Special right triangles

The per share dividend dollar amount paid out by Langs Inc. in 2020 is $0.50.

To calculate the per-share dividend dollar amount, we first need to calculate the total dividends paid out in 2020. We can find this by subtracting the retained earnings at the end of 2019 from the retained earnings at the end of 2020: $4,326,800 - $4,216,800 = $110,000. This $110,000 represents the total dividends paid out in 2020, since the company only uses retained earnings to pay dividends.

Next, we divide the total dividends by the number of shares outstanding at the end of 2020: $110,000 / 220,000 shares = $0.50. This is the per share dividend dollar amount, rounded to the nearest cent.

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Complete Question:

Given the following information for Langs, Inc., calculate per share dividend dollar amount (that is, the dividends per share), rounded to the nearest cent (i.e., $5.21843 = $5.22) that the company paid out in 2020:

The approximate value of [tex]$\int_{0}^{2} x^4 dx$[/tex] using the trapezoidal rule is 16.

Integral : [tex]$\int_{0}^{2} x^n dx$[/tex]

n = 4

Approximation Method : Trapezoidal Rule

The trapezoidal rule is a numerical integration technique used to approximate the definite integral of a function. It works by approximating the region under the curve of the function with a series of trapezoids, and calculating the area of those trapezoids to estimate the integral.

In the given problem, we have to approximate the definite integral of [tex]$x^4$[/tex] from 0 to 2. The trapezoidal rule for approximating a definite integral is given by

[tex]$\int_{a}^{b} f(x)dx \approx \frac{h}{2} \left( f(a) + f(b) + 2 \sum_{i=1}^{n-1} f(x_i) \right)$[/tex]

where [tex]$f(x)$[/tex]  is the function, [tex]$a$[/tex]  and [tex]$b$[/tex]  are the lower and upper limits of the integral, h is the width of each trapezoid, and $n$ is the number of trapezoids.

In our case, [tex]$f(x) = x^4$, $a = 0$, $b = 2$, $h = \frac{2 - 0}{2} = 1$[/tex] and n = 2.

Substituting these values into the formula, we get

[tex]$\int_{0}^{2} x^4 dx \approx \frac{1}{2} \left( 0 + 16 + 2 \left(x_1^4 + x_2^4 \right) \right)$[/tex]

where [tex]$x_1 = \frac{2 - 0}{2} = 1$ and $x_2 = \frac{2 + 0}{2} = 1$[/tex].

Substituting these values, we get

[tex]$\int_{0}^{2} x^4 dx \approx \frac{1}{2} \left( 0 + 16 + 2 \left(1^4 + 1^4 \right) \right) = 16$[/tex]So, the approximate value of [tex]$\int_{0}^{2} x^4 dx$[/tex] using the trapezoidal rule is 16.

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The mistake is in the second step, where the sign of the "4" is changed. The real solution is x = -5

Here we have the solution of the following equation:

4x + 6 = 2x - 4

First, they subtract 2x from both sides, then we will get:

4x + 6 - 2x = 2x - 4 - 2x

2x + 6 = -4

While notice that in the image, in this step they wrote:

2x + 6 = 4

They changed the sign of the 4, there is the mistake.

Now we can keep solving our equation:

2x + 6 = -4

2x = -4 - 6

2x = -10

x = -10/2

x = -5

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The values of x in the expression are -5, -9

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

|x + 7| = 2

Remove the absolute bracket

So, we have the following representation

x + 7 = -2 or x + 7 = 2

Evaluate the like terms

x = -9 or x = -5

Hence, the values are -5 and -9

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The solution is, A = $94193.421., would he save by paying off the loan 8 years early.

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.

here, we have,

We must determine the amount owed for the first 8 years.

The monthly compound interest formula is:

A = P(1+r/12)^12t - Xnt

given,

P = $55,000

r = 6.8%

t = 8

Xnt = $419.84

solving we get,

A = 94193.421

Hence, The solution is, A = $94193.421.

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A graph of the inequality "y is less than negative one fourth times x minus 3" is: D. The graph shows a dashed line passing through the point negative 12 comma 0 and 0 comma negative 3, with shading below the line.

In order to determine the ordered pair that is included in the solution to this linear inequality, we would first of all translate the word problem into an algebraic equation as follows;

y is less than negative one-fourth times x minus 3 ⇒ y < -x/4 - 3.

Next, we would use an online graphing calculator to graphically solve the linear inequality. Therefore, the required solution for the given linear inequality is the shaded area below the dashed line, which is given by the ordered pairs (−12, 0) and (0, -3).

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Answer:

it’s D dotted line with shading below

Step-by-step explanation:

i got it on my quiz

False. It is not reasonable to assume all groups have equal variances in this case.

The standard deviation is a measure of variability or spread of the data. If the standard deviations of different groups are significantly different, then it suggests that the spread of the data within each group is also different. When comparing multiple groups, equal variances are typically assumed so that a valid statistical test can be performed.

If the variances are unequal, then a different type of test, such as Welch's ANOVA, should be used. In this case, the large difference in the standard deviations of the four groups (4974, 6585, 3288, 5488) suggests that it is not reasonable to assume equal variances and a different test should be used.

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Answer: -81x

Step-by-step explanation:

3x times -3= -9x

-9x times -3= 27x

27x times -3= -81x

Answer: 32/243

Step-by-step Explanation: multiply 16/81 by 2/3                                              

(a) The reliability of this system is 0.42

(b) The reliability of this system is 0.8376

As per the given data that the probabilities that components A, B, C, and D function properly are 0.6, 0.7, 0.8, and 0.9 respectively.

Let the name of the components itself denote the event that the corresponding component works properly.

So P(A) = 0.6,  P(B) = 0.7, P(C) = 0.8, P(D) = 0.9.

We have to find the reliability of the given structure of the combination of the above-said components.

The diagrammatic representation(attached at the end of solution(a)) of the system is as follows.

The above system consists of components A and B connected in the sequel. So the reliability of the system is the probability that both A and B function properly.

This probability is given by P(A ∩ B) = P(A)P(B) as the events are independent. Substitute the probabilities in the above formula.

P(A ∩ B) = (0.6)(0.7)

P(A ∩ B) = 0.42

The diagrammatic representation(attached at the end of solution(b)) of the system is as follows.

In the above system, the components A and B and components C and D are connected the in the sequel. The entire system is connected in parallel with A, B and C, D. The reliability of this system is given by the probability.

P(A ∩ B) ∪ P(C ∩ D) = P(A ∩ B) + P(C ∩ D) - P(A ∩ B) ∩ P(C ∩ D)

= P(A)P(B) + P(C)P(D) - P(A)P(B)P(C)P(D)

= (0.6)(0.7) + (0.8)(0.9) - (0.6)(0.7)(0.8)(0.9)

= 0.42 + 0.72 - 0.3024

= 1.14 - 0.3024

= 0.8376

Therefore answer is 0.8376

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the events A and B as follows: A = { x | x < 72.5] and B = { x | x > 52.5].

From the question given above, the following data were obtained:

Angle θ = 71°

Hypothenus = 25

Adjacent = x

Thus, we can obtain the x component of the vector by using the cosine ratio as illustrated below:

Cos θ = Adjacent /Hypothenus

Cos 71 = x/25

Cross multiply for events

x = 25 × Cos 71

x = 25 × 0.3256

x = 8.1

Therefore, the x component of the vector is 8.1

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The probability that point p is closer to point o than to any of the vertices is 0.3737.

The probability that point p is closer to point o than to any of the vertices can be found by considering the ratio of the area of the equilateral triangle xyz to the area of its circumscribed circle. This ratio is [tex]$\frac{3\sqrt{3}}{4\pi}$[/tex]. We can use this ratio to calculate the probability that point $p$ is closer to point o than to any of the vertices:

[tex]$P(op < xp, yp, zp) = \frac{3\sqrt{3}}{4\pi} = 0.3737$[/tex]

Therefore, the probability that point $p$ is closer to point $o$ than to any of the vertices is 0.3737.

