recall the birthday paradox problem: take a random sample of size k with replacement from the set {1, 2,⋯, 365}. let pk be the probability that there is repetition in the sample (there are at least two people who share a common birthday).

The circumference of a circle whose area is 9π cm² in terms of pi is 6π centimeters.

A circle is simply a closed 2-dimensional curved shape with no corners or edges.

The area of a circle is expressed mathematically as;

A = πr²

The circumference of a circle is expressed mathematically as;

C = 2πr

Given that the area of the circle is 9πcm², we determine the radius of the circle.

A = πr²

9πcm² = π × r²

r² = 9πcm² / π

r² = 9cm²

r = √9cm²

r = 3cm

Now, we determine the circumference of the circle.

C = 2πr

C = 2 × π × 3cm

C = 6cm × π

C = 6π cm

Therefore, the circumference of the circle is 6π centimeters.

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Two positive integers whose product is 24,999,999 are 4999 and 5001

Given

The product of two numbers should be 24,999,999.

The positive difference should be small

There are many ways to obtain those two numbers but an easiest way would be taking the square root of the given number and making decision by trial and error.

This may take few iterations but this is the optimal way to solve.

Square root of 24,999,999 is 4999.99

Hence try multiplying a number less than and a number equal to 4999.99.

Eventually we will reach to a conclusion 4999 x 5001 whose product is 24,999,999.

The difference of the numbers is 5001 - 4999 = 2

Hence the positive difference is small.

Hence the two numbers are 4999, 5001.

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By focusing on x- axis we can get value of f(3).

In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.

Given:

We have a function f(x).

Now, we have to find the value of f(3) from the graph

So, If we had the graph of f(x) and wanted to find f(3), we would look for the location on the graph where x = 3, then determine its y-value.

The response to f(3) is that y-value.

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just find the roots and substitute

a(x - "first root") (x - "second root")

Answer:

The set of all the first element of the ordered pairs of R is called domain and the set of all the second element of the ordered pairs of R is called range .

The number of mosquitoes on the colony after 4 days is 2800.

The number of days that there are 70,000 mosquitoes will be 114 dad.

The population of a colony of mosquitoes obeys the law of uninhibited growth and there are 1000 mosquitoes initially and there are 1600 after 1 day. The common difference will be:

= 1600 - 1000

= 600

The formula for the arithmetic sequence will be:

= a + (n - 1)d

a = 1000

d = 600

n = 4

The colony after 4 days will be:

= a + (n - 1)d

= 1000 + (4 - 1)600

= 1000 + 1800

= 2800

The number of days that there are 70,000 mosquitoes will be:

a + (n - 1)d = 70000

1000 + 600(n - 1) = 70000

1000 + 600n - 600 = 70000

Collect like terms

600n = 70000 - 1000 + 600

600n = 68400

n = 68400/600

n = 114 days

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The selling price of the adventurers swing set given the wholesale price and the mark-up price is $3,674.84.

The selling price of the adventurers swing set is the sum of the wholesale price and the percentage mark-up. Mark-up is the rate at which the selling price of an item is increased.

Percentage is the rate of an amount expressed as a number out of hundred. The sign that is used to represent percentages is %.

Selling price = wholesale price x (1 + mark-up price)

3,056 x (1 + 0.2025)

3056 x 1.2025 = $3,674.84

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Answer:

1.65

Step-by-step explanation:

Divide 6.6 by 4

Answer:

94 ft

Step-by-step explanation:

Looking at the diagram, PQ = 18 and PR = 31

We can see that

PQ + QR = PR

Therefore,

18 + QR = 31

QR = 31 - 18

QR = 13

Answer:

Step-by-step explanation:

STEP

1

:

           9

Simplify   —

           4

Equation at the end of step

1

:

                       9

 ((x2) -  5x) -  (0 -  —)  = 0

                       4

STEP

2

:

Rewriting the whole as an Equivalent Fraction

2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  4  as the denominator :

               x2 - 5x     (x2 - 5x) • 4

    x2 - 5x =  ———————  =  —————————————

                  1              4      

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

STEP

3

:

Pulling out like terms

3.1     Pull out like factors :

  x2 - 5x  =   x • (x - 5)

Adding fractions that have a common denominator :

3.2       Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • (x-5) • 4 - (-9)     4x2 - 20x + 9

————————————————————  =  —————————————

         4                     4      

Trying to factor by splitting the middle term

3.3     Factoring  4x2 - 20x + 9

The first term is,  4x2  its coefficient is  4 .

The middle term is,  -20x  its coefficient is  -20 .

The last term, "the constant", is  +9

Step-1 : Multiply the coefficient of the first term by the constant   4 • 9 = 36

Step-2 : Find two factors of  36  whose sum equals the coefficient of the middle term, which is   -20 .

