The students in Karen's class got to choose between pizza and burgers for the celebration on the last day of school. 15 students picked the pizza. If there are 20 students in all in Karen's class, what percentage of the students picked the pizza?

Write your answer using a percent sign (%).

The given differential equation is y" − 2y' + y = 0. To solve this equation, we need to use the characteristic equation, which is given by r^2 − 2r + 1 = 0.

The two solutions to this equation are r = 1 ± i. Thus, the general solution to the differential equation is y(x) = C_1e^(x) + C_2e^(-x)cos(x) + C_3e^(-x)sin(x).

We can use Maxima to verify our solution. To do this, we plug in the boundary conditions, y(π) = e ^π and y'^ (π) = 0, and solve for the constants C1, C2, and C3. This gives us C1 = 1, C2 = -1, and C3 = 0. Thus, the solution to the differential equation is y(x) = e^x - e^(-x)cos(x).

To plot the solution, we can use Maxima's plot2d function with the given solution.

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Answer: 25%

Step-by-step explanation:

To find the percentage, you divide 6 by 24. That equals 0.25, which is a decimal. You need to make it a percent, so multiply 0.25 by 100 which is equal to 25%.

The amount of money in account after 16 years will be $162,823.39. The solution has been obtained by using compound interest.

What is compound interest?

Compound interest considers the principal when determining the interest for the subsequent month, in contrast to simple interest, which does not. In algebra, compound interest is typically denoted by the letter C.I.

We are given the following information:

P = $72000

Rate (r) = 5.1% = 0.051

Time (t) = 16 years

Using the formula, we get

⇒V = P[tex]e^{rt}[/tex]

⇒V = (72000) e⁰°⁰⁵¹ˣ¹⁶

⇒V = (72000) e⁰°⁸¹⁶

⇒V = $162,823.39

Hence, the amount of money in account after 16 years will be $162,823.39.

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The distance between Point A and Point B is 840 meters.

Let's start by using the formula:

distance = speed x time

Since Coco swam at a constant speed of 1.2 m/s, we can find his distance using:

distance(Coco) = speed(Coco) x time

where time is the same for both Coco and Azlinda. Let's call this common time "t".

distance(Coco) =[tex]1.2 m/s \times t[/tex]

Now, let's consider Azlinda's situation. After 5 minutes (or 5/60 = 1/12 hours), she had swum a distance of 420 m and was 37 m away from Coco. Let's call the distance between Point A and Point B "d".

Since Azlinda was swimming towards Point A, she must have covered a distance of (d - 37) m by the time she had swum 420 m. We can use the formula above to find her speed:

speed(Azlinda) = distance(Azlinda) / time

speed(Azlinda) = (d - 37) m / (1/12) h

speed(Azlinda) = 12(d - 37) m/h

Now, we know that Azlinda and Coco were swimming towards each other for a total of 5 minutes (or 1/12 hours), so their total distance apart at that time was:

distance apart = distance(Coco) + distance(Azlinda)

distance apart = [tex]1.2 m/s \times t + 12(d - 37) m/h \times (1/12) h[/tex]

distance apart =[tex]1.2t + d - 37[/tex]

We also know that when they were 37 m apart, Azlinda had swum a distance of 420 m, so we can write:

420 = d - 37 - distance(Coco)

Substituting the expression for distance(Coco) from above, we get:

420 = d - 37 - 1.2t

Now we have two equations with two unknowns (d and t). We can substitute into the other equation and solve for one variable in terms of the other. For example, we can solve the second equation for t:

[tex]1.2t = d - 37 - 420\\1.2t = d - 457\\t = (d - 457) / 1.2[/tex]

When we enter this into the initial equation, we obtain:

distance apart = [tex]1.2t + d - 37[/tex]

distance apart = [tex]1.2((d - 457) / 1.2) + d - 37[/tex]

distance apart = [tex]d - 380.6[/tex]

Now we can substitute this expression for distance apart into the second equation:

[tex]420 = d - 37 - 1.2t\\420 = d - 37 - 1.2(d - 457) / 1.2\\420 = d - 37 - (d - 457)\\420 = -d + 420\\d = 840[/tex]

Therefore, the distance between Point A and Point B is 840 meters.

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Answer:

Angle C = 101 Degree

Angle E = 46 Degree

Step-by-step explanation:

Due to CD and FE are parallel, the adjacent angle between and DCF and angle EFC would sum up to be 180 degrees.

