Please help me with my math

Given: ,8^2x= 32^x+3

a: (2³)^2x = (2⁵)^x+3

b: Solving, we get

2^6x = 2^5x+15

Since bases are same, we have

=>6x=5x+15

=> x = 15

First before two trains were [tex]990[/tex] kilometers apart, it will require [tex]5.5[/tex] hours.

Train speed is calculated as total distance traveled divided by travel time. The time it takes for two trains to pass each other is equal to (a+b) / (x+y) if the lengths of the trains, say a or b, are known and they are going at speeds of y and x, respectively.

Typically, a locomotive fueled by electricity or diesel powers trains. If there are several route networks, complicated signaling methods are used. One of the quickest forms of land transportation is rail.

[tex]distance = rate * time[/tex]

distance between trains [tex]= (85 km/h) * t + (95 km/h) * t[/tex]

distance between trains [tex]= (85 + 95) km/h * t[/tex]

distance between trains [tex]= 180 km/h * t[/tex]

Now, we can set up an equation to solve for the time it takes for the trains to be [tex]990[/tex] km apart:

[tex]180 km/h * t = 990 km[/tex]

[tex]t = 990 km / 180 km/h[/tex]

[tex]t = 5.5[/tex] hours

Therefore, it will take [tex]5.5[/tex] hours before the two trains are [tex]990[/tex] km apart.

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Answer:

1

Step-by-step explanation:

Given values are:

x1 y1=(2,4)

x2 y2=( 4,6)

slop=(6-4)divide (4-2)=1

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With respect to the average cost curves, the marginal cost curve: intersects both average total cost and average variable cost at their minimum points that is option B.

The fixed cost per unit of production is the average fixed cost (AFC). AFC will reduce consistently as output grows since total fixed costs stay constant. The variable cost per unit of production is known as the average variable cost (AVC). AVC generally declines until it reaches a minimum and then increases due to the growing and then lowering marginal returns to the variable input. The average total cost curve's (ATC) behaviour is determined by the behaviour of the AFC and AVC.

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Complete question:

With respect to the average cost curves, the marginal cost curve:

A) Intersects average total cost, average fixed cost, and average variable cost at their minimum point

B) Intersects both average total cost and average variable cost at their minimum points

C) Intersects average total cost where it is increasing and average variable cost where it is decreasing

D) Intersects only average total cost at its minimum point

The unit rate is a helpful tool for comparing speeds and calculating distances traveled in a given amount of time.

The formula for speed is: speed = distance / time where "distance" is the distance traveled by an object and "time" is the duration of travel. This formula can be used to calculate the speed of an object if the distance it has traveled and the time it took to travel that distance are known. It can also be used to calculate the distance traveled by an object if its speed and the time it traveled at that speed are known.

In the given question,

The unit rate describes Sergio's cycling speed by giving the distance he travels in a given amount of time, which is 6 miles per hour. This means that for every hour he cycles, he travels a distance of 6 miles.

By expressing Sergio's cycling speed as a unit rate, we can easily compare it to other speeds and determine how long it will take him to travel a certain distance.

For example, if Sergio increases his target speed to 8 miles per hour, we can use the unit rate to calculate how much farther he must cycle in a given amount of time.

If he wants to cycle for 2 hours, we know that he will travel 6 x 2 = 12 miles at his original speed of 6 miles per hour.

If he wants to cycle for the same 2 hours at a speed of 8 miles per hour, we can use the unit rate to calculate that he will travel 8 x 2 = 16 miles.

This means that he must cycle an additional 4 miles to reach his target distance.

Overall, the unit rate is a helpful tool for comparing speeds and calculating distances traveled in a given amount of time.

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Yes, every odd integer can be written as the difference of two squares.

To prove this, let k be a positive integer. Then the difference of the squares of k+1 and k is (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1, which is an odd integer. Thus, every odd integer can be written as the difference of two squares.

To prove this, we first chose a positive integer, k. We then found the difference of the squares of k+1 and k to be (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1. Since 2k + 1 is an odd integer, it follows that every odd integer is the difference of two squares.

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The calculation shows that 9,000 scientists have seen the deep ocean through Alvin's windows; and

a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.

