Post Test: Polygons 4 Given: ΔMNO Prove: The medians of ΔMNO are concurrent. Given triangle of sides M (x1, y1), N (x2, y2) and O (x3, y3) point to the same triangle on the graph of x-axis and y-axis, with M prime (0, 0), O prime (2t, 0) and N prime (2r, 2s) has a midpoint s formed by connecting Prime PQR Proof: Statements Reasons 1. ∆MNO Given 2. Use rigid transformations to transform ΔMNO to the congruent ΔM'N'O' so that M' is at the origin and M'O' lies on the x-axis in the positive direction. Rigid transformations result in congruent shapes. 3. Any property that is true for ΔM'N'O' will also be true for ΔMNO. Definition of congruence 4. Let r, s, and t be real numbers such that the vertices of ΔM'N'O' are M'(0,0), N'(2r,2s), and O'(2t,0). Defining constants 5. Let P', Q', and R' be the midpoints of M'N', N'O', and M'O', respectively. Defining points 6. P' = (r,s); Q' = (r + t,s); R' = (t,0) ? 7. Definition of slope 8. Applying point-slope formula 9. and intersect at point S . Algebra 10. Point S lies on . The coordinates of S satisfy the equation of . 11. All three lines share point S. Follows from 9 and 10 12. The medians are concurrent. Definition of concurrent lines Which reason completes the proof for step 6? A. Definition of midsegment B. Definition of slope C. Definition of midpoint D. Definition of median

After applying factorization of the given functions the evaluated expression comes out to be (2x + 1)(x² - x - 4.8), under the condition the given function is P(x)=2x × 3- 5.6x +1

In order to factorize the given polynomial P(x) = 2x³ - 5.6x + 1, we can apply the Rational Root Theorem. The theorem projects that in an incident when a  polynomial has integer coefficients, then any rational roots must have a numerator that divides the constant term and a denominator that divides the leading coefficient.
For the given case, the leading coefficient is 2 and the constant term is 1.
Hence, any rational roots must be of the form p/q where p divides 1 and q divides 2. This elaborates that the possible rational roots are ±1/2 and ±1.
We can observe these roots applying synthetic division. Using synthetic division with root -1/2 gives us:
-1/2 | 2   0   -5.6   1
    |    -1    0.8  -0.5
    -------------------
          2   -1   -4.8  0.5
This means that P(x) can be factored as:
P(x) = (2x + 1)(x² - x - 4.8)
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The greatest common factor (GCF) of the three monomial, 21x³y⁵, 63x⁵y² and 45x¹y³ is 3xy²

The greatest common factor (GCF) of the three monomials, 21x³y5, 63x5y² and 45x¹y³ can be obtained as follow:

First, we shall determine the factors of each monomials. This is illustrated below:

21x³y⁵ = 3 × 7 × x × x × x × y × y × y × y × y

63x⁵y² = 3 × 3 × 7 × x × x × x × x × x × y × y

45x¹y³ = 3 × 3 × 5 × x × y × y × y

Finally, we shall determine the greatest common factor (GCF) from the factors listed above. This is shown below:

3 × x × y × y = 3xy²

Thus, we can conlude that the greatest common factor (GCF) of the three monomials is 3xy²

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Answer:

  $15,200

Step-by-step explanation:

You want to know the amount deposited at 1.5% that will earn the same simple interest in a year as $12,000 deposited at 1.9%.

The interest amount is given by the formula ...

  I = Prt

For the two accounts, the interest and the time are the same, so we have ...

  P1·r1 = P2·r2

  12000·1.9% = P2·1.5% . . . . . . . . . . . equate interest amounts for t=1

  P2 = 1.9/1.5·12000 = 15200 . . . . . . divide by 1.5%

Ruth deposited $15,200.

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The relation is a function because (d) Yes because for each output there is exactly one input.

