Prove the following trigonometric identities

Answer:

10 units

Step-by-step explanation:

the longest length is from (11, 1) to (3, 7) and can be clearly shown when graphed

now to find the length you must use the pythagorean theorem

the distance between 11 and 3 is 8 and the distance between 1 and 7 is 6 so

8^2 + 6^2 = c^2

64 + 36 = c^2

100 = c^2

c = 10

that is your answer

Answer:

B) 16

Step-by-step explanation:

12/3 = x/4

48 = 3x  Divide both sides by 3

16 = x

Answer:

C

Step-by-step explanation:

y=mx+b where m is the slope or weekly payments and b is the y-intercept or initial fee.

300 = 30w + 60 fits this

The extreme drop can be said to most likely represent: b. A reflection of the migratory nature of the birds at the lake.

Bird migration is a natural process which can be described as a the movement of different birds by flight across thousands of kilometers in search of ecological conditions and habitats that favors breeding, feeding, and raising of their young.

Migratory birds have well built physiology and morphology that make them better equip to fly fast and across long distances in search of the best ecological conditions, which is dependent on seasons.

Like in the graph given, like we see that at the beginning of January, number of birds were few in the lake until sudden increase occurred. It can be said that during those period of increase, the ecological conditions favored the survival of the bird until November and December. This extreme drop can be said to most likely represent: b. A reflection of the migratory nature of the birds at the lake.

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The function with the greatest rate of change over the interval [0, 2] is: D. a.

In Mathematics, the rate of change of a function over a given interval can be determined by calculating the slope of the straight line which connect its end points.

Mathematically, the slope of any straight line can be calculated by using this formula;

[tex]Slope = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]

For line a, we have:

Points (x, y) = (0, 1) and (2, 4)

Slope = (4 - 1)/(2 - 0)

Slope = 3/2 or 1.5.

For line b, we have:

Points (x, y) = (0, 0) and (2, 2)

Slope = (2 - 0)/(2 - 0)

Slope = 2/2 or 1.0.

For line c, we have:

Points (x, y) = (0, -1) and (2, 0)

Slope = (0 + 1)/(2 - 0)

Slope = 1/2 or 0.5.

For line d, we have:

Points (x, y) = (0, 0.5) and (2, 2.5)

Slope = (2.5 - 0.5)/(2 - 0)

Slope = 2/2 or 1.

In conclusion, we can logically deduce that "function a" has the greatest rate of change over the interval [0, 2].

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The value of 'k' in the quadratic Equation    [tex]4x^{2}+kx+4=0[/tex] having one rational solution is k is 8.

Here, the given equation is  [tex]4x^{2}+kx+4=0[/tex]

By comparing the given equation with the standard form of the quadratic equation we get,

a = 4

b = k

c = 4

The given quadratic is having one rational solution, which means [tex]b^{2} -4ac=0[/tex]

Therefore,

[tex]k^{2} -4 \times 4\times4=0\\k^{2} -64=0\\k^{2} =64\\k=\sqrt{64} \\k=8[/tex]

Therefore, the value of k is 8.

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the operation:

[tex]2*10^1 - 4*10^0 = 20 - 4 = 16[/tex]

Gives a square number.

A perfect square is the product of a number and itself.

As you can see in the example, 16 is a square number. Then let's try to get 16.

To get 16 we can use the difference:

20 - 4

So the first therm needs to be equal to 20, and the second equal to 4.

To write 20 in scientific notation we have:

[tex]2*10^1[/tex]

To write 4 in in scientific notation:

[tex]4*10^0[/tex]

Then the operation:

[tex]2*10^1 - 4*10^0 = 20 - 4 = 16[/tex]

Gives a square number.

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The next step Javier needs to take in order to evaluate the expression is to simplify it to it's lowest term. Details below

From the question given, we were told that Javier submitted 2 for a

Thus, the new expression becomes

2 × 2³ - 1

Recall

Aⁿ × Aᵐ = Aⁿ⁺ᵐ

Thus,

2 × 2³ = 2¹⁺³ = 2⁴

Therefore,

2 × 2³ - 1 = 2⁴ - 1

But

2⁴ = 16

Thus,

2⁴ - 1 = 16 - 1

2⁴ - 1 = 15

Therefore,

2 × 2³ - 1 = 15

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The missing letter is p. The correct option is B. p

From the question, we are to determine the missing letter in the given pattern

The given pattern is

t, s, r, q, _, o, n, m,...

Consider the English alphabet

a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z.

If we study the pattern, we will realize that the alphabets are in the reverse order.

