Solve for x,z,y
-x + y + 3z = -3
X - 2y - 2z = 8
3x - y - 4z = 6

Answer:

3

Step-by-step explanation:

To turn the word problem into an equation, when we read:

"I am thinking of a number" we write "x"

"when I double my number" we write "2x"

"and then subtract the result from 5" we write "5 - 2x"

"I get negative one" we write "-1 = 5 - 2x"

Now we solve for the number, which is x.

Equation: -1 = 5 - 2x

-1 = 5 - 2x

subtract 5 from both sides

-6 = -2x

divide both sides by -2

3 = x

The mathematical expression of the given phrase is 5 - 2x = -1 thus the number will be 3.

A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.

An expression is a mathematical proof of the equality of two mathematical expressions.

Let's say that number is x.

Double 2x

Subtract from 5

5 - 2x = -1

-2x = - 1 - 5

-2x = -6

x = 3

Hence "The mathematical expression of the given phrase is 5 - 2x = -1 thus the number will be 3".

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Answer:

  c. uses the average of the most recent data values in the time series as the forecast for the next period.

Step-by-step explanation:

We assume you want to complete the description of moving averages method.

The moving in moving averages refers to the fact that the data points used to compute the average are some number of most recent data points. As data is accumulated, the data used to compute the average "moves" to include the newest data and exclude the oldest data.

Answer:

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = y intercept

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis

If two lines are perpendicular, it means that the slope of one line is the negative reciprocal of the slope of the other line. The equation of the given line is

y=2x-4

Slope = 2

Therefore, the slope of the perpendicular line is -1/2

It passes through point (4,4)

We would determine the intercept, by substituting m = -1/2 , y = 4 and x = 4 into the slope intercept equation

y = mx + c

4 = -1/2 ×4 + c

4 = -2 + c

c = 4 + 2 = 6

The equation becomes

y = -x/2 + 6

Answer: The speed of Bob is 56.8 km/hr.

Step-by-step explanation:

Let the speed of Bob be 'x'.

Let the speed of John be 'x-8'.

Distance covered = 230 miles

time = [tex]1\dfrac{1}{2}=\dfrac{3}{2}\ hr[/tex]

According to question, we get that

[tex]\dfrac{230}{x}-\dfrac{230}{x+8}=\dfrac{3}{2}\\\\230\dfrac{x+8-x}{x(x+8)}=\dfrac{3}{2}\\\\\dfrac{230\times 8}{x^2+8x}=\dfrac{3}{2}\\\\\dfrac{1840}{x^2+8x}=\dfrac{3}{2}\\\\1840\times 2=3x^2+24x\\\\3680=3x^2+24x\\\\3x^2+24x-3680=0\\\\x\approx 56.8\ km/hr[/tex]

Hence, the speed of Bob is 56.8 km/hr.

Answer: a. 120, b. 386, c. 176, d. 968.

Step-by-step explanation:

For a combination of any number, is given as

C n,r = n!/r!(n-r)!

Please note that "n,r" is a subscript, and the exclamation mark "!" is called factorial.

From the question, n = 10

EXACTLY 3 0s

10 combination 3

r is exactly 3, that is equal 3.

C 10,3= 10!/3!(10-3)! = 10!/3!7!= 120.

For clarification,

10!/3!7!=10×9×8/3×2×1 = 120.

You can also use a calculator to compute the factorials.

MORE 0s than 1s

There will be more 0s than 1s when < 5bits are 0s.

We have r<5

Therefore for r=4

C 10,4 = 10!/4!(10-4)!=10!/4!6!=210

r=3

C 10,3= 10!/3!(10-3)!=10!/3!7!=120

r=2

C 10,2=10!/2!(10-2)!=10!/2!8!=45

r=1

C 10,1=10!/1!(10-1)!=10!/1!9!=10

r=0

C 10,0=10!/0!(10-0)!=10!/0!10!=1

Summing the answers gives us our final answer

210+120+45+10+1= 386.