Angle DCF:

79 + Angle C = 180

Angle C = 101 Degree

Angle EFC:

134 + Angle E = 180 Degrees

Angle E = 46

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Answer:

Albert receives $10,000, Brian receives $20,000, and Charles receives $30,000.

Step-by-step explanation:

1 + 2 + 3 = 6

Next, we can find out what fraction of the estate each person is entitled to:

Albert: 1/6 of the estate

Brian: 2/6 (or 1/3) of the estate

Charles: 3/6 (or 1/2) of the estate

Now, we can calculate the amount each person receives by multiplying their share by the total value of the estate:

Albert: (1/6) x $60,000 = $10,000

Brian: (1/3) x $60,000 = $20,000

Charles: (1/2) x $60,000 = $30,000

Blake needs to now at least 17 lawns in order to earn at least $200 to buy a new stereo system.

To solve this problem, we can set up an inequality that represents the situation. Let's let x be the number of lawns that Blake needs to mow. The inequality will be:
12x >= 200

This inequality states that the amount of money Blake earns from mowing lawns (12x) must be greater than or equal to the amount of money he wants to earn ($200).

To solve for x, we can divide both sides of the inequality by 12:
x >= 200/12

Simplifying the right side of the inequality gives us:
x >= 16.67

Since Blake cannot mow a fraction of a lawn, we need to round up to the next whole number. This means that Blake needs to now at least 17 lawns in order to earn at least $200 to buy a new stereo system.

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In conclusion, Mr. Barnes rode his bike at a rate of 6.67 miles per hour, which was faster than Mrs. Barnes' rate of 6 miles per hour.

We must compute the rate or speed that each participant rode in order to determine who rode faster. The rate is calculated by dividing the distance travelled by the time required. The rates for Mr. and Mrs. Barnes will now be determined:

Rate = Distance / Time = 20 Miles / 3 Hours = 6.67 Miles per Hour, Mr. Barnes

Rate = Distance/Time = 24 Miles/4 Hours = 6 Miles Per Hour, Mrs. Barnes

We can observe from comparing the speeds that Mr. Barnes rode more quickly than Mrs. Barnes. Her speed was 6 miles per hour, compared to his 6.67. As a result, we can conclude that Mr. Barnes was riding more quickly than Mrs. Barnes.

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Answer: The answer is A

Step-by-step explanation:

The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.

We have,

To solve this problem, we can use the concept of a binomial distribution.

The number of students to whom problem 1 is assigned can be modeled as a binomial random variable.

Let's define X as the random variable representing the number of students to whom problem 1 is assigned.

We know that each student has a 3/6 = 1/2 probability of being assigned problem 1.

In a class of 50 students, the probability of a single student being assigned problem 1 is p = 1/2.

The number of students to whom problem 1 is assigned follows a binomial distribution with parameters n = 50 (number of students) and p = 1/2 (probability of success).

The variance of a binomial distribution is given by the formula:

Var(X) = np (1 - p)

Substituting the values, we have:

Var(X) = 50 x (1/2) x (1 - 1/2)

= 50 x (1/2) x (1/2)

= 25 x 1/2

= 25/2

= 12.5

Therefore,

The variance of X, the total number of students to whom problem 1 is assigned, is 12.5.

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Answer: To find the magnitude and direction of the equilibrant of a system of forces, we first need to find the resultant of the forces, and then find the force that will balance the resultant.

For the given system of forces, we can use the Pythagorean theorem to find the magnitude of the resultant:

R = sqrt(32^2 + 48^2)

 = sqrt(1024 + 2304)

 = sqrt(3328)

 ≈ 57.7 N

The direction of the resultant can be found using trigonometry:

tan(theta) = opposite / adjacent

where theta is the angle between the forces, which is 90° in this case. We can choose either force as the adjacent side, and the other force as the opposite side. Let's choose the 32 N force as the adjacent side:

tan(theta) = 48 / 32

theta = atan(48/32)

     ≈ 56.3°

This means that the resultant has a magnitude of approximately 57.7 N and is directed at an angle of approximately 56.3° to the 32 N force.

To find the equilibrant, we need to find a force that has the same magnitude as the resultant but acts in the opposite direction. We can use the same magnitude and opposite direction to find the equilibrant as:

E = -R

 = -57.7 N

This means that the equilibrant has a magnitude of 57.7 N and acts in the opposite direction to the resultant, which is at an angle of approximately 56.3° to the 32 N force.

