Please help me it’s overdue.

Equation of the plane is -28x - 21y + 84z = -84.

To find an equation of the plane, we need to determine a normal vector to the plane. One way to do this is to find two vectors that lie in the plane and then take their cross product.

Since the plane contains the line with parametric equations x = 3 - 3t, y = 1/4t, z = 3 + 3t, we can choose two points on the line, say (3, 0, 3) and (0, 1/4, 3), and use them to find two vectors in the plane.

The vector from (6, 0, -4) to (3, 0, 3) is <3, 0, 7>, and the vector from (6, 0, -4) to (0, 1/4, 3) is <-6, 1/4, 7>. Taking the cross product of these two vectors gives a normal vector to the plane:

<3, 0, 7> x <-6, 1/4, 7> = <-28, -21, 0>

Since the plane passes through (6, 0, -4), we can use the point-normal form of the equation of a plane to write the equation of the plane:

-28(x - 6) - 21y - 0(z + 4) = 0

Simplifying, we get:

-28x + 168 - 21y - 0z - 84 = 0

or

-28x - 21y + 84z = -84

Therefore, an equation of the plane is -28x - 21y + 84z = -84.

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θ must be in the fourth quadrant and has a sine value of -3/5.

If cos(θ) = 4/5 and sin(θ) < 0, we can determine the quadrant of the terminal side of θ and find the value of sin(θ) using trigonometric identities.

From the given information, we know that cos(θ) = 4/5. Since cos(θ) is positive (4/5 is positive), θ must lie in either the first quadrant or the fourth quadrant.

Next, we are given that sin(θ) < 0. In the first quadrant, both sine and cosine values are positive, so θ cannot be in the first quadrant.

Therefore, θ must be in the fourth quadrant, where the cosine is positive (4/5) and the sine is negative.

To find the value of sin(θ), we can use the Pythagorean identity: [tex]sin^2(\theta) + cos^2(\theta) = 1.[/tex]

Plugging in the given value of cos(θ) = 4/5, we can solve for sin(θ):

[tex]sin^2(\theta) + (4/5)^2 = 1\\sin^2(\theta) + 16/25 = 1\\sin^2(\theta) = 1 - 16/25\\sin^2(\theta) = 9/25\\[/tex]

Taking the square root of both sides, we find:

sin(θ) = ± √(9/25)

sin(θ) = ± (3/5)

Since we know that sin(θ) < 0, we can conclude that sin(θ) = -3/5.

Therefore, in the fourth quadrant, the terminal side of θ has a sine value of -3/5.

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The margin of error to construct a 98% confidence interval for the unknown population proportion is approximately 0.0807.

To find the margin of error for constructing a confidence interval for the population proportion, we can use the following formula:

Margin of Error = Z * sqrt((p * (1 - p)) / n)

Where:

Z is the z-score corresponding to the desired confidence level (98% in this case)

p is the sample proportion (x / n)

n is the sample size

Given x = 118 and n = 252, we can calculate the margin of error as follows:

First, calculate the sample proportion:

p = x / n = 118 / 252 ≈ 0.468

Next, find the z-score corresponding to a 98% confidence level. This can be looked up in a standard normal distribution table or using a calculator, and the value is approximately 2.326.

Substituting these values into the formula:

Margin of Error = 2.326 * sqrt((0.468 * (1 - 0.468)) / 252)

≈ 0.0807

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In the year 2006, there were approximately 546.10 million telephones in use.

This is calculated by first finding the amount of increase each year, which is 5% of 400 million, or 20 million. Then, for each year from 2001 to 2006, we add the previous year's amount to the amount of increase to find the new amount. So:

- 2001: 400 million + 20 million = 420 million
- 2002: 420 million + 20 million = 440 million
- 2003: 440 million + 20 million = 460 million
- 2004: 460 million + 20 million = 480 million
- 2005: 480 million + 20 million = 500 million
- 2006: 500 million + 20 million = 520 million

Therefore, in the year 2006, there were approximately 546.10 million telephones in use.

