For each of the following, count the number of four-digit numberssatisfying the condition and with digits in {1, 2,3, 4,5, 6}:(a) The digits are distinct(b) The number is even(c) The number is even and the digits are distinct.

Herb spent 55 hours at the library.

Fractions defines a part of a whole and are represented as a numerical value. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.

Total time spent in the Library = 1/6 + 1/4 + 1/2

Total time spent = 11/12 x 60 mins

Total time spent = 55 hours

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The lines L₁ and L₂ given by the vector equations is perpendicular.

In order to prove that the lines are perpendicular,

L₁: r(t) = (-5+ 3t)i + (3 + t)j

L₂: r(s) = (2+2s)i + (4 - 6s)j

To determine whether the lines L₁ and L2 are parallel, perpendicular, or neither, we need to compare their direction vectors.

The direction vector for L₁ is given by <3,1> and the direction vector for L₂ is given by <2,-6>.

The dot product of the direction vectors is (32) + (1-6) = -8. Since the dot product is zero, the vectors are perpendicular, and therefore the lines are perpendicular.

Therefore, the lines are PERPENDICULAR.

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Another point on the line is  (2, 3)

To find another point on the line the equation of the line will be solved as used to get other points

The slope, m of the lines is calculated the formula

m = (y₀ - y₁) / (x₀ - x₁)

where

m = slope

x₂ and x₁ = points in x coordinates

y₂ and y₁ = points in y coordinates

The slope, m of the linear function is calculated using the points (0, 5), (4, 1)

m = (y₀ - y₁) / (x₀ - x₁)

m = (5 - 1) / (0 - 4)

m = (4) / (-4)

m = -1

equation passing through point (0, 5)

(y - y₁) = m  (x - x₁)

y - 5 = -1 (x - 0)

y - 5 = -x

y = -x + 5 equation of the line

for x = 2

y = -x + 5 = -2 + 5 = 3

the point is (2, 3)

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Answer:

First, find out how many guests have ordered chicken: 240 guests * 3535/10000 = 84 guests

Then subtract that number from the total number of guests to find the number of guests who haven't ordered chicken: 240 guests - 84 guests = 156 guests

Then find how many of those guests have ordered gluten-free meals: 156 guests * 12.5% = 19.5 guests

Then round down the number of guests who have ordered gluten-free meals to the nearest whole number to find the exact number of guests who ordered gluten-free meals: 19 guests

Then subtract that number from the number of guests who haven't ordered chicken to find the number of guests who haven't ordered their meals yet: 156 guests - 19 guests = 137 guests

So 137 people haven't ordered their meals yet.

The amount each boy will have to contribute is $56.23

How to calculate the amount that each boy will contribute?

Andrew and his two brother want to contribute money together to buy a new game system

The cost of the game system is $168.69

Each boy want to contribute the same amount

Therefore the amount that each boy will contribute can be calculated as follows

There are 3 boys in total, Andrew and his two brothers

= 168.69/3

= 56.23

Hence each boy will contribute an amount of $56.23

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The piggy bank has 200 dimes and 50 quarters.

To find this, we can set up the following equation:

(number of dimes x $0.10) + (number of quarters x $0.25) = $39.25

We know that the piggy bank contains 250 coins, so we can substitute that in:

(x x $0.10) + (250-x x $0.25) = $39.25

Now we can solve for x, the number of dimes:

x0.10 + (250-x)0.25 = 39.25

x0.10 + 62.5 - x0.25 = 39.25

0.1x + 62.5 - 0.25x = 39.25

-0.15x = -23

x= 153

So there are 153 dimes in the piggy bank, and 250-153 = 97 quarters.

But we know that the piggy bank contains 250 coins, so we can check that 153 dimes + 97 quarters = 250 coins.

(a) The maximum value of g(x) = -8[tex]x^2[/tex] + 5x - 2 is 5/16.

(b) The maximum value of h(x) = -(1/3)[tex]x^2[/tex] + 3x is 9/2.

(c) The maximum value of k(x) = 8[tex]x^2[/tex] - 5x - 7 is 5/16.

(d) The maximum value of j(x) = -(71/99)[tex]x^2[/tex] + 210 is 0.

(e) The maximum value of f(x) = 4[tex]x^2[/tex] - 3x + 7 is 3/8.

(f) The maximum value of F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9 is 0/0. We cannot say anything about that function.

The general form of a quadratic function;

f(x) = a[tex]x^2[/tex] + bx + c and a < 0,

Then the function f has a maximum value when x = -b/2a

(a) The function is g(x) = -8[tex]x^2[/tex] + 5x - 2

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -8 and b = 5

Now putting the value at x = -b/2a

x = -5/2(-8)

x = (-5)/(-16)

x = 5/16

(b) The function is h (x) = -(1/3)[tex]x^2[/tex] + 3x

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(1/3) and b = 3

Now putting the value at x = -b/2a

x = (-3)/2(-1/3)

x = (3×3)/2

x = 9/2

(c) The function is k(x) = 8[tex]x^2[/tex] - 5x - 7

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 8 and b = -5

Now putting the value at x = -b/2a

x = -(-5)/(2×8)

x = 5/16

(d) The function is j(x) = -(71/99)[tex]x^2[/tex] + 210

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(71/99) and b = 0

Now putting the value at x = -b/2a

x = 0/2(-(71/99))

x = 0

(e) The function is f(x) = 4[tex]x^2[/tex] - 3x + 7

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 4 and b = -3

Now putting the value at x = -b/2a

x = -(-3)/(2×4)

x = 3/8

(f) The function is F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9

We can find the maximum value at x = -b/2a.

On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 0 and b = 0

Now putting the value at x = -b/2a

x = 0/0

We cannot say anything about that function.

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The complete question is given below:

The equation in slope-intercept form that models the total cost of attendance is y = 4487x + 750.

We know lines have various types of equations, the general type is

Ax + By + c = 0, and the equation of a line in slope-intercept form is

y = mx + b.

Where slope = m and b = y-intercept.

the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects

the y-axis at x = 0.

To find the equation in the slope-intercept form,

We'll sum all the annual fees and subtract the scholarship amount and 'x' will represent the number of years while the one-time fee will be our constant term.

So, (6125 + 2112 + 1250 - 5000).

= 4487.

Therefore, The equation in slope-intercept form is,

y = 4487x + 750.

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Answer:

$ 148.75

Step-by-step explanation:

       Principle = $ 425

         Rate = 7%

       Time = 5 yeats

[tex]\sf \boxed{Simple \ Interest = \dfrac{Principle*rate*time}{100}}[/tex]

                           [tex]= \dfrac{425*7*5}{100}\\\\=\dfrac{17*7*5}{4}\\\\\\=\dfrac{595}{4}\\\\=148.75[/tex]

Answer:

X = 11

Step-by-step explanation:

The total degrees of a line is 180

They gave us 39 and 13x-2

We can get rid of the 39 by subtracting it from 180

180-39=141

That means 13x-2 = 141

Now to find X. Adding 2 on both sides, gets rid of the 2 on the left side

13x = 143

Dividing 13 on both sides, we can get X alone

X = 11

Therefore, the 3 patterns are
3,6,9,3,6,9            

Explanation:                      

Given that,

we will carry out the multiplications of succession  counting number by 3 and if there is more than a single digit answer .we add the digits.
we are looking for is a pattern for there single digit numerals.
carryout the plan                  

1x3=3→3
2x3=6→6
3x3=9→9
4x3=12→1+2=3
5x3=15→1+5=6
and so on
continue with some additional at terms.
6x3=18→1+8=9
7x3=21→2+1=3

∵ Therefore the 3 patterns are
3,6,9,3,6,9


A pattern is defined in mathematics as a sequence of repeating objects, shapes, or numbers. A pattern can be associated with any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.


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When evaluating the likelihood of an outcome, the claim or hypothesis statement is assumed. The required answer is a hypothesis claim or statement.

It is possible to test one's assumptions regarding a population parameter through the statistical analysis known as hypothesis testing. The association between two statistical variables can then be estimated by this hypothesis.

We start by writing the claim mathematically and determining if it is the null or alternative hypothesis. The null hypothesis is the assertion that there is no change or no relationship between the two variables; otherwise, the alternative hypothesis is the assertion that tells that there is a change or relationship.

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After the raise, Lisa much will pay $9 in federal income taxes Annually.

For the stated question:

Lisa's salary =  $215.20 a week.

After increment of $40 per week.

$215.20  + $40 = $255.2

She gets 3 allowances and is married.

After raise,

Federal income taxes = $9. (taken from the table)

Thus, After the raise, Lisa much will pay $9 in federal income taxes Annually.

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Center of the sphere would be (-4.5, 1, -6)  and radius be 6.086.

Write the equation of the sphere?

A sphere's general equation is (x - a)2 + (y - b)2 + (z - c) 2 = r2, where (a, b, c) represents the sphere's center, r represents the radius, and x, y, and z are the coordinates of points on the sphere's surface.

To complete the square, we need to add and subtract the square of half of the x coefficient and y coefficient, and the square of half of the z coefficient.

