Maricopa's Success scholarship fund receives a gift of $ 85000. The money is invested in stocks, bonds, and CDs. CDs pay 5.5 % interest, bonds pay 5.2 % interest, and stocks pay 6.4 % interest. Maricopa Success invests $ 20000 more in bonds than in CDs. If the annual income from the investments is $ 4780 , how much was invested in each account?
Maricopa Success invested $ in stocks.
Maricopa Success invested $ in bonds.
Maricopa Success invested $ in CDs.

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: The answer is A

Step-by-step explanation:

Answer:

I think you forgot a picture

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.

Answer:

(c+1)(c+2)

Step-by-step explanation:

you need to find the two numbers that get 2 when you multiply them and get 3 when you add them. those two numbers are 2 and 1 and c times c is c^2

you can check the answer by using foil like this:

(c+1)(c+2)

first: c times c= c^2

second: 2 times c= 2c

third: 1 times c= 1c

last: 1 times 2= 2

and all that together you get c^2+2c+1c+2

simply and get C^2+3c+2

The point on Π that is closest to A is (3,-2,0).

(a) The distance between the plane Π and the point A is 3. To find this, we need to find the equation of the plane Π. The equation of the plane is given by:

Ax + By + Cz + D = 0

where (A, B, C) is the normal vector to the plane, which is perpendicular to the line that passes through points A and B. Since the normal vector of the plane is perpendicular to the line, the normal vector (A, B, C) is equal to the cross product of the two vectors of the line AB, given by:

A=(3,0,-2), B=(-1,1,0)

A x B = (A2B3-A3B2, A3B1-A1B3, A1B2-A2B1) = (-5,3,3)

Therefore, the equation of the plane Π is:

-5x + 3y + 3z + D = 0

To find D, we need to plug in the coordinates of the point (1,2,3). Therefore,

-5(1) + 3(2) + 3(3) + D = 0
-5 + 6 + 9 + D = 0
D = -20

Therefore, the equation of the plane Π is:

-5x + 3y + 3z - 20 = 0

To find the distance between the plane Π and the point A, we need to calculate the shortest distance between the plane and the point A. We can do this using the distance formula, given by:

d = |Ax + By + Cz + D|/sqrt(A^2 + B^2 + C^2)

Substituting the equation of the plane Π and the coordinates of point A into the distance formula, we get:

d = |-5(3) + 3(0) + 3(-2) - 20|/sqrt(-5^2 + 3^2 + 3^2)
d = |-15 - 20|/sqrt(34)
d = |-35|/sqrt(34)
d = 3

Therefore, the distance between the plane Π and the point A is 3.

(b) The point on Π that is closest to A is (3,-2,0). To find this, we need to solve the system of equations given by:

-5x + 3y + 3z - 20 = 0
x - 3 = 0
y - 0 = 0
z + 2 = 0

Solving this system of equations, we get x = 3, y = -2, and z = 0. Therefore, the point on Π that is closest to A is (3,-2,0).

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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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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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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.

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 $​

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 correct answer is option A, counter clockwise 90°, for the needle to lean on the outline section, the wheel must be turned counter clockwise 90°. Any other turn of the wheel will not result in the needle leaning on the outline section.

Counterclockwise is an adjective used to describe motion or rotation that is in the opposite direction of a clock's hands. For example, when a person swims in a pool, they may swim counterclockwise around the perimeter of the pool. This is the opposite direction of the way a clock's hands move. Counterclockwise rotation is also known as anti-clockwise rotation.

When an object is placed on a flat surface and the wheel is turned, it will rotate in the same direction as the motion of the wheel. If the needle is leaning on the outline section, the wheel must be turned counter clockwise 90°. This is because when the wheel is turned counter clockwise, the needle will be forced to move in the same direction and end up leaning on the outline section.

When the wheel is turned clockwise, the needle will move in the opposite direction and it will not end up leaning on the outline section. Therefore, a clockwise turn of 120° or 180° is not the correct answer. Similarly, turning the wheel counter clockwise by 180° is also not the correct answer. This is because the needle will end up pointing in the same direction it was pointing before the wheel was turned.

In conclusion, for the needle to lean on the outline section, the wheel must be turned counter clockwise 90°. Any other turn of the wheel will not result in the needle leaning on the outline section.

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Answer:

A. counter clockwise 90°.

Step-by-step explanation:

a. 240 students were surveyed in all. b. 60 students chose blue. c. The number of students who chose purple is 7, green is 10, others is 7 students.

In mathematics, a proportion is a statement that two ratios or fractions are equal. A proportion can be expressed as an equation of the form:

a/b = c/d

where a, b, c, and d are numbers, and b and d are not equal to zero. This equation can also be written in the form of a cross product:

ad = bc

This equation means that the product of the numerator of one fraction and the denominator of the other fraction is equal to the product of the denominator of the first fraction and the numerator of the second fraction.

a. If 20% of the students surveyed chose yellow and 48 students chose yellow, we can set up the following proportion to find the total number of students surveyed (let "x" be the total number of students):

20/100 = 48/x

Solving for x, we get:

x = 240

Therefore, 240 students were surveyed in all.

b. 25% of the students surveyed chose blue, so the number of students who chose blue is:

25/100 x 240 = 60

Therefore, 60 students chose blue.

c. We are given that 3% of the students surveyed chose purple, and 4% chose green. To find the number of students who chose purple or green, we can add the two percentages and find the corresponding portion of the total number of students:

3/100 + 4/100 = 7/100

So 7% of the students surveyed chose purple or green. The number of students who chose purple is:

3/100 x 240 = 7.2 (rounded to the nearest whole number, this is 7)

Similarly, the number of students who chose green is:

4/100 x 240 = 9.6 (rounded to the nearest whole number, this is 10)

We are also given that 3% of the students surveyed chose "other". Therefore, the number of students who chose "other" is:

3/100 x 240 = 7.2 (rounded to the nearest whole number, this is 7)

So 7 students chose "other".

