solve these counting problems using the pigeonhole principle.(1) the smallest number of people in a group needed to guarantee that at least two were born in the same month is .(2) the smallest number of people in a group needed to guarantee that at least two have the same first and last initials is .(3) the smallest number of people in a group needed to guarantee that at least two have the same first initial and were born on the same day of the week is .(4) the smallest number of people in a group needed to guarantee that at least two were born on the same day of the year, assuming that nobody in the group was born on february 29 in a leap year, is

Therefοre, the expressiοn can be rewritten in terms οf the functiοn [tex]cos x[/tex]as: [tex]2cos^2 x + cos x - 2[/tex].

Trigοnοmetry is a branch οf mathematics that deals with the relatiοnships between the sides and angles οf triangles. It is particularly cοncerned with the study οf angles, the trigοnοmetric functiοns (sine, cοsine, tangent, etc.), and their prοperties and applicatiοns.

Trigοnοmetry has many practical applicatiοns, particularly in fields such as physics, engineering, and astrοnοmy, where it is used tο sοlve prοblems invοlving angles, distances, and shapes. Fοr example, trigοnοmetry is used in surveying tο measure distances and heights, in navigatiοn tο determine the pοsitiοn οf ships and airplanes, and in physics tο describe the mοtiοn οf waves and οscillatiοns.

In the given questiοn,

We can use the trigοnοmetric identity [tex]cos^2 x + sin^2 x = 1[/tex] tο rewrite [tex]sin^2 x[/tex] as: [tex]sin^2 x = 1 - cos^2 x[/tex]

Substituting this intο the οriginal expressiοn gives:

[tex]cos^2 x + cos x - 1 + sin^2 x = cos^2 x + cos x - 1 + (1 - cos^2 x) = 2cos^2 x + cos x - 2[/tex]

Therefοre, the expressiοn can be rewritten in terms οf the functiοn [tex]cos x[/tex] as: [tex]2cos^2 x + cos x - 2.[/tex]

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Answer: 36 m²

Step-by-step explanation:

       First, we will solve for the area of the triangles. Since they are congruent, we will multiply the height times the base. Usually, you would have to divide by two. However, since we have two congruent triangles, we will not be doing this.

       A = BH

       A = (2 m)(6 m)

       A = 12 m²

       Next, we will find the area of the rectangle. We will multiply the length by the width.

       A = BH

       A = (6 m)(4 m)

       A = 24 m²

       Lastly, we will add these two values together.

24 m² + 12 m² = 36 m²

nswer:

Step-by-step explanation:

Step-by-step explanation:

We can use the FOIL method to find the product of (4x − 3)(x + 2):

(4x − 3)(x + 2) = 4x(x) + 4x(2) - 3(x) - 3(2)

= 4x^2 + 8x - 3x - 6

= 4x^2 + 5x - 6

Therefore, the product of (4x − 3)(x + 2) is 4x^2 + 5x − 6. Answer: (A)

Step-by-step explanation:

(4x-3)(x+2)

4x*x,4x*2,-3*x,-3*2

4x2+8x-3x,-6

4x2(8x-3x)-6

4x2+5x-6

so the answer is A

The x-intercepts of the given expression which is a rational function is calculated to be 0 and the y-intercepts are ±√25.

The given expression is a rational function and can be written as the quotient of two polynomials.

The numerator of the rational expression is -x⁷ and the denominator is x²−25. The numerator is a degree 7 polynomial and the denominator is a degree 2 polynomial.

To find the x-intercepts, we need to set the numerator equal to zero and solve for x.

-x⁷ = 0

x⁷ = 0

x = 0

To find the y-intercepts, we need to set the denominator equal to zero and solve for x.

x²−25 = 0

x² = 25

x = ± √25

Therefore, the x-intercepts of the given expression are 0 and the y-intercepts are ± √25.

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Question :

Find the intercepts of the given function.

y = (-x⁷)/(x²−25)

Answer:

d. 2,500

Step-by-step explanation:

The conversion formula to convert kg to g is multiply the mass value by 1000

We just multiply 2.5 by 1000

[tex]2.5\times 1000=2500[/tex]

D is correct

Answer:

see below.

Step-by-step explanation:

Use Trigonometric Expansion.