The equation will be : [tex](x + 9/2)^2 + (y - 1)^2 + (z + 6)^2 = (9/2)^2 + 1^2 + 6^2[/tex]

So the standard form of the equation of the sphere is,    

          [tex](x + 4.5)^2 + (y - 1)^2 + (z + 6)^2 = 37[/tex]

The center of the sphere is (x, y, z) = (-4.5, 1, -6) and the radius is √37 ≈ 6.086

Therefore, Center of the sphere would be (-4.5, 1, -6) and radius be 6.086.

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It is proven that the other nth roots of z can be found by multiplying the initial one by the nth roots of unity.

A complex number satisfying the equation [tex]z^{n-1} = 0[/tex] is known as the nth root of unity. There are about n nth roots of a unit, according to the fundamental theorem of algebra. We know that,[tex]1=e^{2\pi i}[/tex]. Then, the nth root is written as [tex]\omega=e^{2\pi i/n}[/tex].

The integer of powers of this root between 0 and n-1 provides the other nth roots. Then, the roots are [tex]1, \omega, \omega^2 ,\cdots \cdots, \omega^{(n-1)}[/tex]. Now, factorize the equation [tex]z^{(n-1)} = 0[/tex] we get the required identity as, [tex]z^{(n-1)} = (z- 1)(z^{(n-1)}+ z^{(n-2)} + \cdots \cdots+ z + 1) = 0[/tex].  

This indicates that the equation [tex]1+\omega+ \omega^2+\cdots \cdots+ \omega^{(n-1)}[/tex] is true for all nth roots of unity other than 1.

Roots of unity also have the great quality of being periodic, which results from the fact that n = 1. It is equal to a power of that power mod n if the power is bigger than n.

When ω is a third root of unity, [tex]\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0[/tex].

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Step-by-step explanation:

it is a parallelogram.

in comparison to a rectangle one side is shifted a bit, but the endpoints of the shifted side are still positioned the same way relative to each other.

and particularly, their relative offset from each other must be the same as for the endpoints of the other, parallel side.

let's look at

(0, -1) and (2, 7).

their relative offset is given by

x + 2 (from 0 to 2)

y + 8 (from -1 to 7)

the missing point must have the same offset from (-5, -3), as this line is parallel to the one above :

-5 + 2 = -3

-3 + 8 = 5

so, the missing point is (-3, 5).

The area of the rectangle is and using the length and breadth

area of rectangle = length × breadth

area of rectangle = 2.5m × (4n +5)

area of rectangle = 10mn + 12.5m

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Question:

The dimensions, in feet, of a rectangle, are represented by expressions, as shown in the diagram below. The area of the rectangle is 10mn + 12.5m. Explain your work.

Answer:

x = 12

4 races

Step-by-step explanation:

2/3 = 8/x

Cross multiply.

2x = 8 × 3

x = 4 × 3

x = 12

Answer: x = 12

0.75 × 45 = 33.75

Since the number of races must be an integer, they must win a total of 34 races to have at least a 75% success rate.

They already won 30 races.

34 - 30 = 4

Answer: 4 races

The equivalent expression of i⁴¹ is i.

Expressions that are equivalent to the same thing, even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).

Given:

An expression i⁴¹.

Simplifying,

i⁴¹ = {(i⁴)¹⁰} i

i⁴¹ = {(1)¹⁰} i   {i⁴ = 1}

i⁴¹ = i

Therefore, the equivalent expression is i.

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The solution of the given equation is X = B(C - A)⁻¹.

To solve for X in the equation AX + B = CX, we can rearrange the equation as follows:

AX - CX = -B

Factor out X on the left side:

(X(A - C)) = -B

To isolate X, we can multiply both sides of the equation by the inverse of (A - C), assuming it exists:

X = -(A - C)⁻¹ × B

X = (C - A)⁻¹ × B

Therefore, the solution for X is:

X = B(C - A)⁻¹

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The sum of four consecutive integers is,

⇒ 4n + 6

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

We have to given that;

⇒ n = first integer of the four.

Now, Let four consecutive integers with n is first integer of the four are,

⇒ n, n + 1, n + 2, n + 3

Hence, The sum of four consecutive integers is,

⇒ S = n + (n + 1) + (n + 2) + (n + 3)

      = 4n + 6

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1) the rate change of function f(x) in given interval =4(b+1)

2)  the rate change of function g(x) in given interval =1

3) the rate change of function j(x) in given interval =[tex]3h^2+9h+9[/tex]

To find out the average rate change of function on given interval is -

Use the graph of f to determine f (a) and f (b) for the provided range [a,b]. Remember that the graph of f's point with the y-coordinate of f (a) and the x-coordinate of an is called f. Similar to that, f ( b ) is the y-coordinate of the point on the f graph whose x-coordinate is b.

