Consider the following system of equations 21 + 23 = 1 40 + 0 + 503 = 3 401 + x2 + 403 2 Use Q1. to solve the system of equations. 3. Decide if each of the following statements is true or false. (a) Every system of linear equations for which the coefficient matrix is square has a unique solution. (b) Every system of equations has a solution.

The three pairs of fraction whose sum is 3/5 are

We have to find pairs of fractions that have a sum of 3/5.

First pair:

1/5 + 2/5

= 3/5

Second pair:

= -2/5 + 1

= -2/5+ 5/5

= 3/5

Third pair:

= -6/5 + 9/5

= 3/5

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The minimum sample size needed to construct a 95% confidence interval with a margin of error of 0.06 for the population proportion supporting candidate A is 268 residents.

To find the minimum sample size for a 95% confidence interval on the population proportion supporting candidate A, we'll need to use the following terms: sample size (n), population proportion (p), margin of error (E), and confidence level (z-score).

First, let's determine the proportion supporting candidate A from the preselected poll:
100 residents - 22 (supporting B) - 14 (supporting C) = 64 (supporting A)
So, the proportion p = 64/100 = 0.64.

For a 95% confidence interval, the z-score is 1.96 (found using a standard normal distribution table or calculator).

Now, we can use the formula for sample size calculation:
n = (z² × p × (1-p)) / E²

Substituting the values:
n = (1.96² × 0.64 × 0.36) / 0.06²
n ≈ 267.24

Since sample size must be a whole number, we round up to the nearest whole number, which is 268.

Therefore, the minimum sample size needed to construct a 95% confidence interval with a margin of error of 0.06 for the population proportion supporting candidate A is 268 residents.

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The value of the sides are;

CB = 13.8

CD = 6. 9

To determine the value of the sides of the triangle, we need to know the different trigonometric identities are;

From the information given, we have that;

Using the sine identity, we have that;

tan 60 = CD/4

cross multiply the values, we have;

CD = 4(1.73)

multiply the values

CD = 6.9

To determine the value;

sin 30 = 6.9/CB

CB = 13.8

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According to Financial experts you should save between 10% to 15% of your annual income for retirement. For a salary of $79,000/year, the minimum saved should be between $790,000 to $948,000.

Financial experts generally recommend that you should aim to save between 10% to 15% of your income each year for retirement. For a salary of $79,000 per year, this means saving between $7,900 to $11,850 annually.

Assuming you have been saving for retirement throughout your working years and are ready to retire, financial experts suggest that you should have saved at least 10 to 12 times your current annual income to maintain your pre-retirement standard of living. Therefore, the minimum you should have saved in retirement accounts is

$79,000 x 10 = $790,000 (using the conservative end of the range)

or

$79,000 x 12 = $948,000 (using the more aggressive end of the range)

Therefore, the minimum you should have saved in retirement accounts if you are currently making $79,000/year is between $790,000 to $948,000, depending on the end of the range you choose to follow.

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Answer:

m = - 3 , b = 5

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (0, 5) and (x₂, y₂ ) = (2, - 1) ← 2 points on the line

m = [tex]\frac{-1-5}{2-0}[/tex] = [tex]\frac{-6}{2}[/tex] = - 3

the y- intercept b is the value of y on the y- axis where the line crosses

that is b = 5

Answer:

b = 5

m = -3

Step-by-step explanation:

y-intercept is where the line intersects the y-axis. So, the line intersects at (0,5).

So, y-intercept = b = 5

       Choose two points on the line: (0,5) and (1,2)

 x₁ = 0    ; y₁ = 5

  x₂ = 1    ; y₂ = 2

Substitute the points in the below formula to find the slope.

                [tex]\sf \boxed{\bf Slope =\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

                            [tex]= \dfrac{2-5}{1-0}\\\\=\dfrac{-3}{1}[/tex]

                        [tex]\boxed{\bf m = -3}[/tex]

The equation of the tangent plane to the given surface at the specified point (3, -2, 18) is z - 18 = 8(x - 3) - 12(y + 2).