From the question, we have the following parameters that can be used in our computation:

The graph

The graph can be represented as

x  g(x)

-5  1

-2  0

-1   -1

0  2

1  3

5   1

The above table of values is a function

This is so because each output values have different input values.

i.e. it would pass the vertical line test when represented on a graph

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The probability that a randomly selected point within the square falls in the red-shaded square is 0.11

From the question, we have the following parameters that can be used in our computation:

The areas of the above shapes are

Red square = 1² = 1

White square = 3² = 9

The probability is then calculated as

P = Red square/White square

So, we have

P = 1/9

Evaluate

P = 0.11

Hence, the probability that a randomly selected point within the square falls in the red-shaded square is 0.11

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Answer:

The exact volume of the cylinder is 72π cubic centimeters, or approximately 226.19 cubic centimeters to two decimal places.

Step-by-step explanation:

The volume of a cylinder can be calculated using the formula:

[tex]\boxed{V = \pi r^2 h}[/tex]

where:

Given values:

Substitute the given values into the formula and solve for V:

[tex]\begin{aligned} \implies V&=\sf \pi \cdot 3^2 \cdot 8\\&=\sf \pi \cdot 9 \cdot 8\\&=\sf 72\pi \;cm^3\\&= \sf 226.19\; cm^3\;(2\;d.p.)\end{aligned}[/tex]

Therefore, the volume of the cylinder is 72π cubic centimeters, or approximately 226.19 cubic centimeters to two decimal places.

The length of BC is determined as  6√31  by applying trigonometry ratio.

The length of BC is calculated by applying the following formulas for trig ratios.

The trig ratio is simplified as;

SOH CAH TOA;

SOH ----> sin θ = opposite side / hypothenuse side

CAH -----> cos θ = adjacent side / hypothenuse side

TOA ------> tan θ = opposite side / adjacent side

The hypothenuse side of the right triangle is given as 12√31 and the opposite side is length BC.

The length of BC is calculated as follows;

sin 30 = BC / 12√31

BC = 12√31 x sin (30)

BC = 12√31 x ¹/₂

BC = 6√31

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The perimeter of the given triangle is: 19.2 units

The formula for the distance between two coordinates is:

D = √[(y₂ - y₁)² + (x₂ - x₁)²]

Thus: (0, 3)(4, -2)

Slant distance on left side of triangle = √[(-2 - 3)² + (-3 - 0)²] = 5.8 units

Slant distance on right side of triangle = √[(-2 - 3)² + (4 - 0)²] = 6.40 units

Length of horizontal part of triangle = 7 units

Thus:

Perimeter of triangle = 5.83 + 6.40 + 7 = 19.2 units

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After considering all the given options we come to the conclusion that the inequality that represents the  values of x for the give triangle ABC is 15/11 < x < 17/5, which is Option A.

Here, the sides of a triangle should gradually satisfy the triangle  inequality theorem which projects that the summation of the lengths considering any two sides of a triangle should be bigger than or equivalent towards the length of the third side.
Then, for the given sides of triangle ABC,
3x+1 + 16 > 8x

Applying this inequality,
x > 15/11
So,
16 + 8x > 3x+1
Applying simplification
x < 17/5
Hence , the inequality that represents the values of x for which triangle ABC exists is 15/11 < x < 17/5.
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Answer:

Step-by-step explanation:

Amount in 4.6% acct= 15000*(1/3)= 5000

Amount in bonds = 1/4*15000= 3750

Amount in stock= 15000-(5000+3750)= 6250

Amount in each account at the end of three years

5000*(1.046)^3= 5722.22668

3750*(1.052)^3=4365.94728

6250*1.03*(1-.08)*1.06= 6277.85

total: 5722.22668+4365.94728+6277.85=16366.02396

gain= 16366.02396-15000=1366.02396

or (16366.02396/15000)-1= 9.1068264%

Given WX is tangent to OS at point V, the statements that must be true include:

A. m∠TVX =m∠TVX

C. m∠TVW = m∠TSV

D. m∠TUV =m∠TVW

In mathematics, a tangent is a line that touches a curve at exactly one point, without intersecting it at that point. The tangent line at a point on a curve represents the slope of the curve at that point. The concept of tangent is important in calculus, where it is used to calculate derivatives of functions.

TUV = 1/2MTU = TVW

TUV = TVW

Also, MTV = TSV = 2TVW

TVW = TSV

The correct options are A, C and D.

