Antiderivative with the equation

The numbers to complete the pythagorean triple in the equation (n² - 1)² + k² = (n² + 1)² are B. 35 and 37

A pythagorean triple is a set of 3 numbers that obey the pythagorean theorem.

Given the equation

(n² - 1)² + k² = (n² + 1)², we need to find the remaining numbers generated by the equation when k = 12. So, we proceed as follows/

Since we have the equation

(n² - 1)² + k² = (n² + 1)²

Subtracting (n² + 1)², from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = (n² + 1)² - (n² + 1)²

(n² - 1)² - (n² + 1)² + k² = 0

Now, subtracting k from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = 0

(n² - 1)² - (n² + 1)² + k² - k² = 0 - k²

(n² - 1)² - (n² + 1)² + 0 = - k²

(n² - 1)² - (n² + 1)² = - k²

Using the difference of two squares a² - b² = (a + b)(a - b)

(n² - 1)² - (n² + 1)² = - k²

(n² - 1 + n² + 1)[n² - 1 - (n² + 1)] = - k²

(n² + n² - 1 + 1)[n² - n² - 1 - 1)] = - k²

(2n² + 0)[0 - 2)] = - k²

(2n²)(- 2) = - k²

-4n² = - k²

4n² = k²

n² = k²/4

Taking square root of both sides, we have that

n = √(k²/4)

n = k/2

Since k = 12, we have that

n = 12/2

n = 6

So, the first number is (n² - 1) = (6² - 1)

= 36 - 1

= 35

The second number is (n² + 1) = (6² + 1)

= 36 + 1

= 37

So, the numbers are B. 35 and 37

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The cross section of the geometric solid is (d) rectangle

From the question, we have the following parameters that can be used in our computation:

The geometric solid

Also, we can see that

The geometric solid is a cylinder

And the cylinder is divided vertically

The resulting shape from the division is a rectangle

This means that the cross section of the geometric solid is (d) rectangle

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The shape of the cross-section for the geometric solid given in the diagram is a rectangle.

The cross section of the geometric solid represents the shape which extends beyond the actual geometric solid which is a cylinder.

A rectangle has opposite side being equal. This means that the width and and length are of different length.

Therefore, the shape of the cross-section is a rectangle.

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The approximate solution to the equation 1/x-1 = |x-2| after three iterations of successive approximation is x ≈ 5/2 or x ≈ 2.5.

To solve the equation 1/x-1 = |x-2| using three iterations of successive approximation, we will start with an initial guess and refine it using an iterative process.

Given that the equation involves absolute value, we will consider two cases:

Case 1: x - 2 ≥ 0

In this case, |x-2| simplifies to x-2, and the equation becomes 1/(x-1) = x-2.

Case 2: x - 2 < 0

In this case, |x-2| simplifies to -(x-2), and the equation becomes 1/(x-1) = -(x-2).

Now, let's perform the successive approximation:

Iteration 1:

Let's start with an initial guess, x = 2.

Case 1: When x - 2 ≥ 0,

1/(2-1) = 2-2,

1/1 = 0,

which is not true.

Case 2: When x - 2 < 0,

1/(2-1) = -(2-2),

1/1 = 0,

which is not true.

Since our initial guess did not satisfy the equation in either case, we need to choose a different initial guess.

Iteration 2:

Let's try x = 3.

Case 1: When x - 2 ≥ 0,

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

1/2 = 1,

which is not true.

Case 2: When x - 2 < 0,

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

1/2 = -1,

which is not true.

Again, our guess did not satisfy the equation in either case.

Iteration 3:

Let's try x = 2.5.

Case 1: When x - 2 ≥ 0,

1/(2.5-1) = 2.5-2,

1/1.5 = 0.5,

which is true.

Case 2: When x - 2 < 0,

1/(2.5-1) = -(2.5-2),

1/1.5 = -0.5,

which is not true.

Our guess of x = 2.5 satisfies the equation in Case 1.

Therefore, the approximate solution to the equation 1/x-1 = |x-2| after three iterations of successive approximation is x ≈ 5/2 or x ≈ 2.5.

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Answer:

A. 6

Step-by-step explanation:

You have 3 molecules of H2SO4, each molecule has two atoms of H and you have 3 molecules, so 3*2=6 atoms of H

Step-by-step explanation:

b. To determine the coordinates of the points on the graph, we need to consider the information given.

