Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it.
lim x → 0
A.x2^x
B. 2^x - 1

The probability of rolling two numbers with a sum less than 7 when rolling two dice simultaneously is 5/12.

From the given table, there are 36 possible outcomes when rolling two dice (6 sides on each die, so 6 x 6 = 36).

Now, let's identify the outcomes where the sum is less than 7:
(1,1) (1,2) (1,3) (1,4) (1,5)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2) (3,3)
(4,1) (4,2)
(5,1)

There are 15 outcomes where the sum is less than 7.

To calculate the probability, we can use the formula:
Probability = (Number of desired outcomes) / (Total number of outcomes)

In this case, the probability of rolling two numbers with a sum less than 7 is:
Probability = 15 / 36 = 5 / 12

So, the probability of rolling two numbers with a sum less than 7 is 5/12.

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Sure, I'd be happy to help you with that. Based on the information provided, we know that HA Leonardo received three books for his birthday and two of them cost a total of 760. To find out the cost of the third book, we need to subtract the cost of the two books from the total amount paid, which is 1125.

To do this, we can use a simple equation:

Total cost of three books - Total cost of two books = Cost of third book

So, we can plug in the values we know:

1125 - 760 = Cost of third book

Solving for the cost of the third book:

365 = Cost of third book

Therefore, the third book cost 365.

In summary, HA Leonardo received three books for his birthday and two of them cost 760. The total amount paid was 1125, so the cost of the third book was 365.

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Answer:

(Credit to guy/girl above) 63 miles 10 1/2 x 6 is 63.

Step-by-step explanation:

pls mark brainliest

The majority of logs in the pile will have a weight close to the mean of 17 pounds, with a smaller number of logs having a weight that is farther away from the mean.

In statistics, the normal distribution is a commonly used continuous probability distribution.

It is also referred to as a Gaussian distribution, and it has a bell-shaped curve that is symmetrical around the mean.

The mean is the center of the distribution, and the standard deviation describes how spread out the data is around the mean.

In this case, the weights of logs in a wood pile are normally distributed with a mean of 17 pounds and a standard deviation of 3.4 pounds.

This means that the majority of logs in the pile will have a weight close to the mean of 17 pounds, with a smaller number of logs having a weight that is farther away from the mean.

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This solution of g(x) has no real solutions, so g⁻¹ (-7) does not exist.

To find g⁻¹ (-7), we need to solve for x in the equation g(x) = -7.

First, we need to determine which part of the piecewise function to use. Since -7 is less than 5, we know that we need to use the first part of the function: g(x) = 5x² if x ≤ -3.

So, we set g(x) = 5x² equal to -7 and solve for x:

5x² = -7

x² = -7/5

This equation has no real solutions since the square of any real number is always nonnegative. Therefore, g⁻¹ (-7) does not exist in the real numbers.

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The length of leg of given triangle is s 13√(2) units.

A 45-45-90 triangle is a special right triangle in which the two legs are congruent and the angles opposite the legs are both 45 degrees. The hypotenuse is the longest side of the triangle and is located opposite the right angle.

In this problem, we are given that the hypotenuse length of a 45-45-90 triangle is 26 units. Our goal is to find the length of one of the legs of the triangle. Let us denote the length of one of the legs as x.

By the Pythagorean theorem, we know that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. In a 45-45-90 triangle, the two legs are congruent, so we can set up the following equation:

x² + x² = 26²

Simplifying the left-hand side of the equation, we get:

2x² = 676

Dividing both sides by 2, we get:

x² = 338

Taking the square root of both sides, we get:

x = √(338)

Since we are asked to give the answer in simplified radical form, we can write:

x = √(2 × 169) = √(2) × √(169) = 13√(2)

Therefore, the length of one of the legs of the 45-45-90 triangle with a hypotenuse length of 26 units is 13√(2) units.

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If A queen-sized mattress is 20 inches longer than it is wide. A king-sized mattress is 1280 square inches.

In mathematics, a variable is a symbol or letter that represents a value that can change or vary in a given context or problem. The area of the queen-sized mattress is x(x + 20) square inches. The equation to determine the area of the king-sized mattress is (x + 16)(x + 20) = x(x + 20) + 1280

Let x be the width of the queen-sized mattress in inches.