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After considering all the given data we conclude that the the probability of rolling a number less than 7 is 1/2, under the condition that the set of equally likely outcomes is (1,2,3,4,5,6,7,8,9,10,11,12).

The evaluated probability of rolling a number less than 7 on a 12-sided die is the number of outcomes that are less than 7 divided by the total number of possible outcomes.

The set of equally likely outcomes is (1,2,3,4,5,6,7,8,9,10,11,12). There are 6 numbers less than 7 in this set (1,2,3,4,5,6) and 12 possible outcomes.

Hence , the probability of rolling a number less than 7 is:

6/12 = 1/2

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Answer:

Capital expenditures are typically one-time large purchases of fixed assets that will be used for revenue generation over a longer period. Revenue expenditures are the ongoing operating expenses, which are short-term expenses used to run the daily business operations.

The total area of the composite figure is 17 square centimeters.

Remember that the area of a rectangle is equal to the product between its dimensions.

We can separate the figure into two rectangles, one of these rectangles is of 1cm by 2cm, then its area is:

A = 1cm*2cm = 2cm²

The other rectangle is of 3cm by 5cm, then its area is:

A' = 3cm*5cm = 15cm²

Adding that we get a total of:

total area = 2cm² + 15cm² = 17cm²

That is the area of the figure.

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The area of the regular decagon with a radius of 14 units is approximately [tex]195.7 unit^{2}[/tex]

To find the area of a regular decagon, we need to know either the side length or the apothem (the perpendicular distance from the center to a side). In this case, you have provided the radius of the decagon, which is 14 units.

For a regular decagon, the apothem (a) is related to the radius (r) by the formula:

[tex]a = r * cos(\pi/10)[/tex]

Let's calculate the apothem using the given radius:

[tex]a = 14 * cos(\pi/10) =12.149[/tex]

Now, we can calculate the area (A) of the regular decagon using the formula:

[tex]A = (5/2) * a^2 * tan(\pi/10)[/tex]

[tex]A = (5/2) * (12.149)^2 * tan(\pi/10) =195.7[/tex]

Rounding to the nearest tenth, the area of the regular decagon with a radius of 14 units is approximately [tex]195.7 unit^{2}[/tex].

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Answer:

31 years

Step-by-step explanation:

A = P(interest rate)^number of years (n)

if the money is to quadruple, A = 4 X 22,000 = 88,000.

P = investment amount = 22,000.

interest rate is 4.5%. so we have the whole (100%) + 4.5% = 1.045%.

n is number of years.

22,000 (1.045)^n = 88, 000

divide both sides by 22, 000:

(1.045)^n = 4

take logs for both sides:

log (1.045)^n = log 4

simplify using log laws (by bringing down the n):

n log (1.045) = log 4

divide both sides by log (1.045):

n = (log 4) / log (1.045)

= 31.49 (31 years to nearest year)

This given function does not have an end behavior since it is not a polynomial.

The end behavior of a function describes the behavior or trend of the function as the input values (x) approach positive or negative infinity. It helps us understand what happens to the function's values as x becomes very large or very small.

We have three possible cases of end behavior of a function which are;

The end behavior of a function is often determined by using the leading term of the function, which is the term with the highest power of x.

In the given problem;

[tex]f(x) = \frac{8x^6 - 16x^3 + 8}{4x^2 - 4x - 24}[/tex]

The end behavior of this function does not exist because this function is not a polynomial

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Answer:

98 is a whole number

Step-by-step explanation:

0.86 is a decimal

2/5 is a fraction

35% is a percentage

The measurement is closest to the value of d inches is 14. 28

It is important to note that the Pythagorean theorem is stated as;

The Pythagorean theorem states that the square of the longest leg or hypotenuse side of a triangle is equal to the sum of the squares of the other two sides.