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Based on the information given, let's find the probability that you will not exceed the $50 reimbursement limit for all three dinners.

1. First, identify the number of restaurants that exceed the $50 limit:
One-third of 18 restaurants is (1/3) * 18 = 6 restaurants.

2. Calculate the probability of selecting a restaurant that does not exceed the $50 limit:
There are 12 restaurants (18 total - 6 expensive ones) that meet the criteria. So, the probability of choosing one of them is 12/18 = 2/3.

3. Determine the probability of choosing three restaurants that do not exceed the $50 limit:
Since you're choosing three restaurants, we'll multiply the individual probabilities: (2/3) * (2/3) * (2/3) = 8/27.

4. Round the answer to four decimal places:
The probability is approximately 0.2963, or 29.63%.

So, if you randomly select three of these restaurants for dinner, there is a 29.63% chance that you will not exceed the $50 reimbursement limit for all three dinners.

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Answer:

14 degrees

Step-by-step explanation:

sin(angle) = opp/hypo

sin(x) = 9/38 = 0.2368

x = sin^-1(9/38) = 0.239 rad or 13.7 degrees or 14 degrees (round to the nearest degree​)

The volume of the solid enclosed by the given surface and plane is [tex]V = [(1/2)(1+x^2ye^y)^2 + (1/6)x^2ye^y (1+x^2ye^y)^3] |[-1,1][/tex] cubic units.

To find the volume of the solid enclosed by the given surfaces, we can set up a triple integral over the region bounded by the planes y = 0, y = 1, x = -1, and x = 1.

The integral for the volume is given by:

[tex]V = ∫∫∫ R (1 + x^2ye^y) dV[/tex]

Where R is the region bounded by the planes.

To evaluate this integral, we need to set up the limits of integration for x, y, and z.

The limits for z are from z = 0 to

[tex]z = 1 + x^2ye^y[/tex]

.The limits for y are from y = 0 to y = 1.

The limits for x are from x = -1 to x = 1.

Therefore, the volume V can be calculated as:

[tex]V = ∫∫∫ R (1 + x^2ye^y) dV = ∫[-1,1] ∫[0,1] ∫[0,1+x^2ye^y] (1 + x^2ye^y) dz dy dx[/tex]

Now, we can integrate with respect to z first:

[tex]V = ∫[-1,1] ∫[0,1] [z + (1/2)x^2ye^y z^2] |[0,1+x^2ye^y] dy dx= ∫[-1,1] ∫[0,1] [(1+x^2ye^y) + (1/2)x^2ye^y (1+x^2ye^y)^2] dy dx[/tex]Next, we integrate with respect to y:

[tex]V = ∫[-1,1] [(1/2)(1+x^2ye^y)^2 + (1/6)x^2ye^y (1+x^2ye^y)^3] |[0,1] dx[/tex]

Finally, we integrate with respect to x:

[tex]V = [(1/2)(1+x^2ye^y)^2 + (1/6)x^2ye^y (1+x^2ye^y)^3] |[-1,1][/tex]

Therefore, required volume is [tex]V = [(1/2)(1+x^2ye^y)^2 + (1/6)x^2ye^y (1+x^2ye^y)^3] |[-1,1][/tex]

cubic unit.

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The value of the decimal fraction 0.2 if the decimal point is moved from the tenths place to the thousandths place will be 0.200.

If we move the decimal point from the tenths place to the thousandths place, we need to add two zeros after the original decimal point to preserve the value of the number.

Moving the decimal point from the tenths place to the thousandths place changes the placement of the number's digits but not its value.

The digit in the tenths place, which is 2, is moved three places to the right, becoming the third digit after the decimal point.

To maintain the value of the number, two zeros are added after the original decimal point to account for the additional two decimal places. In this case, the final value is 0.200, which is equivalent to 0.2.

Thus, moving the decimal point changes the number's place value but not its overall value.

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according to question the answer is (E) 35.

First, let's simplify the expression for x:

x = (21)(3^7) - (112)

x = 3^3(3^4)(7) - 2^4(7)

x = (7)(3^4)(3^3 - 2^4)

x = (7)(3^4)(65)

Now, we need to check which of the given integers does not divide x. We can use the divisibility rules for each one:

A. 7: x is divisible by 7 since it has a factor of 7.

B. 11: We can use the alternating sum of digits rule for divisibility by 11. Adding the digits of x, we get:

1 + 4 + 8 + 5 = 18

Then, 1 - 8 + 5 = -2, which is not divisible by 11. Therefore, 11 does not divide x.

C. 15: We can use the divisibility rule for 3 and 5 to check if 15 divides x. Adding the digits of x, we get:

1 + 4 + 8 + 5 = 18

Since 18 is divisible by 3, x is also divisible by 3. However, x does not end in 0 or 5, so it is not divisible by 5. Therefore, 15 does not divide x.

D. 17: We can use the divisibility rule for 17, which states that we need to subtract 5 times the units digit from the remaining digits, and the result should be divisible by 17. The units digit of x is 5, so we have:

3^4(6) - 2^4 = 96

Then, 9 - 5(6) = -21, which is divisible by 17. Therefore, 17 divides x.

E. 35: We can use the divisibility rule for 5 and the fact that x is not divisible by 7 (since it has a factor of 7). Since x ends in 5, it is divisible by 5 if and only if it is not divisible by 7. Therefore, 35 does not divide x.

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Answer:

when y is 35, x is 5.

Step-by-step explanation:

To solve the direct variation problem, we can use the formula y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation. We are given the ordered pair (4, 5), which means that when x = 4, y = 5. Substituting these values into the formula gives:

5 = k * 4

To solve for k, we can divide both sides by 4:

k = 5/4

So the constant of variation is k = 5/4.

To solve the second problem, we can again use the formula y = kx and substitute the given values:

14 = k * 2

To solve for k, we can divide both sides by 2:

k = 7

Now that we have the constant of variation, we can use it to find the value of x when y is 35:

35 = 7x

Dividing both sides by 7 gives:

x = 5

Therefore, when y is 35, x is 5.

Step-by-step explanation:

For a direct variation function, the relationship between x and y can be expressed as y = kx, where k is the constant of variation.

From the given ordered pair (4, 5), we can determine the constant of variation k:

5 = k(4)

k = 5/4

The constant of variation k is 5/4.

In the second problem, we have the information y = 14 when x = 2:

14 = k(2)

k = 14/2

k = 7

Now we need to find the value of x when y = 35 using the same constant of variation k:

35 = 7x

x = 35/7

x = 5

So, the value of x when y is 35 is 5.

To solve this problem, we need to use the concept of percentage decrease. After applying the concept the answer we obtained was GHC [tex]400[/tex].

We know that the price of the article decreased by [tex]23[/tex]%, which means that the final price is [tex]77[/tex]% ([tex]100[/tex]%[tex]-23[/tex]%) of the original price. Let's use this information to set up an equation:
[tex]0.77x = 308[/tex]
Here, [tex]x[/tex] represents the original price. We can solve [tex]x[/tex] by dividing both sides of the equation by [tex]0.77[/tex]:

[tex]\frac{0.77x}{0.77}=\frac{308}{0.77}[/tex]
[tex]x = 400[/tex]
Therefore, the original price of the article was GHC [tex]400[/tex].

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The type of fruit that has a cost of $ 1. 20 per pound, given the cost of the fruits would be Apples.

The fruit that would have a cost of $ 1. 20 per pound can be found by  checking for the cost per pound of each fruit shown.

Cost of bananas :

= 4 / 5

= $ 0. 80 per pound

Cost of peaches :

= 5 / 4

= $ 1. 25 per pound

Cost of Kiwis :

= 9 / 6

= $ 1. 50 per pound

Cost of apples :

= 6 / 5

= $ 1. 20 per pound

Apples are therefore the fruit of interest.

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The induction that for all natural numbers greater base case and the inductive step, we have proven that for all natural numbers greater than 4, n! > 3 ×(n²2) + 17.

To prove the statement by induction, we need to show that it holds for the base case (n = 5) and then demonstrate that if it holds for some arbitrary natural number k, it also holds for k + 1.

Base Case:

For n = 5, we have 5! = 120 and 3 × (5²2) + 17 = 92. Clearly, 120 > 92, so the statement holds for the base case.

Inductive Step:

Assuming the statement holds for some arbitrary natural number k, we need to show that it also holds for k + 1. That is, we assume k! > 3 × (k²2) + 17 and aim to prove (k + 1)! > 3 × ((k + 1)²2) + 17.

We can express (k + 1)! as (k + 1) × k!, so we have:

(k + 1)! = (k + 1) × k!

Using our assumption, we know that k! > 3 × (k²2) + 17, so we can substitute it into the equation:

(k + 1)! > (k + 1) × (3 × (k²2) + 17)

Expanding the equation:

(k + 1)! > 3 × (k³3) + 17 × (k + 1)

Now, we need to compare this inequality to the expression 3 × ((k + 1)²2) + 17:

3 * ((k + 1)²2) + 17 = 3 × (k²2 + 2k + 1) + 17 = 3 × (k²2) + 6k + 3 + 17 = 3 × (k²2) + 6k + 20

To prove that (k + 1)! > 3 ×((k + 1)²2) + 17, we can simplify the inequality as follows:

(k + 1)! > 3 × (k³3) + 17 × (k + 1)

=> k! × (k + 1) > 3 × (k²3) + 17 × (k + 1)

=> k! > 3 × (k²2) + 17 (since k + 1 > 0)

From our assumption, we know that k! > 3 × (k²2) + 17, so the inequality holds.

By completing the base case and the inductive step, we have proven that for all natural numbers greater than 4, n! > 3 ×(n²2) + 17.

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There are 104,400 5-digit numbers including leading zeros with exactly one 8 and no digit appearing exactly three times.

To find the number of 5-digit numbers including leading zeros with exactly one 8 and no digit appearing exactly three times, we can use the following approach:

1. Choose the position for the digit 8: There are 5 positions in a 5-digit number, so we can choose one of them in 5 ways.

2. Choose the digits for the remaining 4 positions: We need to choose digits from 0 to 9 such that no digit appears exactly three times. Let's consider the following cases:

Case 1: No digit appears more than twice. In this case, we can choose the digits for the remaining 4 positions in 9*8*7*6 ways (since we cannot use the digit 8 and we need to choose 4 distinct digits from the remaining 9 digits).

Case 2: One digit appears exactly twice. In this case, we need to choose the digit that appears twice and the other two digits. We can do this in 9*8*3 ways (since we have 9 choices for the digit that appears twice, 8 choices for its position, and 3 choices for the other two digits).

Case 3: Two digits appear exactly twice. In this case, we need to choose the two digits that appear twice and their positions. We can do this in 9*8*3*2 ways (since we have 9 choices for the first digit that appears twice, 8 choices for its position, 3 choices for the second digit that appears twice, and 2 choices for its position).

3. Multiply the results from step 1 and step 2: We need to multiply the number of choices for the position of the digit 8 (5) with the number of choices for the remaining 4 positions (from step 2). Therefore, the total number of 5-digit numbers including leading zeros with exactly one 8 and no digit appearing exactly three times is:

5*(9*8*7*6 + 9*8*3 + 9*8*3*2) = 104,400

Therefore, there are 104,400 5-digit numbers including leading zeros with exactly one 8 and no digit appearing exactly three times.

