recipe for sparkling grape juice calls for 1 1/2 quarts of sparkling water and 3/4 quart of grape juice, how much grape juice would you need for 15/4 quarts of sparkling water

Derivative of six inverse trigonometric functions are as follow:

d/dx(sin⁻¹x) = 1 / √1 - x²

d/dx(cos⁻¹x) = -1 / √1 - x²

d/dx(tan⁻¹x) = 1 / √1 + x²

d/dx(cosec⁻¹x) = -1 / |x|√ x² - 1

d/dx(sec⁻¹x) = 1 / |x|√ x² -1

d/dx(cot⁻¹x) = -1 / √1 + x²

As given in the question,

Explanation of the derivative of six inverse trigonometric functions are as follow:

1. y = sin⁻¹x

⇒ x = siny

⇒d/dx(x) = d/dx(siny)

⇒ 1 = cosy dy/dx

⇒dy/dx = 1/cosy

⇒dy/dx = 1/√1-sin²y

⇒dy/dx = 1/√1-x²

⇒d/dx( sin⁻¹x) = 1/√1-x²

2. y = cos⁻¹x

⇒ x = cosy

⇒d/dx(x) = d/dx(cosy)

⇒ 1 =-siny dy/dx

⇒dy/dx = -1/siny

⇒dy/dx = -1/√1-cos²y

⇒dy/dx = -1/√1-x²

⇒d/dx( cos⁻¹x) = -1/√1-x²

3. y = tan⁻¹x

⇒ x = tany

⇒d/dx(x) = d/dx(tany)

⇒ 1 =sec²y dy/dx

⇒dy/dx = 1/sec²y

⇒dy/dx = 1/√1+ tan²y

⇒dy/dx = 1/√1 + x²

⇒d/dx( tan⁻¹x) = 1/√1+x²

4. y = cosec⁻¹x

⇒ x = cosecy

⇒d/dx(x) = d/dx(cosecy)

⇒ 1 =-cosecy coty dy/dx

⇒dy/dx = -1/cosecy coty

⇒dy/dx = -1/cosecy√cosec²y-1

⇒dy/dx = -1/|x|√x²-1

⇒d/dx( cosec⁻¹x) = -1/|x|√x²-1

5.   y = sec⁻¹x

⇒ x = secy

⇒d/dx(x) = d/dx(secy)

⇒ 1 =secy tany dy/dx

⇒dy/dx = 1/secy tany

⇒dy/dx = 1/secy√sec²y-1

⇒dy/dx = 1/|x|√x²-1

⇒d/dx( sec⁻¹x) = 1/|x|√x²-1

6. y = cot⁻¹x

⇒ x =coty

⇒d/dx(x) = d/dx(coty)

⇒ 1 =-cosec²y dy/dx

⇒dy/dx = -1/cosec²y

⇒dy/dx = -1/√1+ cot²y

⇒dy/dx = -1/√1 + x²

⇒d/dx( cot⁻¹x) = -1/√1+x²

Therefore, the derivative of the six inverse trigonometric functions are given below:

d/dx(sin⁻¹x) = 1 / √1 - x²

d/dx(cos⁻¹x) = -1 / √1 - x²

d/dx(tan⁻¹x) = 1 / √1 + x²

d/dx(cosec⁻¹x) = -1 / |x|√ x² - 1

d/dx(sec⁻¹x) = 1 / |x|√ x² -1

d/dx(cot⁻¹x) = -1 / √1 + x²

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Answer: 2.12 miles.

Step-by-step explanation: To find the width of Hardee County, we need to determine the length of one of its sides. We can use the fact that the area of a rectangle is equal to its length times its width to find the length of one of its sides.

The area of Hardee County is 637 square miles, and we know that its width is 30 miles. We can use this information to write the following equation: 637 square miles = 30 miles * length

We can solve this equation for the length by dividing both sides by 30: 637 square miles / 30 miles = length

This simplifies to: 21.2 square miles = length

Since the length of a rectangle is one of its sides, the width of Hardee County is 21.2 miles.

