Three teams, A, B and C, play in a competition.
games won by A: games won by B = 4:3
games won by B: games won by C = 5:2
Team B has won 30 games.
In total, how many games have the three teams won?

Maximum number of identical boxes with no. of supply items in each box will be:

a. 78 boxes with 1 pencil and 1 eraser in each box.

b. 195 boxes with 1 notebook and 1 folder in each box.

c. 65 boxes with 1 eraser, 1 marker and 2 folders in each box.

First Lana will make 78 boxes with 1 pencil and 1 eraser in each box. After that she'll be left with

143 - 78 = 65 erasers.

Secondly she will make 195 boxes with 1 notebook and 1 folder in each box. After that she'll be left with

330 - 195 = 135 folders.

Next she will make 65 boxes with 1 eraser, 1 maker and 2 folders in each box.

By doing this she will be able to make maximum number of identical boxes.

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Using the provided questions with the values of sets A,B and C, Applying the cartesian product, The answer is {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)} and so on.

A mathematical expression is a combination of numbers, variables, and mathematical operations that defines a specific value or a set of values. Examples of mathematical expressions include 2+3, x^2+y^2, and sin(x) + cos(y). These expressions can be evaluated to find a specific numerical value or they can be simplified, rearranged or manipulated using algebraic rules.

In set theory, the Cartesian product of two sets A and B, denoted A x B, is the set of all ordered pairs (a, b) where a belongs to A and b belongs to B. It is the set of all possible combinations of elements from the two sets. For example, if A = {1, 2} and B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}. The Cartesian product is a fundamental concept in set theory and is used in many branches of mathematics, including linear algebra, topology, and graph theory. It can be extended to more than two sets, for example, the Cartesian product of three sets A, B, and C is denoted A x B x C and is the set of all ordered triples (a, b, c) where a belongs to A, b belongs to B, and c belongs to C.

1. A = {1,2,3}, B = {4,5,6} C = {a, b}

a. A x B = {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}

b. B x C = {(4,a),(4,b),(5,a),(5,b),(6,a),(6,b)}

c. A x C = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}

2. A = (3,6,9), B = (4,5,6)

a. A x B = {(3,4),(3,5),(3,6),(6,4),(6,5),(6,6),(9,4),(9,5),(9,6)}

b. Required relation = {(3,4),(3,5),(3,6)}

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The area of the trapezoid is \(\frac{1}{2}(b1+b2)*h\)

Therefore the area is:

   \(\frac{1}{2}*(3+5)*9 =\frac{1}{2}*(8)*9=4*9=36km^2\)

Area: 36 square kilometers

Hope that helps!

Using the volume of the sphere, it is found that 445.1 in³ of the ball is occupied by water.

The volume of a sphere of radius r is given by:

\(V = \frac{4\pi}{3}r^3\)

In this problem, the radius is of r = 5 in, hence the volume in cubic inches is given by:

\(V = \frac{4\pi}{3}(5)^3 = 523.6\)

85% of this is filled by water, hence the space occupied by water, in cubic inches, is of:

0.85 x 523.6 = 445.1.

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Answer:

To solve this problem, we will use the following steps:

Step 1: Let's assume that the larger number is x and the smaller number is y, we know that the difference between the two numbers is 9, so we can write the equation: x-y = 9

Step 2: We know that Four times the larger number plus one-half the smaller is 45, so we can write the equation: 4x + 0.5y = 45

Step 3: Now we have two equations with two unknowns, we can solve for one of the unknowns by isolating it in one of the equations.

x-y = 9

x = y+9

Step 4: Now we can substitute the value of x in the second equation

4(y+9) + 0.5y = 45

4y + 36 + 0.5y = 45

4.5y + 36 = 45

4.5y = 9

y=2

Step 5: We can substitute the value of y in one of the equation, to find the value of x

x-2 = 9

x = 11

Final Answer: The two numbers are 11 and 2.

Certainly! The problem can be solved using the Pythagorean theorem,

which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In this case, the ladder acts as the hypotenuse, and we need to find the length of the vertical side (height) it reaches up the wall.

The ladder forms the hypotenuse, and its length is given as 12 meters. The distance from the foot of the ladder to the base of the wall represents one side of the triangle, which is 4.5 meters.