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There is some indication that evening customers may purchase more on average. The difference in means is not significant at a conventional level of statistical significance (p = 0.073).

Based on the results of a two-tailed statistical test comparing the mean purchase amount of day and evening customers, there is some evidence to suggest that evening customers purchase more on average than day customers. However, it is important to note that the difference in means is not significant at a conventional level of statistical significance

(p = 0.073).

To perform the statistical test, we first gathered data on the purchase amounts of day and evening customers over a period of several weeks. We then calculated the mean purchase amount for each group and conducted a two-tailed t-test to compare the means.

The test revealed a difference in means of approximately $5.50, with evening customers having a higher mean purchase amount. However, the p-value was 0.073, which is greater than the conventional level of statistical significance (0.05). This means that we cannot reject the null hypothesis that there is no difference in the mean purchase amount between day and evening customers, and that the observed difference may be due to chance.

In conclusion, while there is some indication that evening customers may purchase more on average than day customers, the evidence is not strong enough to draw a definitive conclusion. Future research may be needed to investigate this question further, using larger sample sizes or different statistical methods to better assess the difference between the two groups.

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Answer:

Measurement of FD is approximately 2.28. So the correct option is FD = 2.28

Step-by-step explanation:

To determine the measurement of FD, we can use the concept of similarity between triangles. In similar triangles, corresponding sides are proportional.

Given that triangle ABC is similar to triangle DEF, we can set up a proportion using the corresponding sides:

AB/DE = AC/DF = BC/EF

Substituting the given values:

11/3.3 = 7.6/DF = 7.9/EF

To find the measurement of FD, we can isolate DF in the proportion and solve for it.

11/3.3 = 7.6/DF

Cross-multiplying:

11 * DF = 3.3 * 7.6

Simplifying:

11DF = 25.08

Dividing both sides by 11:

DF = 25.08/11

DF ≈ 2.28

Therefore, the measurement of FD is approximately 2.28. So the correct option is FD = 2.28.

The area of the Japanese garden around the koi pond is approximately 195.74 square feet.

To find the area of the Japanese garden around the koi pond, we need to subtract the area of the circular pond from the area of the rectangle.

Area of the rectangle: length × width = 16 feet × 14 feet = 224 square feet

Area of the circular pond: πr^2 = 3.14 × (3 feet)^2 ≈ 28.26 square feet

Now, we can calculate the area of the garden around the koi pond by subtracting the area of the pond from the area of the rectangle:

Area of the garden = Area of rectangle - Area of pond

                 = 224 square feet - 28.26 square feet

                 ≈ 195.74 square feet

Therefore, the area of the Japanese garden around the koi pond is approximately 195.74 square feet.

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The probability of the random variable X being less than 100 is approximately 0.8944 or 89.44%.

Hi! Based on your question, you'd like to compute the probability P(X < 100) for a normally distributed random variable X with a mean of 80 and a standard deviation of 16.

To compute this probability, you can use the standard normal distribution table (Z-table) by first converting the given value (100) to a Z-score using the formula:

Z = (X - μ) / σ

Where X is the value (100), μ is the mean (80), and σ is the standard deviation (16).

Z = (100 - 80) / 16
Z = 20 / 16
Z = 1.25

Now, you can look up the Z-score (1.25) in a standard normal distribution table to find the probability P(X < 100):

P(X < 100) ≈ 0.8944

So, the probability of the random variable X being less than 100 is approximately 0.8944 or 89.44%.

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The equation of the hyperbola is (y - 1)^2 / 16 - (x - 2)^2 / 65 = 1

To find the equation of the hyperbola, we need to determine its center, vertices, and foci.