This is represented as;

x²= y²+ z²

Now, substitute the values, we get;

x² = 11. 9² + 7.9²

Find the square values, we have;

x² = 141. 61 + 62. 41

add the values, we have;

x² = 204. 02

Now, find the square root of both sides, we get;

x = √204. 02

x = 14.28

Know that d = x = 14. 28

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The simplest form of the expression by the use of the distributive property is;

[tex]12x^2 + 24x + 9[/tex]

In mathematics, the distributive property, which explains how multiplication interacts with addition or subtraction, is a fundamental concept. By allocating or multiplying a number or term to each term enclosed by a set of parenthesis, it enables you to simplify expressions.

The distributive property is frequently employed in algebraic expressions, equation simplification, and polynomial operations. By spreading out the multiplication over addition or subtraction, it aids in simplifying complex statements and makes calculations easier to understand.

We know that;

[tex]3(4x^2 + 8x + 3)\\12x^2 + 24x + 9[/tex]

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The distance that the end of the pendulum travels when it swings is given by the relation D = 27.3 cm

Given data ,

A pendulum of length 18 cm is swinging from a fixed point

Now ,  it swings through an angle of 87°

So , Central Angle = ( s x 360° ) / 2πr

where s is the length of the arc

Now , D = ( θ/360 ) x 2πr

On simplifying , we get

D = ( 87 / 360 ) x 2 ( 3.14 ) ( 18 )

D = ( 0.241667 ) ( 113.04 )

D = 27.3 cm

Hence , the pendulum travels a distance of 27.3 cm

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A set of two or more equations with the same variables.

A system of linear equations with EXACTLY ONE solution.

A system of linear equations with INFINITELY MANY solutions.

A system of equations with NO solution.

A number representing how many times the Base number should be multiplied by itself.

A chart of the different values for an equation or graph.

Systems of Equations:

A set of two or more equations with the same variables.

These equations are solved simultaneously to find the values of the variables.
Consistent Independent:

A system of linear equations with EXACTLY ONE solution.

This means that the lines represented by the equations intersect at a single point.
Consistent Dependent (Coincident):

A system of linear equations with INFINITELY MANY solutions.

In this case, the equations represent the same line or overlapping lines, and every point on the line is a solution.
Inconsistent:

A system of equations with NO solution.

This occurs when the lines represented by the equations are parallel and never intersect.
Exponent:

A number representing how many times the base number should be multiplied by itself.

For example, in the expression[tex]2^3,[/tex] the exponent is 3, which means 2 should be multiplied by itself 3 times [tex](2 \times  2 \times  2 = 8).[/tex]
T-Table:

A chart of the different values for an equation or graph.

It typically includes two columns: one for the input values (usually x) and one for the corresponding output values (usually y).

T-tables are helpful for understanding the relationship between the variables in an equation and for plotting points on a graph.

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Median: 212

Mode: 212

The median and mode of the data set are 273 and 212 respectively.

The median is the value that splits the mathematical numbers or expressions in half. The median value is the middle number of data points. To find the median first arrange the data points in ascending order.

To find the median, we need to arrange the numbers in order from least to greatest:

3, 4, 5, 212, 212, 334, 414, 823

There are 8 numbers in total, so the median is the average of the two middle numbers, which are 212 and 334.

Median = (212 + 334)/2 = 273

To find the mode, we need to identify the number(s) that occur most frequently. In this case, 212 appears twice, so it is the mode.

Mode = 212

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The piecewise rule for the pattern is given as follows:

A piece-wise function is a function that has different definitions, depending on the input of the function.

The pattern for this problem is given as follows:

1,6,9,1,6,9.

The period of the pattern is given as follows:

3 units.

The pattern of the function is constant, meaning that all the outputs for an input that is a multiple of 3 are of 1, if the remainder of the division is of 1 the output is of 6, and if the remainder of the division is of 2 the output is of 9.

Hence the pattern is defined as follows:

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Answer:20%

Step-by-step explanation:

The statements that will be true about parallelogram ABCD are: A, B, D, and E.

Properties of a Parallelogram -

Diagonals of a parallelogram bisect each other into congruent segments.

Opposite angles and sides of a parallelogram are always congruent.

Alternate interior angles are always congruent in measure.

Thus, the following would be true of parallelogram ABCD:

AP ≅ CP (congruent segment's of a bisected diagonal)

BC ≅ AD (congruent opposite sides)

∠CAD ≅ ∠ACB (congruent angles)

∠BPC ≅ ∠APD (congruent angles)

Therefore, the statements that will be true about parallelogram ABCD are: A, B, D, and E.

