What is the shortest network (of straight lines) connecting a given set of three points? If the three points are collinear, the answer is obvious. If not, then the answer has two parts: Let A,B and C be the three points. Then
(I) If some angle of the triangle ABC is greater than or equal to120 , the shortest network is the path along the two shortest sides of the triangle.
(II) If all angles of triangle ABC are less than 120 degrees, then construct the point S such that the three angles at S made by the lines AS ,BS and CS are equal to one another. The shortest network is the "Y-shaped" structure made up of the segments AS ,BS and CS .
Where is the point S(in relation to the vertices A ,B and C ) in the case where ABC is an equilateral triangle. What is the total length of the network in this case? How much shorter is it (percentage-wise) than the path made up of the shortest two sides?

78 ft is the value of x in Triangles.

What is a triangle and what are its types?

Triangles are three-sided shapes. The various triangle varieties go under various names. The size of the angles and side lengths determine what kind of triangle it is (corners).

                         According on how long their sides are, triangles can be classified as equilateral, isosceles, or scalene.

Sin45° = x/110

  1/√2 = x/110

  √2 x  = 110

       x = 110/√2

       x = 77.78

        x = 78 ft

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Answer:

The description that matches the translation of the equation y = |x - 5| + 2 is "Moves 5 units to the right and 2 units up".

This is because the equation involves the absolute value of the quantity (x - 5), which means that the expression inside the absolute value bars will always be positive or zero. This means that the graph of the equation will be a "V" shape, centered at x = 5, and opening upwards. The "+2" outside the absolute value bars indicates that the entire graph will be shifted upward by 2 units.

Therefore, the graph of the equation y = |x - 5| + 2 will be obtained by taking the graph of y = |x|, shifting it 5 units to the right to obtain y = |x - 5|, and then shifting it 2 units up to obtain y = |x - 5| + 2.

Step-by-step explanation:

pa brainliest pls

The solution in terms of a and b is x=(-a/6)+(b/6) and y=(2a/3)-(b/3).

To find the solution to the systems using the inverse of the matrix, we can use the formula:
[xy]=[A^-1][b]
where [A] is the coefficient matrix, [b] is the constant matrix, and [A^-1] is the inverse of the coefficient matrix.

For system (a), we have:
[A]=[21​41​]
[b]=[12​]
To find the inverse of [A], we can use the formula:
[A^-1]=1/(ad-bc)[d -b​-c a​]
where a=2, b=1, c=4, and d=1.
So, [A^-1]=1/(-6)[1 -1​-4 2​]=[-1/6 1/6​2/3 -1/3​]
Now, we can find the solution by multiplying [A^-1] and [b]:
[xy]=[-1/6 1/6​2/3 -1/3​][12​]=[-1/6+1/6 2/3-1/3​]=[-1/3​]
So, the solution to system (a) is x=-1/3 and y=1/3.

For system (b), we have:
[A]=[21​41​]
[b]=[20​]
We can use the same inverse of [A] that we found for system (a) and multiply it by [b] to find the solution:
[xy]=[-1/6 1/6​2/3 -1/3​][20​]=[-2/6+0 4/3-0​]=[-1/3​2​]
So, the solution to system (b) is x=-1/3 and y=2.

For the system [21​41​][xy​]=[ab​], we can use the same inverse of [A] and multiply it by [ab]:
[xy]=[-1/6 1/6​2/3 -1/3​][ab​]=[-a/6+b/6 2a/3-b/3​]
So, the solution in terms of a and b is x=(-a/6)+(b/6) and y=(2a/3)-(b/3).

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Answer:

56

Step-by-step explanation:

Twenty Six, perhaps?

The value of ST in the given triangle is is √73.

A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.

Here, we have

Given: In a triangle

TR = 8

SR = 3

ST =?

We apply here Pythagoras theorem and we get

SR² + TR² = ST²

3² + 8² = ST²

9 + 64 = ST²

73 = ST²

ST = √73

Hence, the value of ST is √73.

