The Pigeonhole Principle is a counting principle that states that if n items are put into m containers, with n > m, then at least one container should give more than one item.
This principle is often used in counting problems to find the minimum number of items or containers needed to guarantee a certain outcome.
1. The smallest number of people in a group needed to guarantee that at least two were born in the same month is 13. This is because there are 12 months in a year, so if there are 12 people in the group, it is possible that each person was born in a different month. However, if there are 13 people in the group, by the Pigeonhole Principle, at least two people must have been born in the same month.
2. The smallest number of people are needed to guarantee that at least two have the same first and last initials is 53. There are 26 letters in the given alphabet, so there are
possible two-letter combinations for initials. If there are 52 people in the group, it is possible that each person has a unique pair of initials. However, if there are 53 people in the group, at least two people must have the same initials by the Pigeonhole Principle.
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 8. There are 7 days in a week, so if there are 7 people in the group, it is possible that each person was born on a different day of the week with a different first initial. However, if there are 8 people in the group, by the Pigeonhole Principle, at least two people must have been born on the same day of the week with the same first initial.
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 367. There are 366 possible days of the year (excluding February 29), so if there are 366 people in the group, it is possible that each person was born on a different day of the year. However, if there are 367 people in the group, by the Pigeonhole Principle, at least two people must have been born on the same day of the year.
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cos² (x)(1+tan²(x)) = 1, establish the identity
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
Please help with number 3!! Thank you so much!
Therefοre, the expressiοn can be rewritten in terms οf the functiοn [tex]cos x[/tex]as: [tex]2cos^2 x + cos x - 2[/tex].
What is trigοnοmetry?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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Help with math problems
(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.
What is the solution to the inequality 7 ≤ n + 5?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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A bicycle is originally priced at 80 then the owner gives a discount now it is priced at 60 enter the percent discount
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%..
Aiden models a can of ground coffee as a right cylinder. He measures its radius as 3 4 4 3 in and its volume as 12 cubic inches. Find the height of the can in inches. Round your answer to the nearest tenth if necessary.
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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Find the product of (4x − 3)(x + 2).
4x2 + 5x − 6
4x2 − 5x − 6
4x2 + 11x − 6
4x2 − 11x − 6
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
How do I solve those explain it to me please I need help ASAP thanks
PROGRESSIVE
Employee: "When I started here, there were 500 employees. Since then, we have
added 35% more employees, so we now have a total of
employees."
175
350
535
635
Shanna Bedgood
675
Since the question is asking about total employees, we must add 175 to 500 to get 675. Therefore, the correct answer is 675.
What does increase mean?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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The distance d(in miles) that you can see to the horizon with your eye level h feet above the water is given by d = √3h/2.
How far can you see when your eye level is 35 feet above the water?
With an eye level of 35 feet above the water, you can see approximately 7.24 miles to the horizon.
What is multiplication ?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.
According to the given information: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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if p(Bc)=0.9, find P(B)=
Find X + 50 pts and brainliest.
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
Decompose the composite figure to find its total area.
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
not sure what the answer is but I need the interpets
The x-intercepts of the given expression which is a rational function is calculated to be 0 and the y-intercepts are ±√25.
What does rational function mean?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)
YOU ARE GIVEN A MAGIC PENNY ON THE DAY YOU ARE BORN. YOU ARE TOLD THAT EACH YEAR ON YOUR BIRTHDAY, THE PENNY WILL MAGICALLY DOUBLE IF YOU DONT SPEND IT . HOW MUCH MONEY WILL YOU HAVE ON UR 10TH BIRTHDAY?
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
Express as a product "1 - sin(a) - cos (a)"
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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Which one of these transformations would produce an image that is not congruent to its pre-image? (5 points) Group of answer choices A dilation followed by a reflection A rotation of 90 degrees about the origin A translation of (x − 1, y − 3) A rotation followed by a translation
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.
What is geοmetric transfοrmatiοn?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]
A dilation followed by a reflection A rotation of 90 degrees about the origin A translation of (x − 1, y − 3) A rotation followed by a translation5 Cindy uses leather cord to make necklaces. She has a piece of leather cord that has a length of 3 yards. Each necklace will have a length of 28 inches. How many necklaces can Cindy make? Show your work.
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.
help pls show ur work
The height of the first floor above the second floor would be = 20 ft.
How to calculate the height of the first floor from the second floor?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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A certain drug is made from only two ingredients: compound A and B. There are 5 milliliters of compound A used for every 7 milliliters of compound B. If a chemist wants to make 1104 milliliters of the drug, how many milliliters of compound a needed?
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.
I js need help with this question
a. The numeric values of the function are given as follows:
x = -1, y = -10.x = 5, y = 2.b. The function rule is given as follows: y = 2x - 8.
How to define a linear function?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:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses the y-axis.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:
x = -1: y = 2(-1) - 8 = -2 - 8 = -10.x = 5: y = 2(5) - 8 = 10 - 8 = 2.More can be learned about linear functions at https://brainly.com/question/24808124
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I need help with this please
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
Please help Quick!! Will mark Brainliest too!
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]
find the length of the arc
(70 points)
can mark brainly
The calculated length of the arc has a value of 22.34 yards
Finding the length of the arcTo 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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4 2/3 x 1 1/2 x 2 1/4
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]
Written as a simplified polynomial in standard form, what is the result when
(z-7)² is subtracted from 62?
The simplified polynomial in standard form that represents the result of subtracting (z-7)² from 62 is -z² + 14z + 13.
What is the simplified form of the polynomial?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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What do I need to fix??
If we simplify [tex]3(2v-4)-4(v+4)[/tex], our answer will be [tex]-5v - 20[/tex]
How do we simplify the equation?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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The answer is 4 but I don’t understand why the graph is starting at -2.
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.
AmplitudeThe 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.
Period/FrequencyThe 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.
Horizontal shiftThere 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.
What are two expressions that are equivalent to the expression 5(6j + 3 + j)
Answer:
5(7j+3) AND 35j +15 are both equal to the expression 5(6j + 3 + j)
use the info given in the figure to find x, m
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.
A parabola opening up or down has vertex (0, -4) and passes through (10, 1). Write its
equation in vertex form.
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: