of a kind, and 384 hands contain at least 1 king.

To answer these probability related questions, we must look at the combinations which we can draw. A standard deck contains 52 cards and the hand in question consists of 1 card from 2 different decks, so:

A. The total number of different hands possible is 52 * 52 = 2704.

B. There are 4 aces in each deck so there are 4 * 4 = 16 hands that consist of a pair of aces.

C. A deck has 12 face cards and since 2 cards are being drawn, the number of hands with all face cards is 12 * 12 = 144.

D. The number of hands with exactly 1 king is found by multiplying the number of kings in a deck (4) by the number of non-kings in a deck (52-4). So, 4 * 48 = 192, but we must consider this happening in both decks: so 2 * 192 = 384.

E. For two of a kind hands (2 aces, 2 kings etc.), there are 13 kinds, thus 13 * 4 * 4 = 208.

F. For at least 1 king, either the first card or second card can be a king, so we use similar mathematics as used in D, which is 2 * 192 = 384.

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Answer:

−x2+3x−11

Step-by-step explanation:

−3x2+4x+2x2−x−11

(−3x2+2x2)+(4x−x)−11

−x2+3x−11

Answer:

Step-by-step explanation:

Hope this Helps ;)

Answer:

The dimensions of the rectangle that will allow for the most economical fence to be built are 30x15 feets, where two sides of 30 feets long cost $33 each one per foot, one side of 15 feets costs also $33 and the remaining side costs $99

Step-by-step explanation:

If x and y were the dimensions of the rectangle (in feets), then we have that x*y = 450. Therefore, y = 450/x.

Note that the rectangle as a result is formed by 2 sides with length x and 2 other sides with length 450/x. Lets suppose that x is the length of the 2 sides that costs both $33 and the other two sides, which have length 450/x, one costs also $33 and the other costs $99.

The cost, in $, function f,in terms of x, is given as follows

[tex] f(x) = 2 * 33 * x + 33*\frac{450}{x} + 99*\frac{450}{x} = 66x + \frac{59400}{x} [/tex]

We want to minimize f, so we will derivate it and equalize the derivate to 0:

[tex] f'(x) = 66 - \frac{59400}{x^2} [/tex]

[tex] f'(x) = 0 \leftrightarrow 66 = \frac{59400}{x^2} \leftrightarrow x^2 = \frac{59400}{66} = 900 \leftrightarrow x = \sqrt{900} = 30 [/tex]

(Note that x cant be negative, so in the equation we didnt count the opposite of the square root of 900)

We concluded that one dimension is 30 feets, and the other should be 450/30 = 15.

Answer:

Loans

Step-by-step explanation:

I don´t know how to explain it,and I hope my answer is correct though.

Answer:

banks create money by issuing loans and opening checking accounts  

Step-by-step explanation:

Answer:

(0.0657,0.0943)  is the 90% confidence interval for the population proportion of visitors that click on the advertisement.

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 978

Percentage of users that clicked on advertisement = 8%

Sample proportion:

[tex]\hat{p} = 0.08[/tex]

90% Confidence interval:

[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

[tex]z_{critical}\text{ at}~\alpha_{0.10} = 1.645[/tex]

Putting the values, we get:

[tex]0.08\pm 1.645(\sqrt{\dfrac{0.08(1-0.08)}{978}})\\\\= 0.08\pm 0.0142\\\\=(0.0658,0.0942) = (6.57\%,9.43\%)[/tex]

(0.0657,0.0943)  is the 90% confidence interval for the population proportion of visitors that click on the advertisement.

Final answer:

The sales manager should allocate approximately $341,310 for advertising to achieve the target sales volume of $200,000. This amount is determined by using the linear equation provided and solving for the advertising expenditures.

Explanation:

To find out how much the sales manager should allocate for advertising expenditures to achieve a target sales volume of $200,000, we use the given linear equation:

Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)

First, we convert the target sales volume to thousands of dollars - which would be $200 (since $200,000 is in thousands), and then plug it into the equation:

200 = 46.41 + 0.45(Advertising Expenditures)

Next, we solve for Advertising Expenditures:

200 - 46.41 = 0.45(Advertising Expenditures)
153.59 = 0.45(Advertising Expenditures)

Advertising Expenditures = 153.59 / 0.45
Advertising Expenditures = 341.31

Therefore, the sales manager should allocate approximately $341,310 for advertising in the budget to achieve the target sales volume of $200,000. This value is rounded to the nearest dollar as requested.

