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The expression for r in terms of R is: r = (1-R²) / R²

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 multiplied by itself gives 9:

3 x 3 = 9

The square root symbol is √, and the number under the symbol is called the radicand. So, √9 is read as "the square root of 9."

Starting with the given equation:

R = √(r-1) / √(r+1)

Let's first clear the square roots by squaring both sides:

R² = (r-1) / (r+1)

Now, let's multiply both sides by (r+1) to eliminate the denominator on the right-hand side:

R²(r+1) = r-1

Expanding the left-hand side:

R²r + R² = r-1

Subtracting R²r from both sides:

R² - R²r = -1

Factoring out R² on the left-hand side:

R²(1-r) = -1

Dividing both sides by (1-r):

R² = -1 / (1-r)

Finally, we can take the square root of both sides and multiply by -1 to isolate r:

r = -1 / R² + 1

Therefore, the expression for r in terms of R is:

r = (1-R²) / R²

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Make r the subject of the following:

R= √(r-1) / √(r+1)

The probability that the number of lost-time accidents occurring over a period of 8 days will be no more than 4 is 0.2027, or approximately 20.27%.

To solve this problem, we can use the Poisson distribution formula, which is as follows:

P(X ≤ 4) = ∑(k=0 to 4) [(e^-λ * λ^k) / k!]

where λ is the mean rate of lost-time accidents per day, and X is the number of lost-time accidents occurring over a period of 8 days.

Substituting the given values, we get:

λ = 0.7 * 8 = 5.6

P(X ≤ 4) = ∑(k=0 to 4) [(e^-5.6 * 5.6^k) / k!]

Using a calculator, we can evaluate this probability as:

P(X ≤ 4) = 0.2027 (rounded to four decimal places)

In conclusion, the Poisson distribution can be used to calculate the probability of a certain number of events occurring over a given time period, given the mean rate of occurrence per unit time.

In this case, we used the Poisson distribution to calculate the probability of a certain number of lost-time accidents occurring over an 8-day period, given the mean rate of lost-time accidents per day.

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When the Rhoads family fills their above-ground pool with approximately 3,205 gallons of water, they add a weight of approximately 503,432.61 pounds to the pool.

We can use the given weight of water per gallon to find the total weight of water added to the pool:

Weight of 1 gallon of water = 8.34 pounds

Number of gallons of water in the pool = 3,205 gallons

Radius of the pool = 6 feet (half of the diameter)

Volume of the pool = π × (Radius)^2 × Depth

The depth of the pool is not given, so let's assume it is 4 feet (a common depth for above-ground pools):

Volume of the pool = π × (6 feet)^2 × 4 feet

Volume of the pool ≈ 452.39 cubic feet

Number of gallons of water in the pool = Volume of the pool ÷ 7.48

Number of gallons of water in the pool ≈ 60,381.71 gallons

Total weight of water added to the pool = Weight of 1 gallon of water × Number of gallons of water in the pool

Total weight of water added to the pool ≈ 503,432.61 pounds

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

A gallon of water weighs 8.34 pounds. The Rhoads family has a round, 12-foot diameter, above-ground pool. How much weight is added to the pool when it is filled with 3, 205 gallons of water?

Answer:

Y-intercept → -6

Slope → -1/4

Step-by-step explanation:

#1 Put your information into slope-intercept form or  "y=mx+b" form where

and so:

y= m x + b

y=-6 x  - [tex]\frac{1}{4}[/tex]

GOOD LUCK!!!!

Step-by-step explanation:

Let's start by defining some variables:

y: the maximum amount Geno can spend

x: the number of months he will have the gym membership

T: the total cost of the membership at Total Fitness

G: the total cost of the membership at Gymania

Using these variables, we can set up the following system of inequalities:

T = 30x + 100 (Total Fitness charges $30 per month plus an initial fee of $100)

G = 50x + 25 (Gymania charges $50 per month plus an initial fee of $25)

Geno can spend no more than y dollars, so we can add the following constraint:

T ≤ y

G ≤ y

Now we can solve this system of inequalities to find out which company offers the better deal. We can start by substituting the expressions for T and G:

30x + 100 ≤ y

50x + 25 ≤ y

Next, we can simplify these inequalities:

30x ≤ y - 100

50x ≤ y - 25

Finally, we can solve for x:

x ≤ (y - 100) / 30

x ≤ (y - 25) / 50

The better deal is the gym membership that has the smaller total cost, so we want to find the values of x that satisfy both inequalities. Therefore, we need to take the smaller of the two right-hand sides:

x ≤ min((y - 100) / 30, (y - 25) / 50)

So, the system of inequalities we can use to determine which company offers the better deal is:

x ≤ min((y - 100) / 30, (y - 25) / 50)

To create an 80% alcohol solution, the pharmacist must therefore mix 2.17 cm3 of pure alcohol with 10 cm3 of the 8% alcohol solution.