1) If on each of the 4,500 dives Alvin carried a new pilot and two new scientists, then the total number of scientists who have seen the deep ocean through Alvin's windows is:

4,500 dives x 2 scientists per dive = 9,000 scientists

Therefore, 9,000 scientists have seen the deep ocean through Alvin's windows.

2) To calculate the total number of people who traveled in Alvin, we can add the number of pilots and scientists on each dive and multiply by the number of dives:

4,500 dives x (1 pilot + 2 scientists)

= 4,500 x 3

= 13,500 people

Therefore, a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.

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Step-by-step explanation:

a) Area=144m²

side²= 144

side=12m

b) perimeter=32m

4×side=32

side=32/4

side=8m

Answer: x = -2

Step-by-step explanation:

Since both equations are in the y-slope form,

we can use substitution for y in finding x.

Hence,

4x=x-6

4x-x=x-x-6

Subtract x from both sides to get x on one side and integer on one side.

[tex]\frac{3x}{3} =\frac{-6}{3}[/tex]

Divide 3 to find the value of x

x=-2

The value of the expression z + 3x4 using arithmetic operation where z = 15, is 27. The answer is A) 27.

The given expression is z + 3x4, where z = 15. To evaluate this expression, we substitute 15 for z and perform the multiplication. First, we multiply 3 and 4, which gives us 12. Then, we add 15 and 12 to get the final result of 27.

z + 3x4 = 15 + 3x4

= 15 + 12

= 27

Therefore, the value of the expression when z = 15, is 27. In other words, using arithmetic operation of multiplication and addition, which gives us the final answer of 27. So, the correct answer is option A).

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____The given question is incomplete , the complete question is given below:

Evaluate the expression, where z = 15

z + 3x4

A. 27

B. 32

C. 56

D. 1,304

The speed (in knots) at which the distance between the ships A and B is changing at 6 PM is given as 36 knots or 36 nautical miles per hour.

Consider that the ship A is in the west direction and the ship B is in the north direction and both the ships are in regular motion of speed which is 16 knots and 15 knots and the distance between them is 50 nautical miles.

Using the Pythagoras theorem, the relation of the distance x which represents the distance between ships at 6PM to the distances that each ship has travelled can be given as follows:

x^2 = (50 + 16t)^2 + (15t)^2

where, t is the number of hours that has passed since noon.

Differentiating both sides of the above equation with respect to time, we get:

2x*(dx/dt) = 2(50 + 16t)*(16) + 2*(15t)*(15)

t = 6, at 6 PM, therefore substituting the value and solving, we get:

2x(dx/dt) = 2[(50 + 16(6)]*(16) + 2*[15(6)]*(15)

2x(dx/dt) = 4194

dx/dt = 2097/x

Now substituting the value of x that corresponds to 6 PM:

x^2 = (50 + 16(6))^2 + (15(6))^2

x^2 = 3385

x = √3385 ≅ 58.19

Putting this value in dx/dt, we get:

dx/dt = 2097/58.19 ≅ 36.00 knots

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Answer:

-1/3

Step-by-step explanation:

Answer:

-1/3

Step-by-step explanation:

Using central limit theorem, the probability that the class of 20 students will have a mean completion time greater than 60 minutes on the unit 5 test is 0.00017332

We can use the Central Limit Theorem (CLT) to approximate the distribution of the sample mean completion time for the class. According to CLT, the distribution of the sample mean is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

In this case, the population mean is given as 48 minutes, the population standard deviation is given as 15 minutes, and the sample size is 20. Therefore, the mean of the sample mean completion time is also 48 minutes, and the standard deviation of the sample mean completion time is 15/√20 ≈ 3.3541 minutes.

To find the probability that the class mean completion time is greater than 60 minutes, we can standardize the distribution of the sample mean completion time using the z-score formula:

z = (x - μ) / (σ / √n)

where x is the value we want to find the probability for (in this case, x = 60), μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (60 - 48) / (15 / √20) = 3.5777

Using a standard normal distribution table (or calculator), we can find the probability that a z-score is greater than 3.5777.

P = 0.00017332

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The types of inferences that will be made about population parameters are causation, estimation, and regression on the basis of relationship.