At the beginning of the second lap, one minute and three seconds had passed. Since each lap is 4.7 kilometers long, we can calculate the distance traveled at the beginning of the lap using the formula:

Distance = (Number of laps - 1) × Length of each lap

For the beginning of the second lap:

Distance = (2 - 1) × 4.7 km = 4.7 km

At the beginning of the third lap, one minute and three seconds had passed. Therefore, the distance traveled at the beginning of the third lap is:

Distance = (3 - 1) × 4.7 km = 9.4 km

The coordinates of the points on the graph would be:

Point A: (4.7 km, 4.7 km)

Point B: (9.4 km, 9.4 km)

On the graph, the x-axis would represent the distance traveled at the beginning of the lap, and the y-axis would represent the distance traveled at the end of the lap.

c. To find the equation of the line between the two points, we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope and b is the y-intercept.

Using the coordinates of the two points (4.7 km, 4.7 km) and (9.4 km, 9.4 km), we can calculate the slope:

m = (y2 - y1) / (x2 - x1)

= (9.4 km - 4.7 km) / (9.4 km - 4.7 km)

= 4.7 km / 4.7 km

= 1

The slope of the line is 1.

Since the line passes through the origin (0, 0), the y-intercept (b) is 0.

Therefore, the equation of the line is:

y = 1x + 0

y = x

This line represents a one-to-one relationship between the distance traveled at the beginning of the lap and the distance traveled at the end of the lap.

d. To estimate how long it would take Fernando to finish the entire race if he continues to average the same speed, we need to find the time it takes for one lap.

In the given scenario, Brandon timed Fernando's lap for one minute and three seconds. Since there are 3 laps in total, we can divide the total time by the number of laps to find the average time per lap:

Average time per lap = Total time / Number of laps

= 1 minute and 3 seconds / 3

= 21 seconds

Since each lap is 4.7 kilometers long, we can calculate the speed:

Speed = Distance / Time

= 4.7 km / (1 minute + 21 seconds)

To estimate the time it would take Fernando to finish the entire race, we divide the total race distance by the speed:

Total race time = Total race distance / Speed

= 500 km / (4.7 km / (1 minute + 21 seconds))

To calculate the time, we need to convert 1 minute and 21 seconds to a decimal form:

1 minute + 21 seconds = 1 minute + (21 seconds / 60 seconds)

= 1 minute + 0.35 minutes

= 1.35 minutes

Substituting the values:

Total race time = 500 km / (4.7 km / 1.35 minutes)

= 500 km × (

1.35 minutes / 4.7 km)

= 500 km × 0.28723...

≈ 143.615 minutes

Therefore, it would take approximately 143.615 minutes for Fernando to finish the entire race if he continues to average the same speed.

The inequality representing the situation is "y ≥ 0.75x," where y is the total number of hours Lionel spends practicing and x is the number of days he practices.

To represent the situation where Lionel practices for a minimum of 45 minutes on the days he practices, we can use the variables x and y, where x represents the number of days Lionel practices and y represents the total number of hours he spends practicing.

We know that Lionel practices for a minimum of 45 minutes on each day. Since there are 60 minutes in an hour, this is equivalent to 0.75 hours. Therefore, for each day Lionel practices, he spends at least 0.75 hours.

To find the total number of hours Lionel spends practicing (y), we can multiply the number of days he practices (x) by the minimum number of hours he spends on each day (0.75). This gives us the equation:

y ≥ 0.75x

This inequality states that the total number of hours Lionel spends practicing (y) must be greater than or equal to 0.75 times the number of days he practices (x). It ensures that Lionel practices for a minimum of 45 minutes (0.75 hours) on each day he practices.

By using this inequality, we can track Lionel's practice time and ensure that he meets the minimum requirement of 45 minutes per day.

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Answer:

Step-by-step explanation:

We must simplify and rearrange the variables to divide x in order to determine the comparable inequality to the supplied inequality, -4(x + 7) 3(x - 2).

Below are the steps to solve the problem:

-4(x + 7) < 3(x - 2)

Expanding the equation:

-4x-28<3x-6

Gather the variable term on one side and constants on the other:

Adding -4x and 28 on both sides:

-4x + 4x - 28 + 28 < 3x + 4x - 6 + 28

=>0 < 7x + 22

To make the coefficient of x positive, we divide the entire inequality by 7:

0/7 < (7x + 22)/7

=> 0 < x + 22/7

The correct statement is D. Hannah's unit rate is 1 fewer word per minute than Becky's unit rate.

To compare the unit rates, we need to determine the rate at which each person types words per minute.

