Do the first three questions with Percise steps on how to do it

The statement that "σ(n) < 2n holds true for all n of the form n = p²" has been proved.

Let p be any prime number, and let σ(n) be the sum of all positive divisors of the integer n.

As p is a prime number, and 2 is the smallest prime number, so, p[tex]\geq[/tex]2

So, the positive divisors of the integer n are: 1,p,p².

As σ(n) represents the sum of all positive divisors of the integer n.

σ(n)=1+p+p²

In order to prove that σ(n) < 2n,for all n of the form n = p².

1+p+p²<2p²

p²-p-1>0

It is know that, p[tex]\geq[/tex]2.

So, p²-p-1[tex]\geq[/tex]1

Thus, σ(n) < 2n holds true for all n of the form n = p².

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Answer: Option (3)

Step-by-step explanation:

[tex]\frac{1}{6a^2}-\frac{5b}{3a^3}+\frac{b^2}{4a}\\\\=\frac{2a}{12a^3}-\frac{20b}{12a^3}+\frac{3a^2 b^2}{12a^3}\\\\=\frac{3a^2 +b2 +2a-20b}{12a^3}[/tex]

Answer:

2160°

Step-by-step explanation:

[tex]180(14-2)=2160^{\circ}[/tex]

Answer:

([tex]\frac{5}{2}[/tex] , - [tex]\frac{13}{4}[/tex] )

Step-by-step explanation:

8x + 4y = 7 → (1)

6x - 8y = 41 → (2)

multiplying (1) by 2 and adding to (2) will eliminate y

16x + 8y = 14 → (3)

add (2) and (3) term by term to eliminate y

22x + 0 = 55

22x = 55 ( divide both sides by 22 )

x = [tex]\frac{55}{22}[/tex] = [tex]\frac{5}{2}[/tex]

substitute this value into either of the 2 equations and solve for y

substituting into (1)

8([tex]\frac{5}{2}[/tex] ) + 4y = 7

20 + 4y = 7 ( subtract 20 from both sides )

4y = - 13 ( divide both sides by 4 )

y = - [tex]\frac{13}{4}[/tex]

solution is ( [tex]\frac{5}{2}[/tex] , - [tex]\frac{13}{4}[/tex] )

Using the law of cosines and sines, the measure of angle B is: 38.4°.

Law of cosines is: c = √[a² + b² ﹣ 2ab(cos C)]

Law of sines is: sin A/a = sin B/b = sin C/c

Use the law of cosines to find c:

c = √[12² + 18² ﹣ 2(12)(18)(cos 117)]

c ≈ 25.8

Use the law of sines to find angle B:

sin B/b = sin C/c

sin B/18 = sin 117/25.8

sin B = (sin 117 × 18)/25.8

sin B = 0.6216

B = sin^(-1)(0.6216)

B = 38.4°

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Using the Factor Theorem, the equation of h(x) is given as follows:

h(x) = -2(x³ - 2x² - 5x + 6)

The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:

[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]

In which a is the leading coefficient.

Looking at the table, considering the values of x when h(x) = 0, the roots of h(x) are given as follows:

[tex]x_1 = -2, x_2 = 1, x_3 = 3[/tex]

Then the rule is:

h(x) = a(x + 2)(x - 1)(x - 3)

h(x) = a(x² + x - 2)(x - 3)

h(x) = a(x³ - 2x² - 5x + 6)

The h-intercept is of -12, as when x = 0, h = -12, hence this is used to find a as follows:

6a = -12

a = -12/6

a = -2.

Hence the function is given by:

h(x) = -2(x³ - 2x² - 5x + 6)

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Answer: 56.1 kilograms

Step-by-step explanation:

Solution 1: Multiply 51 by 10% and we'll get 5.1. Then, add 51 and 5.1 and we'll get 56.1

Solution 2: Multiply 51 by 110% and we'll get 56.1

(we can multiply 51 by 1 + 10% because multiplying 51 by 1 is the same thing as multiplying 51 by 100%. So you can just use this slicker formula for solving problems like this)

Answer:

1  (double root)

Step-by-step explanation:

Given quadratic equation:

[tex]9x^2+6x+1=0[/tex]

To find the solutions to the given quadratic equation, factor the equation.

To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to ac and sum to b.