AT LEAST 7 1s

To get this combination, the value of r will be greater than or equal to 7

r>=7

We have,

r=7

C 10,7=10!/7!(10-7)!=10!/7!3!=120

r=8

C 10,8=10!/8!(10-8)!=10!/8!2!=45

r=9

C 10,9=10!/9!(10-9)!=10!/9!1!=10

r=10

C 10,10=10!/10(10-10)!=10!/10!0!=1

120+45+10+1= 176

AT LEAST 3 1s

the value for r will be greater than or equal to 3:

We can the values of r from 3 to 10.

r=3

C 10,3=10!/3!(10-3)!=120

r=4

C 10,4=10!/4!(10-4)!=10!/4!6!=210

r=5

C 10,5=10!/5!(10-5)!=10!/5!5!=252

r=6

C 10,6=10!/6!(10-6)!=10!/6!4!=210

r=7

C 10,7=10!/7!(10-7)!=10!/7!3!=120

r=8

C 10,8=10!/8!(10-8)!=10!/8!2!=45

r=9

C 10,9=10!/9!(10-9)!=10!/9!1!=10

r=10

C 10,10=10!/10!(10-10)!=10!/10!0!=1

Adding our answers gives 968.

The bits can be either 1 or 0. The total number of bit string for each specified case is:

Since the items are indistinguishable, their arrangements doesn't matter.

They can be chosen in [tex]^nC_r = \dfrac{n.(n-1).(n-2)...(n-(r+2)).(n-(r+1))}{r.(r-1).(r-2)...3.2.1} \: \rm (r \leq n)[/tex]

The bit string is of length 10.

Each bit can be in one of the two states, viz 0 or 1.

Evaluating the count of bit strings for given cases:

Think of it as if there are 10 seats and 3 people to sit on. They're going to be 0s. 3 seats can be chosen from 10 seats in [tex]^{10}C_3 = \dfrac{10\times 9\times 8}{3 \times 2\times 1} = 120[/tex] ways.

The three 0s are identical, so no intra-arrangement between them matters.

Thus, total 120 such strings exist.

It means, 0s can be 6,7,8,9, or 10 places.

Just similar to above case, 0s on x places out of 10 places can be in [tex]^{10}C_x[/tex] ways.

Thus, total such strings of 0s being more than 1s and being 10 bit strings are:

[tex]^{10}C_6 + ^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =210+120+45+10+1=386[/tex]

At least seven 1s means either 7, 8, 9, or 10 ones.

Total count of such strings are:

[tex]^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =120+45+10+1=176[/tex]

They are three or more ones. Total count of such strings is:

[tex]^{10}C_3 + ^{10}C_4 + ^{10}C_5+^{10}C_6 + ^{10}C_7 + ^{10}C_8 + ^{10}C_9 + ^{10}C_{10} =120 + 210 + 252 + 210+120+45+10+1=386+582=968[/tex]

Thus, the total number of bit string for each specified case is:

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Answer:

  p(x) = x(x +5)(x -9)

Step-by-step explanation:

If r is a root, then (x -r) is a factor of the polynomial. For the given roots, the factorization is ...

  p(x) = (x -0)(x -(-5))(x -9)

  p(x) = x(x +5)(x -9)

Answer:The rate per week =

20.48/8 = 2.56 kilometers per week

Step-by-step explanation:

A construction crew has just built a new road. It took them 8 weeks to build 20.48 kilometers of road. To determine the rate at which the road was built, we would divide the total length of road that was built by the construction company by the number of weeks or days or even hours used in the construction.

The rate per week =

20.48/8 = 2.56 kilometers per week

If we want to find the rate per day,

1 week = 7 days

8 weeks will be 8×7 = 56 days

So the rate per day =

20.48/56 = 0.366 kilometers per day.

You have 31 nickels and 44 dimes.

Step-by-step explanation:

Total coins = 75

Worth of coins = $5.95 = 5.95*100 = 595 cents

1 nickel = 5 cents

1 dime = 10 cents

Let,

Number of nickels = x

Number of dimes = y

According to given statement;

x+y=75   Eqn 1

5x+10y=595    Eqn 2

Multiplying Eqn 1 by 5

[tex]5(x+y=75)\\5x+5y=375\ \ \ Eqn\ 3\\[/tex]

Subtracting Eqn 3 from Eqn 2

[tex](5x+10y)-(5x+5y)=595-375\\5x+10y-5x-5y=220\\5y=220[/tex]

Dividing both sides by 5

[tex]\frac{5y}{5}=\frac{220}{5}\\y=44[/tex]

Putting y=44 in Eqn 1

[tex]x+44=75\\x=75-44\\x=31[/tex]

You have 31 nickels and 44 dimes.