Step-by-step explanation:

Answer:

g(x) = f(x + 2) = 2^(x + 2) + 2

Step-by-step explanation:

To shift the function f(x) = 2^x + 2 two units to the left, we need to replace x with (x + 2) in the equation. This will shift the graph horizontally to the left by two units.

So, the new function g(x) that represents the shifted graph is:

g(x) = f(x + 2) = 2^(x + 2) + 2

Step-by-step explanation:

Seven people are in the pool who each have expected payouts of $8,600 for the year. One more person is added to the pool whose expected payout is $14,000 per year. How much is the average expected payout for each member of the group of eight? Expected Payout for each member of the group is $​

The equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by y = 5x + 30.


m = (f(x2) - f(x1)) / (x2 - x1)

f(x) = 3x from x1 = 0 to x2 = 5

f(x) = x2 + 2x from x1 = 3 to x2 = 5

m = (f(5) - f(0)) / (5 - 0)

m = (53 + 2(5)) - (03 + 2(0)) / (5 - 0)

m = 25 / 5

m = 5

Therefore, the average rate of change of the function from x1 to x2 is 5.

The equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by the equation:


y = mx + b



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



We can find the slope of the line, m, from the average rate of change of the function from x1 to x2 which is 5.



We can find the y-intercept, b, by substituting the coordinates (-8, -10) in the equation of the line.



y = 5x + b



-10 = 5(-8) + b



b = 30



Therefore, the equation of the line passing through (-8, -10) and passing through (x1, f(x1)) is given by y = 5x + 30.

5.6 kilograms.

1000 grams makes ONE kilogram and you have 5000 grams. Then the extra 600 grams are 6/10 of 1000 so it would be .6.

The equation of the line passing through the points (-2,7) and (0,-3) in slope-intercept form is y = -5x + 7.

Let's first find the slope of the line using the two given points:

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

slope = (-3 - 7)/(0 - (-2))

slope = (-3 - 7)/(0 + 2)

slope = -10/2

slope = -5

Now that we have the slope, we can use the point-slope form of a line to find its equation:

Where m is the slope and (x1,y1) is any point on the line, y - y1 = m(x - x1).

Let's use the point (-2,7) as (x1,y1):

y - 7 = -5(x - (-2))

y - 7 = -5(x + 2)

y - 7 = -5x - 10

y = -5x + 7

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C= (1.17,9.93,−32.25

D= (3.33,14.9,−47).

For Exercise 5, the coordinates of the segment PQ are P = (2,3,1) and Q = (0,5,7). To calculate the distance from the midpoint to the origin, use the midpoint formula:  M = [(P + Q) / 2].

In this case, M = [(2,3,1) + (0,5,7)] / 2 = (1,4,4).

Then calculate the distance from the midpoint to the origin by using the distance formula:  d = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the midpoint coordinates and (x2, y2, z2) is the origin coordinates. In this case, d = √[(1-0)2 + (4-0)2 + (4-0)2] = √17.

For Exercise 6, the coordinates of the segment PQ are P = (1,0,3) and Q = (3,2,5). To calculate the distance from the midpoint to the origin, use the midpoint formula:  M = [(P + Q) / 2]. In this case, M = [(1,0,3) + (3,2,5)] / 2 = (2,1,4). Then calculate the distance from the midpoint to the origin by using the distance formula:  d = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the midpoint coordinates and (x2, y2, z2) is the origin coordinates. In this case, d = √[(2-0)2 + (1-0)2 + (4-0)2] = √21.

For Exercise 7, let A = (−1,0,−3) and E = (3,6,3). To find points B, C, and D on the line segment AE such that d(A,B)=d(B,C)=d(C,D)=d(D,E)= 41d(A,E), first calculate the distance between A and E using the distance formula: d(A,E) = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2], where (x1, y1, z1) is the coordinates of A and (x2, y2, z2) is the coordinates of E. In this case, d(A,E) = √[(3-(-1))2 + (6-0)2 + (3-(-3))2] = √122.

To find the coordinates of points B, C, and D, use the following formula: B = A + (d(A,B)/d(A,E))(E-A), where d(A,B) is the distance from A to B, d(A,E) is the distance from A to E, A is the coordinates of A, and E-A is the vector pointing from A to E. Using this formula, the coordinates of B can be calculated as B = (−1,0,−3) + (41/122)((3,6,3) - (−1,0,−3)) = (−1,4.97,−17.5). Similarly, the coordinates of C and D can be calculated as C = (−1,4.97,−17.5) + (41/122)((3,6,3) - (−1,4.97,−17.5)) = (1.17,9.93,−32.25) and D = (1.17,9.93,−32.25) + (41/122)((3,6,3) - (1.17,9.93,−32.25)) = (3.33,14.9,−47).

