4 friends share 2 granola bars each friend gets _____ of the granola bar ​

Answer:

? = 77°

Step-by-step explanation:

the 3 angles in a triangle sum to 180° , that is

? + 35° + 68° = 180°

? + 103° = 180° ( subtract 103° from both sides )

? = 77°

Answer:77

Step-by-step explanation:

180-103=77

The diver can go 44.11 feet deep without having to resort to use of the artificial light .

The intensity function L(x) satisfy the differential equation : dL/dx = -kL  ;

On solving the differential equation by Integrating both sides, we get:

⇒ln |L| = -kx + C ; where C is the constant of integration.

Taking exponential of both sides,

we have ;

⇒ |L| = [tex]e^{-kx+C}[/tex] = [tex]e^{C}\times e^{-kx}[/tex] ;

⇒ L = [tex]Ce^{-kx}[/tex] ;  where C is a constant of integration.

We know that when x = 19, the intensity (L) is halved. That means :

⇒ L(19) = (1/2)L(0)

Substituting L = [tex]Ce^{-kx}[/tex]  into this equation, we get:

⇒ [tex]Ce^{-k\times19} = (\frac{1}{2} )Ce^{0}[/tex]

Simplifying further , we get ;

⇒ [tex]e^{-19k} = \frac{1}{2}[/tex] ;

Taking natural log(ln) on both sides,

⇒ -19k = ln(1/2)

⇒ k = ln(2)/(19)    ....equation(1)

We also know that the diver must stop diving when L = (1/5)L(0).

Which means :

⇒ L(x) = (1/5)L(0)

Substituting L = [tex]Ce^{-kx}[/tex] into this equation, we get:

⇒ [tex]Cx^{-kx} = (\frac{1}{5}) C[/tex]  ;

⇒ [tex]e^{-kx}[/tex] = 1/5 ;

Taking natural log(ln) on both sides,

⇒ -kx = ln(1/5)

⇒ x = -ln(1/5)/k

Substituting k = ln(2)/(19)  from equation(1) , we get:

⇒ x = [19×ln(5)]/ln(2) .

⇒ x = 44.11 feet  .

Therefore, the diver can go approximately 44.11 feet deep .

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Answer:

Step-by-step explanation:

translation:  (x+4, y-6)

original    image

A(5,4)     A'(9,-2)

B(-9,6)    B'(-5,0)

C(-6,2)   C'(-2,-4)

The mass and volume of two metals is 93.98g, 165.56g and 39.37cm3,81.89cm3.

Suppose that a finite amount of substance is there having its properties as:

mass of substance = m kg

density of substance = d kg/m³

volume of that substance = v m³

Then, they are related as:

[tex]d = \dfrac{m}{v}[/tex]

Given that;

Mass of 100cm3 of metal = 254g

Mass of b volume with 37cm3 and 64cm3 is 100g and 208g.

Now,

m=pv

254=p*100

p=2.54g/cm3

m=2.54v

Part A;

V1 =37cm3

V2=64cm3

m1= 2.54 x 37= 93.98g

m2= 2.54 x 64= 165.56g

Part B:

V1=m1/2.54 = 100/2.54 = 39.37cm3

V2=208/2.54 = 81.89cm3

Therefore, the volumes will be 39.37cm3 and 81.89cm3

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The probability of drawing the ace of spades is 1/52 for the first card and 1/51 for the second card. P(B|As) ≈ 0.1176.

The likelihood of drawing the trump card on the primary draw is for sure 1/52. Be that as it may, when the principal card is drawn, there are presently 51 cards remaining, and only one of them is the trump card. In this way, the likelihood of drawing the trump card on the subsequent draw, considering that the principal card isn't the trump card, is 1/51.

With regards to the issue, the occasion As is characterized as the trump card being picked, whether or not it is the first or second card. Since the issue determines that the two cards are drawn without substitution, there are two cases to consider: either the trump card is drawn first, with likelihood 1/52, or it is drawn second, considering that the principal card isn't the trump card, with likelihood (51/52)*(1/51) = 1/52. Accordingly, the likelihood of occasion Similar to the amount of these two probabilities, which is 1/52 + 1/52 = 1/26.

Utilizing Bayes' recipe, we have:

P(B|As) = P(As|B) * P(B)/P(As)

We previously determined P(As) to be 1/26. To ascertain P(As|B), note that in the event that the two cards are aces, the trump card should be one of them, so P(As|B) = 1. At last, to compute P(B), note that there are (4 nCr 2) = 6 methods for picking two aces out of the four in the deck, and there are (52 nCr 2) = 1326 methods for picking two cards out of the deck without substitution, so P(B) = 6/1326 = 1/221.

Subbing these qualities into Bayes' recipe, we get:

P(B|As) = 1 * (1/221)/(1/26) = 26/221 ≈ 0.1176

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Nine cuboidal boxes and eight cuboidal boxes can each hold 153 and 136 toy cars, respectively.

How is the average calculated?

The ratio of the sum of all provided observations to the total number of observations is the arithmetic mean, which is another name for the average formula. So, any sample of data may have its arithmetic mean determined using the average formula.

What does arithmetic mean?

The mean or arithmetic average are terms that are frequently used to refer to the arithmetic mean. It is determined by adding up all the numbers in a given data collection, then dividing that total by the number of items in the data set. For uniformly distributed integers, the middle number is the arithmetic mean (AM).

Given: There are two storage options for 288 toy cars: 16 and 18 cuboidal boxes.

supposing there were 288 toy cars stored in 16 toy cars.

There are 18 toy cars in a box of 288 (288 / 16) toy cars.

if 288 toy cars were housed in 18 toy cars.

There are 16 toy cars in a box that holds 288 toy cars.

Average: 18 + 16 / 2 = 34 / 2 = 17 toy cars.

Nine boxes of toy cars include 153 toy cars (9 * 17).

eight boxes of toy cars include 136 toy cars (8 * 17)

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Answer:

Step-by-step explanation:

The expression for the team time is:

[tex]11\frac{3}{5} =3\frac{3}{10} +2\frac{4}{5} +2\frac{1}{10} +X\\[/tex]

Where X= Cindy's Time

Isolating X from the equation:

[tex]X=11\frac{3}{5}-(3\frac{3}{10} +2\frac{4}{5}+2\frac{1}{10})\\\\ X=11\frac{3}{5}-(8\frac{1}{5})\\ \\ X=3\frac{2}{5}[/tex]

Cindy's time was [tex]3\frac{2}{5}[/tex] minutes.

The algorithm to write the binary representation of the positive integer "64" is explained below and the binary representation of 64 is 1000000 .

Step(i) : 64 divided by 2 gives a quotient of 32 with a remainder of 0.

Step(ii) : 32 divided by 2 gives a quotient of 16 with a remainder of 0.

Step(iii) : 16 divided by 2 gives a quotient of 8 with a remainder of 0.

Step(iv) : 8 divided by 2 gives a quotient of 4 with a remainder of 0.

Step(v) : 4 divided by 2 gives a quotient of 2 with a remainder of 0.

Step(vi) : 2 divided by 2 gives a quotient of 1 with a remainder of 0.

Step (vii) : 1 divided by 2 gives a quotient of 0 with a remainder of 1.

So , we observe that  the remainders, read from bottom to top, are 1000000.

Therefore, the binary representation of "64" is 1000000 in binary.

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The given question is incomplete , the complete question is

Find the binary representation of the positive integer "64" by working through an algorithm by hand .

The number of marshmallows and graham crackers will Janna use to make 13 s'mores is 26 and 36, the correct option is 36.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

We are given that;

S'mores Marshmallows Graham Crackers 4 8 12 13

Now,

8/4 = 2 Marshmallows for each S'mores,

12/4 = 3 Graham Crackers for each S'mores.

For, 13 S'mores,

2*13=26

3*13=39

Therefore, by algebra the answer will be 26 and 39.

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a) The result of the limit as x tends to +7/4 as required is; -∞.

b). The result of the limit as x tends to -7/4 as required is; -21/8.

Since the limit expression which is to be evaluated is; (-21x / 7 -4x).

a). Therefore, when x tends to; +7/4; we have;

= (-21 • 7/4) / ( 7 - (4• 7/4) )

= (-147/4 ) / 0

= -∞

b) When x tends to; -7/4 ; we have that;

= (-21 • -7/4) / ( 7 - (4 • -7/4) )

= ( -147/4 ) / 14

= -21 / 8.

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1) Multiply both sides of the equation by 2:

2(A)=2[1/2h(b+B)

2A=2/2h(b+B)

2A=1h(b+B)

2A=h(b+B)

2) Divide both sides of the equation by (b+B):

(2A)/(b+B)=[h(b+B)]/(b+B)

2A/(b+B)=h

h=2A/(b+B)

Answer: [tex]h=2\frac{A}{(b+B)}[/tex]

hope this helped!!

Answer:

[tex]A)81x^{-12}y^{20}\\\\B)5\\\\C) x^\frac{2}{3}[/tex]

Step-by-step explanation:

[tex]A) (3x^{-3}y^5)^4=(3)^4 \,\cdot\, (x^{-3})^4\,\cdot\,(y^5)^4=81x^{-12}y^{20}\\\\B) 5x^0=5(1)=5\\\\C) \frac{x^{\frac{5}{6} }}{x^{\frac{1}{6} }} =x^{\frac{5}{6} -\frac{1}{6} }=x^{\frac{5-1}{6} }=x^{\frac{4}{6} }=x^\frac{2}{3}[/tex]

Answer:

do you have an image of the question?

Answer:

6 Shirts

Step-by-step explanation:

if you buy 1 pair of jeans you can buy 6 pairs of shirts since 6 x 11 = 66

66 + 40 = 106

Brayden's total earning by walking lap around a track is $35, if he walks 2+1/2 = 5/2 laps per day for 7 days.

Earnings are the net benefits of a corporation's operation. Earnings is also the amount on which corporate tax is due. For an analysis of specific aspects of corporate operations several more specific terms are used as EBIT and EBITDA. Many alternative terms for earnings are in common use, such as income and profit.

here, we have,

Per day walking of brayden = 5/2 laps

7 days walking of brayden

= 5/2 × 7

= 35/2 laps

As per the question statement he earns $1 per lap if he walk less than 16 laps and $2 per lap if hi walks 16 or more lap.

Since the total walking of brayden is more than 16 laps, so he will earn at the rate of $2 per lap.

So total earning will be,

= 2× 35/2

= $35

Hence, Brayden's total earning by walking lap around a track is $35, if he walks 2+1/2 = 5/2 laps per day for 7 days.

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Answer:

14.5 laps



There is an even number of data points. So the median is the mean of the two middle numbers.

The parametric equation of a line segment joining two points (x1, y1) and (x2, y2) is given by [tex]x = x1 + t(x2 - x1) and y = y1 + t(y2 - y1)[/tex].

The parametric equation of a line segment joining two points (x1, y1) and (x2, y2) is given by [tex]x = x1 + t(x2 - x1) and y = y1 + t(y2 - y1)[/tex]. Here, t is a real variable and its value lies between 0 and 1. This is because when t = 0, the equation gives the coordinates of the first point (x1, y1) and when t = 1, the equation gives the coordinates of the second point (x2, y2). For example, if the two points are (3, 4) and (7, 8), then the parametric equation of the line segment joining them can be obtained as [tex]x = 3 + t(7 - 3) = 3 + 4t and y = 4 + t(8 - 4) = 4 + 4t[/tex]. This equation can be used to easily find the coordinates of any point on the line segment joining the two points.

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If events A and B are independent, then they remain independent when conditioned on event C.

To prove that if events A and B are independent, then they are also independent conditioned on event C, we can use the definition of conditional probability.

Recall that two events A and B are independent if and only if P(A and B) = P(A) * P(B). And the conditional probability of A given B can be calculated as P(A | B) = P(A and B) / P(B).

It follows that P(A and B) = P(A) * P(B) if occurrences A and B are independent. Substituting this into the expression for conditional probability, we get:

[tex]P(A | B) = P(A) * P(B) / P(B) = P(A)[/tex]

Similarly, we can prove that P(B | A) = P(B). This means that the probability of A and B given event C are equal to the probabilities of A and B, respectively, given that A and B are independent. Hence, events A and B are also independent conditioned on event C

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The point (2,5) satisfies only one of the two inequalities in the system, it is not included in the solution area for the system.

To see if the point (2,5) is in the system's solution area, we need to see if it satisfies both inequalities in the system:

4x + 2y ≤ 16

4(2) + 2(5) ≤ 16

8 + 10 ≤ 16

18 ≤ 16

Because this inequality is false, point (2,5) fails to satisfy the first inequality.

x + y ≥ 4

2 + 5 ≥ 4

7 ≥ 4

Because this inequality holds, the point (2,5) satisfies the second inequality.

Because point (2,5) satisfies only one of the system's two inequalities, it is not included in the system's solution area. Landon cannot, therefore, spend $16 on two sandwiches and five bags of chips to feed his four teammates.

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When interpreting pairwise confidence intervals to ANOVA test with either Fisher, Tukey or Bonferroni methods, we determine that there are differences between two treatments whenever  F. the interval does NOT include zero.