Keywords: linear equations, subtraction

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Answer:

Step-by-step explanation:

Pythagorean theorem is given as

a² +b² = c²

a² = c² - b²

a² = 252 – 202

a² = 625 – 400

a² = 225

a = √225

a = 15

Length of the missing side is 15

To find the value of the six trigonometric function

1) sin x = a/c

        = 15/25

     sin x   = 0.6,    x = sin⁻¹ 0.6 = 36.86

2) cos x = b/c

         = 20/25

       cos x  = 0.8,     x = cos⁻¹ 0.8 = 36.86

3) tan x = a/b

         = 15/20

     tan x   = 0.75,     x = tan⁻¹ 0.75 = 36.86

∴ Θ = 36.86°

4) csc x = c/a

         = 25/15

    csc x    = 1.67     x = csc⁻¹ 1.67

5) sec x = c/b

         = 25/20

      sec x   = 1.25     x = sec⁻¹ 1.25

6) cot x = b/a

        = 20/15

   cot x  = 1.33        x = cot⁻¹ 1.33

Answer:

  y = 2^(x–2) + 3

Step-by-step explanation:

The equation above is the one that is graphed. You can pick it from the offered choices by recognizing that the horizontal asymptote on the graph is y=3. That is 3 units above the horizontal asymptote of the parent exponential function. Hence, you must have ...

  y = (some exponential) +3

_____

Please note that the exponent indicator (^) and the grouping parentheses on the exponent are essential. Without those, the equation is that of the line y=2x+1, which is not what is graphed.

The statement is false. Multiple planes can be perpendicular to the same line through a given point.

The statement is False. If a line is perpendicular to a plane, it does not mean that it is the only plane perpendicular to the line.

There can be multiple planes perpendicular to the same line through a given point. For example, consider a line s passing through point Q and a plane M perpendicular to s at point Q. Now, we can also have another plane N perpendicular to line s at point Q, which is different from plane M. Therefore, the statement is false.

Answer:

the answer is 3

7x + 9= 30

7x = 21

x = 3

Answer:

x = 3

Step-by-step explanation:

Collect like terms;

7x + 9 = 30

Subtract 9 from both sides;

7x = 21

Divide both sides by 7;

x = 3

Answer: (39.424, 61.576)

Step-by-step explanation:

When population standard deviation([tex]\sigma[/tex]) unknown ,The confidence interval for population mean is given by :-

[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]

, where n= Sample size

[tex]\overline{x}[/tex] = sample mean.

s= sample standard deviation

[tex]t^*[/tex] = Critical t-value (two-tailed)

Given : n= 15

Degree of freedom= 14  [df=n-1]

[tex]\overline{x}=\ $50.50[/tex]

[tex]s^2=400\\\\\Rightarrow\ s=\sqrt{400}=20[/tex]

Significance level = [tex]\alpha=1-0.95=0.05[/tex]

For [tex]\alpha=0.05[/tex] and df = 14, the critical t-values : [tex]t^*=\pm2.1448[/tex]

Then the 95% confidence interval for population mean will be  :

[tex]50.50\pm (2.1448)\dfrac{20}{\sqrt{15}}\\\\=50.50\pm(2.1448)(5.1640)\\\\\approx50.50\pm11.076\\\\=(50.50-11.076,\ 50.50+11.076)\\\\=(39.424,\ 61.576)[/tex]

Hence, a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall. : (39.424, 61.576)

The 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall is calculated using the sample mean ($50.50), sample size (15), sample standard deviation (20), and Z-value for a 95% confidence interval (1.96). The calculated interval is (-$1.11, $102.11).