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The real zeros of the given polynomial P(x) = x³ - 9x² + 20x - 12 are 3 and 2. The real zeros of the given polynomial P(x) = x³ - 9x² + 20x - 12 can be found by factoring the polynomial and setting each factor equal to zero.

Step 1: Factor the polynomial
P(x) = x³ - 9x² + 20x - 12
= (x - 3)(x - 2)(x - 2)

Step 2: Set each factor equal to zero and solve for x
x - 3 = 0 => x = 3
x - 2 = 0 => x = 2
x - 2 = 0 => x = 2

Step 3: The real zeros of the polynomial are the values of x that make each factor equal to zero.
So, the real zeros of the polynomial are x = 3 and x = 2.

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Using a graphing calculator or algebraic methods, we can find that x ≈ 71.6. Therefore, the model projects the civilian labor force to be 152 million in the year 1950 + 71.6 = 2021.6, or approximately in the year 2022.

According to the given model, the size of the civilian labor force (in millions) for selected years from 1950 and projected to 2050 can be modeled by y = - 0.0001 x^3 + 0.0058 x^2 + 1.43x + 56.7, where x is the number of years after 1950.

To find the projected civilian labor force in 2013, we need to plug in the value of x as 2013-1950 = 63 into the given model:

y = - 0.0001 (63)^3 + 0.0058 (63)^2 + 1.43(63) + 56.7
y = - 0.0001 (250047) + 0.0058 (3969) + 90.09 + 56.7
y = - 25.0047 + 23.0002 + 90.09 + 56.7
y = 144.7855

Therefore, the model projects the civilian labor force to be approximately 144.7855 million in 2013.

To find the year when the model projects the civilian labor force to be 152 million, we need to set y = 152 and solve for x:

152 = - 0.0001 x^3 + 0.0058 x^2 + 1.43x + 56.7
0 = - 0.0001 x^3 + 0.0058 x^2 + 1.43x - 95.3

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The height of the cone is approximately 13.5 feet.

The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.

In this problem, we are given the volume of the cone, which is 1959.36 cubic feet, and the radius, which is 12 feet. We need to find the height, h.

We can start by plugging in the given values into the formula for volume:

1959.36 = (1/3)π(12)²h

Simplifying, we get:

1959.36 = 144πh

To solve for h, we need to isolate it on one side of the equation. We can do this by dividing both sides by 144π:

h = 1959.36 / (144π)

Using a calculator to evaluate this expression, we get:

h ≈ 13.5

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The answer to the question "(1) 42mod11" is 9.

The answer to the question "Modular Arithmetic: Please solve the following questions. (1) 42mod11" is:

The concept of "Modular Arithmetic" is a system of arithmetic that involves only integers. In modular arithmetic, numbers "wrap around" after they reach a certain value called the modulus.

To solve the given question, we can use the following steps:

Step 1: Find the modulus of the given number. In this case, the modulus is 11.

Step 2: Divide the given number by the modulus. In this case, 42/11 = 3 with a remainder of 9.

Step 3: The remainder is the answer to the question. So, 42mod11 = 9.

Therefore, the answer to the question "(1) 42mod11" is 9.

I hope this answer helps you! If you have any further questions, feel free to ask.

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Answer:

1 3/4

Step-by-step explanation:

To turn 7/4 into a mixed number, you have to look at the numerator and the denominator. 4 can go into 7 once (with 3 left over), so it would be 1 and 3/4

The scale factor from triangle DEF to triangle D”E”F” is the ratio of the corresponding sides in the two triangles.

The coordinates of F” can be calculated using the scale factor.

The value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.

Scale factor is a number which is used to describe the relationship between two objects or two measurements. It is used to compare two objects of different sizes in order to calculate the ratio of their size. It is also used to express a proportional relationship between two different measurements, such as their lengths, widths, heights or angles.

The corresponding sides are DE and D”E”, and EF and E”F”. We can calculate the ratio of the two sides by dividing the lengths of the two corresponding sides. For example, the ratio of DE to D”E” would be 12/16. We can do this for all three corresponding sides, and then take the average of the three ratios to get the scale factor from triangle DEF to triangle D”E”F”.