Answer:Y=6

Step-by-step explanation:

I will be using the substitution method:

x+y=9 y=x+3

X + (x + 3) = 9

2x + 3 = 9

*subtract 3 from both sides*

2x = 6

*divide by 2 on each side of the equal sign*

X = 3

find y:

y = 3 + 3

y = 6

check:

3 + 6 = 9

9 = 9 (correct)

The value of x is 3 and the value of y is 6

Answer:

The mass of the 30,000 molecules is 1.59\times 10^{-18}1.59×10

−18

Given that,

There are 30,000 molecules.

And, the mass of one molecule of oxygen i.e. 5.3\times 10^{-23}5.3×10

−23

gram.

Based on the above information, the calculation is as follows:

\begin{gathered}= 30,000 \times 5.3\times 10^{-23}\\\\= 3\times10^4 \times 5.3\times 10^{-23}\\\\= 15.9\times 10^{-23+4}\\\\= 15.9\times 10^{-19}\\\\= 1.59\times 10^{-19}\times 10\\\\= 1.59\times 10^{-18}\end{gathered}

=30,000×5.3×10

−23

=3×10

4

×5.3×10

−23

=15.9×10

−23+4

=15.9×10

−19

=1.59×10

−19

×10

=1.59×10

−18

Therefore we can conclude that the mass of the 30,000 molecules is 1.59\times 10^{-18}1.59×10

−18

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Answer:

top

Step-by-step explanation:

In the bottom, y=5x for all of the columns. In the top, y=2x for all except third column, so it is not proportional.

Answer: Note that log(a) + log(b) is the same thing as log(a × b) This gives us: log7(x(x +5)) = log7(14)

Answer:

Step-by-step explanation:

its x= 1

13-12=1

7-6=1

and i multiplied 13(1)-12 which equals one and 7(1)-6 which also equals one, the answers just have to match with each other

The value of [x] is equivalent to 9.5.

Given are two triangles ABC and DEF as shown.

Since AB is congruent to DE, then -

AB = DE

2x - 1 = 18

2x = 19

x = 9.5

Therefore, the value of [x] is equivalent to 9.5.

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The logarithmic expression as a single logarithm is [tex]log_{2}[/tex](a[tex]b^{2}[/tex]/[tex]c^{3}[/tex])

A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction.

[tex]log_{2}a + 2log_{2}b - 3log_{2}c[/tex]

[tex]2log_{2}b[/tex] may be simplified as [tex]log_{2}b^{2}[/tex]

[tex]3log_{2}c[/tex] may be simplified as [tex]log_{2}c^{3}[/tex]

Which reduces the expression as

[tex]log_{2}a[/tex] +  [tex]log_{2}b^{2}[/tex] -  [tex]log_{2}c^{3}[/tex]

factor out log2 since it is common

and replace addition sign with multiplication and subtraction sign with division

[tex]log_{2}(\frac{a * b^{2} }{c^{3} })[/tex] which finally becomes [tex]log_{2}(ab^{2} /c^{3})[/tex]

In conclusion, the simplified form of the logarithmic expression is  [tex]log_{2}(ab^{2} /c^{3})[/tex]

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After printing the long pdf Document of 47 pages , the remained balance in Fathi's printing account is -$4.90 ..

In mathematics, a basic algebraic operation is any of the common arithmetic operations, including addition, subtraction, multiplication, division, raising integers to powers, and taking square roots.

We have given that,

Total or initial balance of fathi's printing account

= $1.10

Each sheet of paper reduce his balance by $0.25..

Total number of pages in pdf document

= 47

He wants to print this pdf document.

When he print both sides of paper then it reduce the counts of paper used for printing the pdf.

That's if he print only one side then he will print 47 pages but now, he print on both sides .

so, total pages used to print pdf Document= 47/2

= 23 both sided printed + 1 one side printed = 24 pages

Using all provided details, we get

After Fathi print, the balance in his printing account = Initial balance - 24 × balance reduce by each page print

= $1.10 - 24 × $0.25 = $1.10 - $ 6

= - $4.90

Hence, - $4.90 balance in his printing account.

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The possible values of (m,n) are (1,2) and (-17,14).

A system having more than two equations is known as a system of equations.

Given that, the solution of the simultaneous equations is x = 2 and y = 3.