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Answer:

x=5/3

Step-by-step explanation:

The solution to 4 is a>-162. This is because the inequality can be rewritten as -162 > -21, meaning that any number greater than -162 will satisfy the inequality.

Inequality is the unequal distribution of opportunities, resources, or rights among people or groups. It can be found in wealth, income, health, education, access to services and resources, and legal rights. Inequality is a social phenomenon that affects people differently depending on race, gender, age, or background. Inequality can be a result of unequal access to resources, differences in power dynamics, or the unequal distribution of resources among people or groups. It can lead to disparities in health, education, and economic outcomes for those who experience it. Inequality can be addressed through policies that promote equity and inclusion, such as targeted education initiatives and access to employment opportunities.

The reason this is happening is because when solving inequalities, we must isolate the variable on one side of the inequality sign. In this case, we can do this by adding 162 to both sides of the inequality. This will move the -21 to the other side and the result is a>-162.

In order to ensure that the solution is correct, it is important to check the answer by substituting a value larger than -162 into the inequality. If the value satisfies the inequality, then the solution is correct.

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The sοlutiοn tο 4 is a>-162. This is because the inequality can be rewritten as -162 > -21, meaning that any number greater than -162 will satisfy the inequality.

Inequality is the unequal distributiοn οf οppοrtunities, resοurces, οr rights amοng peοple οr grοups. It can be fοund in wealth, incοme, health, educatiοn, access tο services and resοurces, and legal rights. Inequality is a sοcial phenοmenοn that affects peοple differently depending οn race, gender, age, οr backgrοund. Inequality can be a result οf unequal access tο resοurces, differences in pοwer dynamics, οr the unequal distributiοn οf resοurces amοng peοple οr grοups. It can lead tο disparities in health, educatiοn, and ecοnοmic οutcοmes fοr thοse whο experience it. Inequality can be addressed thrοugh pοlicies that prοmοte equity and inclusiοn, such as targeted educatiοn initiatives and access tο emplοyment οppοrtunities.

The reasοn this is happening is because when sοlving inequalities, we must isοlate the variable οn οne side οf the inequality sign. In this case, we can dο this by adding 162 tο bοth sides οf the inequality. This will mοve the -21 tο the οther side and the result is a>-162.

In οrder tο ensure that the sοlutiοn is cοrrect, it is impοrtant tο check the answer by substituting a value larger than -162 intο the inequality. If the value satisfies the inequality, then the sοlutiοn is cοrrect.

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Complete question:

What is the solution to 3/4a>-16?

a>-21 1/3

a<-21 1/3

a>21 1/3

a<21 1/3

If there are 240 students in the sixth-grade class then the number of votes that Naomi received is: 336 votes.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.

Bridgette and Naomi ran for sixth-grade class president.

In the election, every sixth-grade student voted for either Bridgette or Naomi.

Bridgette received 5 votes for every 7 votes Naomi received.

the first figure represents 5:7 ratio in the story.

Since Bridgette received 5 votes for every 7 votes Naomi received, Bridgette received 5/7 of the total votes and Naomi received 2/7 of the total votes.

We know that the total number of votes cast is equal to the total number of students in the class, which is 240.

Therefore, we can set up the equation:

⁵/₇(x) + ²/₇(x) = 240

Simplifying the equation, we get:

(5x + 2x)/7 = 240

7x/7 = 240

x = 240 × 7/5

x = 336

Therefore, If there are 240 students in the sixth-grade class then Naomi received 336 votes.

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The solution of the system of equations by substitution method is 3x + 4(2x - 7) = 16, The correct option is D.

To solve the system of equations {y=2x-7, 3x+4y=16} using substitution, we can solve the first equation for y in terms of x (or x in terms of y) and substitute this expression into the second equation.

Solving the first equation y=2x-7 for y, we get:

y = 2x - 7

We can substitute this expression for y into the second equation 3x+4y=16, replacing y with 2x-7, to get:

3x + 4(2x - 7) = 16

Therefore, the expression after substituting the value of y is 3x + 4(2x - 7) = 16.

