1. Choose the correct answer.
Which theorem or postulate could be used to prove the congruence of the pair of triangles?

The sequence converges and its limit is zero

A sequence is an ordered set of numbers governed by a rule.

The limit of a sequence is the value the sequence approaches

The convergence of a sequence is when the sequence approached a definite value

The divergence of a sequence is when the sequence does not approach a  definite value.

Since we have the sequence (√3 - √1), (√4 - √2), (√5 - √3), (√6 - √2), We need to find the general term of the sequence.

Observing this, we find that the general term Uₙ = √(n + 2) - √n

To determine if the limit converges or diverges, we take the limit of the sequence as n → ∞.

So,

[tex]\lim_{n \to \infty} U_n = \lim_{n \to \infty} \sqrt{n + 2} - \sqrt{n} \\= \lim_{n \to \infty} \sqrt{n + 2} - \lim_{n \to \infty} \sqrt{n} \\= \sqrt{\infty - 2} - \sqrt{\infty} \\= \sqrt{\infty} - \sqrt{\infty} \\= \infty -\infty[/tex]

Since this is an indeterminate form, we use L'hopital's rule to find the limit.

Uₙ = √(n + 2) - √n

= √n{√[(n + 2)/n] - 1}

= √n÷ 1/{√[(1 + 2/n] - 1}

Differentiating, we have

lim n → ∞ Uₙ = lim n → ∞ d√n/dn ÷ d(1/{√[(1 + 2/n] - 1})/dn

= 1/(2√n)/-√[(1 + 2/n] - 1})⁻² × 1/2√(1 + 2/n] × (-2/n²)

= n²[√(1 + 2/n] - 1})²/(2√n)√(1 + 2/n]

= n²[√(1 + 2/n] - 1})²/(2√n)√(1 + 2/n)]

Dividing through by n², we have

= n²[√(1 + 2/n] - 1})²/n² ÷ (2√n)√(1 + 2/n)]/n²

= [√(1 + 2/n] - 1})² ÷ (2(√n)³√(1 + 2/n³)]

Dividing both the numerator and denominator by (√n)³, we have

= [√(1 + 2/n] - 1})²/(√n)³ ÷ (2(√n)³√(1 + 2/n³)]/(√n)³

= (1 + 2/n] - √(1 + 2/n] + 1)/(√n)³ ÷ [2√(1 + 2/n³)]

= (1(√n)³ + 2/n²] - √{(1 + 2/n]/n)³} + 1/(√n)³) ÷ [2√(1 + 2/n³)]

= (1(√n)³ + 2/n²] - √{(1/n³ + 2/n⁴} + 1/√n)³) ÷ [2√(1 + 2/n³)]

So, lim n → ∞ Uₙ = lim n → ∞  (1(√n)³ + 2/n²] - √{(1/n³ + 2/n⁴} + 1/√n)³) ÷ [2√(1 + 2/n³)]

=  (1(√∞)³ + 2/∞²] - √{(1/∞³ + 2/∞⁴} + 1/√∞)³) ÷ [2√(1 + 2/∞³)]

=  (0 + 0] - √{(0 + 0} + 0) ÷ [2√(1 + 0)]

= (0 - 0 + 0)/2

= 0/2

= 0

The sequence converges and its limit is zero

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Answer:

it is basic and .........

Answer:The area A of a rectangle is represented by the formula A=lw where l is the length and w is the width. The length of the rectangle is 5.

Write an equation that makes it easy to find the width of the rectangle if we know the area and the length.

Step-by-step explanation:

Answer:

25%

Step-by-step explanation:

Answer:

Step-by-step explanation:

Total number of candidates is:

Probability of a girl to be chosen as prefect:

Answer:

Probability = 15/23

Step-by-step explanation:

Now we have to,

→ find probability that prefect is a girl.

The total members are,

→ 15 girls + 8 boys

→ s = 15 + 8

→ [ s = 23 ]

Then the probability will be,

→ P = 15/s

→ [ P = 15/23 ]

Hence, the answer is 15/23.

The amount of chocolate left is three and seven-twelfths pounds

The amount of chocolate left is found by addition and subtraction of fraction

The fraction required are:

Eva bought seven and one-half pounds of chocolate chips = 7 1/2

She used three and three-fourths pounds = 3 3/4

two and five-sixths pounds in some muffins = 2 5/6

total amount used = addition of amount used

3 3/4 + 2 5/6

= 3 11/12

the amount left is gotten by subtraction of amount used from amount Eva bought

= 7 1/2 - 3 11/12

= 3 7/12

this is three seven-twelfths

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The equation of the trend line in the scatter plot is y = - 4x + 24

What is the slope-intercept form?

Writing the equation of a line in the slope-intercept form simply makes the slope (steepness) and y-intercept (where the line crosses the vertical y-axis) obvious.