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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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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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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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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:

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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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 polynomial function will be x³ + (3-√2)x² - 3√2x = 0.

What is a polynomial function?

A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc., that only uses non-negative integer powers or only positive integer exponents of a variable.

The simplest straightforward and typical mathematical equation is one with polynomial functions. It can be described using a polynomial. The polynomial function is modelled by a polynomial equation. A polynomial is typically written as P. (x). The degree of a polynomial is the largest exponent of the variable P(x). Understanding a polynomial's degree is crucial since it explains how function P(x) behaves as the value of x increases. Polynomial functions only apply to real numbers in their domain (R).

Given : zeros of polynomial = 0, -3 and √2

So, the polynomial function can be found :

  = (x-0) (x+3)(x-√2)

  = x (x+3) (x-√2)

  = (x²+3x) (x-√2)

  = x³ -√2x² + 3x² - 3√2x

  = x³ + (3-√2)x² - 3√2x

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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.

Answer:

x=5/3

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

The given statement is true for all matrices A.

1) For any 2x2 matrices A and B, (AB)2 = A2B2 - False. A counter-example to this claim would be if A and B are non-invertible matrices, such as A = [1, 0; 0, 0] and B = [1, 0; 0, 1]. (AB)2 = [1, 0; 0, 0], but A2B2 = [1, 0; 0, 1].

2) For any 2x2 matrices A, B, and C, if AB = AC, then B = C - True. This statement is true, as long as A is an invertible matrix. This can be seen from the properties of matrix multiplication; if AB = AC, then B = A-1AC = C.

3) For any 2x2 matrix A, if A2 = A, then either A = 0 or A = I - False. A counter-example to this claim would be if A = [1, 0; 0, 0], then A2 = A = [1, 0; 0, 0], but A ≠ 0 and A ≠ I.

1) Counter-example:Let the matrices A and B be given by\[A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, B=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\].Now,\[(AB)^2=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\].and\[A^2B^2=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}^2\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\]Hence\[(AB)^2\ne A^2B^2\]2) Let AB=AC. Then, we need to show that B=C. Let A, B and C be given as follows:\[A=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, B=\begin{pmatrix} 3 \\ 4 \end{pmatrix}, C=\begin{pmatrix} 5 \\ 6 \end{pmatrix}\].Now, AB=\[\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 3 \\ 4 \end{pmatrix}=\begin{pmatrix} 11 \\ 4 \end{pmatrix}\]and AC=\[\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 5 \\ 6 \end{pmatrix}=\begin{pmatrix} 11 \\ 6 \end{pmatrix}\]Thus,\[AB=AC\]But, B=\[\begin{pmatrix} 3 \\ 4 \end{pmatrix}\]and C=\[\begin{pmatrix} 5 \\ 6 \end{pmatrix}\].Therefore,\[B\ne C\]Thus, AB=AC does not imply that B=C.3) True.According to the given statement, \[A^2=A\]Then,\[A^2-A=0\]⇒\[A(A-I)=0\]Either A=0 or A=I. Thus, the given statement is true for all matrices A.

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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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Yes, dog will catch the frisbee. The coordinates of the point or points where they meet are (0,6) and (6,18). So, option(a) is correct answer for this question.

I have a training your dog to catch a frisbee. The path equation of frisbee is y= –x² + 8x + 6 --(1)

and the path equation of the dog is modeled by

y = 2x + 6 --(2)

We have to check dog catch the frisbee or not and also determine the coordinates. The dog will catch the frisbee, when both equations are intersect to each other. That is intersection point of equation (1) and (2) are the required coordinates. Now, we determine the intersection points between the equation (1) and (2).

Substituting the value of y from equation (2) to equation (1), y = 2x + 6 in y = - x² + 8x + 6

=> 2x + 6 = -x² + 8x + 6

Simplify, x² - 8x + 2x = 6-6

=> x² - 6x = 0

=> x( x - 6) = 0

either x = 0 or x -6 = 0

=> x = 0, x= 6

from equation (1), y = 2(0) + 6 = 6

when x = 6, y = 2(6) + 6 = 18

Hence, required coordinates are (0,6) and (6,18).

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Complete question:

You are training your dog to catch a frisbee. You are playing in a large field, and you are standing next to your dog when you throw the frisbee. If the path of the frisbee is y= –x² + 8x + 6 and the path of the dog is modeled by y = 2x + 6, will the dog catch the frisbee? If so, what are the coordinates of the point or points where they meet?

a) Yes, they intersect at the coordinates (0, 6) and (6, 18)

b) Yes, they intersect at the coordinates (1, 13) and (2, 10)

c) Yes, they intersect at the coordinates (2, 18) and (3, 12)

d)No, the paths do not cross

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 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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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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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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