O is the centre of this circle.
What is the size of angle x?
Fully justify your answer.

Answer:

x=1.9

Step-by-step explanation:

[tex]\frac{x}{4.6} =\frac{4.6}{11}[/tex]

[tex]11x=21.16[/tex]

[tex]X=1.9[/tex]

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 0, x = 0, and x = 1 about the line x = 2 is (5/6)π cubic units.

To find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line, we use the method of cylindrical shells. The formula for the volume of a cylindrical shell is given as,
Volume = 2πrh * Δx

Where, r is the distance of the shell from the axis of rotation, h is the height of the shell, and Δx is the thickness of the shell. The factor 2π accounts for the entire circumference of the shell.
Example problem: Find the volume of the solid generated by rotating the region bounded by the curves y = x^2, y = 0, x = 0, and x = 1 about the line x = 2.
Solution:
Step 1: Draw the graph of the region to be rotated
Step 2: Identify the shell radius and height
For a shell at position x, the radius is given by r = 2 - x (the distance of the line x = 2 from the axis of rotation)
The height of the shell is given by h = x^2 (the difference between the top and bottom curves)
Step 3: Write the volume formula
Volume = 2πrh * Δx
Step 4: Integrate to find the total volume
The limits of integration are from 0 to 1 since the curves intersect at (0,0) and (1,1).
[tex]∫ 2π(2 - x)(x^2) dx from 0 to 1[/tex]
= [tex]2π [∫ 2x^2 - x^3 dx from 0 to 1][/tex]
= [tex]2π [(2/3) - (1/4)][/tex]
= 2π [5/12][tex]2π [5/12][/tex]
= (5/6)π cubic units
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The Variance Inflation Factor (VIF) measures the degree of correlation or the collinearity of the explanatory variables.

VIF quantifies how much each explanatory variable is correlated with a linear combination of the other variables in a multiple regression model. This can help identify variables that are redundant, irrelevant, or harmful to the model's accuracy. The VIF is calculated for each explanatory variable by dividing the variance of the regression coefficient estimates by the variance of the regression coefficient estimates when that variable is excluded from the model.

If the VIF is greater than 1, it indicates that the variance of the regression coefficient estimate for that variable is inflated by the presence of the other variables, which reduces the model's accuracy. Therefore, a VIF greater than 1 is considered to be an indication of collinearity or multicollinearity in the explanatory variables. The VIF measures the degree of correlation or the collinearity of the explanatory variables, and it can identify variables that are redundant, irrelevant, or harmful to the model's accuracy.

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Answer:

[tex]x = 4\sqrt{3}[/tex]

Step-by-step explanation:

The triangle is right-angled

In a right-angled triangle, the following equality holds

[tex]\tan x = \dfrac{\text{Side Opposite x}}{\text{Side adjacent to x}}\\\\\tan 30 = \dfrac{4}{x}\\\\[/tex]

[tex]\tan 30 = \dfrac{1}{\sqrt{3}}\\\\\dfrac{1}{\sqrt{3}} = \dfrac{4}{x}\\\\[/tex]

Cross multiply:
[tex]1 \times x = 4 \times \sqrt{3}\\\\x = 4\sqrt{3}[/tex]

Answer:

x = -13/7, y = 54/7

Step-by-step explanation:

- Set equations equal to each other

[tex]5x + 17 = -2x +4[/tex]

Add 2x to both sides

[tex]7x +17 = 4[/tex]

Subtract 17 from both sides

[tex]7x = -13[/tex]

divide both sides by 7

[tex]x = -13/7[/tex]

Then substitute x into one of the equations. I picked the first one.

[tex]y = 5 (-13/7) + 17[/tex]

I multiplied 5 by -13 to get -65/7. Then I turned 17 into sevenths: 17 x 7 = 119, so that's 119/7. 119 + -65 = 54, so 54/7 for y.

Let me know if this helped by hitting thanks or brainliest. If not, please comment and I'll get back to you ASAP!

The simplified form of the combination is (3n² - 5n + 3)/3

Now let's return to the problem at hand. We have the expression ⁿ⁺¹C₃ − ⁿ⁻¹C₃, which involves two combinations with slightly different values of n. To simplify this expression, we need to first expand each combination using the formula we just discussed.

Starting with ⁿ⁺¹C₃, we have:

ⁿ⁺¹C₃ = (n + 1)!/(3!(n - (3 + 1))!)

