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The exact value of the trigonometric relation sin(θ/2) is √[(1 + √(17)/5)) / 2]

Given data:

The trigonometric relation is :

sin(θ) = (2√2) / 5 and π/2 < θ< π

Using the half-angle identity for sine:

sin(θ/2) = ± √[(1 - cos(θ)) / 2]

To determine the sign, look at the quadrant in which theta lies. Since π/2 < θ< π, θis in the second quadrant, where the sine function is positive. Therefore, we take the positive sign.

Now, let's calculate the cosine of θ. Using the Pythagorean identity for sine and cosine, we have:

sin²(θ) + cos²(θ) = 1

(2√2/5)² + cos²(θ) = 1

8/25 + cos²(θ) = 1

cos²(θ) = 1 - 8/25

cos²(θ) = 17/25

Taking the square root:

cos(θ) = ± √(17)/5

Since theta is in the second quadrant, take the negative sign:

cos(θ) = -√(17)/5

Now, substituting the values into the half-angle identity, we get:

sin(θ/2) = ± √[(1 - cos(θ)) / 2]

sin(θ/2) = ± √[(1 - (-√(17)/5))) / 2]

sin(θ/2) = ± √[(1 + √(17)/5)) / 2]

Since θ is in the second quadrant, where the sine function is positive, take the positive sign:

sin(θ/2) = √[(1 + √(17)/5)) / 2]

Therefore, the exact value of sin(θ/2) is √[(1 + √(17)/5)) / 2].

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There are 0 intersections between the railroad and the highway and there are 2 intersections between the railroad and the turnpike

From the graph, we have the following points on the railroad

(x, y) = (3, 3) and (0, 5)

The slope is calculated as:

m = (y2 - y1)/(x2 - x1)

This gives

m = (5 -3)/(0 - 3)

Evaluate

m = -2/3

The equation is then calculated as:

y = mx + b

This gives

y = -2/3x + b

Substitute (0, 5)

5 = -2/3 * 0 + b

This gives

b = 5

Substitute b = 5 in y = -2/3x + b

y = -2/3x + 5

Multiply through by 3

3y = -2x + 15

Rewrite as:

2x + 3y = 15

The equation of the highway's path is

2x+3y=21

So, we have:

2x+3y=21

2x + 3y = 15

Subtract the equations

2x - 2x + 3y - 3y = 21 - 15

Evaluate

0 = 6

The above equation is false because 0 and 6 are not equal.

This means that the system of equations have no solution

Hence, there are 0 intersections between the railroad and the highway.

Here, we have

y = x^2 and 2x + 3y = 15

Substitute y = x^2 in 2x + 3y = 15

2x + 3x^2 = 15

Rewrite as:

3x^2 + 2x - 15 = 0

Calculate the discriminant using

d =b^2 - 4ac

This gives

d = 2^2 - 4 * 3 * -15

Evaluate

d = 184

The discriminant d is greater than 0.

This means that the equation has 2 real solutions

Hence, there are 2 intersections between the railroad and the turnpike

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John get money $630 and bill get money $270.

Given that John and bill shared $900 in the ratio 7:3 respectively.

The ratio is defined as the comparison of two quantities of equal units, indicating the quantity of one quantity contained in the other quantity.

The total ratio is 7+3=10

For john:

John has money seven out of 10 for the $900, we get

J=900×(7/10)

J=90×7

J=630

For bill:

Bill is three out of ten for $900, we get

B=900×(3/10)

B=90×3

B=270

Hence, john get money $630 and bill get $270 when they both shared $900 in the ratio 7:3 respectively.

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Answer:

7.6

Step-by-step explanation:

mean = [weights] / amount of weights

m = 10 + 7 + 6 + 10 + 13 + 10 / 6

m = 46 / 6

m = 7.66666666 (recurring)

m = 7.6

Multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple.

Pythagoras's principle follows the rule:

c² = a² + b²

where

Therefore, Pythagorean triples are a² + b² = c² where a, b and c are the three positive integers. The most known and smallest triplets are (3,4,5).