Hence, the missing letter is p. The correct option is B. p

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Answer:

28, 4 or 4, 28 (order doesn't matter, both are true answers)

Step-by-step explanation:

l + w = 32

l*w = 112

l = 112/w

112/w + w = 32

w = 28 or 4

l = 28 or 4

Answer:

4,28

Step-by-step explanation:

So let's look at the sentence to take any necessary information out. It says the length of a rectangle plus it's width is 32. Since we do not know what the length and width are, let's just represent them as L (length) and W (width). Using this information we get the equation: [tex]L + W = 32[/tex] and since we're given the area of the rectangle, we can also define another equation which is: [tex]LW = 112[/tex] since the length * width = area.

So now all we have is a systems of equations. We can solve for L or W in the area equation and plug it into the addition equation. We can also solve for L or W in the addition equation and then plug it into the area equation, either should work, but in this example I'll solve for L in the addition equation:

Original Equation:

[tex]L + W = 32[/tex]

Subtract 32 from both sides

[tex]L = 32-W[/tex]

Now let's use this definition of L to plug into the second equation:

[tex]LW = 112 \implies (32-W)W = 112[/tex]

Distribute the W

[tex]-W^2+32W=112[/tex]

Now if you notice, we simply have a quadratic equation and we can subtract 112 from both sides and then use the quadratic formula: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

Subtract 112 from both sides

[tex]-W^2+32W-112=0[/tex]

Use quadratic formula

[tex]W = \frac{-32\pm\sqrt{32^2-4(-1)(-112)}}{2(-1)}[/tex]

Simplify stuff in radical

[tex]W = \frac{-32\pm\sqrt{576}}{-2}[/tex]

Calculate the square root

[tex]W=\frac{-32\pm24}{-2}[/tex]

Use positive sign:

[tex]W=\frac{-32+24}{-2}[/tex]

Simplify numerator

[tex]W = \frac{-8}{-2}[/tex]

Simplify fraction

[tex]W=4[/tex]

Take negative sign
[tex]W = \frac{-32-24}{-2}[/tex]

Simplify numerator

[tex]W=\frac{-56}{-2}[/tex]

W = [tex]28[/tex]

So we have two solutions, and the reason for this is because had we solved for W in the original equation and substituted that into here, there would be no different, it's not bound by some variable name. So these are actually the two sides of the rectangles

4 + 28 = 32

4 * 28 = 112

4, 28

Answer: 4

Step-by-step explanation:

Slope is rise over run (up over sideways) from two points.

The two points shown are (0,4) and (-2,-4).

By counting, you can find the slope, which is 8/2 = 4.

Answer:

84

Step-by-step explanation:

Total students he surveyed: 27

Amount that found it something OTHER than easy: 14 (11 moderate + 3 difficult)

This means that [tex]\frac{14}{27}[/tex] found it something other than easy

Multiply that by 162 to find the proportion for the larger population:

162 * (14/27), and we get 84.

Answer:

925.5

Step-by-step explanation:

Hope it helps!

<3

Hello !

[tex](x {}^{a} ) {}^{b} = x {}^{a \times b} [/tex]

[tex](b {}^{5} ) {}^{2} = b {}^{ 5\times 2} = b {}^{10} [/tex]

→ Answer D

Answer:

D

Step-by-step explanation:

using the rule of exponents

[tex](a^{m}) ^{n}[/tex] = [tex]a^{mn}[/tex] , then

[tex](b^5)^{2}[/tex] = [tex]b^{5(2)}[/tex] = [tex]b^{10}[/tex]

Answer:

No largest no could be 100

Step-by-step explanation:

Yes, the sum of the three numbers is 200, and the largest number can indeed be 75.

Let's assume the three numbers are:

Smallest number = 25

Middle number = x (unknown)

Largest number = 75

The sum of the three numbers is 200, so we can set up the equation:

25 + x + 75 = 200

Now, let's solve for x:

x = 200 - 25 - 75

x = 100

So, the middle number is 100.

Now, we can check if the largest number (75) is correct:

25 + 100 + 75 = 200

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Answer:

1. The zeros are at x = 0 and x = –10

2. The zeros are at x = 3 and x = 4

Step-by-step explanation:

1. f(x) = x (x + 10)

Multiplying any number by 0 = 0,
so f(x) = 0 when...

• x = 0

or

• x + 10 = 0 (if you subtract 10 from both sides, you get x = –10)

2. f(x) = x (x - 3) – 4 (x - 3)

Use distributive property to expand

x² – 3x – 4x + 12

Combine like terms

x² – 7x + 12

Factor

(x – 3) (x – 4)

(You could have also just noticed in the original function that both x and –4 were multiplied by (x – 3) and factored then.)

So f(x) = 0 when...

• x – 3 = 0  (add 3 to both sides to get x = 3)

or

• x – 4 = 0 (add 4 to both sides to get x = 4)

I hope that helped!