[tex]\implies ac=9 \cdot 1=9[/tex]

[tex]\implies b=6[/tex]

Therefore, the two numbers are: 3 and 3.

Rewrite the middle term as the sum of these two numbers:

[tex]\implies 9x^2+3x+3x+1=0[/tex]

Factor the first two terms and the last two terms separately:

[tex]\implies 3x(3x+1)+1(3x+1)=0[/tex]

Factor out the common term (3x + 1):

[tex]\implies (3x+1)(3x+1)=0[/tex]

Therefore:

[tex]\implies(3x+1)^2=0[/tex]

This means that the curve has one root with multiplicity 2. So the curve touches the x-axis and bounces off the axis at one point. (See attached graph).

Apply the zero-product property:

[tex]\implies 3x+1=0 \implies x=-\dfrac{1}{3}[/tex]

Therefore there is one real solution of the given quadratic equation.

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The count of numbers in the list a, a+1, a+2.... b is b - a + 1

The list of numbers is given as:

a, a+1, a+2.... b

From the above list, we can see that the numbers are consecutive numbers.

This means that, the count of numbers in the list is

Count = Highest - Least + 1

Where

Highest = b

Least = a

Substitute the known values in the above equation

Count = b - a + 1

Hence, the count of numbers in the list a, a+1, a+2.... b is b - a + 1

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Answer:

7,647.4?

Step-by-step explanation:

Step-by-step explanation:

M(x) = 4x^3 - 3

let M(x) = Y

Y = 4x^3 - 3...swap places of x & y.

X = 4y^3 - 3

look for x in terms of y.

4y^3 = x + 3

Y^3 = x + 3 / 4...put both sides under cube root.

y = ( x + 3 / 4 )^1/3

Answer:

I think it would be D

The features that exist present in this polar graph is that D. Symmetry about the polar axis θ = 0.

A polar curve exists as a form constructed utilizing the polar coordinate system. Polar curves exists determined by points that exist at a variable distance from the origin (the pole) depending on the angle calculated off the positive x-axis.

The polar graph can be said to contain a symmetry about the polar axis which indicates that theta or θ exists equivalent to 0.

This exists because the graph is symmetric about the x-axis. This indicates that all the lines of symmetry can be seen intersecting the graph at x = 0. This demonstrates that there exists indeed symmetry in the polar graph.

Therefore, the correct answer is option D. symmetry about the plane θ = 0

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The graph is shown in the attached image.

The answer to the question is B. The test statistic is not between the critical values, so we reject the null hypothesis. There is sufficient evidence to support a claim of correlation between percentage of vote and income level.

This is the plot that shows the relationship between two different variables along a straight line. All of the points that are known to have a relationship between these variables would fall under this line. Other parts that fall outside the line are regarded as the outliers in the plot.

In this particular question, the outlier is seen to be outside of the critical values so we have to conclude that the solution is B. We fail to reject the null hypothesis.

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A simultaneous equation is a set of equations that has to be solved in relation to each other at the same time. Thus the required number of stamps are:

4¢ stamps = 16

5¢ stamps = 7

A simultaneous equation is a set of equations that has to be solved in relation to each other at the same time. This process is required so as to determine the values of two unknowns e.g x and y.

From the given question, let the number of 4¢ stamps be represented by n, and that of the 9¢ stamp be represented by m.

So that,

n + m = 23 ............ 1

But 100¢ = $1, so that;

4¢ = x

x = [tex]\frac{4}{100}[/tex]

  = $0.04

also,

9¢ = x

x = [tex]\frac{9}{100}[/tex]

  = $0.09

Thus, we have;

0.04n + 0.09m = 1.27 ......... 2

From equation 1, make n the subject of the formula, such that;

n = 23 - m ........... 3

Substitute equation 3 into equation 2

0.04(23 - m) + 0.09m = 1.27

0.92 - 0.04m + 0.09m = 1.27

collect like terms to have;

0.05m = 1.27 - 0.92

          = 0.35

m = [tex]\frac{0.35}{0.05}[/tex]

m = 7

Now substitute the value of m into equation 3

n = 23 - m ........... 3

  = 23 - 7

n = 16

Therefore the number of 4¢ stamps is 16, while that of the 5¢ stamps is 7.

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If the cost of goods manufactured of $918,700 and cost of goods sold is $955,448. The average manufacturing cost per unit assuming 18,374 units were produced is $102 per unit.