We can simplify this further by noting that (n - 4)! is the same as (n - 3 - 1)! and pulling out the common factors:

(n + 1)!/(3!(n - 4)!) = (n + 1)(n)(n - 1)/(3 x 2 x 1)

Next, let's expand ⁿ⁻¹C₃:

ⁿ⁻¹C₃ = (n - 1)!/(3!(n - (3 - 1))!)

Again, we can simplify by noting that (n - 2)! is the same as (n - 1 - 1)!:

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

Now we can substitute these expressions back into the original equation:

ⁿ⁺¹C₃ − ⁿ⁻¹C₃ = (n + 1)(n)(n - 1)/(3 x 2 x 1) - (n - 1)(n - 2)(n - 3)/(3 x 2 x 1)

We can simplify this by finding a common denominator and combining like terms:

[(n + 1)(n)(n - 1) - (n - 1)(n - 2)(n - 3)]/(3 x 2 x 1)

Simplifying further, we can expand the terms:

[(n³ - n) - (n³ - 7n² + 11n - 6)]/(6)

Which gives us:

(6n² - 10n + 6)/6

We can simplify this by factoring out a 2 from each term in the numerator and denominator:

2(3n² - 5n + 3)/2(3)

And finally, we can cancel out the 2s to get:

(3n² - 5n + 3)/3

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Complete Question:

Simplify:  

ⁿ⁺¹C₃ − ⁿ⁻¹C₃

The rate of filling water in the tank is 23450t gallons per hour.

Let's assume that the time taken to fill the tank is t hours.

The volume of water filled into the tank at time t is given by the expression V(t) = 23450t.

The tank is originally empty, which means its volume = 0 gallons.

After 5 hours,

t= 5 hours

The volume of water filled in is given by [tex]V(5) = 23450 * 5= 1,17,250[/tex] gallons of water.

Therefore, 1,17,250 gallons of water are filled in the tank after 5 hours.

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Answer:

see explanation

Step-by-step explanation:

(a)

the scale factor is the ratio of corresponding sides, shape B to shape A

scale factor = [tex]\frac{12}{15}[/tex] = [tex]\frac{4}{5}[/tex]

(b)

t is then [tex]\frac{4}{5}[/tex] of 7 = [tex]\frac{4}{5}[/tex] × 7 = [tex]\frac{28}{5}[/tex]

Lightbulbs act as resistors. Janine is building a circuit that contains two lightbulbs in parallel. One of the lightbulbs has a resistance of 120 ohms, but the resistance of the second lightbulb is unknown. Therefore, The inverse of Janine's equation is r = -120/(t - 120).

Equation:

An equation is described as a formula that expresses the equality of two expressions, by connecting them with the equals sign. An equation is a formula that expresses that two expressions are equal by joining them with the equal sign =.The word comparison and its cognates in other languages ​​may have subtly different meanings; for example, in French, an equation is defined to contain one or more variables, and in English any well-formed formula composed of two expressions linked by equal signs is an equation.

According to the Question:

In algebra, an equation is a mathematical statement that determines whether two mathematical expressions are equal.

The type of comparison is identity or conditional comparison.

For each potential value, the variable provides an identifier. Only certain combinations of variable values ​​allow conditional comparisons to be true.

In order to  find the inverse of Janine's equation, we will to solve for r in terms of t.

First, we can rearrange the equation to get

⇒ t - 120 = 1/r.

divide both sides of the equation by 1/r to get

⇒ t/r - 120/r = 1.

divide both sides of the equation by -120/r to get

⇒ r = -120/(t - 120).

Therefore, the inverse of Janine's equation is r = -120/(t - 120).

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The probability that x is less than 2 is approximately 0.4866. The probability that x is greater than 3 is approximately 0.3528. The probability that x is less than 5 is approximately 0.6321. The probability that x is between 2 and 5 is approximately 0.1455.