If we reduced the length of the side of the triangle with the greatest common factor, it should give us 3, 4, 5.

Hence,

21, 28 and 35 has greatest common factor as 7

21 / 7 , 28 / 7 and 35 / 7

3, 4, 5

Hence, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple.

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Answer:

About 71.77 degrees

Step-by-step explanation:

The starting leg is 2.3 km and the finishing leg is 6.2 km.

Using the law of cosines C^2 = A^2 + B^2 -2AB*cos(c) where A = 2.3 km, B = 6.2 km, C = 5.9 km, and angle c is the opposite angle to side C, we get:

5.9^2 = 2.3^2 + 6.2^2 -2(2.3)(6.2)*cos(c)

cos(c) = -(5.9^2 - 2.3^2 - 6.2^2)/(2*2.3*6.2)

c = 71.77 degrees

Answer:

72°

Step-by-step explanation:

when we have all the sides and need an angle, we use the law of cosine (the extended Pythagoras) :

c² = a² + b² - 2ab×cos(C)

where c is the side opposite of the angle C.

so, since the starting leg is 2.3 km, and the finishing leg is 6.2 km, we know that 5.9 km is the side opposite of the angle between the starting and finishing legs.

so, we have

5.9² = 2.3² + 6.2² - 2×2.3×6.2×cos(C)

34.81 = 5.29 + 38.44 - 28.52×cos(C)

-8.92 = -28.52×cos(C)

cos(C) = -8.92/-28.52 = 0.312762973...

C = 71.77418076...° ≈ 72°

Answer:

[tex]254 \frac{12}{17}[/tex] or approximately 254.71

Step-by-step explanation:

First, you need to find out how many gallons it will take to travel 866 miles. Because the car's avg miles/gallon is 13.6, you divide 866 by 13.6. However, this gets you an irrational number. To avoid this, make 13.6 into a fraction. Now divide 866 by [tex]13 \frac{3}{5}[/tex]. This becomes 866 ÷ [tex]\frac{68}{5}[/tex]. Convert this into a multiplication problem by converting [tex]\frac{68}{5}[/tex] into its reciprocal. This gets you 866 × [tex]\frac{5}{68}[/tex]. This gets you [tex]\frac{2165}{34}[/tex]. Now that you have the amount of gallons it will take to get to your destination, multiply it by 4. [tex]\frac{2165}{34}[/tex] × [tex]4[/tex][tex]= \frac{4330}{17}[/tex]. This becomes [tex]254 \frac{12}{17}[/tex] or approximately 254.71.

Answer:

x = 4.

Step-by-step explanation:

Each angle in an equilateral triangle is 60 degrees, so we have:

10x + 20 = 60

10x = 40

x = 4.

Answer:

X = 4

Step-by-step explanation:

The angles of a triangle add up to 180 degrees and this is an equilateral triangle which means that all the angles are the same. which means you set up the equation:

10x + 20 + 10x + 20 + 10x + 20 = 180

Simplify: 30x + 60 = 180

                        - 60   - 60

You have : 30x = 120

(Divide by 30)

You end up with: X = 4

The best estimation is 42.

Finding the most accurate estimate for:

[tex]-14\frac{1}{9}[/tex]  x  [tex]-2\frac{9}{10}[/tex]

To obtain, we round each integer to the next full number:

-14 x -3

Remember that adding two negatives produces positive results.

So, 14 x 3

Finally, this provides

42

Therefore, the final answer is 42.

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Answer: its 42

Step-by-step explanation:

Answer:

Step-by-step explanation:

You have 2 x's.  I think the first one is for multiplication and the second one is a variable.

2x2 -1 = 7x  

4-1 = 7x (first multiple 2x2)

3=7x  (second 4-1)

0.428571429 = x (Divide both sides by 7.  On the left side, 3/7 is 0.428571429 and 7x/7 is just x because the 7s cancel out.