For Hannah, the equation y = 11x represents her typing speed, where y is the number of words and x is the number of minutes. This means that Hannah types 11 words per minute (11 words/minute).

Looking at Becky's graph, we can determine her unit rate by calculating the change in the number of words divided by the change in time.

The change in words is 48 - 2 = 46, and the change in time is 3 - 2 = 1. So, Becky's unit rate is 46 words per minute (46 words/minute).

Comparing the unit rates:

Hannah's unit rate: 11 words/minute

Becky's unit rate: 46 words/minute

Therefore, Hannah's unit rate is 35 words per minute less than Becky's unit rate.

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To solve these problems, we can use Coulomb's Law, which states that the magnitude of the electric field produced by a charged particle is directly proportional to the charge and inversely proportional to the square of the distance from the particle.

a) Without finding the charge of the particle, we can use the inverse square relationship. If the electric field magnitude is 2.0 N/C at a distance of 50 cm, then at half the distance (25 cm), the electric field would be four times stronger. Therefore, the electric field at 25 cm would be 4 * 2.0 N/C = 8.0 N/C.

b) Doubling the charge would result in doubling the electric field magnitude. So, if the electric field at a distance of 50 cm was initially 2.0 N/C, it would become 4.0 N/C after doubling the charge.

c) To determine the magnitude of the particle's charge, we need to use the equation for the electric field:

E = k * (|q| / r^2)

where E is the electric field magnitude, k is the electrostatic constant, |q| is the magnitude of the charge, and r is the distance from the particle.

Using the known values of E = 2.0 N/C and r = 50 cm (or 0.5 m), we can rearrange the equation to solve for |q|:

|q| = E * r^2 / k

Substituting the values and the known value of k, we can calculate the magnitude of the charge.

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The length of the arc in terms of π and the actual length is as follows;

8.17π m²

25.68 m²

Let's find the length of the arc of the circle as follows:

Therefore,

length of an arc = ∅ / 360 × 2πr

where

Hence,

∅ = 180 - 88 = 92 degrees

r = 16 m

length of an arc MA = 92 / 360 × 2 × π × 16

length of an arc MA = 2944π / 360

length of an arc MA = 8.17π

Therefore,

length of an arc MA = 25.68 m²

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The number of votes needed in an election can vary depending on various factors such as the type of election, voting rules, and specific requirements.

Without additional context or information about the specific election, it is challenging to provide an exact number of votes needed.The number of votes needed in an election is typically determined by factors such as the majority threshold, minimum vote requirement, or any specific criteria outlined in the election rules.

For example, in some elections, a candidate may need a simple majority (more than half) of the votes cast to win, while in others, a candidate may need a specific number or percentage of votes to secure victory.To determine the number of votes needed, it is essential to refer to the specific guidelines or rules established for that particular election.

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The pattern between the numbers 3, 20, 110, and 1715 can be found using a mathematical method. In order to get the following number from each, there is a sequence that must be applied.Let's take a look at the sequence that was used to generate these numbers:

The first number is multiplied by 2 and then increased by 14 to get the second number. For example:

3 x 2 + 14 = 20

Then, the second number is multiplied by 3 and 20, and 110 is added.

20 x 3 + 110 = 170

The third number is multiplied by 4 and then increased by 110.

110 x 4 + 110 = 550

Finally, the fourth number is multiplied by 5 and then increased by 110.

550 x 5 + 110 = 2825

Therefore, using the above formula, the next number in the sequence can be calculated:

1715 x 6 + 110 = 10400

As a result, the sequence of numbers 3, 20, 110, 1715, 10400 can be calculated using the mathematical formula stated above.

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The probability of drawing blue, red, and white balls consecutively is 120/3375 with replacement and 120/2730 without replacement.

To calculate the probability of drawing three balls from a box containing six white balls, five red balls, and four blue balls, we need to consider two scenarios: with replacement and without replacement.

(i) With replacement:

When each ball is replaced after it is drawn, the total number of balls remains the same for each draw. Therefore, the probability of drawing a specific color on each draw remains constant.

The probability of drawing a blue ball on each draw is 4/15, the probability of drawing a red ball is 5/15, and the probability of drawing a white ball is 6/15 (assuming all colors are equally likely to be drawn).

To find the probability of drawing a blue, red, and white ball consecutively, we multiply the probabilities together since the events are independent:

P(Blue, Red, White) = (4/15) * (5/15) * (6/15) = 120/3375

(ii) Without replacement:

When the balls are not replaced after being drawn, the total number of balls decreases for each subsequent draw. This affects the probability of each color being drawn on subsequent draws.