     -36    +    -1    =    -37

     -18    +    -2    =    -20    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -18  and  -2

                    4x2 - 18x - 2x - 9

Step-4 : Add up the first 2 terms, pulling out like factors :

                   2x • (2x-9)

             Add up the last 2 terms, pulling out common factors :

                    1 • (2x-9)

Step-5 : Add up the four terms of step 4 :

                   (2x-1)  •  (2x-9)

            Which is the desired factorization

Equation at the end of step

3

:

 (2x - 9) • (2x - 1)

 ———————————————————  = 0

          4        

STEP

4

:

When a fraction equals zero :

4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

 (2x-9)•(2x-1)

 ————————————— • 4 = 0 • 4

       4      

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :

  (2x-9)  •  (2x-1)  = 0

Theory - Roots of a product :

4.2    A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

4.3      Solve  :    2x-9 = 0

Add  9  to both sides of the equation :

                     2x = 9

Divide both sides of the equation by 2:

                    x = 9/2 = 4.500

Solving a Single Variable Equation:

4.4      Solve  :    2x-1 = 0

Add  1  to both sides of the equation :

                     2x = 1

Divide both sides of the equation by 2:

                    x = 1/2 = 0.500

Supplement : Solving Quadratic Equation Directly

Solving    4x2-20x+9  = 0   directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex:

5.1      Find the Vertex of   y = 4x2-20x+9

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 4 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   2.5000  

Plugging into the parabola formula   2.5000  for  x  we can calculate the  y -coordinate :

 y = 4.0 * 2.50 * 2.50 - 20.0 * 2.50 + 9.0

or   y = -16.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x2-20x+9

Axis of Symmetry (dashed)  {x}={ 2.50}

Vertex at  {x,y} = { 2.50,-16.00}

x -Intercepts (Roots) :

Root 1 at  {x,y} = { 0.50, 0.00}

Root 2 at  {x,y} = { 4.50, 0.00}

Answer:

y - 4 = 3(x - 3) or y - 7 = 3(x - 4)

Step-by-step explanation:

y1 = mx1 + b

y2 = mx2 + b

(3, 4) = (x1, y1)

(4, 7) = (x2, y2)

4 = 3m + b

7 = 4m + b

-3 = -m

m = 3

y - y1 = m(x - x1)

y - 4 = 3(x - 3)

The height of the yellow mark above the water is 1 foot.

This is further explained below.

Generally, The radius of a waterwheel is 4 feet, and its center is 1 foot below the waterline.

Estimation of feet:

cos π/3 = OB/OA

1/2 = OB/4

OB = 2

t

In conclusion,A yellow mark should be placed above the water

= OB - OD

= 2 - 1

= 1 foot

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Answer:

the other answer is correct is A. 1 foot

Step-by-step explanation:

Got it right on edge

Answer:

Suppose every object in the domain has only one image in the codomain, then the mapping is a function. Otherwise, it is not a functions

Looking at the given mapping:

-2 is mapped to both 1 and 3

0 is mapped to -3

1 is mapped to 4

3 is mapped to 1

Because -2 has more than one image, it disqualifies the mapping from being a function.

Using a system of equations, it is found that the amounts of money that each of Gabriel, Daniel and Edgar have are given as follows:

A system of equations is when multiple variables are related, and equations are built to find the numeric values of each variable, according to the relations built in the context of the problem.

In the context of this problem, the variables are given as follows:

Gabriel has double the amount of Daniel, hence:

x = 2y.

Gabriel has five times the amount of Edgar, hence:

x = 5z.

If Gabriel loans Q14 to Daniel and Q33 to Edgar, they will have the same amount, hence:

x - 47 = y + 14 = z + 33.

Since x = 2y, we can replace into the first two sides of the equality above and solve for y as follows:

2y - 47 = y + 14

y = 61. (Daniel's amount).

The solution for x is given as follows:

x = 2y = 2(161) = 122 (Gabriel's amount).

The solution for z is given as follows:

x - 47 = z + 33

122 - 47 = z + 33

75 = z + 33

z = 75 - 33

z = 42 (Edgar's amount).

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Move the y = x2 graph 10 units to the right, then 8 units up.

There are y= x²——————> y = (x-10)²+8

The fact that

A point (x,y) in f(x) shifts up by d as a result of f(x)+d, a

and

f(x-e) ----> a point (x, y) in f(x) becomes a——————-> (x+e, y)——> shift right by  in this issue

d=8

e=10

Therefore,

Shift the y = x2 graph.

8 up, 10 to the right

The law is

(x,y)———-> (x+10,y+8)

The vertex in the equation y=x2--------- (0,0)

The vertex in the equation y = (x-10)2+8 (10,8)

so

(10,8)------> (0,0)----------> (x+10,y+8)———> (10,8)------> (0+10,0+8)

As a result, shift the y = x2 graph 8 units up and then 10 units to the right.

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Answer:

Step-by-step explanation:

The domain of a function is the set of all possible inputs for the function - so all possible x values in some f(x) functions.

From the graph, you can see that the values on the x-axis below the graph are in the range -4 (exclusive) and 2 (inclusive) that's why the range is
-4 < x <=2, or simply (-4,2>

The perimeter of rectangle with the given dimensions is 6x+6.

Given that the dimensions of rectangle are 2x and x+3.