To find the equation of the tangent plane to the given surface at the specified point (3,-2,18), we first need to find the partial derivatives of z with respect to x and y:

∂z/∂x = 4(x-1)
∂z/∂y = 12(y+3)

Then, we can evaluate these partial derivatives at the given point (3,-2,18):

∂z/∂x = 4(3-1) = 8
∂z/∂y = 12(-2+3) = -12

Next, we can use these partial derivatives and the point (3,-2,18) to write the equation of the tangent plane in point-normal form:

z - z0 = ∂z/∂x(x - x0) + ∂z/∂y(y - y0)

Plugging in the values we found:

z - 18 = 8(x - 3) - 12(y + 2)

Simplifying:

8x - 12y - z = -22

Therefore, the equation of the tangent plane to the given surface at the point (3,-2,18) is 8x - 12y - z = -22.
To find an equation of the tangent plane to the given surface z = 2(x - 1)^2 + 6(y + 3)^2 + 4 at the specified point (3, -2, 18), follow these steps:

1. Calculate the partial derivatives of the function with respect to x and y:
∂z/∂x = 4(x - 1)
∂z/∂y = 12(y + 3)

2. Evaluate the partial derivatives at the specified point (3, -2, 18):
∂z/∂x(3, -2) = 4(3 - 1) = 8
∂z/∂y(3, -2) = 12(-2 + 3) = -12

3. Use the tangent plane equation to find the tangent plane at the specified point:
z - z0 = ∂z/∂x(x - x0) + ∂z/∂y(y - y0)
where (x0, y0, z0) = (3, -2, 18)

4. Plug in the values and simplify the equation:
z - 18 = 8(x - 3) - 12(y + 2)

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We can say with 99% confidence that the true mean price of all transactions is between $2,799.16 and $3,000.84.

To obtain a 99% confidence interval for the mean price of all transactions, we can use the formula:

CI =  ± z*(σ/√n)

Where:
= sample mean price = 2900 dollars
σ = population standard deviation = 290 dollars
n = sample size = 30
z = z-score for a 99% confidence level = 2.576 (from the standard normal distribution table)

Substituting these values into the formula, we get:

CI = 2900 ± 2.576*(290/√30)
CI = 2900 ± 100.84
CI = (2799.16, 3000.84)

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Answer:

1/4

Step-by-step explanation:

it came to me in a dream.

1/4 or 25% is the probability that the randomly selected point is in the bullseye.

Probability is a number that expresses the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

The area of the bullseye is the area of the inner circle with a radius of 4 cm. Similarly, the area of the entire target is the area of the outer circle with a radius of 8 cm.

The area of a circle is given by the formula A = πr², where A is the area and r is the radius.

Therefore, the area of the bullseye is:

A_bullseye = π(4 cm)² = 16π cm²

And the area of the entire target is:

A_target = π(8 cm)² = 64π cm²

So, the probability that the randomly selected point is in the bullseye is the ratio of the area of the bullseye to the area of the target:

P(bullseye) = A_bullseye / A_target

P(bullseye) = (16π cm²) / (64π cm²)

P(bullseye) = 1/4

Therefore, the probability that the randomly selected point is in the bullseye is 1/4 or 25%.

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The two atoms are of elements in the same group in the periodic table include the following: D. Atom 1 and Atom 2.

In Chemistry, a periodic table can be defined as an organized tabular array of all the chemical elements that are typically arranged in order of increasing atomic number (number of protons), in rows.

In Chemistry, valence electrons can be defined as the number of electrons that are present in the outermost shell of an atom of a specific chemical element.

In this context, we can reasonably infer and logically deduce that both Atom 1 and Atom 2 represent chemical elements that are in the same group in the periodic table because they have the same valence electrons of six (6).

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Complete Question:

A student is studying the ways different elements are similar to one another. Diagrams of atoms from four different elements are shown below.

Which two atoms are of elements in the same group in the periodic table?

The likelihood of randomly selecting a terrier from the shelter would be unlikely. That is option B

The formula that can be used to determine the probability of a selected event is given as follows;

Probability = possible event/sample space.

The possible sample space for terriers = 15%

Therefore the remaining sample space goes for Labradors and Golden Retrievers which is = 75%

Therefore, the probability of selecting the terriers at random is unlikely when compared with other dogs.

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Answer: 2

Step-by-step explanation:

Answer: its 144 i think

Step-by-step explanation: Math

The description of the parabola of the quadratic function is:

It opens downwards and is thinner than the parent function

The general formula for expressing a quadratic equation in standard form is:

y = ax² + bx + c

Quadratic equation In vertex form is:

y = a(x − h)² + k .