Using the given techniques, the pairwise confidence intervals and test can be easily generated. These can state whether the disparity among the two groups' means is statistically significant. To compare the means of two groups, pairwise confidence intervals using Fisher, Tukey, or Bonferroni procedures are created as a result of ANOVA testing.

The disparity is statistically significant at the selected level of confidence if the confidence interval for the difference among the means of the two groups does not include zero. This suggests that the two therapies being compared significantly differ from one another. If the confidence interval includes zero, the variation is not statistically significant, and we should not claim that there is a purposeful disparity among the two treatments.

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The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.

Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,

[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]

Let 'x' be the number of miles and 'y' be the tread thickness. Then the two points are (15, 7.2) and (35, 4.8). Then the equation is given as,

(y - 7.2) = [(7.2 - 4.8) / (15 - 35)] (x - 15)

y - 7.2 = -0.12(x - 15)

y - 7.2 = - 0.12x + 1.8

y = - 0.12x + 9

The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.

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According to the information, he gets 6 pieces in total after cutting his boards.

To find the number of pieces Nathan gets, we must divide the total length of the boards by the length of the pieces. Below is the procedure:

According to the above, from one table he gets 4 2-feet pieces, from the other he gets 2 4-feet pieces. Additionally, in total he would have 6 pieces.

Note: This question is incomplete because there is some information missing. Here is the complete information:

Nathan has two 8-foot boards. He cuts one board into 2-foot pieces, and he cuts the other board into 4-foot pieces.

How many pieces does he get?

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A box plot is a special type of diagram that shows the quartiles in a box and the line extending from the lowest to the highest value.

What is Box plot?

Box plots, also known as box-and-whisker plots or box-whisker plots, provide a clear pictorial representation of the distribution of the data. They also demonstrate how remote the extreme numbers are from the majority of the data. Five values are used to create a box plot: the minimum value, the first quartile, the median, the third quartile, and the maximum value. These numbers are used to gauge how closely other data points adhere to them. Use a horizontal or vertical number line along with a rectangular box to create a box plot. The axis' ends are identified by the smallest and greatest data values. The first quartile designates one end of the box, while the third quartile designates the opposite end.

The box plot distribution will demonstrate how skewed, tightly packed, and symmetrical the data are.

In the box and whisker plot:

The box's upper and lower quartiles serve as its ends, allowing it to cross the interquartile range. The vertical line inside the box denotes the median, and the two lines outside the box serve as its whiskers, extending to the highest and lowest observations.

Hence, A box plot is a special type of diagram that shows the quartiles in a box and the line extending from the lowest to the highest value.

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(a) Revolving about the x-axis, the volume of the solid is 3Pi

(b) Revolving about the y-axis, the volume of the solid is 18 Pi

(c) Revolving about the line x=3, the volume of the solid is 0

(d) Revolving about the line x=6, the volume of the solid will be 32/3 Pi

The region bounded by y=x, y=0, and x=3 is a triangle in the first quadrant.

To find the volume of the solid generated by revolving this region about an axis, we can use the disk or washer method.

(a) Revolving about the x-axis:

Each cross-section of the solid perpendicular to the x-axis is a disk with radius x and thickness dx.

The volume of each disk is [tex]\pi x^2 dx[/tex].

The limits of integration are 0 and 3, the x-coordinates of the intersection points of the two curves.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi x^2 dx\\\\\&= \left[\frac{\pi}{3}x^3\right]_0^3\\\\\&= \frac{9\pi}{3}\\\\\&= 3\pi.[/tex]

(b) Revolving about the y-axis:

Each cross-section of the solid perpendicular to the y-axis is a washer with outer radius 3 and inner radius x, and thickness dx.

The volume of each washer is [tex]\pi(3^2-x^2)dx.[/tex]

The limits of integration are 0 and 3.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi(3^2-x^2)dx\\\\\&= \left[\pi(9x-\frac{x^3}{3})\right]_0^3\\\\\&= \pi(27-\frac{27}{3})\\\\\&= 18\pi.[/tex]

(c) Revolving about the line x=3:

Each cross-section of the solid perpendicular to the line x=3 is a washer with outer radius 3-x and inner radius 3-x-|x|, and thickness dx.

The volume of each washer is[tex]\pi((3-x)^2-(3-x-|x|)^2)dx[/tex].

The limits of integration are -3 and 3.