The given equations are:

mx + ny = 8

m²x - n²y = -10

Substitute x = 2 and y = 3 into the above equations:

2m + 3n= 8       (1)

2m² - 3n² = -10     (2)

Rewrite equation (1):

2m + 3n= 8

2m = 8 - 3n

m = (8 - 3n)/2          (3)

Substitute it into equation (2):

[tex]2\frac{(8-3n)^2}{4}-3n^2 = -10\\\\2(64 +9n^2-48n)-12n^2+40 =0\\\\128 +18n^2-96n-12n^2+40=0\\\\6n^2-96n+168=0\\\\n^2-16n+28=0\\\\n^2-14n-2n+28=0\\\\n(n-14)-2(n-14)=0\\\\(n-2)(n-14)=0[/tex]

n = 2 or n =14

Now, substitute n = 2 into equation (3):

m = (8-3×2)/2

m = 1

Further, substitute n = 14 into equation (3):

m = (8-3×14)/2

m = -17

Hence, the possible values of (m,n) are (1,2) and (-17,14).

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The equations that have the same solution as 2x-5=15 include :

2x = 20

2x - 20 = 0

4x - 10 = 30

The options are B, D, E.

We have,

2x - 5 = 15

Here,

The solution is 10.

2x = 15 + 5

2x = 20

x = 20/2

x = 10

We have,

A. 2x = 10

x = 10/2

x = 5

B. 2x = 20

x = 20/2

x = 10

C. 2(x - 5) = 15

2(x - 5) = 15

x - 5 = 15/2

x = 15/2 + 5

x = 12.5

D. 2x - 20 = 0

2x = 20

x = 20/2

x = 10

E. 4x - 10 = 30

4x = 30 + 10

4x = 40

x = 40/4

x = 10

F. 15 = 5 - 2x

15 - 5 = -2x

10 = -2x

x = -10/2

x = -5

The correct options are B, D and E.

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Complete question

What equations have the same solution as 2x - 5 = 15?

A. 2x = 10

B. 2x = 20

C. 2(x - 5) = 15

D. 2x - 20 = 0

E. 4x - 10 = 30

F. 15 = 5 - 2x

The domain of the rational function  [tex]\frac{x^3 + 5x^2 + 3x - 9}{x^2 -x - 2}[/tex] is:

D: {x ∈ R| x ≠ -1, 2}

The domain will be the set of all real numbers, minus these ones that make the denominator equal to zero.

The denominator will be zero when:

x² - x - 2  =0

Using the quadratic formula we will get:

[tex]x = \frac{1 \pm \sqrt{(-1)^2 - 4*1*-2} }{2*1} \\\\x = \frac{1 \pm 3}{2}[/tex]

The two solutions are:

x = (1 + 3)/2 = 2

x = (1 - 3)/2 = -1

If the numerator is not zero on these values, then we need to remove them from the domain.

When x = 2 the numerator is:

2³ + 5*2² + 3*2 - 9 = 25

When x = -1

(-1)³  + 5*(-1)²  + 3*-1 - 9 = -8

Neither of these is zero, so we need to remove x = -1 and x = 2 from the domain, then the domain is:

D: {x ∈ R| x ≠ -1, 2}

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The type of study was cross-sectional study.

An example of an observational study design is the cross-sectional study design. In a cross-sectional study, the researcher simultaneously assesses the participants exposures and outcomes.

After choosing the study subjects, the researcher conducts the study in order to evaluate the exposure and results.

For population-based surveys and to determine the prevalence of illnesses in clinic-based samples, cross-sectional designs are employed.

These studies are typically affordable and reasonably quick to carry out.

An illustration of a cross-sectional study might be a medical investigation into the incidence of cancer in a particular community.

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Answer:

Step-by-step explanation:

ben

Answer:

Not sure, maybe it's 5?

Step-by-step explanation:

[tex]26-11=15[/tex]

[tex]15[/tex] ÷ [tex]3=5[/tex]

Distance from Detroit to columbus is 220 miles

It is given in the question,

The average speed Mrs. smith drove = 60 mph.

Returning, her average speed was 55 mph.

Time taken =  1/3 of an hour longer on the return trip

We have to Find total distance between Detroit and Columbus

So, Let the distance between Detroit and Columbus be "x" miles

Average speed = 60

So, time taken by her to cover distance of x miles with speed of 60 mph

Time = Distance / speed

=> t = x / 60

She Took x/60 hours to reach Columbus

While returning her speed was 55 mph.

