Find the area of the regular polygon with the given radius or apothem.

​(Simplify your answer. Type an exact answer using radicals as​ needed.)

Answer:

x = 5

Step-by-step explanation:

10^5=

(10 x 10) x 10 x 10 x 10=

(100 x 10)x 10 x 10=

(1000 x 10) x 10=

(10,000 x 10) = 100,000

To solve the equation 10^x = 100,000, we need to isolate x on one side of the equation. We can do this by taking the logarithm of both sides of the equation with base 10:

log(10^x) = log(100,000)

Using the logarithmic property log(a^b) = b*log(a), we can simplify the left side of the equation:

x*log(10) = log(100,000)

Since log(10) = 1, we can simplify further:

x = log(100,000)

Using a calculator, we can evaluate the logarithm to get:

x = 5 + log(10)

x = 5 + 1

x = 6

Therefore, the solution to the equation 10^x = 100,000 is x = 6.

On solving the query we can say that This stage consists just of arithmetic multiplication and does not include any specific properties of numbers.

Mathematics is concerned with numbers and their variations, equations and related structures, shapes and their places, and possible placements for them. The relationship between a collection of inputs, each of which has an associated output, is referred to as a "function". An relationship between inputs and outputs, where each input yields a single, distinct output, is called a function. Each function has a domain and a codomain, often known as a scope. The letter f is frequently used to represent functions (x). X is the input. The four main types of functions that are offered are on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

The expression's step is justified by the distributive property. The distributive property of multiplication over addition asserts that for any three integers a, b, and c, a(b + c) = ab + ac. This is the particular property we are employing.

Here are the facts:

2(5 + 17) = 2(5) + 2(17)

which is equivalent to:

2(5 • 17) = (2 • 5) 17

The distributive principle of multiplication over addition is used in this phase.

Then, we further simplify by multiplying 2 by 5 to get 10, which results in:

(2 • 5) 17 = 10 • 17 = 170.

This stage consists just of arithmetic multiplication and does not include any specific properties of numbers.

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On solving the query we can say that This stage consists just of arithmetic multiplication and does not include any specific properties of numbers.

What is function?

Mathematics is concerned with numbers and their variations, equations and related structures, shapes and their places, and possible placements for them. The relationship between a collection of inputs, each of which has an associated output, is referred to as a "function". An relationship between inputs and outputs, where each input yields a single, distinct output, is called a function. Each function has a domain and a codomain, often known as a scope. The letter f is frequently used to represent functions (x). X is the input. The four main types of functions that are offered are on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

The expression's step is justified by the distributive property. The distributive property of multiplication over addition asserts that for any three integers a, b, and c, a(b + c) = ab + ac. This is the particular property we are employing.

Here are the facts:

2(5+17)=2(5) + 2(17)

which is equivalent to:

2(5+17)= 44

The distributive principle of multiplication over addition is used in this phase. Then, we further simplify by multiplying 2 by 5 to

get 10, which results in:

(2×5) 17 = 10× 17 = 170.

This stage consists just of arithmetic multiplication and does not include any specific properties of numbers.

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Answer:

84.5 square units.

Step-by-step explanation:

The formula to calculate the area of a trapezoid is:

Area = (base 1 + base 2) * height / 2

Using the measurements you provided, we can calculate the area of the trapezoid as follows:

Area = (12 + 14) * 6.5 / 2

Area = 26 * 6.5 / 2

Area = 169/2

Area = 84.5

Therefore, the area of the trapezoid is 84.5 square units.