In both forms, y is the y -coordinate, x is the x -coordinate, and a is the constant that tells you whether the parabola is facing up ( + a ) or down ( − a ), (h, k) are coordinates of the vertex

In this case, a is negative and as such it indicates that it opens downwards and is thinner than the parent function

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Step-by-step explanation:

Using z-score table the value is    .9793     (97.93 %)

The probability of puling out

In order to calculate the probability of extracting each shape from the bag, a formula can be employed:

Probability = Number of times the shape was taken out / Total number of times shapes were taken out

Given below are the frequency of each shape:

Triangle: 3 times

Circle: 12 times

Square: 9 times

Total number of times shapes were taken out = 3 + 12 + 9 = 24

Probability of taking out a Triangle

= 3 / 24

= 1/8

Probability of taking out a Circle

= 12/24

= 1/2

Probability of taking out a Square

= 9/24

= 3/8

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P(X = k) = (1 - p)^4   for k = 0
P(X = k) = 4p(1 - p)^3   for k = 1
P(X = k) = 6p^2(1 - p)^2   for k = 2
P(X = k) = 4p^3(1 - p)   for k = 3
P(X = k) = p^4   for k = 4

Note that these probabilities add up to 1, as they should for any probability distribution.

(a) The distribution of X can be modeled as a binomial distribution with parameters n = 4 and p, where p is the probability that the friend correctly identifies a milk-first cup as milk-first. Each cup that the friend tastes can either be identified correctly (success) or incorrectly (failure), and there are 4 cups that were poured milk-first in the experiment.

(b) To find the probability mass function (PMF) of X, we need to find the probability of each possible value of X. Since X is a binomial random variable, the PMF of X is given by:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

where (n choose k) is the binomial coefficient, given by:

(n choose k) = n! / (k! * (n - k)!)

where n! denotes the factorial of n.

In this case, n = 4 and there are 4 cups that were poured milk-first, so we have:

P(X = 0) = (4 choose 0) * p^0 * (1 - p)^4 = (1 - p)^4

P(X = 1) = (4 choose 1) * p^1 * (1 - p)^3 = 4p(1 - p)^3

P(X = 2) = (4 choose 2) * p^2 * (1 - p)^2 = 6p^2(1 - p)^2

P(X = 3) = (4 choose 3) * p^3 * (1 - p)^1 = 4p^3(1 - p)

P(X = 4) = (4 choose 4) * p^4 * (1 - p)^0 = p^4

Since X can only take on values between 0 and 4, the PMF of X is given by:

P(X = k) = (1 - p)^4   for k = 0
P(X = k) = 4p(1 - p)^3   for k = 1
P(X = k) = 6p^2(1 - p)^2   for k = 2
P(X = k) = 4p^3(1 - p)   for k = 3
P(X = k) = p^4   for k = 4

Note that these probabilities add up to 1, as they should for any probability distribution.

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The answers are:

(a) f is decreasing on (-∞, 0) and increasing on (0, ∞).

(b) f is concave up on (-∞, ∞).

(c) Local minimum value at x = 0, local maximum value DNE.

(d) Global minimum value is -2 at x = -1, global maximum value is 22 at x = 2.

(e) There are no points of inflection.

(a) To find where the function is increasing or decreasing, we need to find the critical points and test the intervals between them:

[tex]f(x) = x^4 + 3x^3 + 3x^2\\f'(x) = 4x^3 + 9x^2 + 6x[/tex]

Setting f'(x) = 0, we get:

[tex]0 = 2x(2x^2 + 3x + 3)[/tex]

The quadratic factor has no real roots, so the only critical point is x = 0.

We can test the intervals (-∞, 0) and (0, ∞) to find where f is increasing or decreasing:

For x < 0, f'(x) is negative, so f is decreasing.

For x > 0, f'(x) is positive, so f is increasing.

Therefore, f is decreasing on (-∞, 0) and increasing on (0, ∞).

(b) To find where the function is concave up or concave down, we need to find the inflection points:

f''(x) =[tex]12x^2 + 18x + 6[/tex]

Setting f''(x) = 0, we get:

0 = [tex]6(x^2 + 3x + 1)[/tex]

The quadratic factor has no real roots, so there are no inflection points.

Since the second derivative is always positive, f is concave up everywhere.

(c) To find the local extreme values, we need to find the critical points and determine their nature:

f'(x) = [tex]4x^3 + 9x^2 + 6x[/tex]

At x = 0, f'(0) = 0 and f''(0) = 6, so this is a local minimum.

There are no local maximum values.