Causation is the process of showing the cause-and-effect relationship between two variables. In this case, one variable influences the other variable. This type of inference is significant when making decisions because it helps us understand how a change in one variable leads to a change in another variable.

Estimation: In statistical analysis, estimation refers to determining the possible value of an unknown population parameter. It is impossible to calculate the population parameters directly, and hence we use sample statistics to estimate them.

Regression analysis is the statistical technique used to identify the relationship between two variables. It involves estimating the coefficients of the model that best fit the data.

This type of inference helps us predict the value of a dependent variable based on an independent variable.

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Answer:

54

Step-by-step explanation:

Answer:

y=54

as x=3

so y=2*x^3

y= 2*3^3

y=2*27

y=54

This procedure is repeated for each interval until the overall frequency of 35 is reached.

A data summary called a frequency distribution displays the frequency, or amount of occurrences, of each value or range of values in a data set. It is frequently displayed as a table or graph, with the values enumerated along one axis and the frequencies of those values listed along the other. The pattern or shape of a data collection can be described using frequency distributions, which can also be used to spot outliers or other unusual values. They can also shed light on the data's central trend and variability.

given

To complete a cumulative frequency table:

Class interval Frequency Cumulative frequency

0-10               5                                   5

10-20       8                                  13

20-30       12                                 25

30-40        7                                 32

40-50         3                                35

Total        35                               35

The number of the first class interval is 5.

For the second interval, we multiply the total frequency of 5 plus the frequency of 8 to get 13.

This procedure is repeated for each interval until the overall frequency of 35 is reached.

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The missing value in the cumulative table is 36.

The missing value in the cumulative table can be determined by examining the frequencies given in the table. Let's analyze the frequencies step by step:

1. The frequency for scores less than 145 is given as 16.
2. The frequency for scores less than 150 is given as 26.
3. The frequency for scores less than 155 is given as 36.
4. The frequency for scores less than 160 is not explicitly given in the table, but we can determine it by subtracting the frequency for scores less than 155 (36) from the frequency for scores less than 150 (26).

This gives us a value of [tex]26 - 36 = -10.[/tex]

Since a frequency cannot be negative, we can conclude that there is an error in the given table. The cumulative frequency for scores less than 160 should be 36 instead of -10.

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Answer:3.79*10^14

Step-by-step explanation:

370000000000000+9000000000000=379000000000000

=3.79 x 10^14

Answer:

(3.79×10^14)

Step-by-step explanation:

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The 12 months of age) as percentiles of the CDC growth chart reference population.

The most likely description of Sarah's weights (at 3, 6, 9, and 12 months of age) as percentiles of the CDC growth chart reference population is: 85th percentile at 3 months; 85th percentile at 6 months; 90th percentile at 9 months; 95th percentile at 12 months.What is percentile in statistics?In statistics, a percentile is a value below which a specific percentage of observations in a group falls. It is used to split up data into segments that represent an equal proportion of the entire group, resulting in a data set split into 100 equal portions, with each portion representing one percentage point. Sarah's weight is in the 85th percentile at 3 months, 85th percentile at 6 months, 90th percentile at 9 months, and 95th percentile at 12 months is a most likely description of her weights (at 3, 6, 9, and 12 months of age) as percentiles of the CDC growth chart reference population.

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Victor bought 10 packages of the largest-grain sandpaper.

Total money spent (20+15+9+6) = 50; total money left over (30+15+6+0) = 51. Balance plus spent always equals 50, but balance added to balance does not always equal 50. So, it is always necessary to include simply the amount spent or the expenditure rather than the balance.

Thus, we know how much he spent:

2S + (61 - 2S) = 61 - S

$1 on sandpaper with the biggest grit. We can condense this phrase as follows:

61 - S = 61 - 2S

Adding S to both sides, we get:

S = 0

Then we know that he spent:

2L + Ly = 61

Spending money on sandpaper. Additionally, since he purchased two packages of the finest sandpaper, the price of those two packages is:2S = 2L

We can substitute 2L for 2S in the first equation:

2L + Ly = 61

Simplifying, we get:

2L + L(2/3)L = 61

Multiplying both sides by 3/2, we get:

3L² = 91.5

Taking the square root of both sides, we get:

L ≈ 5.27

determine how many packages of the coarsest sandpaper Victor purchased:

2L + Ly = 61

2(5.27) + 5.27y = 61

10.54 + 5.27y = 61

5.27y = 50.46

y ≈ 9.56

Rounding to the nearest whole number, we get:

y = 10

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If an angle of measurement of 140° then; a circle is centered at the vertex of the angle, then the arc subtended by the angle's rays is 0.0233 cm times as long as 1/360th of the circumference of the circle. Also if a circle is centered at the vertex of the angle, and 1/360th of the circumference is 0.06 cm long then length of the arc subtended by the angle's rays 8.4 cm. Another circle is centered at the vertex of the angle then arc subtended by the angle's rays is 70 cm long,Therefore the circumference of the circle is 180 cm.

a.) To find the fraction of the circle's circumference subtended by the angle's rays, we divide the angle measure by 360 degrees:

fraction of circle's circumference = 140/360

Simplifying this fraction, we get:

fraction of circle's circumference = 7/18

To find the length of the arc subtended by the angle's rays, we multiply the fraction of the circle's circumference by the circumference of the circle. Let's call the circumference of the circle "C":

length of arc = (7/18)*C

We're also told that the length of 1/360th of the circumference is equal to 0.06 cm. So, we can write:

(1/360)*C = 0.06

Multiplying both sides by 360, we get:

C = 360*0.06 = 21.6 cm

Now, we can substitute this value of C into the expression for the length of the arc:

length of arc = (7/18)*C

length of arc = (7/18)*(21.6)

length of arc = 8.4 cm (rounded to one decimal place)

Therefore, the length of the arc subtended by the angle's rays is 8.4 cm.

b.) We're given that 1/360th of the circumference of the circle is 0.06 cm long. To find the length of the arc subtended by the angle's rays, we need to multiply 140/360 by 0.06:

length of arc = (140/360)*0.06

length of arc = 0.0233 cm (rounded to four decimal places)

Therefore, the length of the arc subtended by the angle's rays is approximately 0.0233 cm.

c.) We're told that the length of the arc subtended by the angle's rays is 70 cm. To find the circumference of the circle, we need to find the length of 1/360th of the circumference first. We can do this by dividing 70 by 1/360:

(1/360)*C = 70

Multiplying both sides by 360, we get:

C = 70*360 = 25,200 cm

Therefore, the circumference of the circle is 25,200 cm. We can also verify this by dividing the length of the arc by the fraction of the circumference subtended by the angle's rays:

length of arc = (7/18)*C

C = (18/7)*length of arc

C = (18/7)*70

C = 180 cm (rounded to one decimal place)

This is a different value than we got earlier, so we need to check our calculations. It turns out that the previous calculation was incorrect - we made a mistake when multiplying 7/18 by 21.6. The correct calculation gives us:

length of arc = (7/18)*C

length of arc = (7/18)*(21.6)

length of arc = 8.4 cm (rounded to one decimal place)

Now, we can calculate the circumference of the circle:

length of arc = (7/18)C

C = (18/7) *length of arc

C = (18/7) *70

C = 180 cm (rounded to one decimal place)

Therefore, the circumference of the circle is 180 cm.

Also, If an angle of measurement of 140° then; a circle is centered at the vertex of the angle, then the arc subtended by the angle's rays is 0.0233 cm times as long as 1/360th of the circumference of the circle.

b. A circle is centered at the vertex of the angle, and 1/360th of the circumference is 0.06 cm long.The length of the arc subtended by the angle's rays 8.4 cm

c. Another circle is centered at the vertex of the angle.

The arc subtended by the angle's rays is 70 cm long,Therefore the circumference of the circle is 180 cm.

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The proportion of students who can roll their tongues will be estimated and the margin of error for a 95 percent confidence interval for the true proportion of tongue rollers among students will be determined. There were 317 tongue rollers out of a sample of 400 students.

      As a result, the sample proportion is 317/400 = 0.7925.

      We'll compute the margin of error next. The margin of error (E) for a 95 percent confidence interval is:

                E = zα/2 * sqrt[p(1 - p) / n]

                              where zα/2 is the z-score that corresponds to the level of confidence α/2, p is the sample proportion, and n is the sample size.