Then the length of the queen-sized mattress is x + 20 inches.

The length of the king-sized mattress is the same as the length of the queen-sized mattress, which is x + 20 inches.

We can use the formula for the area of a rectangle to find the area of each mattress:

Area of queen-sized mattress = length x width = (x + 20) x x = x^2 + 20x

Area of king-sized mattress = length x width = (x + 20) x (x + 16) = x^2 + 36x + 320

Area of king-sized mattress = Area of queen-sized mattress + 1,280

Substituting the expressions we found for the areas, we get:

x^2 + 36x + 320 = x^2 + 20x + 1280

Simplifying and solving for x, we get:

16x = 960

x = 60

So the width of the queen-sized mattress is 60 inches, and its length is 80 inches.

The width of the king-sized mattress is 76 inches, and its length is 80 inches.

The area of the queen-sized mattress is:

60^2 + 20(60) = 4,800 square inches

The area of the king-sized mattress is:

76^2 + 36(76) + 320 = 6,080 square inches

And we can verify that the area of the king-sized mattress is indeed 1,280 square inches more than that of the queen-sized mattress:

6,080 - 4,800 = 1,280

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Mike will need 28 feet of fencing.

To solve the problem, we can use the formula for the area of a rectangle:

A = L × W

where A is the area, L is the length, and W is the width.

We know that the area of the patio is 192 ft^2, so we can write:

192 = L × W

We also know that the length is 4 feet longer than the width, so we can write:

L = W + 4

Substituting L = W + 4 into the equation for the area, we get:

192 = (W + 4) × W

Expanding the right side of the equation, we get:

192 = W^2 + 4W

Rearranging, we get a quadratic equation in standard form:

W^2 + 4W - 192 = 0

We can solve for W by factoring or using the quadratic formula, but in this case, we can recognize that 12 and -16 are two numbers that multiply to -192 and add up to 4. Therefore, we can write:

W^2 + 4W - 192 = (W + 16) × (W - 12) = 0

This gives us two possible values for W: W = -16 or W = 12. Since the width cannot be negative, we reject the solution W = -16 and choose W = 12.

Using the equation L = W + 4, we find that the length is L = 16.

Finally, we can calculate the amount of fencing Mike will need by adding up the lengths of the three sides that need to be fenced. The two lengths are L = 16 feet each, and the width is W = 12 feet. Therefore, Mike will need a total of 16 + 16 + 12 = 44 feet of fencing. However, since one side of the patio is adjacent to the back of his house, he only needs to fence three sides.

Therefore, he will need 44 - 16 = 28 feet of fencing.

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Therefore, the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high is approximately 28.7 feet.

In order to find the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high, we will use the angle of elevation and the tangent function.

1. Given the lowest angle of elevation of the sun in winter is 20 degrees, we will use this angle in our calculations.

2. Set up a right triangle with the fence as the vertical side (opposite side), the distance x as the horizontal side (adjacent side), and the angle of elevation (20 degrees) at the point where the fence meets the ground.

3. Use the tangent function to find the distance x:

tan(angle) = opposite side / adjacent side

4. Plug in the values we have:

tan(20) = 10.5 / x

5. Solve for x:

x = 10.5 / tan(20)

6. Calculate the value of x:

x ≈ 28.7 feet

Therefore, the distance x is approximately 28.7 feet.

Note: The question is incomplete. The complete question probably is: In one area, the lowest angle of elevation of the sun in winter is 20 degrees. Find the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high. Round your answer to the tenths place when necessary.

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The possible outcomes of sample space is 12.

To calculate the total number of outcomes in a sample space, multiply the number of serving options with the number of flavors.

There are 4 flavors that are lime, cherry, blueberry, and  watermelon and 3 serving options that are served plain, mixed with ice cream, or as a drink.

Hence, the possible outcomes will be:

4 x 3 = 12

The outcomes can be represented as lime Italian ice mixed with ice cream, cherry Italian ice served as a drink, Watermelon Italian ice mixed with ice cream, Blueberry Italian ice served plain and likewise.

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Answer:

  20°

Step-by-step explanation:

You want the measure of angle G in triangle GPQ with side lengths GP=74, PQ=31, QG=88 meters.