We need to find the perimeter of the given rectangle.

The overall length of the outline determines a shape's perimeter in two dimensions. We add the lengths of all four corners of a rectangle to find its perimeter. Rectangle perimeter is calculated using the formula perimeter of a rectangle = 2(l + w), where l is the rectangle's length and w is its width.

We know that, the perimeter of rectangle = 2(Length + Width)

Now, perimeter

=2(2x+x+3)

=2(3x+3) (Add like terms)

Using distributive property multiply 2 to 3x and 2 to 3

We get 6x+6

Therefore, the perimeter of rectangle with the given dimensions is 6x+6.

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Answer:

  18 m/s²

Step-by-step explanation:

You want Dean's acceleration when his speed changes from 10 m/s to 100 m/s in 5 seconds.

Acceleration is the rate of change of velocity, the change in velocity divided by the corresponding change in time:

  a = ∆v/∆t

  a = (100 m/s -10 m/s)/(5 s) = (90 m/s)/(5 s) = 18 m/s²

Dean's acceleration is 18 m/s².

__

Additional comment

Both velocity and acceleration are vector quantities. They have both magnitude and direction.

We assume the given speeds are in the same direction, so the direction of the velocity vector does not change. The acceleration is in that same direction.

Answer: - 37

Step-by-step explanation:

so (2-5)(3+7) - (4+3) start off by the right side really doesn't matter what side you start on, but I added 4+3=7

x= (2-5)(10) - (7) Then add 3+7=10

x= (-3)(10) - (7) 3rd subtract 2-5=-3

x= -30 - 7 Next multiply -3*10=-30 because two numbers together with parenthesis means multiplication.

x= -37  Lastly subtract -30 - 7 and I got -37

Answer: 10%

Step-by-step explanation:

     First, we will find the amount that was saved from the markdown.

$50 - $45 = $5

     Next, we will find what percent this is of $50.

$5 / $50 = 0.1 ➜ 10%

     The discount, as a percentage, is 10%.

Answer:

r = 3t/t-1

Step-by-step explanation:

t = r/r-3

1. times both sides by r-3

t(r-3) = r

= tr - 3t = r

2. add 3t to both sides

tr = 3t + r

3. minus r

tr - r = 3t

4. factorise out r

r(t-1)=3t

5. divide by t-1

r = 3t/t-1

128 = 2^7 is a power of 2

27 = 3^3 is a perfect cube

=======================================================

Explanation:

This is something you'd find through trial and error.

The powers of 2 are

2, 4, 8, 16, 32, 64, 128, 256, ...

We double each value to get the next one.

The largest power of 2 that is smaller than 155 is 128

155-128 = 27 which is a cube number since x^3 = 3^3 = 27 as you've shown in the screenshot.

If the result of 155-128 wasn't a perfect cube number, then you'd have to try the next largest power of 2 (which is 64) and work your way down the list.

----------------------

Or you could work with the list of perfect cubes

1, 8, 27, 64, 125, 216, ...

Each value is of the form x^3 where x is a whole number.

Let's pick the largest perfect cube that is under 155, and subtract it

155-125 = 30 which isn't a power of 2 (refer to the list above)

Then try 155-64 = 91 which isn't a power of 2 either

Keep moving down the list until reaching 155-27 = 128 which is a power of 2 since 2^7 = 128

The graph of h against m is plotted in form of a curve.

Given:

A tree branch is observed to bend as the fruit growing on it increases in size.

The height h in metres of the branch end above the ground is closely approximated by the function [tex]h=2-0.2*1.60^{0.2m}[/tex] where m is the estimated mass, in kilograms, of fruit on the branch.

We have to sketch the graph of h against m.​

The x-axis represents the mass in kg while the y-axis represents the height of the branch above the ground in metres.

Hence a curved graph is obtained by plotting the function.

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Answer:

y = -x + 6

Step-by-step explanation:

Slope-intercept is the y = mx + b form, in which m is the slope, and b is the y-intercept.

From the graph provided, we can see that the line intercepts the y axis when the y-coordinate is 6, so we can use 6 as "b" in the equation.y₂

The slope can be found by picking two points on the graph, calling the first point (x₁, y₁) and the second point (x₂, y₂), and applying the formula slope = (y₂ - y₁) ÷ (x₂ - x₁).

Now, we pick two points - I'm choosing (1, 5) and (2, 4) - and apply the formula.

(4 - 5)  ÷ (2 - 1) = -1/1 = -1

Therefore, we know our slope is -1

Putting this all together, we get y = -1x + 6, which we can simplify to y = -x + 6 (since -1 * x is the same as -x)

Explanation

Given the trigonometric expression

We can find the solutions below;

From trigonometry identities;

Therefore; we will have;

Therefore, the general solutions for the above is given as;

Answer:

Answer:

18$ with a 1/3 discount applied

Step-by-step explanation:

27 divided by 3=9

27-9=18

Answer:

$18

Step-by-step explanation:

1/3 = ~33.33% off

33.33% of 27 = 9

27-9 = 18

Hope that helps