The given probability density function is an exponential distribution with a rate parameter of λ = 1/3. The formula for P(x < x_0) is the cumulative distribution function (CDF) of the exponential distribution, which is given by:

F(x_0) = ∫[0,x_0] f(x) dx = ∫[0,x_0] 1/3 * 4 * e^(-x/3) dx

a. Write the formula for P(x < x_0):

Using integration, we can solve this formula as follows:

F(x_0) = [-4e^(-x/3)] / 3 |[0,x_0]

= [-4e^(-x_0/3) + 4]/3

b. Find P(x < 2):

To find P(x < 2), we simply substitute x_0 = 2 in the above formula:

F(2) = [-4e^(-2/3) + 4]/3

≈ 0.4866

Therefore, the probability that x is less than 2 is approximately 0.4866.

c. Find P(x > 3):

To find P(x > 3), we can use the complement rule and subtract P(x < 3) from 1:

P(x > 3) = 1 - P(x < 3) = 1 - F(3)

= 1 - [-4e^(-1) + 4]/3

≈ 0.3528

Therefore, the probability that x is greater than 3 is approximately 0.3528.

d. Find P(x < 5):

To find P(x < 5), we simply substitute x_0 = 5 in the above formula:

F(5) = [-4e^(-5/3) + 4]/3

≈ 0.6321

Therefore, the probability that x is less than 5 is approximately 0.6321.

e. Find P(2 < x < 5):

To find P(2 < x < 5), we can use the CDF formula to find P(x < 5) and P(x < 2), and then subtract the latter from the former:

P(2 < x < 5) = P(x < 5) - P(x < 2)

= F(5) - F(2)

= [-4e^(-5/3) + 4]/3 - [-4e^(-2/3) + 4]/3

≈ 0.1455

Therefore, the probability that x is between 2 and 5 is approximately 0.1455.

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In summary, when repeated samples of size n are taken from a normal population with an unknown variance, the statistic t follows the t-distribution with n-1 degrees of freedom.

If repeated samples of size n are taken from a normal population with an unknown variance, then the statistic t follows the t distribution with n-1 degrees of freedom. When a sample of size n is taken from a normal population with a known variance and mean, the sample mean follows a normal distribution with a mean of μ and a variance of σ2/n. This distribution is known as the sampling distribution of the mean.

However, when the variance is unknown, the sampling distribution of the mean can no longer be calculated using the normal distribution. In this scenario, the sample mean is calculated using a t-distribution instead of a standard normal distribution.The t-distribution is similar to the standard normal distribution. However, it is more spread out and flatter than the standard normal distribution. As a result, the t-distribution has thicker tails than the standard normal distribution, which makes it more suitable for analyzing small samples of data.

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Answer:

12.

Step-by-step explanation:

Given: 3(x + 2) + 5x

Put in x:

3(2 + 2) + (5 x 2)

(3 x 4) + 10

12 + 10

= 12.

Answer:

Point-Slope Form: y - 1 = -1/2(x - 4) or Standard Form: y = -1/2x + 3

Step-by-step explanation:

For a line to be perpendicular, you take the negative inverse of the slope of 2x - y = 4. To do this, rearrange the y to one side and you get y = 2x - 4. The slope of the line is 2. So, taking the negative inverse would be -1/m (with m being slope of the the equation given in the problem. This would give you -1/2.

Using point slope formula, y - y1 = m(x - x1), you can plug in the point given, (4,1) and the slope you found to get y - 1 = -1/2(x - 4). For standard form, isolating the y gets you y = -1/2x + 3.

You can check your answer by using Desmos by putting in the line 2x - y = 4, the point (4,1), and the equation you got as your answer. You will see that the equation is perpendicular to 2x - y = 4 and passes through point (4,1). Your equation of the line is y - 1 = -1/2(x - 4) or y = -1/2x + 3

The two remaining angles are 58°, and the length of the third side of the triangle is 6.5 inch.

In order to determine the other two angles of each triangle as well as the length of the third side, we need to use the Cosine Rule. According to the Cosine Rule, for any triangle with sides of length a, b, and c, and angles of A, B, and C, the following equation holds:

[tex]c^2 = a^2 + b^2 - 2ab cos(C)[/tex]

For the first triangle, we are given that the side of length 5 is adjacent to an angle of 61°. Therefore, a = 5, C = 61°. Using the information provided, we can also determine that b = 4.8. Substituting these values into the Cosine Rule equation, we get:

[tex]c^2 = (5)^2 + (4.8)^2 - 2(5)(4.8) cos(61°)[/tex]

We can solve this equation to get c = 6.5. Therefore, the length of the third side in the first triangle is 6.5. Additionally, we can use the Triangle Angle Sum theorem to determine the other two angles. According to this theorem, the sum of the three angles of a triangle is 180°. Therefore, for the first triangle, the two remaining angles are 180 - 61 - (180 - 61) = 58°.