Here, we have

The yellow points are located at points (6, 0) and (4, 8)

For a linear equation that passes through the points (x₁, y₁) and (x₂, y₂), the slope is:

s = (y₂ - y₁)/(x₂ - x₁)

Then, for the equation, the slope is

s = (8-0)/(4-6) = -4

Then the linear equation become

y = -4x + b

where b is the y-axis intercept.

To find b, we use that when x = 6, y = 0

0 = 6×(-4) + b

b = 24

Then equation become y = - 4x + 24

Hence, the equation of the trend line in the scatter plot is y = - 4x + 24

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Since the first child is a boy, this means we need to find the probability of another boy. This is 1/2 since we assume each gender has the same chance of happening.

the dimensions of the dumpster that will minimize its surface area

length l = 6.32

width w =  1.58

height h = 0.39

Length = l

width = w

height = h

given that length is four times its width

l = 4w

Now, the formula for the surface area of a cube is;

S = 2lh + 2lw + 2wh

substitute l = 4w in the above equation

S = 2(4w)h+2(4w)w+2wh

S = 8wh+8[tex]w^{2}[/tex]+2wh

S = 10wh + 8[tex]w^{2}[/tex]

formula for the volume of a cube is;

V = lwh

now substitute l=4w in the above equation

V = (4w)wh

V = 4[tex]w^{2}[/tex]h

h = 4[tex]w^{2}[/tex]/V

Put V/2w² for h in the surface area equation to get;

S= 8w² + 10w(V/4w²)

 = 8w² + 5V/2w

given that volume V = 25.6 substitute that in the above equation

S = 8w² + 5(25.6)/2w

  = 8w² + 128/2w

Since we want to minimize the surface area, let us find the first derivative of S to get;

S' = 16w - 128/2w²

At S' = 0

0 = 16w-  128/2w²

16w =  128/2w²

32w³ = 128

w³ = 128/32

w³ = 4

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

w = 1.58 yds

since l = 4w substitute the value of w in that

l = 4(1.58) = 6.32

h = 4[tex]w^{2}[/tex]/V = 4([tex]1.58^{2}[/tex])/25.6

h = 0.39

the dimensions are

length l = 6.32

width w =  1.58

height h = 0.39

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Using the Law of Cosines, the value of x is 40.9°.

Cosine formula, cos A [tex]=\frac{b^{2}+c^{2}-a^{2} }{2bc}[/tex].

Here, A = x

          a = 16

          b = 15

          c = 24

cos x [tex]=\frac{15^{2}+24^{2}-16^{2}}{2*15*24}[/tex]

         [tex]=\frac{225+576-256}{720}[/tex]

         [tex]=\frac{801-256}{720}[/tex]

         [tex]=\frac{545}{720}[/tex]

         = 0.756

x [tex]= cos^{-1}0.756[/tex]

 = 40.9

Hence, value of x is 40.9°.

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The common difference of the arithmetic sequence -13, -22, -31, is -9.

An arithmetic sequence is a set of numbers that are ordered and have a common difference between each consecutive term. In the arithmetic sequence 3, 9, 15, 21, 27, for example, the common difference is 6. An arithmetic progression is another name for an arithmetic sequence.

To find a specific term in an arithmetic sequence, we use the nth term formula. Step 1: An = a + (n - 1)d gives the nth term of an arithmetic sequence.

Some arithmetic sequence examples using this expression are:

1, 5, 9, 13, 17, 21, 25, 29, 33, ...

2, 4, 6, 8, 10, 12, 14, 16, 18,...

In this case, the difference will be:

= 2nd term - 1st term

= -22 - (-13)

= -22 + 13

= -9

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Answer:

Step-by-step explanation:

You want the equation of the line with slope 3/5 that passes through point (5, 3).

The point-slope form of the equation of a line with slope m through point (h, k) is ...

  y -k = m(x -h)

You have m=3/5, (h, k) = (5, 3), so your equation is ...

  y -3 = 3/5(x -5) . . . . point-slope form of desired equation

This can be simplified to ...

  y -3 = 3/5x -3

  y = 3/5x . . . . . . . add 3

The sales tax is $2 and the total cost of the item is $42.

Given:

selling price = $40

rate of sales tax = 5%

sales tax = 40*5%

= 40*5 / 100

= 200/100

= $2

Total cost = 40+2

= $42.

Therefore the sales tax is $2 and the total cost of the item is $42.

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Answer:

the value of x is 135 i believe

Step-by-step explanation:

sorry if im wrong

Answer:

9(x+4)= 6

Step-by-step explanation:

I could be wrong but im not sure

The number of solutions in the system is 1

From the question, the system of equations is given as

Also, we understand that the linear equation is a vertical straight line.