Verify the following identity:

[tex]cos(x)^2(tan(x)^2+1)=1[/tex]

Write tangent as [tex]\frac{sine}{cosine}[/tex]

[tex]cos(x)^2(\frac{sin(x)}{cos(x)} ^2+1)=1[/tex]

[tex]cos(x)^2(\frac{sin(x)}{cos(x)} ^2+1)=cos(x)^2(\frac{sin(x)^2}{cos(x)^2+1} )[/tex]

[tex](cos(x)^2(\frac{sin(x)^2}{cos(x)^2} +1))=1[/tex]

Put sin(x)^2/cos(x)^2 +1 over the common denominator cos(x)^2 : sin(x)^2/cos(x)^2+1 = cos(x)^2+sin(x)^2 /cos (x)^2:

cos(x)^2(cos(x)^2+sin(x)^2/cos(x)^2)=1

Cancel cos(x)^2 from the numerator to the denominator.

cos(x)^2(cos(x)^sin(x)^2)/cos(x)^2=cos(x)^2(cos(x)^2+sin(x)^2)/cos(x)^2=cos(x)+sin(x)^2:

(cos(x)^2+sin(x)^2=1

Substitute cos(x)^2 + sin (x)^2 = 1:

1 =1

Answer:

  graph 4 (reflected vertically)

Step-by-step explanation:

You want the graph that represents a cosine function with no horizontal shift, an amplitude of 2, and a period of 2π/3.

The amplitude of a sine or cosine function is the difference between the positive peak and the centerline. All of the graphs show functions with an amplitude of 2.

The period of the function is the difference of x-values between the first instance of the function and the first repeat: f(x) ≡ f(x+p) has a period of p.

Graphs 1 and 3 show periods of 2π/9; graphs 2 and 4 show periods of 2π/3, as required.

There is an interaction between horizontal shift and vertical reflection. Sine or cosine functions are effectively inverted (reflected over the x-axis) if they are shifted horizontally by 1/2 period—and vice versa.

The function in graph 3 is a sine function, or a horizontally shifted cosine function.

The function in graph 4 is a cosine function that has been shifted 1/2 perior, OR a cosine function that has been reflected over the x-axis.

Graph 4 is the only viable choice for a cosine function with the correct period, so we have to assume it starts at -2 because it has been reflected vertically over the x-axis. (There seems to be no restriction on that in the problem statement.)

__

Additional comment

Ordinarily, you might not be expected to make any assumptions about vertical reflection. It seems a bit of a trick question.

Answer:

5(7j+3) AND 35j +15 are both equal to the expression 5(6j + 3 + j)

The calculated length of the arc has a value of 22.34 yards

To find the length of an arc, we can use the formula:

L = (θ/360) x 2πr

where L is the length of the arc, θ is the central angle in degrees, r is the radius of the circle, and π is approximately 3.14.

In this case, the central angle is 160 degrees and the radius is 8 yards. So we can plug in these values to get:

L = (160/360) x 2π(8)

Evaluate

L = 22.34 yards

Therefore, the length of the arc is approximately 22.34 yards

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The height of the can is 0.238 inches.

What is the volume of cylinder?

The density of a cylinder is determined by its volume, which represents how much material may be immersed in it or carried inside of it. The formula πr²h, where r is the radius of the circular base and h is the height of the cylinder, determines the volume of a cylinder.

Here, we have

Given: Aiden models a can of ground coffee as a right cylinder. He measures its radius as 3 4/3 ​ in and its volume as 12 cubic inches.

We have to find the height of the can.

the volume of the can of ground coffee = πr²h

12 = 3.14×(12/3)²h

12/50.24 = h

h = 0.238inches.

Hence, the height of the can is 0.238 inches.

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If we simplify [tex]3(2v-4)-4(v+4)[/tex], our answer will be [tex]-5v - 20[/tex]

Simplification is the process of replacing a mathematical expression by an equivalent one, that is simpler, usually shorter.

The question is [tex]3(2v-4)-4(v+4)[/tex], our step of solution is as follows:

==>>3(2v-4)-4(v+4)

Open the bracket with multiplication

= 6v - 12 - 4v - 16

= 2v - 28

= 2(v - 14)

= -5v - 20

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Answer:

460 milliliters

Step-by-step explanation:

We know that the ratio of A to B is 5:7. This means for every 5 milliliters of A, we need 7 milliliters of B. Let's use this ratio to find how many milliliters of A we need for 1104 milliliters of the drug.

First, we need to find out how many sets of 5:7 we have in 1104 milliliters. We can do this by dividing 1104 by the total number of parts in the ratio, which is 5 + 7 = 12.

1104 / 12 = 92

This means we have 92 sets of 5:7 in 1104 milliliters of the drug. To find how many milliliters of A we need, we simply multiply the number of sets by the amount of A in each set, which is 5 milliliters.

92 x 5 = 460

Therefore, we need 460 milliliters of compound A.

Answer:

20

Step-by-step explanation:

You have to multiply 10 by 2, which the ten represents ur 10th birthday, and the 2 represents the amount doubled. 10+2=20

The height of the first floor above the second floor would be = 20 ft.

To calculate the height of the first floor from the second floor, the formula from SOHCAHTOA is used.