Using this formula to determine the average manufacturing cost per unit

Average manufacturing cost per unit= Total cost/Number of units produced

Where:

Total cost=$918,700+$955,448=$1,874,148

Number of units produced=18,374 units

Let plug in the formula

Average manufacturing cost per unit=$918,700+$955,448/18,374

Average manufacturing cost per unit=$1,874,148/18,374

Average manufacturing cost per unit=$102 per unit

Therefore the average manufacturing cost per unit assuming 18,374 units were produced is $102 per unit.

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Answer:

For weeks are 28 days, so the fraction is 28/8= 3,5 and the fraction is 7/2

Answer:

Step-by-step explanation:

slope-- 1/2

y intercept-- -4

Answer:(3+3)(1+3) = 24

Step-by-step explanation: There is a 1 and 3 3's. We know that added up, the sum is not 24. So it can't be all addition. But, 3 is a factor of 6 which is a factor of 24. There are 2 3's, so we add them up to get 6. Then, we have a 1 and a 3 left. we add these up to get 4. 6x4= 24.

The probability that exactly 7 are men out of 10 sports car owners who are randomly selected is 0.215 given that 60% of people who purchase sports cars are men. This can be obtained by using binomial distribution formula.

This question can be solved using binomial distribution formula.

The formula for binomial distribution is the following,

P(X) = ⁿCₓ pˣ qⁿ⁻ˣ

where,

n = number of trials(or the number being sampled)

x = number of success desired

p = probability of getting a success in one trial

q = 1 - p = probability of getting a failure in one trial

Here in the question it is given that,

⇒ 60% of people who purchase sports cars are men

This statements clearly means that probability of men purchase sports cars is 60%.

⇒ P(men purchasers) = p = 60% = 0.6

From this we can find the probability of women who purchase sports cars,

⇒ P(women purchasers) = q = 1 - p = 1 - 0.6 = 0.4

So we can find the probability that exactly 7 are men out of 10 sports car owners who are randomly selected

It is a binomial case with n = 10

By using the  formula for binomial distribution we get,

P(X = 7) = ¹⁰C₇ × 0.6⁷ × 0.4³

P(X = 7) = 120 × 0.0279936 × 0.064

P(X = 7) = 0.21499

P(X = 7) = 0.215

Hence the probability that exactly 7 are men out of 10 sports car owners who are randomly selected is 0.215 given that 60% of people who purchase sports cars are men.

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The required two-digit number is 75.

In mathematics, it deals with numbers of operations according to the statements.

let the number in tenth place be x.
And has two fewer units than tens.
= x - 2

The two-digit number can be given as
= 10x + (x - 2)

= (11x - 2)

Reversing the digits
= (x - 2)10 + x
= 11x -20

The difference between twice the number and the number reversed is 93.

[tex]2(11x - 2) - (11x - 20) = 93\\22x-4-11x+20 = 93\\11x+16 = 93\\11x = 93-16\\11x = 77\\x = 7[/tex]

Now the required two-digit number is,
=  11x -2
=  11*7 - 2
=  77 -2
= 75

Thus the required two-digit number is 75.

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Answer:60 ml

Step-by-step explanation:20 % of 300 = 60 and snce the rest of the the l are coffee thats 20%

Numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.

Given a paragraph in which there are blanks:

A _____ is a number whose only factors are 1 and itself. If a number has  factors besides 1 and itself, it is called a _____. You can use divisibility rules or _______ to help you determine whether a number is prime or composite.

We are required to fill the blank with appropriate options.

We have to fill "prime numbers" in the first blank.

We have to fill "composite numbers" in the second blank.

We have to fill "the sieve of Eratosthenes" in the third blank.

Hence numbers whose only factors are 1 and itself is known as prime numbers , numbers whose factors besides 1 and itself are composite numbers, we can use the sieve of Eratosthenes to determine whether the number is prime or composite.

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Answer:

Area = 33 inches squared

Step-by-step explanation:

The formula for the area of a kite is:

Area = ½ × (d)1 × (d)2

To find the first diagonal (d1):

d1 = 7 in + 4 in = 11 in

To find the second diagonal (d2):

The triangles on the right of the kite must be 3 4 5 triangle since it is a right triangle with a hypotenuse of 5 and a side of 4, we know half of a diagonal would be 3 inches.

d2 = 3 in + 3 in = 6 in

Now we can plug the diagonals into the area formula.