The law of cosines tells you the relevant relationship is ...

  PQ² = GP² +GQ² -2·GP·GQ·cos(G)

Solving for angle G gives ...

  G = arccos((GP² +GQ² -PQ²)/(2·GP·GQ))

  G = arccos((74² +88² -31²)/(2·74·88)) = arccos(12259/13024)

  G ≈ 19.735° ≈ 20°

The golfer must play the shot within an angle of about 20°.

Answer:

86 degrees fahrenheit

Step-by-step explanation:

(30°C × 9/5) + 32 = 86°F

a) We can reject the null hypothesis and cthat theronclude is significant evidence that the probability of Kerrich's coin coming up heads is not 0.5. b) we get a confidence interval of 0.495 to 0.517.

a) To test the hypothesis that the probability of Kerrich's coin coming up heads is not 0.5, we can use a one-sample proportion test at the 5% level of significance. The null hypothesis is that the true proportion of heads is 0.5, and the alternative hypothesis is that it is not equal to 0.5.

The test statistic can be calculated as (5067-0.510000)/(sqrt(100000.5*0.5)) which simplifies to 5.401. The corresponding P-value can be found using a standard normal distribution table or a calculator to be approximately 3.3x10^-8, which is much smaller than 0.05. Therefore, we can reject the null hypothesis .

b) To construct a 95% confidence interval for the true proportion of heads, we can use the formula p ± z*sqrt((p(1-p))/n), where p is the sample proportion, z is the z-score corresponding to a 95% confidence level (which is 1.96), and n is the sample size. Substituting the values, we get a confidence interval of 0.495 to 0.517, which means that we can be 95% confident that the true proportion of heads falls within this range.

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To use the Comparison Test, we need to find a series that we know is either always greater than or always less than our given series

. Since cosine values oscillate between -1 and 1, we know that 2 + cos(n) is always greater than or equal to 1. Therefore, we can compare our given series to the series 1/n, which we know is a p-series with p = 1 and is divergent.

Using the Comparison Test, we can say that since 1/n is divergent and 1/n ≤ 2 + cos(n) for all n ≥ 71.1, then the series 2 + cos(n) n 71.1 must also be divergent. Therefore, the series 2 + cos(n) n 71.1 is divergent.
Hi! To determine whether the series Σ(2 + cos(n)), where n ranges from 1 to infinity, is convergent or divergent using the Comparison Test, we need to compare it with another series whose convergence behavior is known.

Consider the series Σ(2), which is a constant series where every term is 2. This series diverges, as its partial sums keep increasing without bound. Since cos(n) is bounded between -1 and 1, we know that for every n, 1 ≤ (2 + cos(n)) ≤ 3.

Now, using the Comparison Test, we can compare Σ(2 + cos(n)) with Σ(2). Since Σ(2) is divergent and 2 + cos(n) is always greater than or equal to 2, we can conclude that the series Σ(2 + cos(n)) is also divergent.

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The arc length of the polar curve [tex]r = e^{8\theta}[/tex] from θ = 0 to θ = 5 is[tex]\int_0^5 \sqrt{(64e^{16\theta}+1)} d\theta[/tex].

To find the arc length of a polar curve, we use the formula:

L = [tex]\int_a^b \sqrt{[r(\theta)^2+(dr(\theta)/d\theta)^2]} d\theta[/tex]

where r(θ) is the equation of the polar curve, and a and b are the starting and ending values of θ, respectively.

In this case, the equation of the polar curve is[tex]r = e^{8\theta}[/tex], so we have [tex]r(\theta) = e^{8\theta}[/tex]}. To find dr(θ)/dθ, we use the chain rule of differentiation:

dr(θ)/dθ = d/dθ ([tex]e^{8\theta}[/tex]) = [tex]8e^{8\theta}[/tex]

So now we have r(θ) and dr(θ)/dθ, which we can plug into the formula for arc length:

L = [tex]\int_0^5 \sqrt{[e^{16\theta}+(8e^{8\theta})^2] }[/tex]dθ

Simplifying the expression inside the square root, we get:

L = [tex]\int_0^5 \sqrt{(64e^{16\theta}+1) }[/tex]dθ

Unfortunately, this integral cannot be evaluated in terms of elementary functions, so we leave the answer in this form. We can, however, approximate it using Simpson's method and it comes out to be approximately 1.3526 * 10⁸.