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The original price of the shirt was $68

We know that the price after a discount of the 20% and an added $3.80 in taxes is $58.20

Then we can write:

$58.20 = P*(1 - 0.2) + $3.80

Where the factor (1- 0.2) corresponds to a discount of 20%.

Solving that for P we will get:

P = ($58.20 - $3.80)/0.8

P = $68

That is the original price of the shirt.

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The volume of the given cylinder would be = 1.33mm³

The formula that can be used to calculate the volume of the cylinder = v = πr²h

Where,

π = 3.14

r= 0.5mm

h= 1.7 mm

V = 3.14×0.5²×1.7

V= 3.14× 0.25×1.7

V= 1.3345

V= 1.33mm³ ( to the nearest tenth)

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The side lengths of Trapezoid are 4 cm, 5 cm, 4 cm and 4 cm.

The perimeter of a form is the space surrounding its edge.

A closed path that covers, encircles, or outlines a one-dimensional length or a two-dimensional shape is called a perimeter.

Given:

Perimeter of Trapezoid = 17 cm

So, the Perimeter of Trapezoid

17 = Sum of parallel sides + Sum of non parallel side

17 = x + x+ 1 + 2√x + 2√x

17 = 2x + 4√x + 1

2x + 4√x = 16

2x + 4√x - 16 = 0

x + 2√x = 8

2√x = 8- x

Now, Squaring both side

4x = 64 + x² - 16x

x² - 16x -4x + 64 = 0

x² - 20x + 64 = 0

x² - 16x - 4x + 64 = 0

x(x-16) -4( x- 16)= 0

x= 16, 4

So, the sides lengths are

x= 4 cm, x+1 = 5 cm, 2√x = 4 cm, 2√x = 4 cm.

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Answer:

The answer should be should be 51.45°

a) The length of the third side of the triangle is; 6n - 9

b) The lengths of the three sides of the triangle when n = 5 are;

8, 7 and 22.

c) The lengths of the three sides of the triangle when n = 7 are;

12, 13 and 33.

The perimeter of a triangle is defined as the total length of the boundary of the triangle.

The parameters are;

Perimeter of triangle = 7n - 19

Length of side 1 = 2n - 2

Length of side 2 = 3n - 8

a) Thus;

Length of third side = 7n - 19 - (2n - 2 + 3n - 8)

= 7n - 19 - n + 10

= 6n - 9

b) When n = 5, the lengths of the three sides are;

Length of side 1 = 2(5) - 2 = 8

Length of side 2 = 3(5) - 8 = 7

Length of third side = 6(5) - 9 = 22

c) When n = 7, the lengths of the three sides are;

Length of side 1 = 2(7) - 2 = 12

Length of side 2 = 3(7) - 8 = 13

Length of third side = 6(7) - 9 = 33

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Answer: z=4

Step-by-step explanation:

5(z+6) - 10 = 10z

The first step is to distribute the 5 to the inside of the parenthesis. So,

5*z = 5z , 5*6 = 30 .

Now the equation has become 5z+30-10=10z. Next, the like terms can be reduced. So,

30-10= 20

The equation is: 5z+20=10z, 5z will be subtracted on both sides, canceling one side out.

5z-5z=0 , 10z-5z=5z

: 20=5z

The last step is to isolate the variable by dividing 5 on both sides.

5/5 = 0 , 20/5 = 4

Therefore, z=4

if you are using a simple-moving average and you want your forecast to ignore occasional noise in the data, Only A & C are of the following is true .

What is the meaning of the simple moving average?

The average of a range of prices is determined by the number of periods that fall within the range using simple moving averages.

                             A technical indicator called a simple moving average can help forecast whether a price trend—be it bullish or bearish—will continue or reverse.

What is not necessary for simpler forecasting techniques?

various types and amounts of data are needed for various forecasting methodologies. Less information is necessary for simpler techniques, such as the time series techniques outlined below.

                                       Complex models require a lot of data. It might be expensive to create precise forecasts.