               E = 1.96 * sqrt[0.7925 * (1 - 0.7925) / 400]E

                  = 1.96 * sqrt[0.7925 * 0.2075 / 400]E

                  = 1.96 * sqrt(0.00040875)E

                  = 1.96 * 0.0202E

                  = 0.0395

    The margin of error is approximately 0.04 or 4 percent. Hence, the correct option is 0.04.

The margin of error for a 95% confidence interval for the true proportion of tongue rollers among students is closest to 0.04

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Answer:

5/37

Step-by-step explanation:

There are 37 possible outcomes when rolling a 37-sided die, so the probability of rolling any one specific number is 1/37.

To find the probability of rolling any of the given numbers (35, 25, 33, 9, or 19), we need to add the probabilities of rolling each individual number.

Probability of rolling 35: 1/37

Probability of rolling 25: 1/37

Probability of rolling 33: 1/37

Probability of rolling 9: 1/37

Probability of rolling 19: 1/37

The probability of rolling any one of these numbers is the sum of these probabilities:

1/37 + 1/37 + 1/37 + 1/37 + 1/37 = 5/37

So the probability of rolling any of the given numbers is 5/37, which is approximately 0.1351 when rounded to four decimal places.

The answer of the given factor and multiplication term for the positive integers are 3.

Positive, negative, and zero digit whole numbers are all included in the category of integers. They can be stated without any fractional or decimal components and are a subset of the real numbers. On a number line, integers can be visualized as positive numbers to the right of zero and negative numbers to the left.

In mathematics, a multiplication term is a mathematical expression that involves the multiplication of two or more factors. The factors can be numbers, variables, or a combination of both. Multiplication terms are commonly represented using the multiplication symbol "×" or the dot "." symbol.

For example, in the expression 2x × 3y, the multiplication term is 2x × 3y, which involves the multiplication of the factors 2x and 3y.

If you want to find the smallest possible value of the given term we have that,The smallest factor of 15 is 1.

The smallest multiple of 3 is 3so, the smallest answer is 1x3 = 3.

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Answer:

A random sample of size 64 is used to test the null hypothesis that for certain age group the mean score on an achievement test is less than or equal to 40 against the alternative that it is greater than 40. The scores are assumed to be normally distributed with variance 0? 256 _ Consider the hypotheses Ha: L <40 versus HA Lt > 40 and suppose the null hypothesis is to be rejected if and only if the sample mean X exceeds 43.5. What is the size of this test? Compute the probability of type Il error at L = 42

Step-by-step explanation:

As a result, the Styrofoam collar has a volume of roughly 179.594 cubic inches.

The quantity of space occupied by a three-dimensional object is measured by its volume. Units like cubic meters (m3), cubic centimeters (cm3), or cubic inches (in3) are frequently used to quantify it. Depending on the shape of the item, different formulas can be used to determine its volume. For instance, the volume of a cube can be calculated by multiplying its length, breadth, and height, while the volume of a cylinder can be calculated by dividing the base's area (typically a circle) by the cylinder's height.

given

We must apply the calculation for the volume of a cone's frustum in order to determine the volume of the Styrofoam collar:

[tex]V = (1/3)\pi h(R^2 + Rr + r^2)[/tex]

where h is the height of the frustum, r is the small radius, and R is the large radius.

Given the numbers, we can determine:

R = 5 in.

3 centimeters is r.

24 inches tall

With these numbers entered into the formula, we obtain[tex]V = (1/3)\pi (24)(5^2 + 5*3 + 3^2)\\\\ 179.594 cubic inches[/tex]

As a result, the Styrofoam collar has a volume of roughly 179.594 cubic inches.

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The answer is c): 17 and one-half quarts

To answer the question, we need to find the total amount of juice Sam made by adding the given quantities. However, the given quantities are in different units, which makes the addition difficult. Therefore, we need to convert all quantities to the same unit before adding them.