To construct a 95% confidence interval for the average amount that the department store's credit card customers spent on their first visit to their new store, we would use the formula for a confidence interval:

CI = X-bar ± (Z-value * (s/√n)),

where X-bar is the sample mean = $50.50, n is the sample size = 15, s is the sample standard deviation = √400 = 20, and Z-value is the critical value from the Z-table which, for a 95% confidence interval, equals 1.96.

Plug these values into the formula,

CI = 50.5 ± (1.96 * (20/√15))

Using a calculator, the confidence interval comes out to (-$1.11, $102.11).

So, we are 95% confident that the average amount its credit card customers spent on their first visit to the chain's new store in the mall lies between $-1.11 and $102.11.

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To find the difference between the number of tetras and minnows, set up a system of equations using the given ratios. Solve the system to find the number of tetras, guppies, and minnows. Finally, subtract the number of minnows from the number of tetras to find the difference.

To determine the difference between the number of tetras and minnows in the fish tank, we need to first find the number of each type of fish. We can do this by setting up a system of equations using the given ratios. Let T represent the number of tetras, G represent the number of guppies, and M represent the number of minnows.

From the first ratio, we have T/G = 4/2. Simplifying this equation, we get T = 2G.

From the second ratio, we have M/G = 1/3. Simplifying this equation, we get M = (1/3)G.

Since we know there are a total of 60 fish in the tank, we can create the equation T + G + M = 60. Substituting the previous equations into this equation, we get 2G + G + (1/3)G = 60. Solving for G, we find G = 9. Plugging this value into the equations for T and M, we get T = 2(9) = 18 and M = (1/3)(9) = 3.

Therefore, there are 18 tetras and 3 minnows in the fish tank. The difference between the number of tetras and minnows is 18 - 3 = 15.

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Answer:

f(g(x)) = g(f(x))  = x and f and g are the inverses of each other.

Step-by-step explanation:

Here, the given functions are:

[tex]f(x) = x^2 - 3, g(x) = \sqrt{({3+x)} }[/tex]

To Show:  f (g(x))  = g (f (x))

(1)  f (g(x))

Here, by the composite function:

[tex]f (g(x)) = f (\sqrt{3+x} )  = \sqrt{(3+x)} ^2 - 3  =  (3 + x) - 3  =  x[/tex]

⇒ f (g(x))  = x

(2) g (f(x))

Here, by the composite function:

[tex]g(f(x)) = g(x^2 -3)   = \sqrt{3 +(x^2 -3) }  = \sqrt{x^2}   = x[/tex]

⇒ g (f(x))  = x

Hence, f(g(x)) = g(f(x))  = x

⇒ f and g are the inverses of each other.

Answer:

The required probability is [tex]\frac{3}{4}[/tex].

Step-by-step explanation:

Consider the provided information.

There are 8 leaders.

Thus, the total number of ways to arrange 8 leaders are 8!.

Assume that Obama and Berlusconi is one person.

Therefore the total number of leaders are 7 (As Obama and Berlusconi is one person).

The number of ways in which 7 leader can be arranged: 7!

Although Obama and Berlusconi is one unit but they can interchange their place in 2 ways. Like Obama and Berlusconi or Berlusconi and Obama

That means the total number of ways : 7!×2

The number of ways in which they are not next to each other = Total number of ways - The number of ways in which they are next to each other

Number of ways they are not next to each other = 8!-7!×2

The probability that they are not next to each other = [tex]\frac{8!-7!\times2}{8!}=\frac{3}{4}[/tex]

Hence, the required probability is [tex]\frac{3}{4}[/tex].

Answer:

B

Step-by-step explanation:

just multiply and add

Answer:

Average rowing velocity of boat in still water is 4 miles per hour and average velocity of the current is 1 mile per hour.

Step-by-step explanation:

We are given the following in the question:

Let x be the average rowing velocity of boat in still water and y be the the average velocity of the current.

[tex]\text{Speed} = \displaystyle\frac{\text{Distance}}{\texr{Time}}[/tex]

The boat rowed 7.5 miles​ downstream, with the​ current, in 1.5 hours.

Velocity with the current =

[tex]=\text{average rowing velocity of boat in still water} + \text{ average velocity of the current} = x + y[/tex]

Thus, we can write the equation:

[tex]7.5 = (x+y)1.5\\x+y = 5[/tex]

The return trip​ upstream, against the​ current, covered the same​ distance, but took 2.5 hours.