Hence, the volume of the solid is

[tex]V &= \int_{-3}^3 \pi((3-x)^2-(3-x-|x|)^2)dx\\\\\&= \left[8\pi\int_0^3 (3-x-|x|)dx\right]\\\\\&= 8\pi\int_0^3 (3-2x)dx\\\\\&= 8\pi\left[3x-x^2\right]_0^3\\\\\&= 8\pi(9-9)\\\\\&= 0.\\\\[/tex]

(d) Revolving about the line x=6:

Each cross-section of the solid perpendicular to the line x=6 is a washer with outer radius |3-x| and inner radius |x-6|, and thickness dx.

The volume of each washer is[tex]\pi(|3-x|^2-|x-6|^2)dx.[/tex]

The limits of integration are 0 and 3.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi(|3-x|^2-|x-6|^2)dx\\\\\&= \left[\frac{32\pi}{3}\right]\\\\\&= \frac{32}{3}\pi.[/tex]

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answer :

P(t) = 3000 / (1 + 29*e^(-0.637t))

steps

logistic function or logistic curve is a common S-shape curve (sigmoid curve) with equation:

P(t) = K / (1 + A*e^(-rt))

1. There is a type of animal or plant

species that started with 100

individuals.

2. The environment can support up to

3000 individuals, which is called the

"carrying capacity".

3. After 4 years, the population has

grown to 600 individuals.

4. We need to find a "logistic function"

that can show how the population

changes over time.

A species with an initial population of 100 is growing in an environment where the carrying capacity is 3000.

After 4 years, the population is up to 600.

We want to find the logistic function that models this population as a function of time.

The logistic function is a model of population growth that takes into account the carrying capacity of the environment. It is given by the formula:

P(t) = K / (1 + A e^(-r(t-t0)))

Where:

P(t) is the population size at time t

K is the carrying capacity of the environment

r is the growth rate of the population

t0 is the time at which the population starts to grow

A is a constant that determines the initial population size

To find the logistic function that models the population described in the problem, we need to determine the values of K, r, t0, and A.

We know that the initial population size is 100, so A = 100.

After 4 years, the population is up to 600, so P(4) = 600. We can use this information to solve for r:

600 = K / (1 + A e^(-r(4-t0)))

600 = K / (1 + 100 e^(-4r))

600(1 + 100 e^(-4r)) = K

K = 600 + 60000 e^(-4r)

We also know that the carrying capacity is 3000, so K = 3000.

3000 = 600 + 60000 e^(-4r)

2400 = 60000 e^(-4r)

0.04 = e^(-4r)

ln(0.04) = -4r

r = ln(0.04) / -4

r ≈ 0.693

Now we can use the value of r to solve for t0. We know that the initial population size is 100, so we can use this to find the value of A at t0:

100 = K / (1 + A)

100 = 3000 / (1 + A)

1 + A = 30

A = 29

We can now use this value of A to solve for t0:

100 = 3000 / (1 + 29 e^(-r(t0)))

1 + 29 e^(-r(t0)) = 30

e^(-r(t0)) = 1/29

ln(1/29) = -r(t0)

t0 = ln(1/29) / -r

t0 ≈ 2.48

Now we have all the values we need to write the logistic function:

P(t) = 3000 / (1 + 29 e^(-0.693(t-2.48)))

This is the logistic function that models the population as a function of time. It predicts that the population will grow exponentially at first, but then level off as it approaches the carrying capacity of the environment.

We can model the population growth of the species using the logistic equation:

dP/dt = rP(1 - P/K)

where P is the population size, t is time, r is the growth rate, and K is the carrying capacity.

To find the logistic function that models this population as a function of time, we need to determine the values of r and K. We can use the information given in the problem to solve for these values:

The initial population size is 100, so P(0) = 100.

The carrying capacity is 3000.

After 4 years, the population size is 600, so P(4) = 600.

Using these values, we can solve for r and K:

P(t) = K / (1 + A*e^(-rt))

where A is a constant determined by the initial population size, and e is the base of the natural logarithm.

From the initial population size, we know that:

A = (K - P(0)) / P(0) = (3000 - 100) / 100 = 29

We can use the population size after 4 years to solve for r and K:

600 = K / (1 + 29*e^(-4r))

Multiplying both sides by the denominator:

600 + 29600e^(-4r) = K

Substituting K = 3000:

3000 = 600 + 29600e^(-4r)

Dividing both sides by 600:

5 = 29*e^(-4r)

Taking the natural logarithm of both sides:

ln(5) = ln(29) - 4r

Solving for r:

r = (ln(29) - ln(5)) / 4

r ≈ 0.637

Now that we have r and K, we can plug them into the logistic equation to get the logistic function:

P(t) = 3000 / (1 + 29*e^(-0.637t))

This function models the population of the species as a function of time, where P(t) is the population size at time t.