For the second triangle, we use the same process, but with the given side lengths reversed. That is, we set a = 4.8, b = 5, and C = 61°. Again, substituting these values into the Cosine Rule equation, we get:

[tex]c^2 = (4.8)^2 + (5)^2 - 2(4.8)(5) cos(61°)[/tex]

We can solve this equation to get c = 6.5. Therefore, the length of the third side in the second triangle is also 6.5. We can use the Triangle Angle Sum theorem again to determine the other two angles. Again, for the second triangle, the two remaining angles are 180 - 61 - (180 - 61) = 58°.

In conclusion, for each triangle, the two remaining angles are 58°, and the length of the third side is 6.5 inch.

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The position of C on the number line is 1/8th position.

A number line is a visual representation of the real numbers as points or marks on a straight line. The number line is usually represented horizontally, with zero in the middle and positive numbers to the right of zero and negative numbers to the left of zero.

From the given number line, there are total 8 points between 0 to 1

That means, it's fractions of 8.

Location of point C is on the 1st point,

So, position of C on this number line = (1/8) ÷ (1-0) = 1/8

Therefore, In the number line, C is located in the 1/8th place.

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Answer:

I gave a couple solutions as I wasn't sure if you were asking for graphing purposes or substituting y=-x+6 into the second equation 2x-y=-9. So I gave both solutions just in case.

for the first equation y=-x+6, y intercept is (0,6)

for equation two 2x-y=-9, y intercept is (0,9)

In both of the equations the x value is 1.

Solving for y without graphing. Y=9+2x

and x=-1

Step-by-step explanation:substitute i

HOWEVER, if you are saying that the top equation is the value of y, then you substitute it into the bottom equation. 2x--x+6=-9 which would be x=-5

It really depends on what is expected of the question. I wasn't sure which one, so I gave a couple different approaches. If you could give more information, such as, are you graphing, that would be great. I'll keep an eye out for any comments.

The proof is completed using two column proof as follows

Statement                               Reason

QT ⊥ PT                                   Given

∠ QRP ≅ ∠ SRT = 90              definition of perpendicularity

∠ QPR ≅ ∠ STR                       Given

Δ PQR is similar to Δ TSR       AA similarity theorem

The AA similarity theorem, also known as the Angle-Angle Similarity Theorem, states that if two triangles have two corresponding angles that are congruent, then the triangles are similar.

In the given triangle, the two angles given to be equal are

Hence the triangles are similar

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Answer:

:)

Hopefully this helps you guys :)

good luck :D

B) Quick Sort is algorithms sorts data by recursively splitting the array into smaller sub-arrays, until eventually reaching single-element sections, and then joins the sub-arrays back together in sorted order

Merge Sort is an effective algorithm for sorting the data items of a list or an array. This algorithm's fundamental idea is to break the unsorted list into many sub-lists until every sub-list contains only one item.

The merging procedure then recombines all of the sublists, resulting in a sorted list. Divide-and-conquer is the name given to this merging process. Merge Sort divides the original list into N sublists, each of which contains one element.

The Merge Sort algorithm is faster than other sorting algorithms when it comes to time complexity. The time complexity of Merge Sort is O (n log n), which is better than Bubble Sort and Insertion Sort.

It is a more efficient sorting algorithm than other algorithms, such as bubble sort, which has a higher time complexity of O (n²).

Therefore, we can say that Merge Sort is the most efficient algorithm to sort an array in ascending or descending order.

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Answer:

I think it's answer is no.B

The value of z that should be used when making a 93% confidence interval for p is Z=1.81 that is option D.

The z score is a measure of how many standard deviations a raw score is below or above the population mean. It will be positive if the value is more than the mean and negative if it is less than the mean. It is often referred to as the standard score. It represents the number of standard deviations an entity has from the mean.

To utilise a z-score, both the mean and the population standard deviation must be known. The z score calculates the likelihood of a score occuring within a standard normal distribution. It also allows us to compare scores from various samples. A table for the values of, representing the values of the cumulative distribution function of the normal distribution, is known as a

At 93% confidence, the significance level is,

ɑ= 1-0.93

ɑ= 0.07

Divide alpha by 2,

ɑ/2= 0.07/2= 0.035

Now, from the ‘normal probability’ table, the z value corresponding to the inverse probability of 0.035 is 1.81.