The woman's shadow is moving away from the pole at the speed of 4 feet/sec.

The height of a pole-mounted street light = 18 ft

The height of a lady = 6 ft

The woman is moving quickly and directly away from the pole. Her speed = 4ft/sec.

The woman's distance from the pole at the precise moment question want to know the speed of the woman's shadow's leading edge = 45ft

A woman standing 6 feet tall moves away from the pole in a straight route at a pace of 4 feet per second. If we assume that t is the passing time in seconds and that x is the distance in feet between the pole and the lady, we are informed that dx/dt=4 ft/sec.

Therefore, it is concluded that the final answer is 4 ft/sec.

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Answer:

45 numbers

Step-by-step explanation:

The 10-digit numbers are from 10-99. We can manually analyze each tenth digit and see if we can find a pattern from it. First, 10-19 ~ 10 is the only number in the sequence where the tenth digit is greater than the unit digit (because 1 > 0). In every other case, 1 is less than or equal to the unit digit.

Next, 20-29 ~ 20 and 21 are the two numbers where the tenth digit is greater than the unit digit

30-39 ~ 30, 31, 32 follow the property

40-49 ~ 40, 41, 42, 43 follow the property

We can keep going, but we can see that there is a pattern. The number of numbers that correspond with the property for each set of 10 numbers is equal to the value of the tenth digit. We see that for the 40s, there are 4 numbers. For the 20s, on the other hand, there are 2 numbers. This pattern carries through:

50-59 ~ 5 numbers, 60-69 ~ 6 numbers, 70-79 ~ 7 numbers, 80-89 ~ 8 numbers, 90-99 ~ 9 numbers

We need to find how many 2-digit numbers total follow the property. We can just add our results together to get our answer:

1+2+3+4+5+6+7+8+9=45

The goals does the team score in total in the first 9 matches of the competition is 36 and  their mean number of goals after 10 matches is 3.9.  

Given that the mean number of goals a netball team scores per match in the first 9 matches of a competition is 4.

Thus the average of the goals in 9 matches is 4.

Total number of goals is = 9*4 = 36 goals .

Given that the team scores 3 goals in their next match .

Thus the total number of goals scored in 10 matches is 36 + 3 = 39 goals,

Required mean = Total goals / 10 matches .

∴ Required mean = 39/10 = 3.9 .

Therefore, the goals does the team score in total in the first 9 matches of the competition is 36 and  their mean number of goals after 10 matches is 3.9.  

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By the definition of similar polygons, the lengths of corresponding sides are proportional.

The proportion for [tex]x=40[/tex].

As it is given in the definition, similar polygons have congruent angles and proportional sides.

Therefore, by the definition of similar polygons, the lengths of corresponding sides are proportional.

Cross multiply and solve the proportion  [tex]x[/tex] as follows:

The blank should be proportional.

[tex]\frac{12}{8} =\frac{20+x}{x}\\ 12x=160+8x\\4x=160\\x=40[/tex]

Therefore, the proportion for [tex]x=40[/tex].

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Answer:

Step-by-step explanation:

All pyramids have a volume as one-third of a prism with the same base area and the same height.

This will make the formula for the volume [tex]\frac{1}{3}(base*height)[/tex]. We already know the height of the pyramid is 10, but the area of the base for this rectangle would be its length and its width. Let's calculate this:

[tex]7*2.3=16.1\hspace{0.1cm}in^2[/tex]

Let's put these values back into the formula:

[tex]V=\frac{1}{3}(16.1*10)\\ V=\frac{1}{3}(161)\\ V= 53.7\hspace{0.1cm}in^3[/tex]

The statement that can be concluded from the triangle is A. RS/VU=ST/UT and ∠S≅∠U.

The Side-Angle-Side Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

In this problem the included angle is ∠S≅∠U, therefore side RS must be proportional to side VU  and side ST must be proportional to side UT

Hence, RS/VU=ST/UT.

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In Wagnerian music drama, a basic recurring theme representing a person, object, event, or emotion is known as Leitmotif.

What is serious opera called?

What is Wagnerian leitmotif?

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Answer:

9

Step-by-step explanation:

Let x = # of movies Jamie watches in a month. 5 + 2.25x represents the monthly cost of Amazing Primo, and 15 + x represents the monthly cost of Netflakes.

5 + 2.25x > 15 + x

1.25x > 10

x > 8

Jamie must watch more than 8 movies each month for the Netflakes account to be cheaper overall.