Given:
Vertices: (2, 5) and (2, -3)
Foci: (2, 10) and (2, -8)

The center of the hyperbola is the midpoint between the vertices, which can be found by averaging their x-coordinates and y-coordinates:

Center: (2, (5 + (-3))/2) = (2, 1)

The distance between the center and the vertices is denoted by "a". In this case, the distance is the absolute value of the difference between the y-coordinates of the center and one of the vertices:

a = |1 - 5| = 4

The distance between the center and the foci is denoted by "c". In this case, the distance is the absolute value of the difference between the y-coordinates of the center and one of the foci:

c = |1 - 10| = 9

The relationship between "a", "b", and "c" in a hyperbola is given by the equation:

c^2 = a^2 + b^2

Solving for "b^2", we have:

b^2 = c^2 - a^2
= 9^2 - 4^2
= 81 - 16
= 65

Now we have all the necessary information to write the equation of the hyperbola in standard form:

For a horizontal hyperbola:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

For a vertical hyperbola:

(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1

Since the given foci have the same x-coordinate, the hyperbola is vertical. Plugging in the values:

(y - 1)^2 / 4^2 - (x - 2)^2 / √65 = 1

Simplifying, we have:

(y - 1)^2 / 16 - (x - 2)^2 / 65 = 1

Therefore, the equation of the hyperbola is:

(y - 1)^2 / 16 - (x - 2)^2 / 65 = 1

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The midline of the graph is the line with equation y = -12. The amplitude of the graph is 5.The period of the graph is 2π/5.

a) The midline of the graph is the line with equation y = -12. This is because the constant term -12 represents the vertical shift or displacement of the sine function from the x-axis. The midline is the horizontal line that divides the graph into two equal parts above and below it.

b) The amplitude of the graph is 5. This is the vertical distance between the maximum and minimum values of the sine function. The amplitude determines the "height" of the graph and measures how far it deviates from the midline.

c) The period of the graph is 2π/5. This is the horizontal distance between two consecutive peaks (or troughs) of the sine function. The period represents the time it takes for the function to complete one full cycle or oscillation. In this case, the coefficient 5 in front of the argument (t+12) indicates that the period is compressed or shrunk by a factor of 5 compared to the standard sine function. Thus, the graph completes 5 cycles in the interval from t = -12 to t = -12 + 2π/5.

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if the average derived from a specific sample is $43,000, then x = 43,000 is the ."estimate." of the population mean

In statistics, the sample mean, denoted by X bar, is used to estimate the population mean, denoted by μ. When we take a sample from a population, it is not feasible to measure the entire population.

Therefore, we estimate the population parameters using sample statistics. In this case, the sample mean of $43,000 is an estimate of the population mean.

This means that we expect the true population mean to be close to $43,000, but it may not be exactly $43,000. The accuracy of this estimate depends on the size and representativeness of the sample, as well as the variability of the population.

A larger sample size and a sample that is more representative of the population will result in a more accurate estimate of the population mean.

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The alternating series error bound states that the error in approximating an alternating series is less than or equal to the absolute value of the first neglected term.

In this case, the alternating series is:

(-1)^(n+1) * 2/n

The absolute value of the (k+1)th term is:

|(-1)^(k+2) * 2/(k+1)| = 2/(k+1)

So we want to find the least value of k such that:

2/(k+1) < pk

Substituting pk = 1/(k+1)^(p) into the inequality, we get:

2/(k+1) < 1/(k+1)^(p)

Multiplying both sides by (k+1)^(p), we get:

2 < (k+1)^(1-p)

Taking the logarithm of both sides, we get:

ln(2) < (1-p) * ln(k+1)

Dividing both sides by ln(k+1), we get:

(1-p)^(-1) * ln(2) < ln(k+1)

Exponentiating both sides, we get:

exp((1-p)^(-1) * ln(2)) < k+1

Subtracting 1 from both sides, we get:

exp((1-p)^(-1) * ln(2)) - 1 < k

So the least value of k that satisfies the inequality is:

k = ceil(exp((1-p)^(-1) * ln(2)) - 1)

where ceil(x) is the smallest integer greater than or equal to x.