Velocity against the current =

[tex]=\text{average rowing velocity of boat in still water} - \text{ average velocity of the current} = x - y[/tex]

Thus, we can write the equation:

[tex]7.5 = (x-y)2.5\\x-y = 3[/tex]

Solving, the two equations:

[tex]2x = 8\\x = 4, y = 1[/tex]

Thus, average rowing velocity of boat in still water is 4 miles per hour and average velocity of the current is 1 mile per hour.

Answer:

A rotation, then a dilation

Step-by-step explanation:

When two triangles are congruent, the three sides and angles will be the same.

A dilation is a type of transformation that works with scale factors and enlarges or reduces a figure, to create a new figure.

Now, the composition of transformations that will create a pair of similar but not congruent triangles are - a rotation, then a dilation.

A composition of a rotation followed by a dilation will create a pair of similar, but not congruent, triangles, option D.

The question asks which composition of transformations will create a pair of similar, but not congruent, triangles. In the realm of Euclidean geometry, certain transformations maintain the shape and size of geometric figures, while others maintain only the shape but not the size.

Among the choices given, a rotation followed by a reflection, a translation followed by a rotation, and a reflection followed by a translation will all produce congruent triangles, as they are types of isometries which preserve shape and size.

However, a rotation followed by a dilation is the correct composition that will result in triangles that are similar but not congruent. This is because rotation preserves the shape and size, but when followed by dilation, the size is changed while the shape remains the same, satisfying the condition of the question. Option D is correct.

Answer:

15.87% of the invoices were paid within 15 days of receipt

Step-by-step explanation:

An invoice was paid an average of 20 days after it was received.

Mean = [tex]\mu = 20[/tex]

Standard deviation = [tex]\sigma = 5[/tex]

Now we are supposed to find what percent of the invoices were paid within 15 days of receipt i.e.P(x<15)

Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]

At x = 15

Substitute the values

[tex]Z=\frac{15-20}{5}[/tex]

[tex]Z=-1[/tex]

Refer the z table for p value

So, p value = 0.1587

So, 15.87% of the invoices were paid within 15 days of receipt

Priya practiced the violin for 114 minutes

Given that  

Han spent 75 minutes practicing the piano over the weekend

Priya practiced the violin for 152% as much as Han practiced the piano

Need to determine how long did priya practice  

Duration of Han practicing the piano over weekend = 75 minutes

As given that Priya practiced the violin for 152% as much as Han practiced the piano

=> Duration of Priya practicing the piano = 152% of Han practicing the piano

=> Duration of Priya practicing the piano = 152% of 75 minutes

We know that a % of b is written in fraction as [tex]\frac{a}{100} \times b[/tex]

[tex]\Rightarrow \text { Duration of Priya practicing the piano }=\frac{152}{100} \times 75=114 \text { minutes }[/tex]

Hence Priya practiced the violin for 114 minutes

Answer:

  A.  ZY=15

Step-by-step explanation:

Insufficient information is given about angles to make any statement about the value of p or the measures of any angles. (Eliminates B,C,D,E)

Side YZ corresponds to side FG. Since they are congruent, their measures are the same. This means ...

  2n -5 = n +5

  n = 10 . . . . . . . . add 5-n

YZ = ZY = 2·10 -5 = 15 . . . . . . matches choice A

Answer:

A factor

Step-by-step explanation:

Take the equation 2 x 4 = 8 as an example.

2 and 4 are multiplied together to get 8.

2 and 4 are factors, and 8 is the product.

Answer: c

Step-by-step explanation:

The minimum sample size does not depend on the size of the population

The size of the population, N, is NOT required to determine the minimum sample size for estimating a population mean, contrasting with the required elements like the desired confidence level, margin of error, and population standard deviation.

The question asks which factor is NOT required to determine the minimum sample size needed to estimate a population mean. The options are:

The correct answer is C. The size of the​ population, N. When estimating a population mean, the key factors required include the desired confidence level, the desired margin of error, and the value of the population standard deviation (sigma), but not necessarily the size of the population. This is especially true in cases where the population is very large or infinite, and the sample size needed for a specific confidence level and margin of error can be calculated without this information.