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For a matrix A, the determined matrices are

a) Matrix E₁ is [tex]E₁ = \begin{bmatrix} \frac{1}{2} & \frac{ - 3}{2} \\ -1 & 3\end{bmatrix} [/tex]

and it's inverse is equals to zero.

b) Matrix E₂ and it's inverse E₂⁻¹ are

[tex]E₂ = \begin{bmatrix} 1 & -3 \\ - 5& - 1\end{bmatrix} [/tex]

[tex]E₂⁻¹ = \begin{bmatrix} \frac{ - 1}{16} & \frac{3}{16} \\ \\ \frac{5}{16} & \frac{1}{16} \end{bmatrix}[/tex]

We have a matrix 2×2, A such that

[tex]A= \begin{bmatrix} 1 & -3 \\ -1 & 3\end{bmatrix}[/tex]

and we have to determine the matrix E₁ with their inverses. Also, matrix E₁ is generated by multiplying the first row of matrix A by 1/2. As we see elements of first row of matrix are 1 and -3.

[tex]E₁ = \begin{bmatrix} 1 \times \frac{1}{2} & -3 \times \frac{1}{2} \\ -1 & 3\end{bmatrix}

[/tex]

[tex]E₁ = \begin{bmatrix} \frac{1}{2} & \frac{ - 3}{2} \\ -1 & 3\end{bmatrix} [/tex]

The inverse of A is A⁻¹only when AA⁻¹= A⁻¹A = I.

Steps to determine the inverse of a 2x2 matrix:

Determinant of E₁ = ad - bc , here a =1/2 , b = -3/2 , c= -1 and d = 3. So, |E₁ | = 1/2× 3 - (-3/2)×(-1) = 3/2 - 3/2 = 0 . Thus, Inverse of E₁⁻¹ = 0 .

b) When we multiplies the second row of A by -4 then the elementary matrix E₂ is form. [tex]E₂ = \begin{bmatrix} 1 & -3 \\ -1 + (- 4)& 3 + ( - 4)\end{bmatrix}[/tex]

[tex]E₂ = \begin{bmatrix} 1 & -3 \\ - 5& - 1\end{bmatrix}[/tex]

Now, we have determine the Inverse of E₂. The determinant of |E₂| = -1× 1 - (-5)×(-3) = -1 - 15 = -16

So, inverse of matrix is , E₂⁻¹

[tex]E₂⁻¹ = \frac{1}{16}\begin{bmatrix} - 1 & - ( -3) \\ - ( - 5)& 1\end{bmatrix}[/tex]

[tex]E₂⁻¹ = \frac{1}{16} \begin{bmatrix} -1 & 3 \\ 5& 1\end{bmatrix}[/tex]

[tex]E₂⁻¹ = \begin{bmatrix} \frac{ - 1}{16} & \frac{3}{16} \\\\\frac{5}{16} & \frac{1}{16} \end{bmatrix}[/tex]

Hence, we get the required inverse.

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Complete question:

Let A be a 2 x 2 matrix. Given the following descriptions, A = [ 1 -3 ; -1 3]

determine the following elementary matrices and their inverses.

a. The elementary matrix E1 multiplies the first row of A by 1/2 .

b. The elementary matrix E2 adding the second row of A by -4

The value of expression  (√19+√11i)(√19-√11i) is 30.

An expression is combination of variables, numbers and operators.

The given expression is (√19+√11i)(√19-√11i)

Square root of nineteen plus square root of eleven i times to the Square root of nineteen minus square root of eleven times i

We have an algebraic property

(a+b)(a-b)=a²-b²

Take a as √19 and b as √11 i

(√19+√11i)(√19-√11i)=(√19)²-(√11i)²

=19-11(-1)

=19+11

=30

Hence, the value of  (√19+√11i)(√19-√11i) is 30.

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The models that represent the sum 1.2 + 0.3 are

Option 3 and Option 5

For Option 3:

The first square represents 1.2, with all 10 columns shaded in one color. The next two columns represent 0.3, shaded in a different color. The sum of 1.2 and 0.3 is 1.5, which is represented by the combination of the two squares.