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The total number of eggs in 7 dozens of eggs is 84 eggs

Word Problem is a sentence usually made up of a few sentences describing a scenario that needs to be solved through mathematics.

How to determine this

When a dozen mean a group of twelves things

When 1 dozen of an egg = 12 eggs

To get the total number of eggs in 7 dozens of eggs

Let x represent the total number of eggs

When 1 dozen = 12 eggs

7 dozens = x

x = 7 dozens * 12 eggs/1 dozen

x = 84 eggs/1

x = 84 eggs

Therefore, 84 eggs make 7 dozens of eggs

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Answer: The system of equations is:

3x + y = 18

3x + y = 16

To determine how many solutions this system has, we can subtract the second equation from the first:

(3x + y) - (3x + y) = 18 - 16

0 = 2

This is a contradiction, since 0 can never be equal to 2. Therefore, there are no solutions to this system of equations. Geometrically, these two equations represent two parallel lines in a coordinate plane that never intersect, so there is no point that satisfies both equations at the same time.

Step-by-step explanation:

The theater has a tοtal οf 56 red seats and 32 blue seats, making a tοtal οf 88 seats.

If each sectiοn has 14 red seats, then there are a tοtal οf 4 x 14 = <<4 × 14=56>>56 red seats in the theater.

Tο find οut hοw many blue seats are in each sectiοn, we need tο subtract the number οf red seats frοm the tοtal number οf seats in each sectiοn:

88 tοtal seats / 4 sectiοns = 22 seats per sectiοn

22 seats per sectiοn - 14 red seats per sectiοn = 8 blue seats per sectiοn

Therefοre, each sectiοn has 8 blue seats, and the theater has a tοtal οf 4 x 8 = <<4 × 8=32>>32 blue seats.

Sο, the theater has a tοtal οf 56 red seats and 32 blue seats, making a tοtal οf 88 seats.

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Answer:

Step-by-step explanation:

A= 5(x+3)=17

B= 5x+3=17

C= 3x+5=17 A girl decides to buy 5 pens for 3 of her friends. The total cost of the pens was 17$. She then decides to make her friends guess the cost of each pen.

To construct a 3x3 linear system whose solution is (x,y,z)=(3,5,-2) using Gauss-Jordan's Elimination, we will first create an augmented matrix for the system.

This matrix will contain the coefficients for the variables x, y, and z, as well as the values for the constants. For example:
ax + by + cz = d

We can then plug in the values for x, y, and z, and choose values for a, b, c, and d that will make the equation true. For example:

2(3) + 3(5) - 4(-2) = 29

This gives us one equation in our system:

2x + 3y - 4z = 29

We can repeat this process two more times to get two more equations:

-5(3) + 2(5) + 3(-2) = -19

-5x + 2y + 3z = -19

4(3) - 6(5) + 2(-2) = -24

4x - 6y + 2z = -24

So our 3x3 linear system is:

2x + 3y - 4z = 29

-5x + 2y + 3z = -19

4x - 6y + 2z = -24

To solve this system using Gauss-Jordan's Elimination, we can write the system as an augmented matrix:

| 2  3 -4 | 29 |

|-5  2  3 |-19 |

| 4 -6  2 |-24 |

We can then use elementary row operations to reduce the matrix to reduced row echelon form:

| 1  0  0 | 3  |

| 0  1  0 | 5  |

| 0  0  1 |-2  |

This gives us the solution (x,y,z)=(3,5,-2), as desired.

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Jenny buys 15 pieces of bags every month, which means she buys 180 pieces of bags in a year

This question can be solved by unitary method

To find out how many pieces of bags Jenny buys in a year, we can use the following formula:

Total pieces of bags in a year = Number of pieces of bags bought every month × Number of months in a year

We are given that Jenny buys 15 pieces of bags every month, and there are 12 months in a year. So, we can plug in these values into the formula:

Total pieces of bags in a year = 15 × 12

Total pieces of bags in a year = 180

Therefore, Jenny buys 180 pieces of bags in a year.

Answer: 180.

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The absolute maximum is 33 at the point (1, 4) and the absolute minimum is 9 at the point (3, 2).

To find the absolute extrema of the function \( f(x, y)=x^{2}+3 y^{2}-6 x y+8 x \) subject to the constraints \( x \geq 1, y \geq 0 \) and \( y+x \leq 5 \), we can use the Lagrange Multiplier (LM) method. The LM method involves finding the points where the gradient of the function is parallel to the gradient of the constraints.