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The expected cell count for free music downloads in a sample of 200 bottles from a beverage company's prize promotion is 30, based on the company's claimed probabilities of winning.

To find the expected cell count for music downloads in this study, we first need to calculate the total expected counts for each category of the prize promotion.

Given that the company claims that 5% of the bottle-tops will win a free beverage and 15% will win a free music download, the expected counts for each category in a sample of 200 bottles are

Expected count for free beverages = 0.05 * 200 = 10

Expected count for free music downloads = 0.15 * 200 = 30

Expected count for non-winners = 0.80 * 200 = 160

The expected count for music downloads is 30, which is calculated by multiplying the probability of winning a free music download (0.15) by the sample size (200). Therefore, the expected cell count for music downloads in this study is 30.

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The approximate length of the crust when x = 1.2 is 17.28 centimeters. The correct option is D.

To find the total perimeter of Sydney's sandwich, we need to add up the lengths of all four sides. From the given dimensions, we can see that the top and bottom sides each have a length of 2x²8, and the right and left sides each have a length of 2x²9. Therefore, the total perimeter can be expressed as:

2(2x²8) + 2(2x²9)

Simplifying this expression gives:

4x²8 + 4x²9

And further simplifying by factoring out 4x² gives:

4x²(8 + 9)

Which equals:

4x²17

Now, to find the approximate length of the crust when x = 1.2, we simply plug in this value for x into the expression we just found:

4(1.2)²17

Simplifying this expression gives:

4(1.44)17

Which equals:

5.76 + 11.52 = 17.28

Therefore, the approximate length of the crust when x = 1.2 is 17.28 centimeters. The answer is option D, which is 4x²17; 22.76 centimeters.

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Answer:

28 degrees and 62 degrees

Step-by-step explanation:

Set up your equation like this:

x+(x-34)=90

2x-34=90

2x=56

x=28

28+34=62

So the smaller angle is 28 degrees and the larger angle is 62 degrees.

Hope this helps!! :D

Answer:

$4139.93

Step-by-step explanation:

Formula for amount accrued at compound interest annually is given by

A = P( 1+ r)ⁿ

where
P = principal
r = interest rate expressed as a decimal
n = number of years

Given P = 3900, r = 1/100 = 0.01, n = 6 we get

A = 3900(1 + 0.01)⁶ = 3900(1.01)⁶ = 4139.93

So the amount accrued after 6 years = $4139.93

Miguel will wait for 6 minutes in line, there are 6 employees working.

Joni will wait 3 minutes to order when 8 employees are working.

Here are the steps to answer your questions:

1. If Miguel waits 6 minutes in line to order, predict the number of employees working:

We have the trend line equation y = -1.5x + 15, where y is the waiting time in minutes, and x is the number of employees. We are given that Miguel waits for 6 minutes, so we'll plug y = 6 into the equation and solve for x:

6 = -1.5x + 15

To solve for x, first subtract 15 from both sides of the equation:

6 - 15 = -1.5x
-9 = -1.5x

Now, divide both sides by -1.5:

x = -9 / -1.5
x = 6

So, when Miguel waits 6 minutes in line, there are 6 employees working.

2. Joni arrives at the restaurant when 8 employees are working. Predict the amount of time Jodi will wait to order:

We'll use the same trend line equation and plug in x = 8 to find the waiting time for Joni:

y = -1.5(8) + 15

Multiply -1.5 by 8:

y = -12 + 15

Now, add 15:

y = 3

Joni will wait 3 minutes to order when 8 employees are working.

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The type 1 error in this scenario would be rejecting the null hypothesis that the mean level of fluoride in children's drinking water is less than 1.2 ppm, when in reality it is true.

The EPA claims that fluoride in children's drinking water should have a mean level of less than 1.2 ppm to reduce the number of dental cavities. A type 1 error occurs when we reject the null hypothesis when it is actually true. In this case, the null hypothesis (H0) would be that the mean fluoride level is less than or equal to 1.2 ppm, and the alternative hypothesis (H1) would be that the mean fluoride level is greater than 1.2 ppm.