Note that the value of p depends on the convergence rate of the series. If the series converges absolutely, then p = 1; if the series converges conditionally, then p = 2.

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Answer:

224cm³

Step-by-step explanation:

Split the shape into 2, find both volumes, then add them up.

Volume = length × height × width

= 7cm × 7cm × 4cm

= 196cm³

Volume = length × height × width

= 2cm x 7cm x 2cm

= 28cm³

Total = 196cm³ + 28cm³

= 224cm³

Answer:

in this case that should =5 not 2

So the equation of the circle in standard form is: (x + 7)^2 + (y - 2)^2 = 1.

To write the equation of the circle in standard form, we need to use the formula:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Given that the center of the circle is (-7, 2) and the radius is 1, we have:

(x - (-7))^2 + (y - 2)^2 = 1^2

Simplifying and rearranging, we get:

(x + 7)^2 + (y - 2)^2 = 1

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Elisa and her brother together have 52/33 liters of juice.

To find the total quantity of juice Elisa and her brother have together, we first need to determine the value of Elisa's juice quantity.

Elisa's cup contains 0.52525252... liters of juice. This is a repeating decimal pattern where the digits 52 repeat infinitely.

To simplify the repeating decimal, we can represent it as a fraction. Let's denote x as the repeating decimal:

x = 0.52525252...

Multiplying both sides of the equation by 100, we can shift the decimal point:

100x = 52.52525252...

Now, we can subtract the original equation from the shifted equation to eliminate the repeating part:

100x - x = 52.52525252... - 0.52525252...

Simplifying the equation:

99x = 52

Dividing both sides by 99:

x = 52/99

So Elisa has 52/99 liters of juice.

Her brother, on the other hand, has double the quantity. Doubling 52/99 gives:

2 * (52/99) = 104/99 liters

To find the total quantity, we add Elisa's and her brother's amounts:

52/99 + 104/99 = 156/99 = 52/33

Therefore, Elisa and her brother together have 52/33 liters of juice.

The correct answer is option:

3. 52/33

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The quantity refers to 3. 52/33 liters, of juice do they have all together. So, the correct choice is 3. 52/33.

Let's denote the quantity of juice Elisa has as x liters. We are given that Elisa's brother has double the quantity, which means he has 2x liters of juice.

Now, Elisa's cup has a repeating decimal representation of 0.52525252..., where the repeating pattern is 52. To express this decimal as a fraction, we can use the fact that the repeating pattern has two digits. Let's call the repeating pattern "r". Then, we can write:

x = 0.52525252...

100x = 52.52525252... (Multiplying both sides by 100 to shift the decimal two places to the right)

100x - x = 52.52525252... - 0.52525252... (Subtracting the equation x from 100x)

99x = 52

Dividing both sides of the equation by 99, we get:

x = 52/99

So, Elisa has 52/99 liters of juice.

Elisa's brother has double the quantity, which is 2 times (52/99):

2x = 2 * (52/99) = 104/99 liters

To find the total quantity of juice they have together, we add their individual quantities:

52/99 + 104/99 = 156/99

Simplifying the fraction, we get:

156/99 = 52/33

Therefore, the correct option is 3. 52/33 liters, which represents the total quantity of juice Elisa and her brother have together.

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We have verified that a · ∇r = a in Cartesian coordinates.

In Cartesian coordinates, a vector a can be written as:

a = a(I) + b(j) + c(k)

where a(I), b(j), and c(k) are the components of a along the x, y, and z axes, respectively. The dot product of a with the gradient operator ∇r is:

a · ∇r = (a(I) + b(j) + c(k)) · (∂/∂x I + ∂/∂y j + ∂/∂z k)

Expanding the dot product using the distributive property, we get:

a · ∇r = a(I)(∂/∂x) + b(j)(∂/∂y) + c(k)(∂/∂z)

Next, we apply this operator to the position vector r = x(I) + y(j) + z(k):

a · ∇r = a(I)(∂/∂x)(x(I)) + b(j)(∂/∂y)(y(j)) + c(k)(∂/∂z)(z(k))

= a(I) + b(j) + c(k)

= a

Therefore, we have verified that a · ∇r = a in Cartesian coordinates.