ChatGPT

The ratio of students to classrooms  is 26:1

A ratio is defined as the comparison of two or more numbers indicating their sizes in relation to each other.

It  shows how many times one number contains another.

For example, if there are 20 oranges and 15 lemons in a bowl of fruit, then the ratio of oranges to lemons is 20 to 15.

From the information given, we have that;

234 students in 9 different classrooms

The ratio of students to classrooms would be;

= 234: 9

Find the least common factor

= 26: 1

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The value of angle x based on the information will be 131°.

Alternate interior angles are the angles formed when a transversal intersects two coplanar lines.

The two marked angles are on opposite sides of the transversal, so they are called "alternate" angles. They are both between the parallel lines, so they are called "interior" angles.

In this geometry, alternate interior angles are congruent. They have the same measure. x° = 131°.

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The minimum value of the function P over the feasible region is -3, which occurs at the point (1,2).

The expression that established the relationship between the dependent variable and independent variable is referred to as a function. In the function as the value of the independent variable varies the value of the dependent variable also varies.

To find the minimum value of the function P = x - 2y over the feasible region determined by the vertices (1,2), (7,3), and (5,1), we need to evaluate the function at each of these points and find the lowest value.

P(1,2) = 1 - 2(2) = -3

P(7,3) = 7 - 2(3) = 1

P(5,1) = 5 - 2(1) = 3

Therefore, the minimum value of the function P over the feasible region is -3, which occurs at the point (1,2).

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After the final audit of sales results, the West Region needs to have their sales updated for Q3 to $15,740 thus is obtained by clicking D3. Option A is correct answer.

A row and a column are joined to create a cell. It is, in other words, where a row and a column meet. In general, rows are labelled with numbers like 1, 2, and 3, whereas columns are labelled with letters like A, B, and C. Each cell has a unique name or cell address based on its column and row.

A cell range in Microsoft Excel is a collection of cells. It may appear in a formula.

After the final audit of sales statistics, the West Region's sales for Q3 must be revised to $15,740, so the cell that would be clicked is D3.

Hence, after the final audit of sales results, the West Region needs to have their sales updated for Q3 to $15,740 thus is obtained by clicking D3.

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The probability density function is of E(Y1) and E(Y2) is 1/6 and variance is 1/18.

To find the expected value E(Y1+Y2), we can use the linearity of expectation and the fact that the expected value of a constant is that constant:

E(Y1+Y2) = E(Y1) + E(Y2)

To find E(Y1) and E(Y2), we need to integrate y1 and y2 respectively over their joint probability density function:

E(Y1) = ∫∫ y1 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y1) 2y1 dy2 dy1

= ∫0^1 2y1(1-y1)/2 dy1

= ∫0^1 y1-y1^2 dy1

= [y1^2/2 - y1^3/3] from 0 to 1

= 1/6

Similarly,

E(Y2) = ∫∫ y2 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y2) 2y2 dy1 dy2

= ∫0^1 2y2(1-y2)/2 dy2

= ∫0^1 y2-y2^2 dy2

= [y2^2/2 - y2^3/3] from 0 to 1

= 1/6

Therefore, E(Y1+Y2) = E(Y1) + E(Y2) = 1/6 + 1/6 = 1/3.

To find the variance V(Y1+Y2), we can use the formula:

V(Y1+Y2) = E((Y1+Y2)^2) - [E(Y1+Y2)]^2

To find E((Y1+Y2)^2), we need to integrate (y1+y2)^2 over their joint probability density function:

E((Y1+Y2)^2) = ∫∫ (y1+y2)^2 f(y1,y2) dy1 dy2

= ∫0^1 ∫0^(1-y1) (y1+y2)^2 2 dy2 dy1

= ∫0^1 [(2/3)y1^3 + y1^2 + (1/3)y1] dy1

= 5/18

Therefore, V(Y1+Y2) = E((Y1+Y2)^2) - [E(Y1+Y2)]^2 = 5/18 - (1/3)^2 = 5/18 - 1/9 = 1/18

The variance is 1/18.

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The cosine of the angle S is 4/5

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

The triangle

On the triangle and with the use of law of cosines, we have

cos(S) = Adjacent/Hypotenuse

This means that

cos(S) = 8/10

Simplify

cos(S) = 4/5

Hence, the solution is 4/5

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