A value or combination of values that satisfy an equation or system of equations are referred to as solutions in mathematics. For instance, if we substitute x = 2 into the equation, we get 2(2) + 3 = 7, which is a true statement, hence the answer to the equation 2x + 3 = 7 is x = 2. Similar to this, an equation system may have one or more solutions that simultaneously fulfil every equation in the system. Finding answers to equations or systems of equations is a key component of many branches of mathematics and has significant applications.

given

Find out how much pure alcohol is now contained in the 8% solution to start.

An 8% alcohol solution in 10 cm3 contains:

There are 0.8 cm3 of pure alcohol in 0.08 x 10 cm3.

Let's now calculate the amount of pure alcohol that has to be added to achieve an 80% solution using the alligation method.

We must add pure alcohol to the solution to raise the concentration from 8% to 100%. In order to connect 100% to 8% in the left column, we place 100% in the right column. There is a 92% discrepancy between these two percentages.

We put 80% in the middle column because we aim to arrive at an 80% solution. Between 80% and 100%, there is a 20% difference.

Now, we may construct the subsequent equation:

20/92 = x/10

After finding x, we obtain:

[tex]x = 2.17 cm^3[/tex]

To create an 80% alcohol solution, the pharmacist must therefore mix 2.17 cm3 of pure alcohol with 10 cm3 of the 8% alcohol solution.

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As a general rule, a Z-score of - 3.0 to 2.0 recommends that a stock is exchanged inside three standard deviations of its mean.

A Z-Score is a statistical estimation of a score's relationship to the mean in a gathering of scores.

A Z-score can uncover to a merchant on the off chance that worth is normal for a predefined informational collection or on the other hand assuming it is abnormal.

As a general rule, a Z-score of - 3.0 to 2.0 recommends that a stock is exchanged inside three standard deviations of its mean.

Merchants have created numerous techniques that utilize z-score to distinguish connections between's exchanges, and exchanging positions, and assess exchanging systems.

The higher (or lower) a z-score is, the further away from the mean the fact is. This isn't really positive or negative; it just shows where the information lies in a regularly conveyed test. This implies it comes down to inclination while assessing speculation or opportunity. For instance, a few financial backers utilize a z-score scope of - 3.0 to 3.0 on the grounds that 99.7% of regularly conveyed information falls here, while others could utilize - 1.5 to 1.5 in light of the fact that they favor scores nearer to the mean.

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The probability of exactly 3 items out of 15 being defective is 0.184 or approximately 18.4%.

What is Probability ?

Probability can be defined as ratio of number of favourable outcomes and total number outcomes.

To solve this problem, we can use the binomial distribution, which is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

In this case, the probability of a single item being defective is 10%, or 0.1, and the probability of a single item being non-defective is 90%, or 0.9. We want to know the probability of getting exactly 3 defective items out of a sample of 15 items.

Using the binomial distribution formula, we can calculate this probability as follows:

P(X = 3) = (15 choose 3) * [tex]0.1^3[/tex] *[tex]0.9^12[/tex]

where (15 choose 3) is the number of ways to choose 3 items out of 15, which is given by the binomial coefficient:

(15 choose 3) = 15! / (3! * 12!) = 455

Substituting these values into the formula, we get:

P(X = 3) = 455 *  [tex]0.1^3[/tex] *[tex]0.9^12[/tex]

Therefore, the probability of exactly 3 items out of 15 being defective is 0.184 or approximately 18.4%.

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Answer:

C 9:8

Step-by-step explanation:

total students = 36

5+4=9

5:4 = 20:16

20-2=18

ratio for monday = 18:16

=9:8

Answer:

The answer is 2.

Overall, the shape of the magnetosphere is determined by the interaction of the solar wind and the Earth's magnetic field.