The opposite part of the triangle created by the escalator = ?ft

The hypotenuse of the triangle which is the height of the escalator = 40 ft

Sin 30° = X/40

X = sin30° × 40

X = 0.5×40

X = 20 ft

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Answer:

15,75

Step-by-step explanation:

[tex]4 \frac{2}{3} \times 1 \frac{1}{2} \times 2 \frac{1}{4} = \frac{14}{3} \times \frac{3}{2} \times \frac{9}{4} = \frac{14}{2} \times \frac{9}{4} = \frac{7}{2} \times \frac{9}{2} = \frac{63}{4} = 15 \frac{3}{4} = 15.75[/tex]

Answer:

First, we need to convert the length of the leather cord to inches because the length of the necklace is given in inches.

3 yards = 3 x 3 = 9 feet (since there are 3 feet in a yard)

9 feet = 9 x 12 = 108 inches (since there are 12 inches in a foot)

Now we can divide the length of the leather cord by the length of each necklace:

108 inches ÷ 28 inches = 3.857

Since we cannot make a fraction of a necklace, we round down to the nearest whole number:

3 necklaces

Therefore, Cindy can make 3 necklaces with the piece of leather cord she has.

Answer:

x = 9

Step-by-step explanation:

AD is an angle bisector and divides the side opposite ∠ A into segments that are proportional to the other two sides , that is

[tex]\frac{CD}{BD}[/tex] = [tex]\frac{AC}{AB}[/tex] ( substitute values )

[tex]\frac{x}{6}[/tex] = [tex]\frac{18}{12}[/tex] ( cross- multiply )

12x = 6 × 18 = 108 ( divide both sides by 12 )

x = 9

Answer:

This is a symmetrical shape so that means all of the sides are going to match. To find x you need to look at the opposite side of the shape and see that it’s needs to match the three. In order to do that X would need to equal two so that it would be two times two minus one which would give you the three that you need to match the other side.

Answer:

Since the vertex is (0, -4), we know that the equation of the parabola will have the form:

y = a(x - 0)^2 - 4

where "a" is the coefficient that determines whether the parabola opens up or down.

To find the value of "a", we can use the fact that the parabola passes through the point (10, 1):

1 = a(10 - 0)^2 - 4

1 + 4 = 100a

5 = 100a

a = 1/20

Substituting this value of "a" into the equation for the vertex form, we get:

y = (1/20)x^2 - 4

So the equation of the parabola in vertex form is y = (1/20)x^2 - 4, with vertex (0, -4) and passing through (10, 1).

Step-by-step explanation:

(1) The graph that shows the solution of the inequality 7 ≤ n + 5 is the graph of option A.

(2) Eduardo wrote at least 34 pages before today.

(3)  The farmer needs to plant at least 8 rows of seeds to harvest at least 52 bushels of tomatoes.

The solution to the inequality 7 ≤ n + 5 is as follows:

7 ≤ n + 5

collect similar terms together

7 - 5 ≤ n

2 ≤ n

So we can interpret the result of this inequality as, n is greater than and equal to 2. Hence option A is the correct answer.

2. Let p be the number of pages Eduardo wrote before today. Then, the inequality that represents the situation is:

p + 16 > 50

To solve for p, first subtract 16 from both sides of the inequality:

p + 16 - 16 > 50 - 16

p > 34

Therefore, Eduardo wrote at least 34 pages before today.

3. Let r be the number of rows of seeds the farmer needs to plant. Then, the equation that represents the situation is:

6.5r ≥ 52

divide both sides of the equation by 6.5:

r ≥ 8

Therefore, the farmer needs to plant at least 8 rows of seeds to harvest at least 52 bushels of tomatoes.

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1 - sin(a) - cos(a) can be expressed as the product 2(cos(a)(sin)(a)).

What are Trigonometric Identities?

Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions.

We can use the identity:

[tex]cos^2(a) + sin^2(a) = 1[/tex]

Substituting this into the expression 1 - sin(a) - cos(a), we get:

[tex]cos^2(a) + sin^2(a) - sin(a) - cos(a) \\\\cos^2(a) - cos(a) + sin^2(a) - sin(a) \\\\cos(a)( 1 - cos(a)) + sin(a)( 1- sin(a) )\\\\[/tex]      eq2

It is given that

[tex]1 - sin(a) - cos (a)\ \text{this gives us}:\\\\1 - sin(a) = cos (a)\ \text{and} \ 1 - cos (a) = sin(a)[/tex]

Substituting these values in the eq2

[tex]cos(a)( 1 - cos(a)) + sin(a)( 1- sin(a) )\\\\cos(a)(sin(a)) + sin(a)(cos(a) )\\\\[/tex]

[tex]2(cos(a)(sin)(a))[/tex]

Therefore, 1 - sin(a) - cos(a) can be expressed as the product

[tex]2(cos(a)(sin)(a))[/tex]

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Answer:

52800 cm^3

Step-by-step explanation:

Given:

Rectangular prism

h = 6 cm (height)

l = 11 m = 1100 cm (length)

w = 8 cm (width)

[tex]v =a(base) \times h[/tex]

[tex]a(base) = l \times w = 1100 \times 8 = 8800 \: {cm}^{2} [/tex]

[tex]v = 8800 \times 6 = 52800 \: {cm}^{3} [/tex]

The simplified polynomial in standard form that represents the result of subtracting (z-7)² from 62 is -z² + 14z + 13.