As a result, the z value is 1.81 at 93% confidence level.

z= 1.81

The value for z score is 1.81

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The required equation in terms of 'a' is: a = (ac - by)/x.

The number being calculated is the formula's subject. On one side of the equals symbol, it is identifiable as the letter on its own.

To make 'a' the subject of the equation:

Ax + by = ac

we can start by isolating 'a' on one side of the equation.

Ax = ac - by

Then, we can divide both sides by 'x':

a = (ac - by)/x

Therefore, the equation in terms of 'a' is:

a = (ac - by)/x

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Complete questiion:

Find the subject of the equation ax+by=ac.

In summary, to find the null spaces of the matrices given in exercises 9 and 10, use the Gauss-Jordan elimination method and refer to the Remarks that follow example 3 in section 4.2 of the text. This will give the dimension of the null space and the number of free variables.
In exercises 9 and 10, the null space of the given matrices can be found by solving the homogeneous linear system of equations. In order to do this, use the Gauss-Jordan elimination method. Refer to example 3 in section 4.2 of the text for a detailed explanation. Afterwards, use the Remarks that follow the example to determine the dimension of the null space and the number of free variables.

The null space of a matrix is the set of all vectors that produce a zero vector when the matrix is multiplied by the vector. Therefore, to find the null space of a matrix, the homogeneous linear system of equations needs to be solved. The Gauss-Jordan elimination method involves adding multiples of one row to another to get a row with all zeroes. After this is done for all the rows, the equations can be solved for the free variables. The number of free variables will determine the dimension of the null space. Refer to example 3 in section 4.2 of the text for more details.

The Remarks that follow the example are important when determining the dimension of the null space and the number of free variables. In the Remarks, it is mentioned that the number of free variables is equal to the number of columns with a zero row. Therefore, after using the Gauss-Jordan elimination method to get the row with all zeroes, the number of columns with a zero row can be counted. This will give the dimension of the null space and the number of free variables.

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when we Roll a fair 6 66-sided die. then the probability of P (not 5) is 65/66

     The concept used a die is a tool used for games and gambling. The die is a cube with six faces, each of which has a different number of dots from one to six. The roll of a die is a random event, which means that it is unpredictable, and each roll is independent of any other roll.

 Let's solve the question that is P(not 5)

                 Since the die is fair, we have 66 sides on the die. Each of the sides has a probability of 1/66 of being rolled. P(not 5) means that we need to determine the probability of rolling a number that is not 5.

           We know that a 66-sided die has 65 numbers other than 5.

     Therefore, the probability of rolling a number that is not 5 is 65/66.

            P(not 5) = 65/66

   So, the probability of not rolling a 5 is p(not 5) = 65/66.

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a) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

Exponents are mathematical notation used to indicate that a quantity is being multiplied by itself a certain number of times. The exponent is usually written as a superscript to the right of the base number.For example, in the expression 2³, the base number is 2 and the exponent is 3. This means that 2 is being multiplied by itself three times, resulting in a value of 8. Exponents can also be negative or fractional, indicating that the base number is being divided by itself a certain number of times.

In the given question,

a) The value of -42 is different from the value of (-4)². The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

b) The value of -23 is different from the value of (-2)³. The brackets serve to indicate that the exponent applies to the entire quantity within the brackets.

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The family will use 3,600 fewer gallons of water with the new showerheads in a 30-day period.

A family of five recently replaced its 5-gallon-per-minute showerheads with water-saving 2-gallon-per-minute showerheads. Each member of the family averages 8 minutes in the shower per day. In a 30-day period, the family will use 3,600 fewer gallons of water with the new showerheads. How many fewer gallons of water will the family use with the new showerheads?

Calculate the number of gallons used per person per shower before:

5 gal/min x 8 min = 40 gallons per person per shower.

The number of gallons used per person per shower after:

2 gal/min x 8 min = 16 gallons per person per shower.

The number of gallons used per person per day before:

40 gallons x 1 shower = 40 gallons per person per day.

The number of gallons used per person per day after:

16 gallons x 1 shower = 16 gallons per person per day

The number of gallons used per family per day before:

40 gallons x 5 people = 200 gallons per day

The number of gallons used per family per day after:

16 gallons x 5 people = 80 gallons per day

The number of gallons used per month before:

40 gallons x 5 people x 30 days = 6,000 gallons per month

The number of gallons used per month after:

16 gallons x 5 people x 30 days = 2,400 gallons per month

The family will use 6,000 - 2,400 = 3,600 fewer gallons of water with the new showerheads in a 30-day period. Therefore, the answer is 3,600.