Since we know that the coordinates of F are (12,5), we can multiply the x-coordinate (12) by the scale factor and the y-coordinate (5) by the scale factor to get the coordinates of F”.

The angle D” can be used to calculate the values of cos(D”), sin(D”), and tan(D”). Since we know the angle D”, we can use the trigonometric functions to calculate the values of the trigonometric functions for that angle. For example, if the angle D” is 45 degrees, then the value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.

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1.  Based on the given stock-out level, the corresponding Z-value is 1.96. 2. The safety stock level is 12. 3. The pharmacy store should place orders when the inventory level drops to 332 bottles. 4. The pharmacy store changing the stock-out probability to 4%, should place orders when the inventory level drops to 331 bottles.

1. The corresponding Z-value for a stock-out level of 2.5% is 1.96. This can be found by looking at a standard normal table or using a Z-value calculator.

2. The safety stock level can be calculated using the formula:

Safety stock = Z-value * standard deviation of lead time demand.

In this case, the standard deviation of lead time demand is the square root of the variance, or 6 bottles. So the safety stock level is:

= 1.96 * 6 = 11.76

which should be rounded up to the next whole number, or 12 bottles.

3. The pharmacy store should place orders when the inventory level drops to the reorder point, which is calculated using the formula:

Reorder point = Lead time demand + Safety stock.

In this case, the lead time demand is 16 days * 20 bottles/day = 320 bottles.

So the reorder point is 320 + 12 = 332 bottles.

Therefore, the pharmacy store should place orders when the inventory level drops to 332 bottles.

4. If the pharmacy store wants to change the stock-out probability to 4%, the corresponding Z-value is 1.75. Using the same formula as before, the safety stock level is:

= 1.75 * 6 = 10.5, which should be rounded up to the next whole number, or 11 bottles.

The reorder point is then 320 + 11 = 331 bottles. Therefore, the pharmacy store should place orders when the inventory level drops to 331 bottles.

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If S1 and S2 are two sets of vectors that each span V, then S1 ∪ S2 also spans V.

Let S1 and S2 be two sets of vectors that each span V. We need to show that S1 ∪ S2 also spans V.

Let v be any vector in V. Since S1 spans V, there exist vectors u1, u2,..., un in S1 and scalars a1, a2,..., an such that:

v = a1u1 + a2u2 + ... + anun

Similarly, since S2 spans V, there exist vectors w1, w2,..., wm in S2 and scalars b1, b2,..., bm such that:

v = b1w1 + b2w2 + ... + bmwm

Now, since S1 ∪ S2 contains all the vectors in S1 and S2, we can write v as a linear combination of the vectors in S1 ∪ S2:

v = a1u1 + a2u2 + ... + anun + b1w1 + b2w2 + ... + bmwm

Therefore, S1 ∪ S2 spans V.

In conclusion, if S1 and S2 are two sets of vectors that each span V, then S1 ∪ S2 also spans V.

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The correct answer is for the equation is C, "minimizes the sum of the squared distances from the actual y-value to the predicted y-value," as explained below.

The least-squares regression line is a line that best fits the given data by minimizing the sum of the squared distances between the predicted y-value (ŷ) and the actual y-value (y) for each data point. In other words, the line is chosen so that the sum of the squares of the vertical distances between the data points and the line is as small as possible.

In this case, the equation provided represents the least-squares regression line for the given data, which suggests that for every unit increase in x, y increases by 0.5 units.

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Answer:

- ∞ < x < 2

Option 3

Step-by-step explanation:

The given function is
[tex]f(x) =(x - 2)^2 + 3[/tex]

This is the  equation of a parabola in vertex form
The general vertex form equation is f(x) =a(x -h)² + k

where (h, k) is the vertex which can be either a maximum or minimum and a is a constant

If a is positive, the vertex is at a minimum and the parabola opens downward

If a is negative,  the vertex is at a maximum and the parabola opens upward

Here a = 1 is positive
The vertex is (2, 3) and is a minimum

So for all values of x < 2, the parabola is decreasing and for all values of x > 2 the parabola is increasing (see figure)

x = 2 is the turning point for the parabola

therefore the values of x for which the function is decreasing is

- ∞ < x < 2

The matrix C of the linear transformation T(x) = B(A(x)) is:

[tex]\(C=\left[\begin{array}{ll}15 & 45 \\ 38 & 42\end{array}\right]\)[/tex]

We can find the matrix C of the linear transformation T(x) = B(A(x)) by multiplying the matrices A and B.