By substituting the given values into the Pythagorean theorem equation: (12m)^2 = h^2 + (4.5m)^2, we can solve for the unknown height 'h'.

Squaring 12m gives us 144m^2, and squaring 4.5m yields 20.25m^2. By subtracting 20.25m^2 from both sides of the equation, we isolate 'h^2'.

We then take the square root of both sides to find 'h'. The square root of 123.75m^2 is approximately 11.12m.

Therefore, the ladder reaches a height of approximately 11.12 meters up the wall.

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The value of the given problem by using box method is 2546.

Multiplication computations involving numbers greater than ten can be simplified using the grid method, sometimes referred to as the box method, which is also known as the method of boxes. Making a table with the first factor's place values along the top and the second factor's place values down the left is how you multiply two numbers using the box approach. Place values are multiplied, and their products are added to the table. Clear up the table. In arithmetic, there is an alternative to the lengthy, error-prone classical multiplication approach: the box method. Multiplication of two or more digits is made easier with the use of the box approach.

X       60        7

30    1800    210

8       480      56

Therefore, 1800+480+210+56=2546

Therefore, the value of the given problem by using box method is 2546.

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Answer:

0.3

Step-by-step explanation:

3/5x1/2=3/10 = 0.33333

They can buy 120 vans, 60 small trucks, and 80 large trucks.

The truck company plans to spend 10 million on 260 vehicles. Each commercial van cost 25,000 dollars. Each small truck 50,000 dollars and each large truck 50,000 dollars. They needed twice as many van as small truck

Therefore,

let

s = number of small truck

number of van = v

let

l = number of large truck

v + s + l = 260

25,000(v) + 50,000(s) + 50,000(l) = 10,000, 000

v + 2s + 2l = 400

Hence,

v = 2s

So,

2s + 2s + 2l = 400

4s + 2l = 400

2s + s + l = 260

3s + l = 260

2s + l = 200

s = 60

l = 200 - 2(60)

l = 200 - 120

l = 80

v = 2(600 = 120

Therefore, they can buy the following:

number of small truck = 60

number of van = 120

number of large truck = 80

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Answer:

Arionna is incorrect. There are only 3 cubic feet in a cubic yard because there are 3 feet in a yard, and the proportion will remain constant.

Step-by-step expArionna is incorrect. There are only 3 cubic feet in a cubic yard because there are 3 feet in a yard, and the proportion will remain constant.

lanation:

Answer:

The short answer you are looking for is C!

Step-by-step explanation:

I hope you all live a long prosperous life, I wish that you and your family live long and happy, I hope that you will all get through this pandemic and get good flying grades! god bless you if you read this to the end, I love you. ʕっ•ᴥ•ʔっ <3

Answer:

66

Step-by-step explanation:

the difference of 54 and 32 is 54 - 32 = 22

the difference of 8 and 5 is 8 - 5 = 3

then

22 × 3 = 66

Answer:

Step-by-step explanation:

To find the number that Grace thought:

We'll represent the unknown number as "x."

"Added 7": This can be represented as (x + 7).

"Multiplied by 3": This becomes 3 * (x + 7).

"Took away 5": This is represented as 3 * (x + 7) - 5.

"Divided by 4": This gives (3 * (x + 7) - 5) / 4.

"To give an answer of 7": The equation is (3 * (x + 7) - 5) / 4 = 7.

Now we can solve for x:

(3 * (x + 7) - 5) / 4 = 7

Multiply both sides by 4 to eliminate the denominator:

3 * (x + 7) - 5 = 28

Simplify the left side:

3x + 21 - 5 = 28

Combine like terms:

3x + 16 = 28

Subtract 16 from both sides:

3x = 12

Divide both sides by 3:

x = 4

Therefore, the number that Grace thought of is 4.

Answer:

See below

Step-by-step explanation:

Starting with 42, there would be two branches of 2 and 21 connected to 42. Since 2 is prime, there are no more branches. Since 21 is composite, then two more branches are drawn of 3 and 7 connected to 21. Since 3 and 7 are prime, there are no more branches.

Therefore, the prime factorization of 42 is 2*3*7, and its factors would be 1,2,3,6,7,14,21,42.

Therefore, the probability that AT LEAST ONE of them has been vaccinated is 0.98693

If 58% = 58÷100 of the people have been vaccinated, then;

1 - (58/100) = 42% = 42÷100 of the people who have not been vaccinated.