A type 1 error would occur if we incorrectly conclude that the mean fluoride level is greater than 1.2 ppm when, in reality, it is less than or equal to 1.2 ppm. This could lead to unnecessary actions being taken to reduce fluoride levels when they are already at an acceptable level.

In other words, falsely concluding that the mean level of fluoride in the water is above 1.2 ppm and therefore causing harm to the children's dental health by not reducing the number of dental cavities.

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the roots of the function are[tex]x = 1[/tex] and  [tex]x = 3.[/tex] These are also the x-intercepts of the parabola.

This polynomial is a quadratic function and can be graphed as a parabola. a polynomial of degree two.

[tex]f(x) = x^2 - 4x + 3[/tex]

The coefficient of the x^2 term is positive, which means that the parabola opens upwards. The vertex of the parabola is at (2, -1) and the y-intercept is at (0, 3).

The roots of the function can be found by setting f(x) = 0 and solving for x:

[tex]x^2 - 4x + 3 = 0[/tex]

[tex](x - 3)(x - 1) = 0[/tex]

[tex]x = 1, 3[/tex]

Therefore, the roots of the function are[tex]x = 1[/tex] and  [tex]x = 3.[/tex] These are also the x-intercepts of the parabola.

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Angle S,  Angle T and side ST are common to triangle SUT and triangle SVT.

To identify the angle or side that is common to triangle SUT and triangle SVT, based on given information, we can evaluate them as follows:

A. Angle S is common to both triangles SUT and SVT, as it is the vertex where the two triangles share a point.
B. Side SV is not common to both triangles, as it is only a side of triangle SVT.

C. Side ST is common to both triangles SUT and SVT, as it connects the shared vertex S to points U and V in each respective triangle.
D. Angle T is also common to both triangles SUT and SVT, as it is the other vertex where the two triangles share a point.

Therefore,  Angle S, side ST, and Angle T are all common to triangle SUT and triangle SVT.

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A sample size of 77 is needed for a scientist studying premature babies to estimate the mean birth weight of babies born during a week of gestation within 0.5 pounds with 95% confidence, assuming a population standard deviation of 2 pounds.

To determine the minimum sample size needed for the scientist to be confident that her estimate is within a certain margin of error of the true mean birth weight of babies born during the week, we can use the formula

n = (zα/2 * σ / E)²

where

n = sample size

zα/2 = critical value for the desired level of confidence (e.g. 1.96 for 95% confidence)

σ = population standard deviation

E = margin of error (i.e. the maximum amount by which the estimate can differ from the true mean)

Substituting the given values, we have

n = (1.96 * σ / E)²

To determine E, we need to use the fact that the margin of error is equal to the critical value times the standard error, where the standard error is given by

SE = σ / √(n)

Substituting the given values, we have

SE = σ / √(n) = 2 / √(n)

Thus, we have

E = zα/2 * SE = 1.96 * 2 / √(n) = 3.92 / √(n)

Substituting this expression for E into the formula for n, we have

n = (1.96 * σ / (3.92 / √(n)))²

Simplifying, we get

n = (1.96 * σ * √(n) / 3.92)²

n = (0.5 * σ * √(n))²

n = (0.25 * σ² * n)

n = (zα/2)² * σ² / E²

Substituting the given values, we have

n = (1.96)² * 4 / (0.5)² = 76.96

Rounding up to the nearest whole number, we get

n = 77

Therefore, the minimum sample size needed for the scientist to be confident that her estimate is within 0.5 pounds of the true mean birth weight of babies born during the week is 77.

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Answer:

  (c)  y < -x +1

Step-by-step explanation:

You want the inequality that matches the graph.

The boundary line of the shaded area has a negative slope: it falls one unit for each unit to the right, so the slope is ...

  m = rise/run = -1/1 = -1

The line crosses the y-axis at y = 1, so the y-intercept is b = 1.

The equation of the boundary line is ...

  y = mx + b

  y = -x +1

The boundary line is dashed, so its values are not part of the solution set. The shading is below the line, so only y-values less than those on the line are included.