Recall that the matrix product of two matrices

[tex]\(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\)[/tex]    and   [tex]\(B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right]\)[/tex]

is given by  

 [tex]\(C=AB=\left[\begin{array}{ll}a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22}\end{array}\right]\)[/tex].

So, for

[tex]\(A=\left[\begin{array}{ll}3 & 9 \\ 8 & 6\end{array}\right]\)[/tex]       and   [tex]\(B=\left[\begin{array}{ll}5 & 0 \\ 2 & 4\end{array}\right]\)[/tex],

we have:

[tex]\(C=BA=\left[\begin{array}{ll}5 & 0 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}3 & 9 \\ 8 & 6\end{array}\right]=\left[\begin{array}{ll}5(3)+0(8) & 5(9)+0(6) \\ 2(3)+4(8) & 2(9)+4(6)\end{array}\right]=\left[\begin{array}{ll}15 & 45 \\ 38 & 42\end{array}\right]\)[/tex]

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The probability distribution function of Y = |X| is given by:
P(Y ≤ y) = P(|X| ≤ y) = P(-y ≤ X ≤ y) = P(X ≤ y) - P(X < -y)

Since X ~ N(0, 2), we can use the standard normal distribution to find the probabilities:
P(X ≤ y) = Φ(y/√2)
P(X < -y) = Φ(-y/√2)

Therefore, the probability distribution function of Y is:
P(Y ≤ y) = Φ(y/√2) - Φ(-y/√2)

To find E(|X|) and Var(|X|), we can use the properties of the absolute value function:
E(|X|) = ∫_(-∞)^(∞) |x| f(x) dx = 2 ∫_(0)^(∞) x f(x) dx
Var(|X|) = E(|X|^2) - (E(|X|))^2 = 2 ∫_(0)^(∞) x^2 f(x) dx - (2 ∫_(0)^(∞) x f(x) dx)^2

Since f(x) is the probability density function of X, we can substitute it with the standard normal distribution:
f(x) = (1/√(2π)) e^(-(x^2)/4)

Therefore, E(|X|) and Var(|X|) can be found by evaluating the integrals:
E(|X|) = 2 ∫_(0)^(∞) x (1/√(2π)) e^(-(x^2)/4) dx = √(2/π)
Var(|X|) = 2 ∫_(0)^(∞) x^2 (1/√(2π)) e^(-(x^2)/4) dx - (√(2/π))^2 = 2 - (2/π)

So the final answers are:
1. E(|X|) = √(2/π)
2. Var(|X|) = 2 - (2/π)

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The probability that the first sister wins the first prize, the second sister wins the second prize, and the third sister wins the third prize is given as follows:

1/1320.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.

The probabilities for each sister are given as follows:

Hence the probability is given as follows:

1/12 x 1/11 x 1/10 = 1/(12 x 11 x 10) = 1/1320.

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The answer is an ordered pair (x,y)=(-7,2).

To solve the system of equations, we can use the elimination method to eliminate one of the variables and solve for the other.

First, let's multiply the first equation by 2 so that we can eliminate the y variable:

{20x+10y=−120

Next, let's subtract the second equation from the first:

{20x+10y−(20x−10y)==−120−(−80)

Simplifying the equation gives us:

{20y=40

Dividing both sides by 20 gives us:

{y=2

Now, let's plug this value of y back into the first equation to solve for x:

{10x+5(2)==−60

Simplifying the equation gives us:

{10x+10=−60

Subtracting 10 from both sides gives us:

{10x=−70

Dividing both sides by 10 gives us:

{x=−7

So, the solution to the system of equations is the ordered pair (x,y)=(−7,2).