For the first draw, the probability of drawing a blue ball is 4/15, a red ball is 5/15, and a white ball is 6/15.

For the second draw, the probability of drawing a blue ball is 3/14, a red ball is 5/14 (as one blue ball is already drawn), and a white ball is 6/14.

For the third draw, the probability of drawing a blue ball is 2/13, a red ball is 4/13 (as two blue balls and one red ball are already drawn), and a white ball is 6/13.

To find the probability of drawing a blue, red, and white ball consecutively without replacement, we multiply the probabilities together:

P(Blue, Red, White) = (4/15) * (5/14) * (6/13) = 120/2730

Therefore, the probability of drawing blue, red, and white balls consecutively is 120/3375 with replacement and 120/2730 without replacement.

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a) The summation (Ex) of the given data set, we add up all the values

Ex = 77

b) n = 5

c) x = 15.4

a) To compute the summation (Ex) of the given data set, we add up all the values:

19 + 10 + 15 + 17 + 16 = 77

b) The sample size (n) is the total number of data points in the set. In this case, there are 5 data points, so:

n = 5

c) To calculate the sample mean (x), we divide the summation (Ex) by the sample size (n):

x = Ex / n

x = 77 / 5

x = 15.4

Therefore, the answers are:

a) Ex = 77

b) n = 5

c) x = 15.4

The summation is 77, the sample size is 5, and the sample mean is 15.4.

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The midpoint of the line segment joining the pair of points (8, -8) and (-5, -1) is (1.5, -4.5).

To find the midpoint, we need to take the average of the x-coordinates and the average of the y-coordinates of the two given points.

For the x-coordinate: (8 + (-5))/2 = 3/2 = 1.5

For the y-coordinate: (-8 + (-1))/2 = -9/2 = -4.5

Thus, the midpoint of the line segment is (1.5, -4.5).

In more detail, the midpoint formula is derived by averaging the x-coordinates and y-coordinates separately. For a line segment with endpoints (x1, y1) and (x2, y2), the midpoint (xm, ym) is given by:

xm = (x1 + x2)/2

ym = (y1 + y2)/2

In this case, the x-coordinates are 8 and -5, and their average is (8 + (-5))/2 = 1.5. The y-coordinates are -8 and -1, and their average is (-8 + (-1))/2 = -9/2 = -4.5.  

Therefore, the midpoint of the line segment joining the two given points is (1.5, -4.5).

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The hypotenuse (x) of the right triangle is 10 units

From the question, we have the following parameters that can be used in our computation:

The right triangle

The hypotenuse (x) of the right triangle can be calculated using the following sine equation

sin(30) = 5/x

Using the above as a guide, we have the following:

x = 5/sin(30)

Evaluate

x = 10

Hence, the hypotenuse of the right triangle is 10

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The slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

To determine the slope of a line that is perpendicular to the given line, we need to find the negative reciprocal of the slope of the given line.

The equation of the given line is x + 6y = 3.

To find the slope of the given line, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope:

x + 6y = 3

6y = -x + 3

y = (-1/6)x + 1/2

From the equation y = (-1/6)x + 1/2, we can see that the slope of the given line is -1/6.

To find the slope of a line that is perpendicular, we take the negative reciprocal of -1/6.

The negative reciprocal of -1/6 can be found by flipping the fraction and changing its sign:

Negative reciprocal of -1/6 = -1 / (-1/6) = -1 * (-6/1) = 6

Therefore, the slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

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Cameron has 1 book left over after sorting it into group of 5

To determine the number of book left,

we need to first divide 56 by 5         [56 ÷ 5]

Quotient = 11

then find the remainder which is 1

therefore there os only 1 book left over

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Juan could have a maximum of 3 quarters.

1. Let's assume Juan has x quarters.

2. The value of x quarters in dollars would be 0.25x.

3. We are given that Juan has 18 dimes, which have a value of 0.10 * 18 = $1.80.

4. The combined value of Juan's dimes and quarters should be less than or equal to $2.75.

5. We can write the equation: 0.25x + 1.80 ≤ 2.75.

6. Subtracting 1.80 from both sides of the equation, we have: 0.25x ≤ 2.75 - 1.80.

7. Simplifying, we get: 0.25x ≤ 0.95.

8. Dividing both sides of the inequality by 0.25, we have: x ≤ 0.95 / 0.25.

9. Evaluating the expression, we find: x ≤ 3.8.

10. Since Juan cannot have a fraction of a quarter, the maximum number of quarters he could have is 3.

11. However, we need to check if the combined value of the dimes and quarters is at least $0.20.

12. If Juan has 3 quarters (0.25 * 3 = $0.75) and 18 dimes ($1.80), the combined value is $0.75 + $1.80 = $2.55.

13. Since $2.55 is less than $2.75, Juan can have 3 quarters.

14. Therefore, the maximum number of quarters Juan could have is 3.

Note: There are no possible solutions where Juan has more than 3 quarters and still satisfies the conditions.