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Answer:

a. 6

b. 6

Step-by-step explanation:

A. mean proportion

3:x::x:12

product of mean = product of extreme

3 x 12 = x^2

x^2= 36

now take square root on both sides

x = 6

as sqaure root will cancel the square on x and square root of 36 is 6.

B. the geometric mean is the product of all

the numbers in a set, with the root of how many numbers there are. here, there are two numbers, so a square root is used. the value of the geometric mean of 3 and 12 is 6.

Hope's it helps...

Answer:

-----------------------------------

The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0.

To find the y-intercept, substitute 0 for x in the equation y = 4x + 2:

The y-intercept is 2 and matching choice is C.

Answer:

2

Step-by-step explanation:

We need to find out the y intercept of the given equation which is y = 4x + 2 .

y intercept is a point where the graph of the equation cuts the y axis. So at y intercept x coordinate will be 0 , so plug in x = 0 , in the given equation as ;

y = 4x + 2

y = 4(0) + 2

y = 0 + 2

y = 2

Therefore the y intercept of the line is 2 . ( option C )

As we have defined the Matlab function called myta which takes four arguments in the form myta(f,n,a,b).

The purpose of the function is to find the nth Taylor polynomial of the function f(x) at x = a and evaluate it at x = b. Then, it should return the absolute value of the difference between this value and f(b).

Now that we have the nth Taylor polynomial of f(x) at x = a, we can evaluate it at x = b and calculate the absolute difference between this value and f(b).

function result = myta(f,n,a,b)

   syms x; % define x as symbolic variable

   g = f(a); % initialize g as f(a)

   for i=1:n % iterate from 1 to n

       deriv = diff(f,x,i-1); % calculate the ith derivative of f

       term = deriv*(x-a)^(i-1)/factorial(i-1); % calculate the ith term of the Taylor series

       g = g + term; % add the ith term to g

   end

   result = abs(g - f(b)); % calculate the absolute difference between g(b) and f(b)

end

This code calculates the absolute difference between g(b) and f(b) using the "abs" function and assigns it to the output variable "result".

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Over the range [5, 7], the definite integral of f(x) = 1 / (x² + 10x + 25) is around -1/60.

To find the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7], we can use the following formula:

∫[a,b] f(x) dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

First, we need to find the antiderivative of f(x):

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx

To do this, we can use a technique called partial fraction decomposition:

1 / (x² + 10x + 25)

= A / (x + 5) + B / (x + 5)²

Multiplying both sides by the denominator (x² + 10x + 25), we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

A + B = 1

A + 1 = 1

A = 0

Therefore, the partial fraction decomposition of f(x) is:

1 / (x² + 10x + 25) = 1 / (x + 5)²

Now we can find the antiderivative:

∫ f(x) dx = ∫ 1 / (x² + 10x + 25) dx = ∫ 1 / (x + 5)² dx

Using the substitution u = x + 5, du = dx, we get:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

where C is the constant of integration.

Now we can evaluate the definite integral over the interval [5, 7]:

∫[5,7] f(x) dx = F(7) - F(5)

∫[5,7] f(x) dx = [-1 / (7 + 5) + C] - [-1 / (5 + 5) + C]

∫[5,7] f(x) dx = [-1 / 12 + C] - [-1 / 10 + C]

∫[5,7] f(x) dx = -1 / 12 + C + 1 / 10 - C

∫[5,7] f(x) dx = -1 / 60

Therefore, the definite integral of f(x) = 1 / (x² + 10x + 25) over the interval [5, 7] is approximately -1/60.

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The measure of the side MR is given as 20

We have to first find the value of the angle

∠TQR = 180 - (35 + 25)

= 180 - 60

= 120 degrees

180 - 120 = 60 degrees

angle TPR  = 90 degrees

next we have to find ∠PTQ

= 180 - (90 + 60)

= 180 - 150

= 30 degrees

given that TMN = TQR

TNP = PTQ

So if TMN = 35 degrees since TQR = 35 degrees

PTQ = 30, so TNP = 30 degrees

The measure of QR = 4 since MN = 4

NP = pq

np = 6

Hence the measure of MR

= 4 + 6 + 6 + 4

= 20

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