The required volume and total surface area of the pentagonal pyramid are 15√3

Using the Pythagorean theorem, we can find s:

s² = (3√2)² + (s/2)²

s²2 = 18 + (s²/4)

3s²/4 = 18

s² = 24

s = 2√6

Now, we can find the area of each triangle:

Area of triangle = (1/2) * apothem * side

Area of triangle = (1/2) * 3√2 * 2√6

Area of triangle = 3√3

The total area of the base pentagon is 5 times the area of each triangle:

Area of base = 5 * 3√3

Area of base = 15√3

Now, we can find the volume of the pentagonal pyramid:

V = (1/3) * Base Area * Height

V = (1/3) * 15√3 * 3

V = 15√3

To find the surface area of the pentagonal pyramid,
Area of triangular face = (1/2) * side * slant height
The slant height is the distance from the midpoint of one of the sides of the pentagon to the apex of the pyramid.
slant height² = height² + (s/2²
slant height² = 9 + 6
slant height = √15

Now, we can find the area of each triangular face:
Area of triangular face = (1/2) * 2√6 * √15
Area of triangular face = 3√10

There are five triangular faces, so the total area of the triangular faces is:

Total area of triangular faces = 5 * 3√10
The total area of triangular faces = 15√10

Adding the area of the pentagonal base, we get the total surface area:

Total surface area = 15√3 + 15√10

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The measure of the angle ∠6 as 57 degrees.

Same-side interior angles are pairs of angles that are on the same side of the transversal and between the two parallel lines. When two parallel lines are intersected by a transversal, the same-side interior angles are supplementary, which means that they add up to 180 degrees. This is known as the Same-Side Interior Angles Theorem.

Now, let's apply this theorem to the given problem. We are given that ∠4 is 123 degrees and we need to find the measure of ∠6. Since ∠4 and ∠6 are same-side interior angles, we know that they add up to 180 degrees. We can use this information to set up an equation:

∠4 + ∠6 = 180

Substituting the value of ∠4 as 123 degrees, we get:

123 + ∠6 = 180

To solve for ∠6, we can subtract 123 from both sides of the equation:

∠6 = 180 - 123

Simplifying the right-hand side of the equation, we get:

∠6 = 57

Therefore, the measure of ∠6 is 57 degrees.

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Complete Question:

Same-side interior angles always add up to 180°. If ∠4 and ∠6 are same-side interior angles, what is the measure of ∠6 where the value of ∠4 is 123 degrees?

[tex]\textit{Pythagorean Identities} \\\\ \sin^2(\theta)+\cos^2(\theta)=1\implies \cos^2(\theta )=1-\sin^2(\theta ) \\\\ 1+\cot^2(\theta)=\csc^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{1-\sin^2(\theta )}{1+\cot^2(\theta )}~~ = ~~\sin^2(\theta )\cos^2(\theta ) \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1-\sin^2(\theta )}{1+\cot^2(\theta )}\implies \cfrac{\cos^2(\theta )}{\csc^2(\theta )}\implies \cfrac{\cos^2(\theta )}{ ~~ \frac{1}{\sin^2(\theta )} ~~ }\implies \cos^2(\theta )\sin^2(\theta )[/tex]

4√5+2√5

4+2(√5)

6√5

option C

Answer: 65.25 u²

Step-by-step explanation:

I'm going to flip the triangle so c is at the bottom.  i draw a height from <C down to c so that I create a right triangle and I will use trig to find h

they gave me b (hypotenuse) and <A and i want that new h we drew which is opposite <A

sin A = b/h

sin 65 = 18/h    cross multipy to bring h up

h=18/sin65

h=16.31

Area= 1/2 b h     b=base=c=8

=1/2 (8)(16.31)

=65.25 u²

It makes sense to begin with the y-intercept when sketching the graph of y = mx + b because the y-intercept is the point at which the line intersects the y-axis, where x = 0. This point gives us a starting point for the graph and allows us to plot a single point before determining the slope of the line. Once we have the y-intercept, we can use the slope to find additional points on the line, and then connect the points to create the graph of the line.

Answer:

Step-by-step explanation:

When graphing a linear equation of the form y = mx + b, where m is the slope and b is the y-intercept, it makes sense to begin with the y-intercept instead of the slope because the y-intercept gives us a point on the y-axis where the line crosses it.

In the case of the equation y = X + 2, the y-intercept is the point (0,2) on the y-axis, which is a point we can easily plot on the graph. Once we have plotted the y-intercept, we can then use the slope, which is 1 in this case, to find another point on the line by moving one unit to the right along the x-axis and one unit up along the y-axis. We can then draw a straight line through these two points to complete the graph of the line.