Therefore, the answer is an ordered pair (x,y)=(-7,2).

I hope this helps! Let me know if you have any further questions.

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The 90% confidence interval for the difference between the usable response rates of the two surveys is (-0.0124, 0.0912).

The difference between the usable response rates of the two surveys can be estimated using the formula for the difference between two proportions:

d = p1 - p2
Where p1 is the usable response rate for the printed survey, and p2 is the usable response rate for the electronic survey. To calculate these response rates, we can use the formula:

p = x/n

Where x is the number of usable responses, and n is the total number of customers who were sent the survey. For the printed survey, we have:

p1 = 261/631 = 0.4138
And for the electronic survey, we have:

p2 = 155/414 = 0.3744

So the difference between the two response rates is:

d = 0.4138 - 0.3744 = 0.0394

To estimate a 90% confidence interval for this difference, we can use the formula:

CI = d ± z*√[(p1(1-p1)/n1) + (p2(1-p2)/n2)]

Where z is the critical value for a 90% confidence level, which is 1.645. Plugging in the values we have:

CI = 0.0394 ± 1.645*√[(0.4138(1-0.4138)/631) + (0.3744(1-0.3744)/414)]

CI = 0.0394 ± 1.645*√[0.000407 + 0.000581]

CI = 0.0394 ± 1.645*0.0315

CI = 0.0394 ± 0.0518

CI = (-0.0124, 0.0912)

So the 90% confidence interval for the difference between the usable response rates of the two surveys is (-0.0124, 0.0912). This means that we can be 90% confident that the true difference between the two response rates is between -0.0124 and 0.0912.

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The ratio of amount of flour to amount of water is 8 : 3. 5 cups of flour require 15/8 cups of water

An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.

From the recipe, 2 cups of flour require 3/4 cups of water

a) The ratio of amount of flour to amount of water = 2 cups of flour / (3/4) cups of water = 8/3 = 8 : 3

b) 2 cups of flour require 3/4 cups of water

5 cups of flour = 5 cups of flour * 3/4 cups of water per 2 cups of flour = 5 * (3/4)÷2 = 5 * 3/8 = 15/8 cups of water

5 cups of flour require 15/8 cups of water

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To schedule the examination so that no student has two examinations on the same day, we can use graph coloring. Graph coloring is the process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color.

1. First, we can create a graph with the seven subjects as vertices and the pairs of subjects with common students as edges.
2. Next, we can assign a color to each vertex, starting with S1 and moving clockwise around the graph. We can use the smallest available color for each vertex, making sure that no two adjacent vertices have the same color.
3. Once we have assigned a color to each vertex, we can use the colors to schedule the examinations. Each color represents a different day, and all of the subjects with the same color can be scheduled on the same day.

The resulting schedule would look something like this:

Day 1: S1, S5
Day 2: S2, S4
Day 3: S3, S6
Day 4: S7

This schedule ensures that no student has two examinations on the same day, since no two adjacent vertices in the graph have the same color.

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An angle which is supplementary to ∠AGB is ∠BGD

Two angles that sum up to 180 degrees are referred to as supplementary angles. In other words, the sum of the measurements of two additional angles equals 180 degrees.

Angles A and B, for instance, are supplementary if angle A is 70 degrees and angle B is 110 degrees. This is because 70 + 110 = 180.

Given that,

The arrangement which has multiple rays,

And it is known that,

The sum of angles of supplementary angles are 180°

∠AGB & ∠BGD has the sum of 180°,

So it can be state that ∠BGD is supplementary to the angle ∠AGB

Therefore, the supplementary angle is ∠BGD

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The measurement of the missing angle is 33 degrees

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

The triangle

On the triangle, we have the following equation

tan(?) = 9.09/14

Evaluate

tan(?) = 0.6493

Take the arc tan of both sides

? = 33 degrees

Hence, the angle is 33 degrees

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The height of the plane is modeled by the linear equation:

height = 3000ft - 2000ft*t

If the rate at which the plane descends is constant, then the height is modeled by a linear equation.