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Answer:c

Step-by-step explanation:

The estimated standard deviation of the bear weights, based on the given graph and using the 69-95-99.7% rule, is approximately 63.

The 69-95-99.7% (empirical) rule, also known as the 3-sigma rule, is a rule of thumb that applies to data that follows a normal distribution. According to this rule:

- Approximately 68% of the data falls within one standard deviation of the mean.

- Approximately 95% of the data falls within two standard deviations of the mean.

- Approximately 99.7% of the data falls within three standard deviations of the mean.

In the given graph, if we assume the bear weights follow a normal distribution, we can estimate the standard deviation using the 69-95-99.7% rule.

Based on the graph, we know that the midpoint of the distribution (mean) is 200. Assuming the graph is symmetric, we can estimate one standard deviation as half the distance between the mean (200) and either end (74 or 326).

To calculate this, we subtract the mean from one of the endpoints and divide by 2:

Standard Deviation ≈ (326 - 200) / 2 ≈ 126 / 2 ≈ 63

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Answer:

[tex]x^{3}[/tex] - 4[tex]x^{2}[/tex] + 2x - 4

Step-by-step explanation:

(8x + 2[tex]x^{3}[/tex] - 4[tex]x^{2}[/tex]) - (4 + [tex]x^{3}[/tex] + 6x)

= 8x + 2[tex]x^{3}[/tex] - 4[tex]x^{2}[/tex] - 4 - [tex]x^{3}[/tex] - 6x

= 2x + [tex]x^{3}[/tex] - 4[tex]x^{2}[/tex] - 4

= [tex]x^{3}[/tex] - 4[tex]x^{2}[/tex] + 2x - 4

This flexibility of a money market account makes it an ideal option for people who need to save money without risking their inheritance.

After inheriting $25,000 from a distant uncle, you would like to save it for ten years before doing anything with it. Since you want to save the money, there are several options for keeping it safe and earning interest, including a savings account, a certificate of deposit (CD), and a money market account.

Money market accounts, in my opinion, would be the best place to keep the inheritance. The money market account is a low-risk account with high-interest rates, making it an attractive option for someone who wants to save their money. As a result, it would be reasonable to place the inheritance into a money market account that pays a daily compounded rate of 3%.There are several reasons for choosing this option.

Firstly, the daily compounded interest will generate a higher return over the ten-year period than the simple interest or monthly compounded interest offered by other accounts. Second, the account is FDIC-insured, which means that the account holder is guaranteed to receive their money in the event of a bank failure.

Furthermore, the money market account provides easy access to the account holder's money while still earning interest. Most money market accounts have a limit on the number of withdrawals a person can make per month. Still, the account holder can easily transfer funds into a checking account or withdraw money from an ATM if needed.

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Answer:

(f + g)(1) = 5

Step-by-step explanation:

(f + g) means we are going to add f(x) and g(x). But also, the (1) part means we are going to let x be equal to 1. We're going to fill in 1 in place of x. You can do this in either order.

Generally speaking its "easier" to fill in the 1 for x first and then do the adding part.

f(x) = x + 6

f(1) = 1 + 6 = 7

and,

g(x) = x - 3

g(1) = 1 - 3 = -2

add the 7 and -2 together:

7 + - 2

= 5

It works out the same if you add first:

f(x) + g(x)

= x + 6 + x - 3

= 2x + 3

then put the 1 in:

= 2×1 + 3

= 2 + 3

= 5

Hope this helps!

Answer:

A: $51.00

B: $54.57

Step-by-step explanation:

Let amount = $68.00

Part A:

Since the coupon is for 25%, the family pays 75% of $68.00

75% of $68.00 = 0.75 × $68.00 = $51.00

The new price is $51.00

Part B:

The tax is 7% of $51.00

7% of $51.00 = $3.57

The total price is the sum of $51.00 and the amount of tax, $3.57

Total price = $51.00 + $3.57 = $54.57