Since the answer needs to be to two decimal places, I have to decide if 0.428571429 is closer to 0.42 or 0.43 Since the digit after the 2 in the answer is 8, that makes the number closer to 0.43 than 0.42.  So, the final answer is 0.43

Answer:

7k - 7

Step-by-step explanation:

- k - (- 8k + 7) ← distribute parenthesis by - 1

= - k + 8k - 7 ← collect like terms

= 7k - 7

The given limit indicates that there is a vertical asymptote at x = 2.

The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator. When x tends to these values laterally, the limit of f(x) goes to infinity.

In this problem, the function is:

[tex]f(x) = \frac{x^2 - 4x - 12}{x - 2}[/tex]

The vertical asymptote is:

x - 2 = 0 -> x = 2.

Hence:

[tex]\lim_{x \rightarrow 2^+} f(x)[/tex] and [tex]\lim_{x \rightarrow 2^-} f(x)[/tex] are either positive infinity or negative infinity.

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The answer is 100π cm².

Since we are given the volume, equate it to the formula of a sphere to find the radius.

Now, substitute this in the formula to find surface area.

Answer:22

Step-by-step explanation:22

Using it's concept, there is a 0.0308 = 3.08% probability that a prize ticket with a vowel will be chosen, not replaced, and then another prize ticket with a vowel will be chosen. Since the letter is not replaced, the number of outcomes change, hence they are dependent.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

For the first ticket, since 5 out of 26 letters are vogal, the probability is:

5/26.

For the second ticket, since there is no replacement, the probability is:

4/25.

The probability of both tickets is:

[tex]p = \frac{5}{26} \times \frac{4}{25} = 0.0308[/tex]

0.0308 = 3.08% probability that a prize ticket with a vowel will be chosen, not replaced, and then another prize ticket with a vowel will be chosen. Since the letter is not replaced, the number of outcomes change, hence they are dependent.

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Answer:

#3

Step-by-step explanation:

The y-intercept (the point where the line intersects with the y-axis) is -2, and the slope is 2/3 (two units up every three units to the right).

Answer:

C.

Step-by-step explanation:

First, look for the graph that shows a line with a y-intercept of -2. There is only one graph that shows a line with a y-intercept of -2. The graph will be the third graph. Please see attachment for the equation in a graphing calculator.

Hope this helps!

Answer:

Step-by-step explanation:

For two side measures and one angle measure to form one distinct triangle, the given angle must be between the given sides, or opposite the longest side.

When the given measures are ...

The given angle is opposite the longest side, so one distinct triangle can be formed.

No triangle can be formed if the given angle is opposite the shorter given side, and either of ...

__

Additional comment

As we can see from the above, the only ambiguous cases arise when the given angle is opposite the shorter given side. When that happens, the ratio of the shorter to longer given side relative to the sine of the given angle determines the nature of the solution:

Answer:

1

0

Step-by-step explanation:

Answer:

Step-by-step explanation:

If we subtract 3 from the unknown number, we get just as much as if we subtracted 7 from the unknown number and we multiplied the result by 2

Answer:

B

Step-by-step explanation:

15% off means 85 % of original price

400 * .85        now increase this price by 25%

400* .85   *  1.25    = $425

Answer: Width is 7 inches, length is 15 inches.

Step-by-step explanation:

Let the width be w. Then, the length is w+8.

[tex]2(w+w+8)=44\\\\w+w+8=22\\\\2w+8=22\\\\2w=14\\\\w=7\\\\\implies l=15[/tex]

The answer is . one-to-one.

(a) The function is onto because a, b, c, and d are members of its codomain. All points are converted into various points, making it a one-to-one relationship. The response is that it is one-to-one and onto ( A and C).

(b) It is onto because a, b, c, and d are members of the function's codomain. All points are converted into various points, making it a one-to-one relationship. The response is that it is one-to-one and onto (A and B).

(C) The element b is not onto since it is not a part of the function's codomain. All points are converted into various points, making it a one-to-one relationship. One-to-one is the right response.

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The complete question is -

Determine if each of the following functions from {a,b,c,d} to itself is one-to-one and/or onto.

Check ALL correct answers.