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Answer:

The volume of the cylinder is 500 π cubic kilometers.

Step-by-step explanation:

Formula: V = πr²h

r = d/2

r = 20km/2

r = 10 km

h = 5km

Solve for the volume using the formula.

Substitute the given

V = π(10km)²(5km)

V = 500π km³

I hope this helps you

The assumption of a decay constant of k = 0.1, we estimate that there would be approximately 36.788 fossils per cubic meter after 10 years.

To estimate the number of fossils after 10 years, we need to solve the given equation. However, since the equation is not provided, we'll assume a simple exponential decay model for the population of fossils.

Let's assume the equation for the population of fossils is:

N(t) = N0 * e^(-kt)

where:

N(t) is the number of fossils after t years,

N0 is the initial number of fossils,

k is a decay constant, and

e is the base of the natural logarithm (approximately 2.71828).

Since the initial number of fossils is given as 100, we have N0 = 100. Now, we need to find the value of the decay constant (k) to estimate the number of fossils after 10 years.

Without further information, it's challenging to determine the exact decay constant. However, let's assume a hypothetical value of k = 0.1 for this example.

Plugging the values into the equation, we get:

N(10) = 100 * e^(-0.1 * 10)

= 100 * e^(-1)

≈ 100 * 0.36788

≈ 36.788

So, with the assumption of a decay constant of k = 0.1, we estimate that there would be approximately 36.788 fossils per cubic meter after 10 years.

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Answer:

B. 2333.59

Step-by-step explanation:

We have been given the following differential equation:

[tex]\dfrac{dN}{dt}=\dfrac{5000}{6+5t}[/tex]

To find the number of fossils N(t) after t = 10 years, we first need to solve the differential equation using integration and the given parameters to create an equation for N(t) in terms of t.

Solving a differential equation means using it to find an equation in terms of the two variables, without a derivative term.

Rearrange the equation so that all the terms containing N are on the left-hand side, and all the terms containing t are on the right-hand side:

[tex]1\;dN=\dfrac{5000}{6+5t}\; dt[/tex]

Integrate both sides:

[tex]\displaystyle \int 1 \;dN=\int \dfrac{5000}{6+5t}\; dt[/tex]

Use the following integration rules:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Integration rulest}\\\\\\$\displaystyle \int n\:\text{d}x=nx+\text{C}$\;\;(where $n$ is any constant value)\\\\\\$\displaystyle \int \dfrac{n}{a+bx}\:\text{d}x=\dfrac{n}{b}\ln |a+bx|+\text{C}$\\\\ \end{minipage}}[/tex]

Therefore:

[tex]\begin{aligned}\displaystyle \int 1\;dN&=\int \dfrac{5000}{6+5t}\; dt\\\\\implies N&=\frac{5000}{5}\ln \left|6+5t\right|+C\\\\N&=1000\ln \left|6+5t\right|+C\end{aligned}[/tex]

As the initial number of fossils is 100, then N = 100 when t = 0.

Substitute these values into the equation and solve for C:

[tex]100=1000\ln \left|6+5(0)\right|+C[/tex]

[tex]C=100-1000\ln (6)[/tex]

Therefore, the equation for N in terms of t is:

[tex]N=1000\ln \left|6+5t\right|+100-1000\ln(6)[/tex]

Use the quotient log rule to simplify:

[tex]N=1000\ln \left|6+5t\right|-1000\ln(6)+100[/tex]

[tex]N=1000\ln \left(\dfrac{\left|6+5t\right|}{6}\right)+100[/tex]

To estimate the number of fossils after 10 years, substitute t = 10 into the equation:

[tex]N=1000\ln \left(\dfrac{\left|6+5(10)\right|}{6}\right)+100[/tex]

[tex]N=1000\ln \left(\dfrac{56}{6}\right)+100[/tex]

[tex]N=2333.59\; \sf (2\;d.p.)[/tex]

Therefore, the number of fossils after 10 years is 2333.59.