(d) To find the global extreme values on [-1, 2], we need to check the endpoints and the critical points:

f(-1) = -2, f(0) = 0, f(2) = 22

The global minimum value is -2 at x = -1, and the global maximum value is 22 at x = 2.

(e) To find the points of inflection, we need to find where the concavity changes:

Since there are no inflection points, there are no points of inflection.

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Answer:

1(1/2)

Step-by-step explanation:

how you use this is do 7 divided by 11 and 11 divided by 6 which is 1 and 1/2

Answer:

77/66 (simplified would equal 7/6)

Step-by-step explanation:

When multiplying fractions you simply just multiply the numerators together, making the new numerator, then multiply the denominators together, making the new denominator, and  you have your answer.

EXTRA: To simplify the fraction to its simplest you find a number that both the numerator and the denominator can be divided into equally, in this case it would be 11, then divide the numerator and denominator by this number and that would be your answer. Example; 77/66, divide 77 and 66 by 11 and you get 7/6.

Hope this helps (:

In the given scenario, the nominal variables are Taia's income, the price of a digital movie rental, and the price of an americano. The real variables would be Taia's income adjusted for inflation, the real price of a digital movie rental, and the real price of an americano.

To calculate the real value of a variable, we need to adjust it for inflation using a suitable price index. As the question does not provide any information about inflation, we cannot calculate the real value of any variable.

Therefore, none of the options given in the question would give the real value of a variable.
Hi! I'd be happy to help you with this question. In the context of the classical dichotomy, real variables are quantities or values that are adjusted for inflation, while nominal variables are unadjusted values.

In the given scenario, Taia spends her income on digital movie rentals and americanos. We have the following information for 2016:

1. Hourly wage: $28.00 (nominal variable)
2. Price of a digital movie rental: $7.00 (nominal variable)
3. Price of an americano: $4.00 (nominal variable)

To determine the real value of a variable, we need to adjust these nominal values for inflation. However, the question does not provide any information about the inflation rate or a base year for comparison. Thus, we cannot calculate the real values for these variables in this scenario.

In summary, we do not have enough information to determine the real value of any variable in this case. Please provide the inflation rate or base year if you'd like me to help you calculate the real values.

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The probability that the sample mean will be within $500 of the population mean is approximately 0.3982 (or 39.82% when expressed as a percentage) to 4 decimal places.

To find the answers using the z-table, we need to calculate the standard error of the mean and then use it to determine the probability.

a. The standard error of the mean (SE) is calculated using the formula:

SE = σ / sqrt(n),

where σ is the standard deviation and n is the sample size.

Given that the standard deviation is $8,500 and the sample size is 80, we can calculate the standard error of the mean:

SE = 8,500 / sqrt(80) ≈ 950.77.

Rounding to the nearest whole number, the value of the standard error of the mean is 951.

b. To find the probability that the sample mean will be more than $29,000, we need to calculate the z-score and then look up the corresponding probability in the z-table.

The z-score is calculated using the formula:

z = (x - μ) / SE,

where x is the sample mean, μ is the population mean, and SE is the standard error of the mean.

In this case, x = $29,000, μ = population mean (unknown), and SE = 951.

Since the population mean is unknown, we assume that it is equal to the sample mean.

z = (29,000 - 29,000) / 951 = 0.

Looking up the probability in the z-table for a z-score of 0 (which corresponds to the mean), we find that the probability is 0.5000.

However, since we want the probability that the sample mean will be more than $29,000, we need to find the area to the right of the z-score. This is equal to 1 - 0.5000 = 0.5000.

Therefore, the probability that the sample mean will be more than $29,000 is 0.50 (or 50% when expressed as a percentage) to 2 decimal places.

To find the probability that the sample mean will be within $500 of the population mean, we need to calculate the z-scores for the upper and lower limits and then find the area between these z-scores using the z-table.

c. Let's assume the population mean is equal to the sample mean, which is $29,000. We want to find the probability that the sample mean falls within $500 of this value.

The upper limit is $29,000 + $500 = $29,500, and the lower limit is $29,000 - $500 = $28,500.

To calculate the z-scores for these limits, we use the formula:

z = (x - μ) / SE,

where x is the limit value, μ is the population mean, and SE is the standard error of the mean.

For the upper limit:

z_upper = ($29,500 - $29,000) / 951 ≈ 0.526

For the lower limit:

z_lower = ($28,500 - $29,000) / 951 ≈ -0.526

Now, we look up the probabilities associated with these z-scores in the z-table. The area between the z-scores represents the probability that the sample mean will be within $500 of the population mean.