(a) f(a)=d,f(b)=a,f(c)=c,f(d)=b

A. onto.

B. neither one-to-one nor onto.

C. one-to-one.

f(a)=b,f(b)=a,f(c)=c,f(d)=d

A. one-to-one.

B. onto.

C. neither one-to-one nor onto.

f(a)=c,f(b)=d,f(c)=a

A. one-to-one.

B. onto.

C. neither one-to-one nor onto.

The volume will be one-eighth times the volume.

The scale factor k is given as

k = 1/2

Let the initial volume be v

When each side is multiplied by the scale factor, we have the new volume to be

V = k^3 * v

Substitute 1/2 for k in the above equation

V = (1/2)^3 *v

Evaluate the exponent

V = 1/8 *v

This gives

V = 1/8v

Hence, the volume will be one-eighth times the volume.

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The coordinates of M' is M'(-3,-4).

According to the statement

LMNP is rotated 90 degrees clockwise about the origin.

and we have to find the coordinates of M'.

so,

when it is rotated then it take four steps to reach at the end.

So, due to its moving procedure it takes four steps to reach at end.

and due to the four step procedure it's coordinates are changes four time and that's why at the end the coordinates we have to find are become a -3 and -4. So, the coordinate we get are -3 and -4

Due to this procedure the coordinate of m' become M'(-3,-4)

So, The coordinates of M' is M'(-3,-4).

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Answer: B. M'(4, -3) Just took the test

Step-by-step explanation:

by taking the quotient between the volume in the pot and the volume of each serving, we conclude that there are 12 servings.

The number of servings is given by the quotient between the volume in the pot and the volume of each serving, so we have:

N = (3/4)*/(1/16) = 12

So there are 12 servings in the pot.

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Answer: D

Step-by-step explanation:

There are vertical asymptotes at [tex]x=\pm 1[/tex].

This eliminates everything except for D.

The graph is of the rational function  f(x) = 1 / (x-1)(x + 1) Hence, Option D. is correct.

A rational function is an algebraic fraction such that both the numerator and denominator are polynomials.

Here, we have been given the graph.

We have to find which of the given rational functions is graphed in image.

On x-axis, 1 unit = 2 units

Clearly, we can see the graph is not defined at point x = - 4 and at x = 1.

Corresponding to x = - 4, factor is (x+4) and Corresponding to x = 1, factor is (x-1) .

So, this graph is of the rational function

f(x) = 1 / (x-1)(x + 1)

Hence, Option D. is correct

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The equation that will produce the given graph is; 6x² + 6y² = 144

The standard equation of a circle can be expressed as:

(x + a)² + (y - b)² = r²

where;

(a, b) is the center of the circle

r is the radius of the circle.

From the diagram, we can see that the radius of the circle is between 4 and 5.  Thus, the correct equation must have a radius between these values.

From the option, we can see that the only equation that will have a radius between 4 and 5 is;

6x² + 6y² = 144

Divide through by 6 to get;

x² + y² = 24

x² + y² = √24

Thus r = √24 which is greater than 5

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First, find the supplement of the angle that is 105 degrees.

105 + 75 = 180

Now, we can find complete the lower left triangle's angles.

39 + 75 + ? = 180

? = 66 degrees

The 66 degree angle of the lower left triangle and angle x are vertical angles. Vertical angles are congruent.

Therefore, the measure of angle x is 66 degrees.

Hope this helps!

Answer: 18.7

Step-by-step explanation:

Find the lengths of AB, BC, and CA

AB = [tex]\sqrt{(4-(-1))^{2} + (-1-4)^{2} }[/tex] = 5[tex]\sqrt{2}[/tex] ≈ 7.1

BC = [tex]\sqrt{(0-(-1))^{2} + (-3-4)^{2} }[/tex] = 5[tex]\sqrt{2}[/tex] ≈ 7.1

CA = [tex]\sqrt{(0-4)^{2} + (-3-(-1))^{2} }[/tex] = 2[tex]\sqrt{5}[/tex] ≈ 4.5

7.1 + 7.1 + 4.5 = 18.7