Starting with the y-intercept also makes sense because it gives us a sense of the vertical position of the line before we consider its slope. This can be helpful in cases where the y-intercept is a large or small number, as it can affect the scale of the graph and make it easier to read. Additionally, starting with the y-intercept can help us to quickly identify whether the line intersects the y-axis or not, and at what point.

The volume of sphere A is 256π/3 cm³.

The volume of sphere B is  288π cm³.

The volume of sphere C is   2048π/3 cm³.

The volume of each sphere is calculated as follows;

V = ⁴/₃ πr³

where;

Volume of sphere A;

V = ⁴/₃ πr³

V = ⁴/₃ π(4³) = 256π/3 cm³

Volume of sphere B;

V = ⁴/₃ πr³

V = ⁴/₃ π(6³) = 288π cm³

Volume of sphere C;

V = ⁴/₃ πr³

V = ⁴/₃ π(8³) = 2048π/3 cm³

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Answer:

Step-by-step explanation:

These three graphs have a greater rate of change than the equation y=1/2 x+3.

What is equation?

An equation is an expression that states the equality of two things using mathematical symbols. It is a statement of balance between two elements, usually represented by numbers or variables. Equations are used to solve problems in many fields, including biology, chemistry, physics, engineering, and economics. Equations can be written in a variety of ways, such as linear, quadratic, and polynomial equations, and each type has a unique set of properties. Equations are useful for understanding the relationships between different variables and for predicting outcomes.

Graphs A, D, and F all have a greater rate of change than y=1/2 x+3.

Graph A has a slope of 2, which is double that of y=1/2 x+3 (1).

Graph D has a slope of 4, which is quadruple that of y=1/2 x+3 (1).

Finally, Graph F has a slope of 6, which is six times that of y=1/2 x+3 (1).

Therefore, these three graphs have a greater rate of change than the equation y=1/2 x+3.

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The first 3 common denominators of 5/24 and 5/8 are the same as the first 3 lowest multiples of 24, which are;  24, 48, and 72

The lowest common multiples of two numbers are the smallest multiples that are common to two numbers.

The first 3 common denominators of 5/24 and 5/8 can be found from the first three lowest common multiples of 24 and 8 as follows;

The first three lowest multiples of 24 are 24, 48, and 72

The first three lowest multiples of 8 are 8, 16, and 24

Therefore, the first three common denominators of 5/24 and 5/8 are 24, 48, and 72

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When testing the null hypothesis with an alpha value of 0.05, the critical value will be smaller than if you were testing the alpha value of 0.01.

This is because the alpha value represents the maximum level of probability that we are willing to accept for making a Type I error (rejecting the null hypothesis when it is actually true).

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Answer:

Therefore, the appropriate measure of variability for this data set is the IQR, and its value is 2.

Step-by-step explanation:

The range is the simplest measure of variability and is the difference between the largest and smallest values. In this case, the largest value is 20 and the smallest value is 2, so the range is:

Range = 20 - 2 = 18

However, since there are some extreme values (2 and 20), it may be better to use a measure of variability that is less affected by outliers. The interquartile range (IQR) is a good measure of variability that is less affected by extreme values.

To find the IQR, we need to find the median of the data set. The median is the middle value when the data is arranged in order. In this case, the data set has an even number of values, so the median is the average of the two middle values:

Median = (10 + 12) / 2 = 11

Next, we need to find the first quartile (Q1) and the third quartile (Q3). Q1 is the median of the lower half of the data set, and Q3 is the median of the upper half of the data set. In this case, the lower half of the data set is {2, 10, 10} and the upper half of the data set is {12, 12, 20}.

Q1 = Median of lower half = 10

Q3 = Median of upper half = 12

So, the IQR is:

IQR = Q3 - Q1 = 12 - 10 = 2

Therefore, the appropriate measure of variability for this data set is the IQR, and its value is 2.

Answer:

To determine Martha's monthly pension, we need to use the formula provided by the Merrick Oaks School District for calculating the annual pension benefit:

Annual Pension Benefit = (Years of Service x Average of Last Three Annual Salaries) / 55

where 55 is the "pension factor" used by the district.