Step 2: Calculate the average rate of change of  f over the range [a, b] using the average rate of change formula.

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

1)

[tex]f(x)=4x^2-7 \ on\ [1,b]\\\\f(1)=4-7=-3\\\\f(b)=4b^2-7\\\\the \ rate \ change =\frac{4b^2-7+3}{b-1}\\\\=4(b+1)[/tex]

2)

[tex]p(x)=3x+4 \ on\ [2,2+h]\\\\p(2)=10\\\\p(2+h)=10+h\\\\The\ average\ rate \ change= \frac{10+h-10}{2+h-2}\\\\=1[/tex]

3)

[tex]j(x)=3x^3 \ on \ [1,1+h]\\\\j(1)=3\\\\j(1+h)= 3(1+h)^3\\\\\\so, the\ rate\ change = 3h^2+9h+9[/tex]

The complete Question is: -

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

1. f(x)= 4x^2 - 7 on [1, b]

2. p(x)= 3x+4 on [2,2+h]

3. j(x)= 3x3 on [1, 1+h]

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Answer: (-4t - 7) + (3t + 8) cannot be simplified further as the terms have different variables.

(-4r and -7) and (3r and 8) cannot be combined as these terms have different variables.

(-41 and 3r) cannot be combined as these terms have different variables.

Step-by-step explanation:

Answer: 3

Step-by-step explanation: 60 x 5/100 = 3

There were three first-year teachers in the meeting.

In the problem, it is solved that the number of dogs John has is 14

The problem is solved by expression the word problem into a simultaneous equation as follows

let d be the number of dogs and c d number of chickens

d + c = 25 (since all must have a head)

4d + 2c = 78 (2 legs for chicken and 4 legs for dog)

using calculator to solve gives

d = 14 and c = 11

hence the number of dogs is 14

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One can infer some information about a line's slope just by looking at its graph, especially in comparison to other lines plotted on the same coordinate plane.

Rise / run = a slope.

The Complete Question is line's slope.

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To show that the improper integral g(x)dx is divergent, but the average value of g on the interval 4, infinity is finite, we can consider a function g(x) = 1/x, which is mostly used as an example for a divergent integral.

The improper integral of g(x) from 4 to infinity is given by:

[tex]\int\limits {g(x)} \, dx = \int\limits {(1/x)} \, dx[/tex]  from 4 to infinity

= lim (upper limit -> infinity) [ln|x|] from 4 to x

= [ln|x|] from 4 to infinity

= ln(infinity) - ln(4) = infinity - ln(4) = infinity

As the limit of the integral is infinity, the integral is divergent.

However, the average value of g on the interval [4, ∞) is given by:

average value = (1/∞ - 4) * ∫(1/x)dx from 4 to infinity

= (1/∞ - 4) * (infinity - ln(4))

= -4 + ln(4)

= ln(4) - 4

which is a finite value when the integral is divergent

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The problems with Alisha's statistical study are that the difficulty levels of the sudoku puzzles might be different, self timing can lead to inaccurate measures of time and also the experience and skill level of her friends will be different.

In her statistical study, Alisha asked 10 of her friends to try 10 different sudoku puzzles from the internet and time themselves while doing it. In this study there are a few errors. The first one being that the difficulty level of the puzzles that her friends will be solving are going to be different.

Another issue is that since her friends are self-timing, there can be inaccurate measures of time and lastly, the experience and skill of her friends for sudoku is going to create a bias in her study.

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The required set of positive two-digit even numbers less than 36 that do not contain the digit 2 is 11,13,14,15,16,17,18,19,30,31,33,34,35.

An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Zero is not a fraction or decimal of any number. It is neither positive nor negative.

Given:

Set D is the set of positive two-digit even numbers less than 36 that do not contain the digit 2.

According to the given question, we have

2-digit numbers are numbers that have two digits and they start with the number 10 and end with the number 99.

Starting at 11 and going up the two-digit numbers even numbers less than 36 that do not contain the digit 2.

11,13,14,15,16,17,18,19,30,31,33,34,35.

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Python lines to get the output for the probability P(x<0.325) are -

a) import scipy.stats as

st print(st.norm.cdf(0.325, 0, 1))

Hence, option a is the correct answer.

The Python code -

        import scipy.stats as

        st print(st.norm.cdf(0.325, 0, 1))

This line of code imports the scipy.stats module as "st", and then uses the norm.cdf function to calculate the cumulative distribution function (CDF) of a normal distribution with mean 0 and standard deviation 1, evaluated at x=0.325.

The CDF gives the probability that a random variable from the normal distribution is less than or equal to a certain value, so this line of code is outputting the probability P(x<0.325).

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