Using the z-table, we find that the probability corresponding to z = 0.526 is approximately 0.6991, and the probability corresponding to z = -0.526 is approximately 0.3009.

The probability that the sample mean will be within $500 of the population mean is the difference between these two probabilities:

Probability = 0.6991 - 0.3009 ≈ 0.3982.

Therefore, the probability that the sample mean will be within $500 of the population mean is approximately 0.3982 (or 39.82% when expressed as a percentage) to 4 decimal places.

To determine how the probability would change if the sample size were increased to 120, we need the population standard deviation (σ). Unfortunately, the value of the population standard deviation was not provided.

The population standard deviation is a crucial parameter for calculating the standard error of the mean (SE) and determining the probability associated with the sample mean falling within a certain range around the population mean.

Without knowing the population standard deviation, we cannot calculate the new standard error of the mean or determine the exact change in the probability. The population standard deviation is necessary to estimate the precision of the sample mean and quantify the spread of the population values.

In general, as the sample size increases, the standard error of the mean decreases, resulting in a narrower distribution of sample means. This reduction in standard error typically leads to a higher probability of the sample mean falling within a specific range around the population mean.

To determine the specific change in the probability, we would need to know the population standard deviation (σ). Without that information, we cannot provide a precise answer to part (d) of the question.

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The two positive numbers with a product of 242 and the smallest possible sum of one number and twice the second number are 11 and 22.

To find two positive numbers with a product of 242, we can start by finding the prime factorization of 242, which is 2 x 11 x 11. From this, we know that the two numbers we're looking for must be a combination of these factors.

To minimize the sum of one number and twice the second number, we need to choose the two factors that are closest in value. In this case, that would be 11 and 22 (twice 11). So the two positive numbers we're looking for are 11 and 22.

To check that these numbers have a product of 242, we can multiply them together: 11 x 22 = 242.

Now we need to check that the sum of 11 and twice 22 is smaller than the sum of any other combination of factors. The sum of 11 and twice 22 is 55. If we try any other combination of factors, the sum will be larger. For example, if we chose 2 and 121 (11 x 11), the sum would be 244.

Therefore, the two positive numbers with a product of 242 and the smallest possible sum of one number and twice the second number are 11 and 22.

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The average speed of the train is both greater than or equal to 2.3928 km/minute and less than or equal to 3.3567 km/minute.

To find the average speed of the train, we divide the distance traveled by the time taken:

Average speed = distance / time

= 503 km / 180 minutes

= 2.7944... km/minute

Since the distance is measured correct to the nearest kilometer, the actual distance could be as low as 502.5 km or as high as 503.5 km. Similarly, since the time is measured correct to the nearest minute, the actual time taken could be as low as 2.5 hours or as high as 3.5 hours.

To find the maximum average speed, we assume that the distance traveled is 503.5 km and the time taken is 2.5 hours.

Maximum average speed = 503.5 km / 150 minutes = 3.3567... km/minute

To find the minimum average speed, we assume that the distance traveled is 502.5 km and the time taken is 3.5 hours.

Minimum average speed = 502.5 km / 210 minutes = 2.3928... km/minute

Therefore, the average speed of the train is both greater than or equal to 2.3928 km/minute and less than or equal to 3.3567 km/minute.

Rounding to two decimal places, the average speed of the train is 2.79 km/minute.

Reason: Both 2.79 km/minute and the minimum and maximum average speeds are correct to the nearest hundredth of a kilometer per minute and take into account the maximum possible error in the measurements.

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The approximate percentage error is 22.1514%.

Formulate:  (427.53−350)÷350

Calculate the sum or difference: 77.53/350

Multiply both the numerator and denominator with the same integer:

7753/35000

Rewrite a fraction as a decimal: 0.221514

Multiply a number to both the numerator and the denominator:

0.221514×100/100

Write as a single fraction: 0.221514×100/100

Calculate the product or quotient: 22.1514/100

Rewrite a fraction with denominator equals 100 to a percentage:

22.1514%

Percent error is the difference between estimated value and the actual value in comparison to the actual value and is expressed as a percentage. In other words, the percent error is the relative error multiplied by 100.

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The probability that either event will occur is given as follows:

P(A or B) = 0.75.

The formula used to calculate the probability is given as follows:

P(A or B) = P(A) + P(B) - P(A and B).