Given that, ( z - 7 )² is subtracted from 62.

To subtract ( z - 7 )² from 62, we first need to first expand this ( z - 7 )².

( z - 7 )²

( z - 7 )( z - 7 )

z( z - 7 ) - 7( z - 7 )

Apply distributive property

z×z - z×7 - 7×z -7×-7

z² - 7z - 7z + 49

z² - 14z + 49

Now, we can substitute this expression into the original equation:

62 - (z² - 14z + 49)

Simplifying, we get:

13 + 14z - z²

-z² + 14z + 13

Therefore, the simplified polynomial is -z² + 14z + 13.

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With an eye level of 35 feet above the water, you can see approximately 7.24 miles to the horizon.

Multiplication is a mathematical operation that involves combining two or more numbers to find their total or product. It is often denoted by the symbol "×" or "*", and the result of multiplication is called the product. For example, multiplying 3 and 4 gives a product of 12, written as 3 × 4 = 12.

To find out how far you can see when your eye level is 35 feet above the water, we can use the given formula:

d = √(3h/2)

Plugging in h = 35, we get:

d = √(3 * 35/2)

d = √(105/2)

d = √52.5

d ≈ 7.24

So, when your eye level is 35 feet above the water, you can see approximately 7.24 miles to the horizon. Keep in mind that this is an approximation and actual visibility may vary due to factors such as atmospheric conditions and the height of objects obstructing the view.

Therefore, With an eye level of 35 feet above the water, you can see approximately 7.24 miles to the horizon.

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Since the question is asking about total employees, we must add 175 to 500 to get 675. Therefore, the correct answer is 675.

Increase means to grow or make larger in size, amount, degree, or intensity. It is the opposite of decrease, meaning to make or become less.

This is because when there were 500 employees, a 35% increase would be a total of 500 + 35% of 500

= 500 + 175

= 675.

To calculate 35% of 500, we use the formula:

(35/100) × 500 = 175

This means that 35% of 500 is 175, so when we add 175 to 500, we get 675.

Since the question is asking about  total employees, we must add 175 to 500 to get 675.

Therefore, the correct answer is 675.

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Step-by-step explanation:

Sol'n,

Here,

We can see that the Marked Price is Rs.80

a discount of x percent is added such that the new price i.e. SP with discount becomes about Rs.60

Now,

We know that,

Discount Percentage = (MP-SP with discount/SP with discount)×100%

= (80-60/60)×100%

=33.33%

Hence, by this we can conclude the discount percentage to be of about 33.33%..

a. The numeric values of the function are given as follows:

b. The function rule is given as follows: y = 2x - 8.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

From the table, when x = 0, y = -8, hence the intercept b is given as follows:

b = -8.

Considering points (0, -8) and (10, 12), when x increases by 10, y increases by 20, hence the slope m is given as follows:

m = 20/10

m = 2.

Hence the function is given as follows:

y = 2x - 8.

To complete the table, the numeric values are given as follows:

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The οptiοn “a dilatiοn fοllοwed by a reflectiοn” will prοduce an image that is nοt cοngruent tο its pre-image οptiοn, first οptiοn is cοrrect.

It is defined as the change in cοοrdinates and the shape οf the geοmetrical bοdy. It is alsο referred tο as a twο-dimensiοnal transfοrmatiοn. In the geοmetric transfοrmatiοn, changes in the geοmetry can be pοssible by rοtatiοn, translatiοn, reflectiοn, and glide translatiοn.

As we knοw, the translatiοn will shift the οbject.

If rοtate the image with 90 degree abοut οrigin will change the οrientatiοn οf the image but shape will remains same.

A dilatiοn fοllοwed by reflectiοn will change the image because the transfοrmatiοn dilatiοn is used tο resize an item. Dilatiοn is a technique fοr making items appear larger οr smaller.

Thus, the οptiοn "a dilatiοn fοllοwed by a reflectiοn" will prοduce an image that is nοt cοngruent tο its pre-image οptiοn fοurth is cοrrect.

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Complete question:

Which one of these transformations would produce an image that is not congruent to its pre-image? (5 points)

[Group of answer choices]