The possible outcomes are WH, BH, BH, BH, RH, RH, WL, BL, BL, BL, RL, RL

The given parameters are:

Coin = Head (H) and Tail (T)

Marble = W, B, B, B, R, R

See attachment for the tree diagram

From the tree diagram, we have the following possible outcomes

WH, BH, BH, BH, RH, RH, WL, BL, BL, BL, RL, RL

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The five rational numbers are 1/1729, 1/2, 3/2, 2/1, 1729/1 and the numbers should be of form p/q where p or q should be a taxicab number.

According to the question,

To represent 5 rational numbers on line, the numbers should be of form p/q where p or q should be a taxicab number.

A taxicab number is a smallest integer can be expressed as a sum of two positive integer cube.

The most famous taxicab number is 1729. Thus,

1729 = 1³ + 12³

1729 = 9³+ 10³

Now, 2 = 1³ + 1³ so, 2 is the first taxicab number. Therefore, five rational numbers are  1/1729, 1/2, 3/2, 2/1, 1729/1.

In 1/1729 -> 1729 is taxicab number

In 1/2 -> 2 is taxicab number

In 3/2 -> 2 is taxicab number

In 1/1729 -> 2 is taxicab number

In 1729/1  -> 1729 is taxicab number

Hence, five rational numbers are 1/1729, 1/2, 3/2, 2/1, 1729/1 and the numbers should be of form p/q where p or q should be a taxicab number.

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Using a system of equations, it is found that she earns $5.5 for each hat and $7 for each scarf sold.

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

In this problem, the variables are:

One week she earns $114.00 by selling eight hats and 10 scarves, hence:

8x + 10y = 114.

The following week she sold 12 hats and six scarves and earned $108.00, hence:

12x + 6y = 108

Simplifying by 6:

2x + y = 18

y = 18 - 2x.

Replacing in the first equation:

8x + 10y = 114.

8x + 10(18 - 2x) = 114

12x = 66

x = 66/12

x = 5.5

y = 18 - 2x = 18 - 2 x 5.5 = 7.

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The value of X is 1/3 and Y is 1/2 and Z is 1/4.

According to the statement we have given the equations which are

X + Y = 5XY  -(1)

Y + Z = 6YZ  -(2)

Z  + X = 7XZ  -(3)

and we have to find the value of X,Y,Z.

So, solve the equations for the X,Y,Z

From (1) equation

Y = X / (5X-1)  -(4)

From (3) equation

Z = X / (7X-1)  -(5)

Put (4) and (5) in (2) then

X / (5X-1)  + X / (7X-1)  = 6{[X / (5X-1)][X / (7X-1)]}

X(7X-1)  + X(5X-1) =  6{[X / (5X-1)][X / (7X-1)]} /  {[1 / (5X-1)][1 / (7X-1)]}

Then after solving

12(X)^2 -2X = 6(X)^2

6(X)^2 -2X = 0

X (6X-2) = 0

X = 0, X = 1/3

Here X = 0 is neglected.

So, X=1/3  -(6)

Put (6) in (4) and (5) then

From 4

Y = X / (5X-1)

Y = (1/3) / (5(1/3)-1)

Y = (1/3) / (2/3)

Y = 1/2.

From (5) equation

Z = X / (7X-1)

Z = (1/3) / (7(1/3)-1)

Z = (1/3) / (4/3)

Z = 1/4.

So, The value of X is 1/3 and Y is 1/2 and Z is 1/4.

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a. circle 2 = 60°

circle 3 = 120°

circle 4 = 180°

circle 5 = 240°

circle 6 = 300°

b. circle 8 = 15°

circle 9 = 75°

circle 10 = 155°

circle 11 =  215°

circle 12 =  265°

From the information given, we have that;

There are six major circles

Six minor circles

a. There are 6 circles at a point

Note that angles at a point sum up to 360 degrees

Now, let's divide their sum by the six parts

= [tex]\frac{360}{6}[/tex]

= [tex]60[/tex] degrees

Then, circle 1

= 0 × 60

= 0

Circle 2

=  1 × 60

=  60°

Circle 3

= 2 × 60

= 120°

Circle 4

= 3 × 60

= 180°

Circle 5

= 4 × 60

= 240°

Circle 6

= 5 × 60

= 300°

b. We have that each of these circles ( circle 7 - 12) radian measure is

= main circle - [tex]\frac{\pi }{12}[/tex]

The relative position or direction of circle 7 - 12 to their corresponding main circles ( 1 - 6) are all the same, we then have;
For circles 8

= 60° - 45°

= 15°

Circle 9

= 120° - 45°

= 75°

Circle 10

= 180° - 45°

= 155°

Circle 11

= 240° - 45°

= 215°

Circle 12

= 300° - 45°

= 265°

They share the relationship of radian of circle 7 - 12 corresponding to circles 6 - 12 minus π/12

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