The inequality is ...

  y < -x +1 . . . . . choice C

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The area of the shaded sector is 461.7 [tex]yd ^ {2}[/tex]

The shaded region in the given question is a sector. So, we will calculate the area of the sector. The area of a sector is nothing but a fraction of the area of the whole circle. So, we will use the given formula to find the area of a sector of the circle.

area of sector = [tex]\frac{angle of sector}{360} * \pi r^{2}[/tex]

We know that the angle of the sector is 167 degrees and the radius of the circle is given to be 17.8 yd. We will substitute these values in the formula to calculate the area.

area of sector =   [tex]\frac{167}{360} * \pi * (17.8)^{2}[/tex]

area of sector = 461.7 [tex]yd^{2}[/tex]

Therefore the area of the shaded region to the nearest tenth is 461.7 square yards.

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The complete question is "Find the area of the shaded sector.

round to the nearest tenth.

167°

17.8 yd

area = [ ? ]yd?

The image is shown below."

Answer:

angle g is 98 degrees.

Step-by-step explanation:

assuming the figure is a quadrilateral,

angle g + 82 + 104 + 76 = 360 ( Property of Quadrilateral)

262 + angle g = 360

angle g = 98 degrees

The shopkeeper sold an average of 40.5 eggs per day during the given interval.

To calculate the average number of eggs sold per day, add up the total number of eggs sold and divide by the number of days.

Total number of eggs sold = 42 + 49 + 61 + 35 + 27 + 36 + 50 + 34 + 31 + 40 = 405

Number of days = 10

Average number of eggs sold per day = 405 / 10 = 40.5

The shopkeeper sold an average of 40.5 eggs per day during the given interval.

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The shopkeeper sold every day the number of eggs is recorded. At equal intervals group 42, 49, 61, 35, 27, 36, 50, 34, 31, 40

What is the average number of eggs sold per day by the shopkeeper over the given interval?

The number of servings did Pablo make is (B) 26.

Unit conversion is the process of converting a value expressed in one unit of measurement to another unit of measurement that represents the same quantity but is expressed in a different unit.

The given quantities are in different units, so we need to convert them to a common unit to determine the number of servings. For this, we need to convert quarts and gallons to fluid ounces.

Here we have

Pablo mixed 2 1/2 quarts of fruit juice with 1 gallon of seltzer.  

Each serving is 8 fluid ounces

Let's first convert the quantities to the same units.

=> 1 gallon = 4 quarts

Total amount of liquid = 2 1/2 + 4 = 6 1/2 quarts

Now, let's find how many 8-ounce servings are in 6 1/2 quarts:

=> 1 quart = 32 fluid ounces

=> 6 1/2 quarts = 6.5 x 32 = 208 fluid ounces

Given that, 1 serving = 8 fluid ounces

Number of servings = 208/8 = 26

Therefore,

The number of servings did Pablo make is (B) 26.

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Complete Question:

Pablo mixed 2 1 /2 quarts of fruit juice with 1 gallon of seltzer. If each serving is 8 fluid ounces, how many servings did Pablo make?

A: 20    B: 26     C: 32    D: 36

Using the formula V = k*T/P to get the volume, the required volume in the given situation is 1.77L.

The measurement of three-dimensional space is volume. It is frequently expressed quantitatively using SI-derived units, as well as several imperial or US-standard units.

Volume and the notion of length are connected.

The area that any three-dimensional solid occupies is known as its volume.

These solids can take the form of a cube, cuboid, cone, cylinder, or sphere.

So, the volume can be obtained using the equation:

V = k*T/P

The value of the constant k is:

K = PV/T = 16N/cm²*2.2L/340K = 0.104N*L*K⁻¹*cm⁻²

We can now determine the volume when:

T = 408 K

P = 24 N/cm²

V = k*T/P = 0.104N*L*K⁻¹*cm⁻²*408K/21Ncm = 1.77L

Therefore, using the formula V = k*T/P to get the volume, the required volume in the given situation is 1.77L.

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Correct question:

The volume of a gas varies inversely to the pressure and direction of the temperature (in degrees Kelvin). If a certain gas occupies a volume of 2.2 liters at a temperature of 340 K and a pressure of 16 newtons per square centimeter, find the volume when the temperature is 408 K and the pressure is 24 newtons per square centimeter.