1 gallon = 4 quarts (since 1 gallon is equal to 128 ounces, and 1 quart is equal to 32 ounces,

thus 1 gallon = 128/32 = 4 quarts)

1 quart = 2 pints

1 pint = 2 cups

Therefore, 3 gallons = 3 x 4 = 12 quarts

2 quarts = 2 x 1 = 2 quarts

4 pints = 4 / 2 = 2 quarts

6 cups = 6 / 4 = 1.5 quarts

Now, we can add all the quantities in quarts to get the total amount of juice that Sam made:

12 + 2 + 2 + 1.5 = 17.5 quarts

Therefore, Sam made 17 and one-half quarts in all. Thus, the correct option is (c) seventeen and one half.

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The simplified square expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

What is square root ?

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 x 3 = 9.

The square root is denoted by the symbol √, also known as the radical symbol. For instance, the square root of 16 is written as √16 = 4.

The square root can be used to solve various types of equations, including quadratic equations and problems involving areas and volumes. It is also used in various fields such as physics, engineering, and finance.

According to the question:
To simplify the expression [tex]\sqrt{3a^{2}/10b^6}[/tex], we can first separate the numerator and denominator inside the square root:

[tex]\sqrt{3a^2/10b^6} = \sqrt{3a^2}/\sqrt{10b^6}[/tex]

Next, we can simplify the square root of the numerator:

[tex]\sqrt{3a^2} = |3a|,[/tex] where |За| represents the absolute value of За.

Finally, we can simplify the square root of the denominator by factoring out the perfect square[tex]b^2[/tex]:

[tex]\sqrt{10b^6} = \sqrt{10} * \sqrt{b^6} = \sqrt{10} * b^{3}[/tex]

Substituting these values back into the original expression, we get:

[tex]\sqrt{3a^2/10b^6} = |3a| / \(sqrt{10} * b^3[/tex]

Therefore, the simplified expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

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Grace need to buy the jump rope, the game, and the ball $8.

$7 will get you a jump rope.

$10 will get you a board game.

$4 will get you a playground ball.

total amount to be spent: $7 + $10 + $4 = $21

She has ten dollars.

$3 was all her uncle gave her.

13 dollars are all she has.

She needed $8, therefore 21 - 13 = $8.

a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.

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The solutions to the equation are x = -4 and x = 1.

The quadratic formula, which is often employed in the disciplines of mathematics, physics, engineering, and other sciences, is a potent tool for resolving quadratic problems. We must first get the values of a, b, and c from the quadratic equation in order to apply the quadratic formula. To get the answers for x, we then enter these values as substitutes in the formula and simplify.

The given equation is 4x² + 6x - 13 = 3x².

Rearranging the equation we have:

x² + 6x - 13 = 0

The quadratic formula is given as:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values of a = 1, b = 6, and c = -13.

x = (-6 ± √(6² - 4(1)(-13))) / 2(1)

x = (-6 ± √(100)) / 2

x = (-6 ± 10) / 2

x = -8/2 or x = 2/2

x = -4 or x = 1

Hence, the solutions to the equation are x = -4 and x = 1

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To prepare a buffer solution with a given pH, we need to choose a weak acid and its conjugate base, such that the pKa of the weak acid is close to the desired pH.

The pKa is related to the Ka value as follows:

pKa = -log(Ka)

So, for each of the weak acids given, we can calculate the pKa:

Phosphoric acid (H3PO4): Ka = 7.5 x 10^-3, so pKa = -log(7.5 x 10^-3) = 2.12

Acetic acid (CH3COOH): Ka = 1.8 x 10^-5, so pKa = -log(1.8 x 10^-5) = 4.74

Formic acid (HCOOH): Ka = 1.8 x 10^-4, so pKa = -log(1.8 x 10^-4) = 3.74

Now, let's consider the desired pH values and choose the best components for buffer solutions:

For a pH of 2.5, the best choice would be phosphoric acid (pKa = 2.12).

For a pH of 4.5, the best choice would be formic acid (pKa = 3.74) or a mixture of acetic acid and acetate ion (CH3COOH/CH3COO-, pKa = 4.76).

For a pH of 6.5, the best choice would be a mixture of acetic acid and acetate ion (CH3COOH/CH3COO-, pKa = 4.76).

Note that a buffer solution can be prepared by mixing a weak acid and its conjugate base in roughly equal amounts, so the appropriate salt can be added to the acid to form the buffer solution. For example, to prepare an acetate buffer, one could mix acetic acid with sodium acetate.

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