Now:

Probability, P( > 1 ), that at least one of the selected four has been vaccinated is given by;

P( > 1 ) = 1 - P(0)                 -----------(1)

Where;

P(0) = probability that all of the four have not been vaccinated.

P(0) = P(1) x P(2) x P(3) x P(4)×p(5)

Where;

P(1) = Probability that the first out of the four has not been vaccinated

P(2) = Probability that the second out of the four has not been vaccinated

P(3) = Probability that the third out of the four has not been vaccinated

P(4) = Probability that the fourth out of the four has not been vaccinated

P(5)=Probability that the fifth out of the five has not been vaccinated

Remember that 42÷100 of the population has not been vaccinated. Therefore,

P(1) = 42÷100    

P(2) = 42÷100

P(3) = 42÷100

P(4) = 42÷100

P(5)=42÷100

P(0) = (42÷100) x (42÷100) x (42÷100) x (42÷100)×(42÷100)

P(0) = (42÷100)⁵

P(0) = (0.42)⁵

P(0) = 0.013067

Therefore, from equation (1);

P( > 1 ) = 1 -0.013067

P( > 1 ) = 0.98693

Therefore, the probability that AT LEAST ONE of them has been vaccinated is 0.98693

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Answer:

→please refer to the above attachment←

f(x) = 5x - 1 and g(x) = x² + 2

(fog)(x)=5x²+9

(fog)(-2)= 29

★ 29 is the right answer.

Answer:

huh

Step-by-step explanation:

Using linear function you would need to cross the bridge 30 times for the costs of the two toll options to be the same.

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The range of values that we are permitted to enter into our function is known as the domain of a function.

The collection of values that can be plugged into a function's domain are called its parameters. The x values for a function like f are contained in this collection (x). The collection of numbers that the function assumes is known as its range.

The entire range of independent variable values that can exist is referred to as a function's domain.

Let y = f(x) represent a function where x and y are independent variables. The chosen x-value is said to fall within the domain of f if a function f offers a means of effectively producing a single value y while using a value for x to that end.

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Step-by-step explanation:

Let x=distance traveled on 20 gal

One approach is to set up a simple ratio: 15 is to 240 as 20 is to x and this is written as follows:

15/240=20/x cross multiply (actually we are multiplying both sides by 240x)

15x=240*20=4800 divide both sides by 15

x=320 mi------------------------------------distance travelled on 20 gal

Another approach would be to determine the mileage the car gets in miles/gal:

If the car goes 240 mi on 15 gal, then it goes 240/15 or 16 mi on 1 gal.

Soooooooooo our car gets 16 mi/gal and it can go 16*20 or 320 mi on 20 gal

Answer: hope this help

Step-by-step explanation:

She reduced benefit would be approximately: $2,293.52 x (1 - 0.30) = $1,605.46 per month.

Average This is an arithmetic mean calculated by adding a set of numbers and then dividing by the number of those numbers. For example, the average of the numbers 2, 3, 3, 5, 7 and 10 is 30 divided by 6, which is 5.

 If Rachel retires at  age  67, her total Social Security pension can be estimated using the following steps:

Determine Rachel's average indexed monthly income (AIME) by taking the sum of her 35 highest  earnings (in today's dollars) and dividing it by 420 (the number of months over 35 years). GOAL = $3100 x 12 x 35 / 420 = $3100

Use the formula to determine the Primary Sum Assured (PIA), which is the monthly benefit amount payable under AIME at full retirement age. PIA = 90% x $996 32% x ($3100 - $996) = $2293.52

So, if Rachel were to retire at  age  67, her  Social Security pension would be approximately $2,293.52 per month.  If Rachel decides to start collecting her benefits now at age 62, her benefit amount will be reduced by a certain percentage based on the longer period  she receives benefits. The reduction factor is based on how many months before  full retirement age he begins receiving benefits.

In 2022, a person turning 62 will have a full retirement age of 67, which means the discount factor for starting benefits at age 62 is 30%. So, if Rachel started collecting  Social Security  now at age 62, her reduced benefit would be approximately:

$2,293.52 x (1 - 0.30) = $1,605.46 per month.

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Answer:

i go with 55

Step-by-step explanation:

Answer:

25°{vertically opposite angle}

hope it helps.