Area = ½ × 11 × 6 = 33 inches squared

Answer: A = 33 in²

Step-by-step explanation:

Given information:

Longer side = 7 in

Shorter side = 4 in

Shorter Hypotenuse = 5 in

Given formula:

A = (1/2) × (d₁) × (d₂)

Please refer to the attachment below for a graphical understanding

Determine the length of the shorter side (vertical) through the Pythagorean theorem

a² + b² = c² (Pythagorean theorem)

(4)² + b² = (5)² (Substitute values into the equation)

16 + b² = 25

b² = 25 - 16

b² = 9

b = 3  or b = -3 (rej)

Substitute values into the given formula

A = (1/2) × d₁ × d₂

A = (1/2) × (7 + 4) × (3 + 3)

Simplify values in the parenthesis

A = (1/2) × 11 × 6

Simplify by multiplication

A = (11/2) × 6

[tex]\Large\boxed{A=33}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

width = 7

length = 10

Step-by-step explanation:

solve for the one letter that gets talked about the most

which is w or width

width = w

length = 3 + w

2 inch border means

width = w + 2

length = 3 + w + 2

area = length times width

area = ( 3 + w + 2 ) times (w + 2 )

108 = ( 3 + w + 2 ) times (w + 2 )

108 = ( w + 5 ) times (w + 2 )

( w + 5 ) times (w + 2 ) = 108

w^2 + 7w +10 = 108

w^2 + 7w - 98 = 0

going to make w = x

x^2+7x-98=0

(x-7)(x+14)

x = 7

x = -14

w=7

w= -14

cant have a negative measurement so w=7 is used

width = w

length = 3 + w

width = 7

length = 3 + 7 = 10

Answer:

[tex](2,-1.5)[/tex]

Step-by-step explanation:

The midpoint formula goes as follows:

[tex](x,y)=(\frac{x1+x2}{2} , \frac{y1+y2}{2})[/tex]

Lets plug in our values, let x1 be -3, x2 be 7, y1 be 2, y2 be -5.

[tex](\frac{-3+7}{2} , \frac{2+-5}{2})[/tex]

Solve numerators

[tex](\frac{4}{2} , \frac{-3}{2})[/tex]

Solve

[tex](2 , -1.5)[/tex]

Therefore the midpoint must be at [tex](2,-1.5)[/tex]

The value of x which S(x) is a global minimum is x = 1/2

From the question, we have:

This means that:

S(x) = 1/x + 4x^2

From the question, we understand that x is a positive number.

This means that the domain of x is x > 0

As a notation, we have (0, ∞)

Recall that:

S(x) = 1/x + 4x^2

Differentiate the function

S'(x) = -1/x^2 + 8x

Set to 0

-1/x^2 + 8x = 0

Multiply through by x^2

-1 + 8x^3 = 0

Add 1 to both sides

8x^3 = 1

Divide by 8

x^3 = 1/8

Take the cube root of both sides

x = 1/2

To prove the point is a global minimum, we have:

S'(x) = -1/x^2 + 8x

Determine the second derivative

S''(x) = 2/x^3 + 8

Set x = 1/2

S''(x) = 2/(1/2)^3 + 8

Evaluate the exponent

S''(x) = 2/1/8 + 8

Evaluate the quotient

S''(x) = 16 + 8

Evaluate the sum

S''(x) = 24

Because S'' is positive, then the single critical point is a global minimum

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Answer:

b. 30 m²

Step-by-step explanation:

find the area of the rectangle and triangle first

area of rectangle = length x height

aR = 10 x 6

aR = 60

area of triangle = base x height x 1/2

aT = 10 x 3 x 1/2

aT = 30

now, to find the area of the shaded area subtract the triangle from the rectangle

area = rectangle - triangle

a = 60 - 30

a = 30

By the 30 - 60 - 90 theorem the right options of the multiple choice question is:

A. The hypotenuse is twice as long as the shorter leg.

D. The longer leg is √3 times as long as the shorter leg.

In this problem we must look for the right choices in a multiple choice question. According to the Euclidean geometry, 30 - 60 - 90 have the following properties:

Thus, the right options of the multiple choice question is:

A. The hypotenuse is twice as long as the shorter leg.

D. The longer leg is √3 times as long as the shorter leg.

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