We can write:

height = initial height - rate*time

We know that the initial height is 3000ft, and that the plane descends at ar ate of 2,000ft per minute, then we can write the linear equation:

height = 3000ft - 2000ft*t

Where t is the time in minutes.

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Complete question:

"an airplane 3,000 ft above the ground begin descending at a rate of 2,000 ft per minute​, which will be the height of the plane after t minutes?"

1. The Formula used SA = 2w (l +h) + lh

2. The box is 0.37125 inch deep.

3. The SA of box with lid is 294.7972 inch².

The area is the territory covered by a shape or figure, whereas the perimeter is the distance covered by the shape's outside boundary. The unit of area is the square unit or unit², while the unit of perimeter is the unit.

Given:

length = 13.2 inch

Height = 10.5 inch

And, Surface Area used = 295.02 inch²

Using the Formula

SA = 2lw + 2wh + lh

SA = 2w (l +h) + lh

295. 02 = 2w ( 13.2 + 10.5) + (13.2)(10.5)

147.51 = w x 24 + 138.6

8.91 = 24w

w = 0.37125 inch

SA of box with lid

= 2( lw+ wh + lh)

= 2( 13.2 x 10.5 + 10.5 x 0.37125 + 0.37125 x 13.2)

= 2(138.6 + 3.8981 + 4.9005)

= 2 x 147.3986

= 294.7972 inch²

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Ryan, a consumer electronics​ salesperson, earns a base wage of ​$ 1300 per month and a commission of 7​% on the amount of sales he makes. One month Ryan received a $1622 paycheck, which made his sales for the month $4600.

To find the amount of Ryan's sales for the month, we can use the equation:

Total paycheck = Base salary + (Commission rate × Sales)

We can plug in the given values and solve for the amount of sales:

$1622 = $1300 + (0.07 × Sales)

$322 = 0.07 × Sales

Sales = $322 ÷ 0.07

Sales = $4600

Therefore, Ryan's sales for the month were $4600.

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To find the exact value of s in the given interval that has the given circular function value, we can use the properties of the tangent function and the unit circle.

The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In the unit circle, the opposite side is the y-coordinate and the adjacent side is the x-coordinate. Therefore, we can write:

tan s = y/x

Given that tan s = √3, we can write:

√3 = y/x

Cross-multiplying and simplifying, we get:

y = √3x

Now, we can use the Pythagorean theorem to find the value of x:

x^2 + y^2 = 1

Substituting y = √3x, we get:

x^2 + ( √3x)^2 = 1

Simplifying, we get:

4x^2 = 1

Dividing by 4, we get:

x^2 = 1/4

Taking the square root of both sides, we get:

x = ±1/2

Since we are looking for the value of s in the interval [????, 3????/2], we need to find the corresponding value of y for x = 1/2. Substituting x = 1/2 into y = √3x, we get:

y = √3(1/2)

Simplifying, we get:

y = √3/2

Therefore, the point on the unit circle that corresponds to the given value of the tangent function is (1/2, √3/2).

Now, we can use the properties of the unit circle to find the value of s. The angle s is the angle between the positive x-axis and the line connecting the origin and the point (1/2, √3/2). Since the point (1/2, √3/2) is in the first quadrant and the angle is measured counterclockwise, the value of s is:

s = π/3

Therefore, the exact value of s in the given interval that has the given circular function value is π/3.

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The measure of the two angles is x = 69 degrees and y = 70 degrees.

In every triangle, the total of the angles is 180 degrees. The Triangle Sum Theorem.

For triangle ABC it is given as follows:

Angle A + Angle B + Angle C = 180.

Let us suppose the measure of two angles as x and y.

According to the ratio we can write as follows:

x = 6k and y = 7k

The sum of the angles in a triangle is 180°, so we can write:

50° + 6k + 7k = 180°

13k + 50° = 180°

13k = 130°

k = 10°

Substituting the value of k in x and y we have:

x = 6k = 6(10°) = 60°

y = 7k = 7(10°) = 70°

Hence, the measure of the two angles is x = 69 degrees and y = 70 degrees.

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