Using this formula, we can calculate Martha's annual pension benefit as follows:

Annual Pension Benefit = (18 x $100,000) / 55

Annual Pension Benefit = $32,727.27

Therefore, Martha's annual pension benefit is $32,727.27.

To calculate her monthly pension, we simply divide the annual pension benefit by 12:

Monthly Pension Benefit = $32,727.27 / 12

Monthly Pension Benefit ≈ $2,727.27

Therefore, Martha's monthly pension benefit would be approximately $2,727.27 if she retires after 18 years of service.

Step-by-step explanation:

It is proved that a multiple of a pythagorean triple is also a pythagorean triple

When ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:

[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]

where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).

WE can see few pythagorean triple as

[tex]{3}^{2} + {4}^{2} = {5}^{2} {(3 \times 2)}^{2} + {(4 \times 2)}^{2} = {(5 \times 2)}^{2} {6}^{2} + {8}^{2} = {10}^{2}[/tex]

3-4-5 is a Pythagorean triple.

Now Multiplying each of these numbers by 2, we obtain 6-8-10,

This is a Pythagorean triple.

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The commutative property of multiplication is as shown x × y = y × x .

The given values of x and y are:

x = a + bi and y = c + di

By the commutative property which states that it is a binary operation in which ab=ba whenever you change the order of the operands.

Therefore,

x = a + bi and y = c + di

Now, x × y = (a + bi) × (c + di)

 = ac + adi + bci + bdi²

= ac - bd + i(ad + bc)

Now, y × x = (c+ di) × (a + bi)

 = ac + bci + adi + bdi²

= ac - bd + i(ad + bc)

Therefore,  x × y = y × x

and when ab = ba then the property is known as commutative property.

Hence commutative property of multiplication is as shown above.

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Answer:

see explanation

Step-by-step explanation:

the volume (V) of a triangular prism is calculated as

V = Ah ( A is the area of the triangular face and h the depth of the prism )

1

the area of the triangular face is

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )

here b = 7 and h = 5 , then

A = [tex]\frac{1}{2}[/tex] × 7 × 5 = 17.5 in²

the depth of the prism is 12 , so

V = 17.5 × 12 = 210 cm³

2

A = [tex]\frac{1}{2}[/tex] × 13 × 15 = 97.5 ft²

the depth of the prism is 2 , so

V = 97.5 × 2 = 195 ft³

3

A = [tex]\frac{1}{2}[/tex] × 11 × 4 = 22 yd²

the depth of the prism is 6 , so

V = 22 × 6 = 132 yd³

Answer:

Since each letter can be used more than once in any given password, we can use the multiplication rule to find the total number of possible passwords.

There are 6 positions in the password, and each position can be filled with any of the 6 letters (a, e, i, o, u, y). Therefore, the total number of possible passwords is:

6 x 6 x 6 x 6 x 6 x 6 = 46656

So there are 46,656 possible passwords that can be formed if the letters can be used more than once in any given password.

mark me brilliant

well, hmmm keeping in mind that complex roots never come all by their lonesome, their sister always comes along, namely their conjugate, so if we have the complex roots of "i" or namely "0 + i", we also have her sister "0 - i", and if we have "1 - 2i", she also came with "1 + 2i", and we also have the root of 5, and that'd give us the fifth degree polynomial, so

[tex]\begin{cases} x = 0+i &\implies x -i=0\\ x = 0-i &\implies x +i=0\\ x = 1-2i &\implies x -1+2i=0\\ x = 1+2i &\implies x -1-2i=0\\ x = 5 &\implies x -5=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -i )( x +i )( x -1+2i )( x -1-2i )( x -5 ) = \stackrel{0}{y}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{difference of squares} }{( x -i )( x +i )}\implies x^2-i^2\implies x^2-(-1)\implies x^2+1 \\\\[-0.35em] ~\dotfill[/tex]