The total number of events from the Venn's diagram is given as follows:

4 x 9 = 36.

Hence the probability of each outcome is given as follows:

Hence the or probability is given as follows:

P(A or B) = P(A) + P(B) - P(A and B).

P(A or B) = 0.5 + 0.5 - 0.25

P(A or B) = 0.75.

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The calculated test statistic is 0.571. P 0.5, the null hypothesis.

A one-tailed z-test can be used to verify the assertion that there is a higher than 50% chance of the home side winning.

p > 0.5, where p is the percentage of football games won by the home team in the population.

The test statistic is calculated as:

(p - p) / (p(1-p) / n) = z

If n = 53 is the sample size, p = 0.5 is the hypothesized population proportion, and p is the sample fraction of football games won by the home team.

The percentage of the sample is p = 27/53 = 0.5094.

The calculated test statistic is:

z = (0.5094 - 0.5) / √(0.5(1-0.5) / 53) = 0.571

We determine the p-value for this test to be 0.2826 using a calculator or a table of the normal distribution as a reference.

We are unable to reject the null hypothesis since the p-value is higher than the significance level of 0.05. Therefore, at the 5% level of significance, we lack sufficient data to draw the conclusion that there is a better than 50% chance of the home team winning.

The calculated test statistic is:

z = (0.5094 - 0.5) / √(0.5(1-0.5) / 53)

= 0.571

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The coordinates of any point on the flat surface of the washer, and the radius is half of the diameter, which is 3/7 inches.

To describe the flat surface of a washer that is 3.6 inches in diameter and has an inner hole with a diameter of 3/7 inch, we can use the following inequalities:

For the outer circumference of the washer:

[tex]x^2 + y^2[/tex]≤ [tex](3.6/2)^2[/tex]

where x and y are the coordinates of any point on the flat surface of the washer, and the radius is half of the diameter, which is 3.6/2 inches.

For the inner circumference of the washer:

[tex]x^2 + y^2[/tex] ≥ [tex](3/14)^2[/tex]

where x and y are the coordinates of any point on the flat surface of the washer, and the radius is half of the diameter, which is 3/7 inches.

Note that these inequalities represent the circular boundaries of the flat surface of the washer, where the outer circumference is a circle with radius 1.8 inches and the inner circumference is a circle with radius 3/14 inches. The flat surface of the washer is the region bounded by these two circles.

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The coordinate point (8, -15) is lies in fourth quadrant.

The given coordinate point is (8, -15).

Part A: Here, x-coordinate is positive that is 8 and the y-coordinate is negative that is -15.

Quadrant IV: The bottom right quadrant is the fourth quadrant, denoted as Quadrant IV. In this quadrant, the x-axis has positive numbers and the y-axis has negative numbers.

So, the point lies in IV quadrant.

Part B:

Here r²=x²+y²

r²=8²+(-15)²

r²=64+225

r²=289

r=√289

r=17 units

So, the radius is 17 units

Therefore, the coordinate point (8, -15) is lies in fourth quadrant.

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Answer:

99.6%

Step-by-step explanation:

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|| (3,3) - (1,1) || < 2 + r

Simplifying this inequality, we get:

2√2 < 2 + r

r > 2√2 - 2

So, any value of r such that r > 2√2 - 2 will satisfy the condition B2(1,1)∩Br(3,3)≠0.

For the first question, we need to find an r such that the open ball centered at (0,0) with radius 1 (denoted as Br(0,1)) intersects with the open ball centered at (2,0) with radius t (denoted as Bt(2,1)). Since the usual norm is the Euclidean norm, the distance between (0,0) and (2,0) is 2. Thus, we have the inequality:

|| (2,0) - (0,0) || < 1 + t

Simplifying this inequality, we get:

2 < 1 + t

t > 1

So, any value of r such that 1 < r < 3 will satisfy the condition Br(0,1) ∩ Bt(2,1)≠0.

For the second question, we need to find an r such that the open ball centered at (1,1) with radius 2 (denoted as B2(1,1)) intersects with the open ball centered at (3,3) with radius r (denoted as Br(3,3)). Using the Euclidean norm, we have:

|| (3,3) - (1,1) || < 2 + r

Simplifying this inequality, we get:

2√2 < 2 + r

r > 2√2 - 2

So, any value of r such that r > 2√2 - 2 will satisfy the condition B2(1,1)∩Br(3,3)≠0.

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