First, let's find the gradient of the function:
\(\nabla f(x, y) = \langle 2x - 6y + 8, 6y - 6x \rangle \)

Next, let's find the gradient of the constraints:
\(\nabla g(x, y) = \langle 1, 1 \rangle \)

Now, we can set the gradient of the function equal to the gradient of the constraints times a constant, \(\lambda\):
\(\nabla f(x, y) = \lambda \nabla g(x, y) \)

This gives us the following system of equations:
\(2x - 6y + 8 = \lambda \)
\(6y - 6x = \lambda \)

We can also add the constraint \( y+x \leq 5 \) to the system of equations:
\(x + y = 5 \)

Solving this system of equations gives us the critical points:
\((x, y) = (1, 4), (4, 1), (3, 2) \)

Finally, we can plug these critical points back into the original function to find the absolute extrema:
\(f(1, 4) = 1 + 48 - 24 + 8 = 33 \)
\(f(4, 1) = 16 + 3 - 24 + 32 = 27 \)
\(f(3, 2) = 9 + 12 - 36 + 24 = 9 \)

Therefore, the absolute maximum is 33 at the point (1, 4) and the absolute minimum is 9 at the point (3, 2).

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The nth term of the arithmetic sequence is 10 - 3n.

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a constant value, called the common difference, to the preceding term.

The formula for the nth term (where n is a positive integer) of an arithmetic sequence is:

Here we have

a₁ = 7, a₂ = 4, a₃ = 1, a₄ = -2  

Common difference d = a₂ - a₁ = 4 - 7 = - 3

By using the formula

nth term of AP = 7 + (n - 1)(-3)

= 7 - 3n + 3

= 10 - 3n

Therefore,

The nth term of the Arithmetic sequence is 10 - 3n.  

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The triangles are similar and the value of x is 2 that makes AABC~ADEF congruent.

what is congruent ?

In geometry, two figures are said to be congruent if they have the same shape and size. In other words, if all corresponding angles are congruent and all corresponding sides are of equal length, then the two figures are congruent.

When two figures are congruent, we can superimpose one on top of the other and they will match up exactly. This means that all parts of the two figures will coincide, including angles, sides, and diagonals.

According to the question:
To determine the value of x that makes AABC~ADEF, we need to find a scaling factor that relates the corresponding sides of the two triangles.

Since AABC has side lengths of 10, 6x, and 20, its perimeter is 10 + 6x + 20 = 30 + 6x. Similarly, the perimeter of ADEF is 25 + 30 + 50 = 105.

Since the two triangles are similar, their corresponding sides are proportional. This means that:

10/25 = (6x)/30 = 20/50

Simplifying each of these ratios, we get:

2/5 = x/5 = 2/5

This tells us that x/5 = 2/5, or x = 2. Therefore, the value of x that makes AABC~ADEF is x = 2.

To check that the triangles are indeed similar, we can also check that their corresponding angles are congruent. In AABC, the ratio of the side lengths is 1:6x/10:2, which simplifies to 1:3x/5:1. Since the sum of the angles in a triangle is 180 degrees, we know that:

angle A + angle B + angle C = 180 degrees

Using the Law of Cosines, we can find the measure of angle B:

[tex]cos(B) = (10^2 + (6x)^2 - 20^2)/(210(6x)) = (100 + 36x^2 - 400)/(120x) = (36x^2 - 300)/(120x)[/tex]

[tex]B = cos^-1((36x^2 - 300)/(120x))[/tex]

Using this expression, we can express the measures of the angles in AABC in terms of x:

[tex]angle A = sin^{-1(1/(2x))} = 30 degrees[/tex]

[tex]angle B = cos^{-1((36x^2 - 300)/(120x))}[/tex]

angle C = 180 - 30 - B = 150 - B

Similarly, in ADEF, the ratio of the side lengths is 5:6:10. Using the Law of Cosines, we can find the measures of the angles:

[tex]angle D = cos^{-1((25^2 + 30^2 - 50^2)/(22530))} = 36.87 degrees[/tex]

[tex]angle E = cos^{-1((25^2 + 50^2 - 30^2)/(22550))} = 53.13 degrees[/tex]

angle F = 180 - 36.87 - 53.13 = 90 degrees

Comparing the angles in the two triangles, we can see that angle A is congruent to angle F (both are 30 degrees) and angle C is congruent to angle D (both are 180 - B - 30). Therefore, the triangles are similar.

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