The shape of Earth's magnetosphere is modified by the solar wind. The Earth's magnetosphere is a protective magnetic shield around the Earth that protects us from the harmful particles and radiation from the Sun. The magnetosphere is not a perfect sphere, but rather a complex shape that is constantly changing due to the interaction of the solar wind and the Earth's magnetic field.The solar wind is a continuous stream of charged particles, mostly protons and electrons, that are ejected from the Sun's outer atmosphere. When these charged particles come into contact with the Earth's magnetic field, they are deflected around the Earth, forming a bow shock in front of the magnetosphere.

The magnetosphere then acts as a barrier, trapping many of the charged particles and preventing them from reaching the Earth's surface. However, some particles are able to penetrate the magnetosphere and reach the upper atmosphere, where they can cause auroras and other phenomena.The shape of the magnetosphere is constantly changing due to the changing conditions in the solar wind.

For example, during periods of high solar activity, the magnetosphere can become compressed and distorted, leading to more auroras and other phenomena. During periods of low solar activity, the magnetosphere can expand and become more symmetrical.

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According to the solution we have come to find that, The missing sides are t=ST=9 and s=RT=12.

what is right angle triangle?

A right angle triangle, also known as a right triangle, is a triangle that has one angle measuring 90 degrees, which is also known as a right angle. The side opposite to the right angle is called the hypotenuse, and the other two sides are called legs or catheti. The Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, is a fundamental property of right triangles. Right triangles are important in mathematics and physics, and they have many applications in geometry, trigonometry, and calculus.

We can use the fact that the two triangles are similar to set up a proportion and solve for the missing sides.

AB/RS = BC/ST = AC/RT

Substituting the given values:

5/RS = 3/9 = 4/s

Solving for RS:

RS = (5/3) * 9 = 15

Solving for RT:

RT = (4/5) * 15 = 12

Therefore, the missing sides are t=ST=9 and s=RT=12.

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Mr. Fransico had average overtime in a day is 2.1 hours.

In mathematics, the central value of a set of data is expressed as the average of a list of data. It is defined mathematically as the ratio of the total number of data points to the number of units in the list. In terms of statistics, the term "mean" also refers to the average of a certain set of numerical data. The average of 2, 3, and 4 is, for instance, (2+3+4)/3 = 9/3 = 3. The center value of 2, 3, and 4 in this instance is 3, thus. Finding the mean value of a bunch of numbers is the definition of average.

For 5 days Mr fransico had a total of 10. 5 hours of overtime in the office

Then the average daily overtime is-

10.5÷5 = 2.1 hours

Hence Mr. Fransico had total overtime in a day is 2.1 hours.

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The probability of randomly chosen, a 50-year-old or older voter from the winning party is 45.84%

Given the table of values

From the table of values, we have the winning party to be

New Democratic

From the column of New Democratic, we have

Total = 9422

50-year-old or older voter = 4319

So, the required probability is

Probbaility = 4319/9422

Evaluate

Probbaility = 0.45839524517

This gives

Probbaility = 45.839524517%

Approximate

Probbaility = 45.84%

Hence, the probability is 45.84%

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The result is a very large negative number, specifically -2,176,782,192. Therefore: 4a²−b⁶ = -2,176,782,192, when a=6 and b=36.

An equation is a mathematical statement that shows the equality between two expressions. It usually consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).

The expressions on both sides can contain variables, constants, and mathematical operations such as addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), and others. The goal of an equation is to find the values of the variables that make both sides equal.

For example, the equation "2x + 3 = 7" means that the sum of 2 times the variable x and 3 is equal to 7. The solution to this equation is x = 2, because when we substitute x = 2 into the equation, we get 2(2) + 3 = 7, which is a true statement.

by the question.

To evaluate the expression 4a²−b⁶ when a=6 and b=36, we substitute the values of a and b into the expression:

[tex]4(6)^{2} - (36)^{6}[/tex]

Simplifying the expression:

[tex]4(36) - 2,176,782,336\\144 - 2,176,782,336[/tex]e

[tex]= -2,176,782,192,[/tex]

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To calculate the probability that at least 1 of 5 randomly selected American children (ages 3 to 5) is read to every day by someone at home.