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The r-value that represents the strongest correlation is -0.83.

In correlation analysis, the correlation coefficient (r-value) measures the strength and direction of the linear relationship between two variables. The value of the correlation coefficient ranges from -1 to +1.

The magnitude of the correlation coefficient indicates the strength of the relationship, where values closer to -1 or +1 indicate a stronger relationship. In this case, an r-value of -0.83 indicates a strong negative correlation between the two variables.

A negative correlation means that as one variable increases, the other variable tends to decrease, and vice versa. The closer the r-value is to -1, the stronger the negative correlation. Therefore, an r-value of -0.83 suggests a very strong negative correlation, indicating a robust and consistent inverse relationship between the two variables.

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13.) The probability that when a tile is chosen at random it has at least 4 sides and not a hexagon = 5/18

14.) Probability that the tile is not a square and has less than 7 sides= 9/16

To calculate the probability of the given event, the formula that should be used is given as follows:

Probability = possible outcome/sample space

For 13.)

when tile has at least 4 sides;

possible outcome= 12+8 = 20

sample space = 60

p(at least 4 sides) = 20/60 = 1/3

When not a hexagon;

possible outcome = 12+15+8+6+9= 50

p(not a hexagon) = 50/60= 5/6

The probability that when a tile is chosen at random it has at least 4 sides and not a hexagon;

= 1/3×5/6

= 5/18

For 14.)

when tile is not a square;

possible outcome= 60-15= 45

p(not a square)= 45/60= 3/4

p(less than 7 sides)

possible outcome= 12+15+8+10= 45

p(less than 7 sides) = 45/60 = 3/4

Probability that the tile is not a square and has less than 7 sides= 3/4×3/4= 9/16

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Green's Theorem to evaluate the line integral along the given positively oriented curve. is -1215.

To evaluate the line integral using Green's Theorem, we first need to calculate the double integral of the curl of the vector field over the region enclosed by the given curve.

Let's denote the vector field as F(x, y) = (xy^2, 5x^2y). The curl of F is given by ∇ × F = (∂Q/∂x - ∂P/∂y), where P = xy^2 and Q = 5x^2y.

∂Q/∂x = 10xy

∂P/∂y = 2xy

Now, we can compute the line integral using Green's Theorem, which states that the line integral of a vector field F around a positively oriented curve C is equal to the double integral of the curl of F over the region D enclosed by C:

∫C (P dx + Q dy) = ∬D (∂Q/∂x - ∂P/∂y) dA

In this case, the region D is the triangle with vertices (0,0), (3,3), and (3,6). To evaluate the double integral, we can use an appropriate coordinate system, such as Cartesian or polar coordinates, depending on the complexity of the region D.

Since the triangle is simple and the integrals can be easily evaluated in Cartesian coordinates, we can proceed with that. The limits of integration for x are from 0 to 3, and for y, it is from y = x to y = 6.

∬D (∂Q/∂x - ∂P/∂y) dA = ∫[0,3] ∫[x,6] (10xy - 2xy) dy dx

Integrating with respect to y first:

∫[0,3] [(5xy^2 - xy^2) evaluated from y=x to y=6] dx

∫[0,3] [(5x(6)^2 - x(6)^2) - (5x(x)^2 - x(x)^2)] dx

∫[0,3] [180x - 30x^3] dx

[90x^2 - 7.5x^4] evaluated from 0 to 3

[810 - 2025] - [0 - 0]

-1215

Therefore, the line integral along the given curve is -1215.

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