The value of the expression \(2^-^3(3^-^2)\)  is  \(\frac{1}{72}\).

What is value of the expression?

When the expression's variables and constants are given values, the calculation it describes yields the expression's value. In an algebraic expression, numbers can be represented by letters.

You must substitute a number for each variable and carry out the arithmetic operations in order to evaluate an algebraic expression.

In an algebraic expression, numbers can be represented by letters. The process of conducting the operations and inserting a predetermined value for each variable is known as "evaluating the expression."

Consider the given expression, \(2^-^3(3^-^2)\)

Apply the exponent rule: \(a^-^b=\frac{1}{a^b}\)

Here,

\(2^-^3=\frac{1}{2^3} \\3^-^2=\frac{1}{3^2}\)

So,

\(2^-^3(3^-^2)=\frac{1}{2^3}(\frac{1}{3^2})\\2^-^3(3^-^2) = \frac{1}{8}(\frac{1}{9}) \\ 2^-^3(3^-^2) = \frac{1}{72}\)

Hence, the value of the expression is, 1/72.

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Answer:

Yusuf's mistake is from Step 1

Explanation:

Given the equation;

for the first step let us subtract 2 from both sides;

To solve further, let us multiply both sides by 4;

Therefore, Yusuf's mistake is from Step 1

Answer:

f(x) = x³ - 4x + 6

f'(x) = 3x² - 4

a) f'(2) = 3(2²) - 4 = 12 - 4 = 8

6 = 8(2) + b

6 = 16 + b

b = -10

y = 8x - 10

b) 3x² - 4 = 0

3x² = 4, so x = ±2/√3 = ±(2/3)√3

= ±1.1547

f(-(2/3)√3) = 9.0792

f((2/3)√3) = 2.9208

c) 3x² - 4 = 8

3x² = 12

x² = 4, so x = ±2

f(-2) = (-2)³ - 4(-2) + 6 = -8 + 8 + 6 = 6

6 = -2(8) + b

6 = -16 + b

b = 22

y = 8x + 22

f(2) = 6

y = 8x - 10

The equation perpendicular to the tangent is y = -1/8x + 25/4

-The smallest slope on the curve is 2.92

The curve has the smallest slope at the point (1.15, 2.92)

The equations at tangent points are y = 8x + 16 and  y = 8x - 16

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

y = x³ - 4x + 6

Differentiate

So, we have

f'(x) = 3x² - 4

The point is (2, 6)

So, we have

f'(2) = 3(2)² - 4

f'(2) = 8

The slope of the perpendicular line is

Slope = -1/8

So, we have

y = -1/8(x - 2) + 6

y = -1/8x + 25/4

We have

f'(x) = 3x² - 4

Set to 0

3x² - 4 = 0

Solve for x

x = √[4/3]

x = 1.15

So, we have

Smallest slope = (√[4/3])³ - 4(√[4/3]) + 6

Smallest slope = 2.92

So, the smallest slope is 2.92 at (1.15, 2.92)

Here, we set f'(x) to 8

3x² - 4 = 8

Solve for x

x = ±2

Calculate y at x = ±2

y = (-2)³ - 4(-2) + 6 = 6: (-2, 0)

y = (2)³ - 4(2) + 6 = 6: (2, 0)

The equations at these points are

y = 8x + 16

y = 8x - 16

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The area inside the polar curve R = 2cos(3θ) for three full rotations is 6π.

The polar curve R = 2cos(3θ) is symmetric about the x-axis, with three petals that extend from θ = 0 to θ = π/3, π to 4π/3, and 5π/3 to 2π. To find the area inside the curve, we need to integrate the expression for the area element in polar coordinates over the desired region:

dA = (1/2) \(R^{2}\) dθ

The limits of integration for θ are from α = -6π to β = 6π, since we want to find the total area inside the curve for three full rotations. Thus, the integral for the area is:

A = ∫(β=6π, α=-6π) (1/2) \(R^{2}\)dθ

= ∫(β=6π, α=-6π) (1/2) (2cos(3θ))^2 dθ

= 2∫(β=6π/3, α=-6π/3) cos^2(3θ) d(3θ) (using the identity cos^2(3θ) = (1 +    cos(6θ))/2)

= ∫(β=6π/3, α=-6π/3) (1/2 + (1/2)cos(6θ)) d(3θ)

= (1/2)θ + (1/12)sin(6θ) ∣(β=6π/3, α=-6π/3)

= (1/2)(6π - (-6π)) + (1/12)(sin(36π) - sin(-36π))

= 6π

Correct Question :

Find The Area Inside R=2cos(3θ) (Tip: Use Α=−6π And Β=6π )

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Answer:

Dylan is 10 years old, and Genesis is 11.