To find the probability that none of the 5 children are read to, we can use the fact that the probability that a single child is not read to every day is 1 - 0.58 = 0.42. We can then use the multiplication rule to find the probability that none of the 5 children are read to, which is (0.42)⁵ = 0.0075. Finally, we can use the complement rule to find the probability that at least 1 child is read to every day, which is 1 - 0.0075 = 0.9925. Therefore, the probability that at least 1 of 5 randomly selected American children (ages 3 to 5) is read to every day by someone at home is 0.9925.

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Answer:

3.1% of Ramon's earning are spent on electricity.

Step-by-step explanation:

Ramon's monthly salary

= $1,710

Electricity rent

=$53.60

*work show below*

[tex]\frac{53.60}{1,710} *100[/tex]

0.0313450292397661*100=3.134502923976608

3.134502923976608 rounded to the nearest tenth is 3.1%...

Thus my-our answer checks out! have a great day bestie!

Answer:

D

Step-by-step explanation:

Volume: π*[tex]r^{2} *h[/tex]= π*[tex]4^{2}[/tex]*6.2=311.49

The value of the perimeter, p, of the smaller trapezium is 113.

The perimeter of the trapezium is the distance round the trapezium and for this given diagram it can be calculated using congruence theorem.

The Congruence Theorems are a set of geometric principles that state when two geometric figures are congruent, which means they have the same size and shape.

Applying congruence theorem, we will have the following equation;

Side length: x/42

Area: 200/288

Perimeter : p/162

x/42 = p/162 ------ (1)

200/288 = p/162 ---- (2)

from (1), p = (162x)/42 = 3.857x

Substitute the value of p into (2)

200/288 = (3.857x)/162

162(200/288) = 3.857x

112.5 = 3.857x

x = 29.17

p = 3.857 x 29.17

p = 112.5

p ≈113

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P(80 < X < 125) = 0.9525 - 0.0918

P(80 < X < 125) = 0.8607

This means that approximately 86.07% of people have IQ scores between 80 and 125.

To answer this question, we need to calculate the standardized score (also known as z-score) for both 80 and 125:

z-score for 80: (80-100)/15 = -1.33

z-score for 125: (125-100)/15 = 1.67

Once we have the z-scores, we can use a standard normal distribution table or calculator to find the proportion of scores between them. Alternatively, we can use the following formula:

P(80 < X < 125) = P(Z < 1.67) - P(Z < -1.33)

Using a standard normal distribution table or calculator, we can find that P(Z < 1.67) is approximately 0.9525 and P(Z < -1.33) is approximately 0.0918.

Therefore:

P(80 < X < 125) = 0.9525 - 0.0918

P(80 < X < 125) = 0.8607

This means that approximately 86.07% of people have IQ scores between 80 and 125.

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Answer:

The equation 3^4 x 2^3 = 6^4+3 is incorrect.

To evaluate the left side of the equation, we first simplify each term using exponent rules:

3^4 = 3 x 3 x 3 x 3 = 81

2^3 = 2 x 2 x 2 = 8

So 3^4 x 2^3 = 81 x 8 = 648

To evaluate the right side of the equation, we simplify the exponent first:

6^4+3 = 6^4 x 6^3 = 1296 x 216 = 279936

Therefore, the corrected equation should be:

3^4 x 2^3 = 648 = 6^4 - 288

Notice that 6^4 - 288 is equal to the original value of 279936, but the equation has been written correctly by moving the 3 to the other side of the equation and changing the operation from addition to subtraction.

Answer:

Step-by-step explanation:

Since △PQR ~ △STU, their corresponding angles are congruent, and their corresponding sides are proportional.

First, we can find the measure of angle Q as follows:

m∠Q = 180 - m∠P - m∠R = 180 - 70 - 46 = 64 degrees

Next, we can use the fact that the sides of the similar triangles are proportional to set up the following proportions:

frac{ST}{21} = frac{SU}{14} and frac{ST}{28} = frac{TU}{21}

Solving for ST gives us:

ST = frac{21}{14} SU = frac{3}{2} SU

and

ST = frac{28}{21} TU = frac{4}{3} TU

Substituting these values into the second proportion, we get:

frac{3}{2} SU = frac{4}{3} TU

Multiplying both sides by 2/3, we get:

SU = frac{8}{9} TU

Now we can use the fact that the angles in a triangle add up to 180 degrees to find the measure of angle T.

m∠T = 180 - m∠S - m∠U = 180 - m∠S - (180 - m∠P - m∠R)

m∠T = m∠P + m∠R - m∠S = 70 + 46 - m∠S = 116 - m∠S

Finally, we can use the fact that the angles in △STU add up to 180 degrees to find the measure of angle S.

m∠S + m∠T + m∠U = 180

Substituting the previously found values for m∠T and SU into the equation, and solving for m∠S gives us:

m∠S = 52 degrees

Therefore, the missing measures are:

SU = 6 x 8/9 = 16/3

ST = 3/2 x 6 = 9

TU = 4/3 x 9 = 12

m∠S = 52 degrees

m∠T = 116 - 52 = 64 degrees

m∠U = 180 - 52 - 64 = 64 degrees

Wi represents the weight of the course, and xi represents the grade received for that course.

Care must be taken when assigning variables to weights and observations, because the average grade point average (GPA) is the average grade received for each hour of coursework taken.

Therefore, each grade must be weighed against the number of credits for that course.

For example, if two courses are worth 3 credits and one course is worth 6 credits, then the GPA would be calculated by adding the three grades together and then dividing by the sum of the credits (3+3+6=12).

In this case, a grade of A in the 6 credit course would have a greater impact on the GPA than the same grade in the 3 credit course.

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The polynomial equation roots are the values of x that make the equation equal to zero.

Graphing calculator and the system of equations are used to find the roots of a polynomial equation. To find the roots of a polynomial equation, follow the below methods ,Use a Graphing Calculator to Graph the Equation. Graphing the equation is one of the easiest ways to find the roots of a polynomial equation. By looking at the graph, you can see where the equation crosses the x-axis. If it crosses the x-axis, then the value of x where it crosses is a root of the polynomial equation.

Another way to find the roots of a polynomial equation is to use a system of equations. In a system of equations, you have two equations that you solve simultaneously. To use a system of equations, you will need to know the degree of the polynomial equation, the coefficients of the terms, and the values of the constants.

We can also use synthetic division to find the roots of a polynomial equation. Synthetic division is a way to divide a polynomial by a linear factor. If the result is zero, then the value of x that you divided by is a root of the polynomial equation.

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The probability that the number of lost-time accidents occurring over a period of 8 days will be no more than 5 is 0.6695

This scenario can be modeled using the Poisson distribution, which is a probability distribution that describes the number of events that occur in a fixed time period when the events occur independently and at a constant rate.

The mean rate of lost-time accidents per day is given as 0.6. Therefore, the mean rate of lost-time accidents over 8 days is

Mean rate = (0.6 accidents/day) x (8 days) = 4.8 accidents

Let X be the number of lost-time accidents occurring over 8 days. Then, X follows a Poisson distribution with parameter λ = 4.8.

To find the probability that the number of lost-time accidents occurring over a period of 8 days will be no more than 5, we need to calculate

P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

Using the Poisson probability mass function, we get

P(X = k) = (e^(-λ) × λ^k) / k!

where k is the number of lost-time accidents.

Substituting λ = 4.8 and k = 0, 1, 2, 3, 4, 5 in the above formula, we get

P(X = 0) = (e^(-4.8) × 4.8^0) / 0! = 0.0082

P(X = 1) = (e^(-4.8) × 4.8^1) / 1! = 0.0393

P(X = 2) = (e^(-4.8) × 4.8^2) / 2! = 0.0944

P(X = 3) = (e^(-4.8) × 4.8^3) / 3! = 0.1573

P(X = 4) = (e^(-4.8) × 4.8^4) / 4! = 0.1888

P(X = 5) = (e^(-4.8) × 4.8^5) / 5! = 0.1815

Therefore,

P(X ≤ 5) = 0.0082 + 0.0393 + 0.0944 + 0.1573 + 0.1888 + 0.1815 = 0.6695

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Answer:

Step-by-step explanation:

Translate the shape 2 units up and 3 units to the left on a graph and you will find the answer.

Answer:

R= -4,8

S= 2,8

T= 1,6

U= -5,6

Answer: slope=4

Step-by-step explanation:

[tex]Slope=\frac{y2-y1}{x2-x1}[/tex]

[tex]Slope=\frac{4- - 4}{3-2}[/tex]

[tex]Slope=\frac{4+4}{3-2}[/tex]

[tex]Slope=4[/tex]