So, time taken by her to cover the same distance but at a speed of 55 mph :

=> t = x / 55

Now we are given that she took 1/3 of hour more while returning .

So, [tex]\frac{x}{55} - \frac{x}{60} = \frac{1}{3}[/tex]

⇒ x/660 = 1/3

=> x = 660/3

=> x = 220 miles

Hence ,  Detroit is 220 miles far from Columbus

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The probability of rolling a winning combination will be 7/12 as the winning combination is 21 and Total combination will be 36.

The probability is computed by dividing the total number of possible outcomes by the number of possible ways the event might occur. Probability and odds are two distinct ideas. Odds are calculated by dividing the likelihood of an event by the likelihood that it won't. By dividing the number of positive outcomes by the entire number of potential outcomes, probability—the likelihood that an event will occur—is determined. The most basic illustration is a coin toss. There are only two outcomes when you flip a coin: either it comes up heads or tails.

Here,

If the first die is red and the second one is green, then the sample space S is the following:

S=

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2, 1), (2,2), (2,3), (2,4), (2,5), (2,6), (3, 1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5, 1), (5,2), (5,3), (5,4), (5,5), (5,6), (6, 1), (6, 2), (6,3), (6,4), (6,5), (6,6)

Hence n(S)=36.

Let's list the losing combinations:

E={(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (2, 1),

(2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6,5)}

n(E)=15

There are 36 total possiblities.

Out of which, winning probability is 21.

=21/36

=7/12

Given that the winning combination is 21 and the total combination is 36, the chance of rolling a winning combination is 7/12.

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The strength of the linear relationship between two numerical variables may be measured by the  coefficient of correlation

The correlation coefficient is a statistical indicator of how strongly two variables are linearly related. Its values may be between -1 and 1. A correlation coefficient of -1 indicates a complete negative, inverse, or inverse-correlation, where values in one series increase as those in the other fall and vice versa. A coefficient of 1 denotes a perfect positive correlation or a straight line. A correlation value of 0 indicates that there is no linear relationship.

In both science and finance, correlation coefficients are used to measure the degree of relationship between two variables, components, or data sets. For example, because high oil prices benefit crude producers, one might assume that the correlation between oil prices and forward returns on oil stocks is strongly positive.

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The profit increased at an average pace of $600 between Hour 5 to Hour 10.

An indicator of the how one dimension changes in proportion to another is a rate of change.

The table for the price is given as;

Hour                Profit

0                        $0

2.5                     $500

5                        $900

7.5                     $1200

10                        $1500

After 5 hours, the profit is 900, and after another 5 hours, the profit is 1500.

In order to determine that average rate of change, we must take 900 away from 1500.

1500-900 = 600

Thus, an average rate of change for profit during Hour 5 to Hour is therefore $600.

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Answer:

c = 20 degrees

Step-by-step explanation:

7c + c + c = 180

9c = 180

9/9 c = 180/9

c = 20

The total distance is 13 units he ran during these sessions

The length along a line or line segment between two points on the line or line segment.

From the line plot, we have the following frequency table:

Distance   Frequency

0                    0

1/2                  4

1                     3

[tex]1\frac{1}{2}[/tex]                    1

2                    2

[tex]2\frac{1}{2}[/tex]                   1

3                    0

The total distance from the line plot is then represented as

Total distance = Sum of the product of distance and frequency

Total distance = (0×0) + (1/2 × 4) + (1 × 3) + ([tex]1\frac{1}{2}[/tex]  × 1) + (2 × 2) + ([tex]2\frac{1}{2}[/tex]  × 1) + (3 × 0)

Total distance = 13

Hence, the total distance is 13 units he ran during these sessions

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It will take 112.5 minutes to mow a lawn that measures 6000 sq ft .

In the question ,

it is given that ,

the time taken to mow 800 sq ft of the given lawn is = 15 minutes ,

So , the time taken to mow 1 sq ft of lawn is = 15/800 minutes .

to find the time taken to mow 6000 sq ft , we multiply by 6000 ,

We get ,

time taken to mow 6000 sq ft of the lawn is = 6000*(15/800)

on solving , we have

= 90000/8000

Simplifying further ,

we get ,

= 112.5 minutes

Therefore , 6000 sq ft lawn will be mowed in 112.5 minutes .