Step-by-step explanation:

If Genesis and Dylan's age are consecutive integers, and Genesis is older, we can represent their ages as:

Dylan's age: x

Genesis' age: x+1

This would mean Genesis is a year older than Dylan.

The product of their ages is 110.

We can write an equation:

x×(x+1)=110

x²+x=110 (Distribute x)

x²+x-110=0 (Move 110 to the other side)

You can solve this by the quadratic equation, by factoring or by completing the square

I'll solve it by the quadratic equation:

We must first find the coefficients a, b and c, and then plug it into the formula.

\(x = \frac{ - 1 + - \sqrt{ {1}^{2} - 4 \times 1 \times - 110} }{2 \times 1} \\ x = \frac{ - 1 + - \sqrt{1 + 440} }{2} \\ x = \frac{ - 1 + - 21}{2} \)

Since we have a ± symbol, we get 2 real solutions, x1 and x2.

x=-1±21/2

x1=-1+21/2

x1=20/2

x1=10

x2=-1-21/2

x2=-22/2

x2=-11

Since their age can't be negative, x2 can't be a solution, so Dylan's age must be 10, and Genesis' age must be 11.

Hope this helps, and let me know if you need help with another method to solve this problem!

Answer:

I think it she can make 10 7/8 batches

Step-by-step explanation:

7 1/4 ÷ 2/3 = 10 7/8

Please tell me if im wrong sorry if i am

The expressiοn that shοws the result οf that transfοrmatiοn intο the bοx under it are  \(f(x) - 4 = 3^{x} - 4\)

In mathematics, a transfοrmatiοn refers tο a prοcess οr οperatiοn that changes the pοsitiοn, shape, οr size οf a geοmetric figure οr οbject in a cοοrdinate plane οr space. Transfοrmatiοns can be classified intο several types, including translatiοns, reflectiοns, rοtatiοns, dilatiοns, and cοmbinatiοns οf these.

Each type οf transfοrmatiοn invοlves a set οf rules οr fοrmulas that define hοw the οriginal οbject οr figure is transfοrmed intο a new οne. Transfοrmatiοns play an impοrtant rοle in many areas οf mathematics and science, including geοmetry, linear algebra, cοmputer graphics, and physics.

given functiοn : \(f (x ) = 3^{x}\)

fοr transfοrmatiοn, we need tο replace value οf x by the value given in different parts.

part 1 :

-4 f(x) = ?

we have \(f (x ) = 3^{x}\)

sο,  -4 ×  f (x ) = -4 × \(3^x\)

∴  -4 f(x) =  -4 × \(3^{x}\)

part 2 :  

f(-4x) = ?

we have f (x ) = \(3^{x}\)

replacing value οf x by -4x,

∴  f(-4x) = \(3^{-4x}\)

part 3

: f(x-4) = ?

we have \(f (x ) = 3^{x}\)

sο, replacing value οf x by x-4,

∴\(f(x-4) = 3^{x-4}\)

part 4 :

f(x) - 4 = ?

we have \(f (x ) =3^{x}\)

sο, \(f(x) - 4 = 3^{x} - 4\)

Hence the sοlutiοn are

-4 f(x) =  -4 × \(3^{x}\)

f(-4x) = \(3^{-4x}\)

f(x-4) = \(3^{x-4}\)

f(x) - 4 = \(3^{x} - 4\)

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if we notice the tickmarks on the triangle, we can see the all sides are equal, namely is an equilateral triangle, and if all sides are equal, then all interior angles are equal as well.

Let's recall that the sum of all interior angles in a triangle is 180°, now if we divide that by 3, that gives us 60, namely each angle in the equilateral triangle is 60°, that means

\(10x=60\implies x=\cfrac{60}{10}\implies \boxed{x=6} ~\hfill 12y=60\implies y=\cfrac{60}{12}\implies \boxed{y=5}\)