So, we have the system to be

A vertical line can only intersect with a quadratic line at one point

Hence, there is only one solution in the system

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The angles of the triangle will be 50°, 70°, 60° .

let the 3 angles of a triangle be A, B and C

The first acute angle A is given by = x + 20

The second acute angle B is given by = 2x + 10

The third acute angle C is given by = 4x - 60

Now we know that the sum of the angles of a triangle is 180°

So ,

A + B + C + = 180°

x + 20 + 2x + 10 +  4x - 60 = 180°

7x - 30 = 180°

7x = 210°

x = 30°

So, from the value of x , we can calculate the value of each angle

Therefore, the required angles are :

angle A = (x + 20)° = 30° + 20° = 50°

angle B = (2x + 10)° = 60° + 10° = 70°

angle C = (4x - 60)° = 120° - 60° = 60°

verification : sum of the angles = 50°  + 70° + 60°  = 180°

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Answer:

100°

Step-by-step explanation:

The sum of any polygon angles add to  ( n-2)(180)   where n is the number of sides ......for a quadrlateral    ( 4-2)(180 ) = 360

60 + 90 + 110 +  x = 360    shows    x = 100 °

Answer:

sin75^°-sin15^°=1/√2

[tex](\stackrel{x_1}{10}~,~\stackrel{y_1}{4})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{- \cfrac{3}{5}}(x-\stackrel{x_1}{10}) \\\\\\ y-4=- \cfrac{3}{5}x+6\implies {\Large \begin{array}{llll} y=- \cfrac{3}{5}x+10 \end{array}}[/tex]

Answer:

4

Step-by-step explanation:

you can use the formula y=mx+b where b is 4

Answer:

[tex]f(x) = (x+1)(x-2)(x+4)[/tex]

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

The image of the polygon after rotation of 90° about the origin is  A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) , the graph of the polygon and its image is shown below .

In the question ,

it is given that ,

the vertices of the polygon is A(-3,-1), B(2, 2), C(3,-3) .

the graph of the polygon ABC is shown below .

We know when the point (x , y) is rotated 90° in clockwise direction , it s coordinate change to (y , -x) .

So , image of coordinate A(-3,-1) is

A'(-1,-(-3)) = A'(-1, 3)

image of coordinate B(2, 2) is

B'(2 , -2)

image of coordinate C(3,-3) is

C'(-3 , -3)

So , the graph of the image with coordinates A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) is shown below .

Therefore , The image of the polygon after rotation of 90° about the origin is  A'(-1, 3) , B'(2 , -2) , C'(-3 , -3) , the graph of the polygon and its image is shown below .

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the 4.3 represents The number of apples in 1 pound

The 4.3 in the equation represents the number of apples in 1 pound

The solution of an equation refers usually to the values of the variables involved in that equation which if substituted in place of those variable would give a true mathematical statement.

We are given equation y = 4.3x can be used to determine the total cost, y, in dollars, of x pounds of apples.

Here we can see that y = 4.3x

4.3 = y/x

then we can say that 4.3 is slope of striaght line in term os linear equation.

4.3 in the equation represents the number of apples y, in dollars, of x pounds.

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Answer: based on the information that you gave me, I think the answer is :

A. sex is an explanatory variable; starting salary is the response variable ;area of specialization is a possible confounding variable

hope this helps

Step-by-step explanation:

The values of the variables "a", "b", and "c" are 3.06, 169.74, and 0, respectively. The total amount Gary will pay is $769.74.

Gary purchased a S750 TV on a credit card with a 22% annual percentage rate, and he wants to pay it off in payments of $200 per month. The table shows the information for the first four months after Gary used his credit card. We need to calculate the values of the missing variables and the total amount he will pay.

The variable "a" is the interest charged on the left amount in the third month.

a = (366.68 - 200)*0.018333 = 3.06

The variable "b" is the monthly amount paid by Gary. It is less than or equal to $200, depending on the payment amount. Since the amount to be paid is $169.74, which is less than $200, he will pay the whole amount in one go.

b = 169.74

The variable "c" is the interest charged on the left amount in the fourth month. Since then, the entire amount has been paid.

c = 0

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The probability that their next (third) child will have extra digits is 0.5

Polydactyly refers to the presence of extra fingers in the human body. It is a hereditary condition that is passed down from parents to children. The thumb, middle, and little fingers are the most commonly affected by this condition. Polydactyly refers to the presence of extra fingers in the human body. It is a hereditary condition that is passed down from parents to children. The thumb, middle, and little fingers are the most commonly affected by this condition.

A trait is a distinguishing characteristic that a person possesses. Dominant traits are defined as having only one copy of a gene on a chromosome that expresses a specific trait.

Individuals exhibit the dominant traits most prominently. Polydactyly is the dominant condition in this case.

The father has six fingers and six toes, while the mother is in normal condition. In this particular condition, polydactyly is a dominant trait. Twelve percent of the couple's children are expected to have extra digits.

Because the father has the dominant trait and the mother is in normal condition, there is a 50% chance that this deformity will be passed down to children.

The probability that their next child will have extra digits is 0.5

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Answer: 44

Step-by-step explanation:

since OPN and RPQ are similar triangles, you can use ratios to find their sides

[tex]\frac{24}{15} =\frac{x}{15+12.5}[/tex]  or  [tex]\frac{24}{15} =\frac{x}{27.5}[/tex]

cross multiply to get: 24*27.5 = 15x

660/15 = x

x = 44