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18 months of All-Star Library membership cost $76 in total.

For the given equation.

The sign-up fee for the library-all-star-members club = $40.

Let 'x' be the total number of months.

Let 'y' be the total cost.

Monthly fee = $2.

Then, the linear equation models is-

y = 2x + 40

Given that a particular club member has been a part of it for 19 months, we replace x with the number 19 as follows:

y = 2x + 40

y = 2 * 18 + 40

y = = 36 + 40

y = 76.

Thus, 18 months of All-Star Library membership cost $76 in total.

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The complete question is-

3. In order to become a member of the library-all-star-members club, there is a $40 sign-up fee and a $2 monthly fee. Write an equation in slope-intercept form that models this situation. Use equation you wrote in problem 3 to find the total cost of being an all-star library member for 18 months.

Two linear equations in two variables can be solved using the substitution techniques, graphical method or by the elimination method.

Let us consider any two real world linear equations in two variables.

x - y = 0

x + y = 4

Now let us first solve them by substitution method:

equation 1 can be written as : x = y ,

Now we substitute this value of x in the second equation:

x + y = 4

or,  y + y = 4

or, 2y = 4

or, y = 2

At x = 2 , y = 2. Hence the solution is (2,2)

Now let us solve by elimination method:

adding equation 1 and 2 we get:

x - y = 0

x + y = 4

2x = 4

or, x =2

at x=2 , y = 4-2 =2 Hence the solution is (2,2)

Now we will solve the equations graphical method.

In the graph attached below we can clearly see that the straight line representation of the two equations intersect at the point (2,2)

Hence the solution is (2,2) .

Therefore any linear equation in 2 variables can be solved by substitution , elimination and graphically.

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If each side of a square is increasing at a rate of 7 cm/s, then the rate of increasing the area of the the square when the area of the square is 64 cm square is 112 centimeter square per second

The rate of change of length with respect to the time = 7 cm per second

That is

dl / dt = 7 cm per second

The area of the square = l^2

Where l is the length of the square

The rate of change area with respect to the time is

dA / dt = 2l dl/dt

The area is given that 64 centimeter square and the side will be 8 cm

Substitute the values in the equation

dA / dt = 2 × 8 × 7

= 112 centimeter square per second

Therefore, the rate of increasing the area is 112 centimeter square per second

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The value of n in 1:n is 4.5 in the given equivalent ratio.

What is equivalent ratio?

The concept of a ratio in mathematics is the divisional comparison of two quantities, the antecedent and consequent. As an illustration, each ingredient must be added according to a ratio during cooking. So, we may argue that a ratio is employed to represent one quantity as a portion of another. The ratio can be written as a fraction as well. If the ratio a:b is a fraction, its form is a/b. As a result, it is simple to compare two or more equivalent ratios expressed as equivalent fractions.

The given ratio is 8:36.

In fraction form = 8/36

= 1 / 4.5

=1:4.5

Hence, the value of n in 1:n is 4.5.

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Prediction interval is more relevant than confidence interval.

What is Prediction Interval?

A prediction interval is an estimate of the range within which a future observation will fall, with a particular probability, given what has already been observed, in statistical inference, more especially predictive inference.

Short term measures:

Undoubtedly, short-term decisions and actions serve as important starting points for both real accomplishments and symbolic commitments. Taking into account the challenges, assets, and weaknesses that have been identified, a number of immediate activities seem to have the highest chance of fostering intelligent growth in the long run.

Long term measures:

The phrase "Long-Term Performance Measures" refers to the performance metrics developed and employed by the Committee to evaluate the Company's performance and ascertain the quantity of Common Stock to be paid to the Participant at the end of a Restriction Period.

Here,

Prediction intervals must take into consideration both the random variation of the individual values and the uncertainty in predicting the population mean. Consequently, a prediction interval is always larger than a confidence interval. Additionally, as the sample size grows, the prediction interval won't condense to a single value.

Hence, Prediction interval is more relevant than confidence interval.

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