To achieve a target sales volume of $200,000, the company should allocate approximately $341,310 for advertising.

To determine the amount to allocate for advertising to reach a target sales volume of $200,000, we use the provided linear equation:

⇒ Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)

⇒ 200 = 46.41 + 0.45(Advertising Expenditures)

⇒ 200 - 46.41 = 0.45(Advertising Expenditures)

⇒ 153.59 = 0.45(Advertising Expenditures)

⇒ Advertising Expenditures = 153.59 ÷ 0.45

⇒ Advertising Expenditures ≈ 341.31

Therefore, the company should allocate approximately $341,310 (rounded to the nearest dollar) for advertising expenditures to achieve the target sales volume of $200,000.

Complete question:

Suppose that a company wishes to predict sales volume based on the amount of advertising expenditures. The sales manager thinks that sales volume and advertising expenditures are modeled according to the following linear equation. Both sales volume and advertising expenditures are in thousands of dollars.

Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)

If the company has a target sales volume of $200,000, how much should the sales manager allocate for advertising in the budget? Round your answer to the nearest dollar.

Answer:

The answer is option 1.

Step-by-step explanation:

It is given that the area of sector is, A = θ/360 × π × r² where r is the radius of circle. Using the formula, you are able to find the shaded sector :

θ = 90°

r = 6 yd

A = 90/360 × π × 6²

= 1/4 × π × 36

= 9π

= 28.3 yd² (near. tenth)

Answer:

The answer is 7.1yd

Step-by-step explanation:

got the wrong answer so I hope this helps yall and god bless️

Answer:

Step-by-step explanation:

the explanation is attached below

The conditional pdfs of Y given that X is 5 x and X given that Y is 5 y is 20< y<30.

Explain about the probability?

Calculating or estimating how likely something is to occur is what probability is all about. The likelihood of an event occurring can be expressed using words like "certain," "impossible," or "probable." Probabilities are always expressed in mathematics as fractions, decimals, or percentages with values ranging from 0 to 1.

The definition of probability, methods for calculating the probabilities of single and multiple random events, and the distinction between probabilities and odds of an event occurring are all covered in this article. Key conclusions: The probability that an event will occur is determined by probability: P(A) = f / N.

f y/x(y/x) = f(x, y)/f x(x)

 =K(x²+y²)/ 10Kx²+0.05

     0

20<y<30

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Answer: The value of y is 13

Step-by-step explanation: To find the value of y, we will use properties of equality.

Step 1: -7y = -91  (We want to find the value of y, or 1 y)

Step 2: (Use the division property of equality) -7y/-7 = -91/-7

Step 3: (Answer) y = 13

Answer:

y= 13

Step-by-step explanation:

-7y = -91

divide both sides by -7

y = 13

Answer:

Step-by-step explanation:

The first mathematician orders 1 beer, the second orders 1/2 a beer, the third orders 1/4 a beer, the fourth orders 1/8 a beer, the fifth orders 1/16 a beer. . .

Therefore, if the first mathematician orders 6 beers, the second orders 2 beer, the third orders 2/3 a beer, and so on

To find out how many beers the bartender should pour for the infinite series of mathematicians ordering in a pattern starting with 6 beers and each following ordering two-thirds of the previous, we use the sum formula for a geometric series. The sum is 18 beers.

The student is asking a question related to an infinite series in mathematics. In the scenario described, the first mathematician orders 6 beers, the second orders 2 beers, and each mathematician that follows orders a fraction of the previous mathematician's order, specifically two-thirds of the previous amount. To find out how many beers the bartender should pour, we need to find the sum of the geometric series.

The general form of a geometric series is a + ar + ar² + ar³ + ..., where 'a' is the first term and 'r' is the common ratio between terms. In this case, the first term 'a' is 6 (the first order) and the common ratio 'r' is 2/3 (each subsequent mathematician orders two-thirds of the previous one's amount).

To find the sum of this infinite series, we can use the formula S = a / (1 - r), when |r| < 1. Substituting the values from our question, we get S = 6 / (1 - 2/3) = 6 / (1/3) = 6 * 3 = 18 beers. Therefore, the bartender should pour 18 beers.