[tex]( x -1+2i )( x -1-2i )\implies \stackrel{ \textit{difference of squares} }{( [x -1]+2i )( [x -1]-2i )} \\\\\\ (x-1)^2 -(2i)^2\implies (x^2-2x+1)-4i^2\implies (x^2-2x+1)-4(-1) \\\\\\ x^2-2x+1+4\implies x^2-2x+5 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{so we can say}}{a(x^2+1)(x^2-2x+5)(x-5)=y}\hspace{5em}\textit{we also know that } \begin{cases} x=0\\ y=-75 \end{cases} \\\\\\ a(0^2+1)(0^2-2(0)+5)(0-5)=-75\implies -25a=-75 \\\\\\ a=\cfrac{-75}{-25}\implies a=3 \\\\[-0.35em] ~\dotfill[/tex]

[tex]3(x^2+1)(x^2-2x+5)(x-5)=y\implies 3(x^3-5x^2+x-5)(x^2-2x+5)=y \\\\\\ 3(x^5-7x^4+16x^3-32x^2+15x-25)=y \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} 3x^5-21x^4+48x^3-96x^2+45x-75=y \end{array}}~\hfill[/tex]

Check the picture below.

[tex]\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =42\\ r=13 \end{cases}\implies A=\cfrac{(42)\pi (13)^2}{360} \\\\\\ A=\cfrac{1183\pi }{60}\implies \implies A\approx 61.94[/tex]

Answer: Each piece of poster board with be 0.78 cents each.

Step-by-step explanation: In order to find x, you need to figure out why 35 pieces of poster boards came up to a total of $27.30.

You will divide in this situation. If you divide $27.30 by 35 poster boards, you should get 0.78 as the result, representing the cost of each poster board.

Answer:

Yes, you need to find the GCF (greatest common factor) to solve this word problem.

The greatest common factor of 35 and 42 is 7. So you can create 7 teams with equal number of boys and girls on each team without leaving anyone out

Step-by-step explanation:

The probability that the publishing company produces more than 7500 books/magazines in a day is about 0.0062

To solve this problem, we need to use the standard normal distribution formula, which involves converting the given normal distribution into a standard normal distribution with a mean of 0 and a standard deviation of 1.

To convert a normal distribution with mean μ and standard deviation σ into a standard normal distribution, we use the following formula

z = (x - μ) / σ

where x is the value we want to convert, μ is the mean of the distribution, σ is the standard deviation of the distribution, and z is the corresponding value on the standard normal distribution.

Using this formula, we can convert the value of 7500 books/magazines per day to a z-score as follows

z = (7500 - 5500) / 800 = 2.5

We can then use a standard normal distribution table or a calculator to find the probability that the standard normal distribution produces a value greater than 2.5. This probability is approximately 0.006

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Equation to find b is 0.05b = 5.10.

Let's use the variable b to represent the cost of the haircut.

According to the problem, the tip was 5% of the cost of the haircut, which can be written as:

0.05b = 5.10

To solve for b, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.05:

b = 5.10 ÷ 0.05

b = 102

So the cost of the haircut was $102. We can check this by calculating 5% of $102, which gives us a tip of $5.10, as stated in the problem.

Therefore, the equation to find b, the cost of the haircut, is:

0.05b = 5.10

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Approximately 5% of students spent their day off playing outside.

To find the percentage of students who spent their day off playing outside, we first need to determine the total number of students. We can do this by adding the number of votes for each activity:

Sleeping: 24

Playing Games 56

Playing Outside: 8

Shopping: 45

Fun Projects: 17

Total Votes (Students) = 24 + 56 + 8 + 45 + 17 = 150

Now we can calculate the percentage of students who spent their day off playing outside:

Percentage = (Number of students who played outside / Total number of students) * 100

Percentage = (8 / 150) * 100 = 0.0533 * 100 = 5.33%

Now, we'll round to the nearest percent:

Percentage ≈ 5%

So, approximately 5% of students spent their day off playing outside.

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Answer:

Yes, this definition is valid. A regular polygon is a polygon where all sides are equal in length and all angles between the sides are also equal. Therefore, stating that a regular polygon has all sides of the same length is an accurate description of the defining characteristic of such a shape.