For Option 5:

On the number line, the first arrow represents 1.2, starting from 0 and ending 2 marks past 1, which is equal to 1.2. The second arrow represents 0.3, starting from 2 marks past 1 and ending at 5 marks past 1, which is equal to 1.5. The sum of 1.2 and 0.3 is 1.5, which is represented by the end point of the second arrow on the number line.

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The general solution is an equation that describes all possible solutions of the system. Finally, we can use the general solution to find the specific solutions to the system.

1. [tex]\frac{dy}{dt} = t - y, \frac{dx}{dt} = y - t$a.[/tex]

b. [tex]$y^2 - 2ty + x = c$[/tex]

2.[tex]$\frac{dy}{dt} = y - t^2, \frac{dx}{dt} = t^2 - x$a.[/tex]

b. [tex]$y^2 - 2tx + x^2 - t^2 = c$[/tex]

3. [tex]$\frac{dy}{dt} = ty - x, \frac{dx}{dt} = y - tx$[/tex]

a.

b. [tex]$y^2 - 2tx + x^2 = c$[/tex]

4.[tex]$\frac{dy}{dt} = xy, \frac{dx}{dt} = x$[/tex]

a.

b. [tex]$y = cxe^{x^2/2}$[/tex]

In each of the four problems, we are given a system of equations. To solve the system, we first need to draw a direction field and sketch a few trajectories. This gives us a visual representation of the system, which helps us understand it better. Next, we need to find the general solution of the system. To do this, we need to solve the system of equations algebraically. The general solution is an equation that describes all possible solutions of the system. Finally, we can use the general solution to find the specific solutions to the system.

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There are 1049 children went to movie.

Two or more expressions with an Equal sign is called as Equation.

Let the number of children are x and number of adults were y.

Tickets are $5.50 for children and $7 for adults.

1250 are the total number of people attended the movie.

x+y=1250

5.50x+7y=7979

x=1250-y

Now plug in x value in the second equation.

5.5(1250-y)+7y=7979

6875-5.5y=7979

Subtract 6875 on both sides

-5.5y=7979-6875

-5.5y=1104

y=-200.7

Now plug in y value

x+(-200.7)=1250

x=1049

Hence, there are 1049 children went to movie.

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a) The combined number of $3 carwashes for cars, c, and $5 carwashes for trucks or SUVs, t, needed can be represented by the equation y = c + t.

b) If only 5 cars are washed, the number of trucks or SUVs needed to be washed to raise the full $915 is 180.

An equation is a mathematical statement that shows that two mathematical expressions are equal or equivalent.

Equations use the equation symbol (=) to show the equality of expressions.

The total amount needed for the new equipment = $915

The charge for a car wash per car = $3

The charge for cash wash per truck or SUV = $5

Let the number of cars = c

Let the number of trucks or SUVs = t

c + t = y... Equation 1

3c + 5t = 915... Equation 2

If the number of cars, c = 5

From equation 2:

3c + 5t = 915

3(5) + 5t = 915

15 + 5t = 915

5t = 900

t = 180

Thus, from equation 2, we can determine that if cars are 5, the trucks or SUVs must be 180 to meet the target.

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The function A(x) = x(400 - x) gives the area (in square feet) of the fenced-in region as a function of the length of the side of the rectangle perpendicular to the building (x).

The area of a rectangular field enclosed by 400 feet of fence can be calculated using the formula Area = Length x Width. Since one side of the field is against a building, fencing is not required on that side. Thus, the length of the rectangle perpendicular to the building (x) is one of the sides of the rectangle. The other side is calculated by subtracting x from 400 feet. The area (in square feet) of the fenced-in region is then given by the function A(x) = x(400 - x).

To illustrate, if the length of the rectangle perpendicular to the building (x) is 200 feet, then the width is calculated by subtracting 200 from 400, giving a width of 200 feet. The area of the fenced-in region is then given by A(x) = 200 x (400 - 200) = 200 x 200 = 40,000 square feet.