Answer:

She should contribute $ 8369.38 ( approx )

Step-by-step explanation:

Let P be the amount invested by the other partner,

∵ The amount formula in compound interest,

[tex]A=P(1+\frac{r}{n})^{nt}[/tex]

Where,

r = annual rate,

n = number of compounding periods in a year,

t = number of years,

Here, r = 9% = 0.09, n = 4 ( quarters in a year ), t = 2 years,

Then the amount after 2 years,

[tex]A = P(1+\frac{0.09}{4})^{8}[/tex]

According to the question,

A = $ 10,000,

[tex]P(1+\frac{0.09}{4})^{8}= 10000[/tex]

[tex]P(1+0.0225)^8 = 10000[/tex]

[tex]\implies P = \frac{10000}{1.0225^8}\approx \$ 8369.38[/tex]

Answer:

a) 100000

b) 90000

Step-by-step explanation:

We have the possibility of 10 digits

(0,1,2,3,4,5,6,7,8,9)

If there are no restrictions on the digit, there are 10 possibilities for each of the five digits

We then have;

10*10*10*10*10

= 10^5

= 100000

This means 100000 zip codes are possible if there are no restrictions.

b) If the first digit cannot be 3, there are 9 possibilities. This is because the possibility of the first digit being 3 is 1 out of 10. Therefore the possibility of not being 3 is 9 out of 10

The other four digits have 10 possibilities each.

So we have

9*10*10*10*10

= 90000

This means there are 90000 zip codes if the first digit does not start with 3

Answer:

  False

Step-by-step explanation:

If we take this equation at face value, the expected utility bill is ...

  23.40 +0.4×2300

  = 23.40 +920

  = 943.40 ≠ 92.00

The equation does NOT predict a bill of $92.00.

The statement is false. Using the provided regression model (23.40 + 0.4x), the expected utility bill for a house of 2,300 square feet is $923.40, not $92.00.

The subject of this question lies within the field of Mathematics, specifically within statistics and regression analysis. In the given example, we have a study of 30 customers focusing on their utility bills. The regression model for this study is 23.40 + 0.4x, where 'x' denotes the number of square feet in a house. This model shows the relationship between the size of the house (in square feet) and the monthly utility bill.

To address the student's question, we use this model to calculate the expected utility bill for a customer who has a 2,300 square feet house by substituting 'x' with 2300. The calculation becomes: 23.40 + 0.4*2300 = 923.40, not $92.00. Therefore, the expected utility bill is approximately $923.40, not $92.00. So, the statement in the question is False.

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Answer:

Yes. The outcomes can be classified into two categories, the trials are fixed, and the events are independent.

Step-by-step explanation:

Hope this helps!!

deltax = (56-0)/4 = 14

number of intervals 4

The intervals are:

(0,14),(14,28),(28,42),(42,56)

The Midpoint Rule =

f(7)+f(21)+f(35)+f(49)

[0.47577]+[-0.99159]+[-0.35892]+[0.65699]

deltax =14

sum = -0.21774

Multiplying by deltax = -3.0484

To approximate the integral of sin(x^2) dx from 0 to 5 using the Midpoint Rule with n=5, first calculate Δx = 1. Next, perform calculations with x values 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, use the Midpoint Rule formula to calculate the approximated integral.

To solve this problem, you'll need to use the Midpoint Rule, which is a numerical method used to approximate definite integrals. In this case, we want to approximate ∫ from 0 to 5 of sin(x^2) dx with an n value of 5.

The Midpoint Rule can be represented as: Mn = Δx[f(x1) + f(x2) + ... + f(xn)], where Δx = (b-a)/n and each xi = a + (Δx/2) + (i-1)Δx.

Here, we'll first find our Δx = (5-0)/5 = 1, then calculate each xi and plug those into our function. Our xi values will be 0.5, 1.5, 2.5, 3.5, and 4.5. Finally, we plug into our formula and solve to find our approximated integral value.

A detailed and step-by-step solution would involve calculating each f(xi) with the xi values given above, and then adding these up

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