Therefore, the function A(x) = x(400 - x) gives the area (in square feet) of the fenced-in region as a function of the length of the side of the rectangle perpendicular to the building (x).

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What does the problem asks for:

 --> mean and median

Let's calculate the mean:

 [tex]\text{Mean}=\dfrac{\text{sum of all the numbers}}{\text{number of numbers given}}[/tex]

   --> the sum of all the numbers = 21 + 40 + 48 + 26+ 32 + 31 = 198

   --> number of numbers given: 6

       ==> let's calculate

            [tex]\text{Mean}=\dfrac{198}{6} =33[/tex]

Let's calculate the median

 --> Median definition: middle number in an ordered sequence

   ==> let's order the sequence: 21, 26, 31, 32, 40, 48

      --> since there are an even number of numbers, there isn't an exact

            middle number

           --> thus, we must find the mean of the two numbers closest to the

               middle

                         [tex]\text{Median} = \dfrac{31+32}{2} =\dfrac{63}{2} =31.5[/tex]

Answer: 31.5

Step-by-step explanation:

21, 26, 31, 32, 40, 48

Median

31+32= 63

63÷2= 31.5

Mean

21+26+31+32+40+48=198

198÷6=33

To solve for 1/x(x-3) = 2/(x-3) + 3/x, we can simplify and rearrange the terms on the right side:

1/x(x-3) = 2/(x-3) + 3/x

Multiplying both sides by x(x-3):

1 = 2x + 3x(x-3)/x

Expanding the right side:

1 = 2x + 3x^2 - 9x/x

Cancelling x from the right side:

1 = 2x + 3x^2 - 9

Dividing both sides by x^2:

1/x^2 = 2/x + 3 - 9/x^2

Rearranging the terms:

1/x^2 - 9/x^2 = 2/x + 3

Isolating the x terms:

-8/x^2 = 2/x + 3

Multiplying both sides by x^2:

-8 = 2x + 3x^2

Expanding the right side:

-8 = 2x + 3x^2

Rearranging the terms:

3x^2 + 2x + 8 = 0

This is a quadratic equation and can be solved using the quadratic formula or factoring. The solutions are the values of x that make the equation equal to zero.

a. The family pays $336.

b. The family's payment when the dental charge is $4500 is  $2375.

c. The model that relates payment and charges are,

250 + 0.2x, x < 1500 and 250 + 0.5x, x > 1500.

A function that is piecewise defined by numerous subfunctions, each of which has a separate domain interval for which it is applicable.

Piecewise definition is more of an expression of the function than it is a property of the function.

a. Determining the family's payment if the dental charge is $680.

$650 falls in the category of the first $250 paid by the family and the rest 80% paid by the insurance company.

So, $(680 - 250) = $430.

Therefore, The family pays 250 + 0.2x (when x < 1500).

= 250 + 0.2(430).

= 250 + 86.

= $336.

b. The family's payment when the dental charge is $4500 is,

= 4500 - 250.

= $4250.

So, 250 + 0.5(4250)   (when x > 1500).

= 250 + 2125.

= $2375.

c. The model that relates payment and charges are,

250 + 0.2x. (x < 1500) and 250 + 0.5x (x > 1500).

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A linear transformation is a function from one vector space to another that preserves each vector space's underlying (linear) structure. A linear transformation can also be referred to as a linear operator or map.

The correct answers are A, D, and E.

A linear transformation from ℝ3 to ℝ3 is invertible if and only if it is bijective, meaning it is both one-to-one and onto.

The following linear transformations are invertible:

A. Identity transformation: The identity transformation maps every vector to itself and is one-to-one, so it is invertible.

D. Rotation about the x-axis: A rotation about the x-axis is a bijective linear transformation that maps each point in ℝ3 to another point in ℝ3 in a one-to-one manner, so it is invertible.

E. Dilation by a factor of 6: A dilation by a factor of 6 scales every vector by 6, and the scaling factor is nonzero, so this transformation is bijective and invertible.

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