A teacher buys a binder for $1. 75 and a monthly planner for $9. 50 for each student in their classroom. The teacher spends a total of $393. 75 for all of the binders and monthly planners. How many students are in the teacher's class?

Answer:

Equation of parabola in vertex form:
y = -5( x + 1)² + 3

Equation of parabola in standard form:
y = -5x² -10x - 2

Step-by-step explanation:

The vertex form equation for a parabola is

y = a(x - h)² + k

where (h, k) is the vertex and a is a constant

Given (h, k) = (- 1, 3)
h = -1 , k = 3

y = a( x - (-1) )² + 3
y = a(x + 1)² + 3

To find a, we know the parabola passes through point(- 2, - 2). Plug in x = -2, y = -2 and solve for a

- 2 = a( -2 + 1)² + 3

-2 = a(-1)² + 3

-2 = a · 1 + 3

-2 = a + 3

Subtract 3 from both sides:

-2 - 3 = a + 3 -3

-5 = a

or

a = -5

Equation of parabola in vertex form:
y = -5( x + 1)² + 3

The standard form of this parabola is y = ax² + bx + c

Expand (x + 1)² = x² + 2 · 1 · x + 1² = x² + 2x + 1

Therefore

y = -5( x + 1)² + 3 becomes
y = -5( x² + 2x + 1) + 3

y = -5x² -10x - 5 + 3

y = -5x² -10x - 2

Answer:

79° , 65° , 144°

Step-by-step explanation:

65° , x° and 36° lie on a straight line and sum to 180° , that is

65° + x° + 36° = 180°

101° + x° = 180 ( subtract 101° from both sides )

x= 79°

y + 10 and 65° are alternate angles and are congruent , then

y + 10 = 65°

z and 36° are same- side interior angles and sum to 180° , that is

z + 36° = 180° ( subtract 36° from both sides )

z = 144°

(2π)^(-1)E[∫▒〖e^(-√(-1) xy)]dy〗

The characteristic function of a random variable X is defined as ∅(t) = E[e^(√(-1)tx)]. To show that (2π)^(-1) ∫▒〖e^(-√(-1) xt)∅(t)dt〗 is the Lebesgue density of X, we need to demonstrate that it is a probability density function.

We first calculate the integral to obtain (2π)^(-1) ∫▒〖e^(-√(-1) xt)∅(t)dt〗 = (2π)^(-1) ∫▒〖E[e^(-√(-1) xt +√(-1) tx)]dt〗.

Since the expectation is a constant, we can pull it out of the integral to get (2π)^(-1)E[∫▒〖e^(-√(-1) xt +√(-1) tx)]dt〗.

Now, if we substitute t = y-x and rewrite the integral, we obtain (2π)^(-1)E[∫▒〖e^(-√(-1) x(y-x))e^(√(-1) xy)]dy〗.

This simplifies to (2π)^(-1)E[∫▒〖e^(-√(-1) xy)]dy〗.

Because the integrand is 1, the integral is simply y. Then, the expectation becomes E[y] which is the mean of the random variable X. Thus, the Lebesgue density of X is (2π)^(-1)E[y], which is the probability density function of X.

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The answer is: x^(2) + 20/(x-2)

Synthetic division is a method of dividing a polynomial by a linear factor. It is a shorthand version of long division that is quicker and easier to do.

Here are the steps to solve this problem using synthetic division:

1. Write the coefficients of the dividend in a row: 1 -2 0 10
2. Write the constant term of the divisor to the left of the row: 2 | 1 -2 0 10
3. Bring down the first coefficient: 2 | 1 -2 0 10
      |_______
       1
4. Multiply the first coefficient by the divisor and write the result under the second coefficient: 2 | 1 -2 0 10
      |___2___
       1  0
5. Add the second coefficient and the result: 2 | 1 -2 0 10
      |___2___
       1  0
6. Repeat the process with the remaining coefficients: 2 | 1 -2 0 10
      |___2__0_20
       1  0 0 20
7. The final row of numbers represents the coefficients of the quotient and the remainder: (x^(2) + 0x + 0) + 20/(x-2)

So, the answer is: x^(2) + 20/(x-2)

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Answer:

3/5

Step-by-step explanation:

In your problem you have a right triangle.

For this problem, you need to know what sine is. Sin is [tex]\\\frac{opposite}{hypotenuse}[/tex]

In this problem we know that the hypotenuse is 20, but we do not know the opposite. The opposite refers to the side that is opposite to the angle. In this case, the angle we are using is c. Also, in this case, the name of the opposite side is ED. To figure out that side, we need to use the pythagorean theorem.

[tex]a^{2} +b^{2} =c^{2}[/tex]

A is 16, c is 20

Therfore we can plug those values into our formula

[tex]16^{2} +b^{2}=20^{2}[/tex]

Simplify:

[tex]256 + b^{2} =400[/tex]

Simplify even more:

[tex]b^{2} = 144[/tex]

Square root both sides:

[tex]b = 12[/tex]

Now we Know all 3 sides of our triangle: 20, 16, and 12.

Sin refers to OPPOSITE/HYPOTENUSE

Our opposite is 12, and our hypotenuse is 20.

Plugging those values into our formula gives us:

12/20

Simplify:

3/5.

THERFORE, THE ANSWER IS 3/5

**Also, this whole process seems long, but in reality it is very short and you'll master it in no time!

The x number for which the following functions are equivalent is determined to be 3 by the preceding statement.

The term "function" refers to the correlation between a set of inputs and outputs. Simply defined, a function is an input-output relationship where each input is coupled to exactly one output.

The given functions are f(x) = x - 2 and f(x) = -2x + 7.

Assign the functions the following equivalences to determine the value of x:

variable x - 2 = -2x + 7

variable x + 2x = 7 + 2

Therefore, x = 3

Hence, the specified value for x is 3.

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Inverse of the given matrix is

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

the matrix in question is

−1 0 1

2 2 0

1 0 1

now, we are given that inverse of the matrix exists.
A⁻¹ = adjoint of A/|A|

The determinant of the matrix A is:
det(A) = (-1) [(2)(1) - (0)(0)] - 0 [(2)(1)-(0)(1)] + 1 [(2)(0) - (2)(1)] = -4

The adjoint of the matrix A is:

2 2 −2

0 −2 −2

−2 −2 −2

Therefore, the inverse of the matrix A is:

A⁻¹ =  1/(-4)  [2 2 −2]

                  [0 −2 −2]

                  [−2 −2 −2]

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

So, the inverse of the matrix A is:

A⁻¹  =     [0.5   0.5 -0.5]
            [ 0    -0.5  -0.5]
            [-0.5 -0.5  -0.5]

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To decode the words 0011, 0101, 0110 and 1111, we can find the corresponding codeword in the standard array. The codewords for 0011, 0101, 0110 and 1111 are 0011, 0101, 0110 and 1111, respectively.

To construct a standard array for this code, we can start by finding the generator polynomial of the code. This can be done by multiplying the generator matrix with its transpose and finding the resultant matrix. This results in the generator polynomial: 110101001.

To construct the standard array, we first construct a parity check matrix, which is the transpose of the generator matrix: 0101 1001 1100 1000.

From the parity check matrix, we can then construct the standard array. The standard array consists of all the possible codewords given the generator polynomial and the parity check matrix. It can be constructed by multiplying the generator matrix with all possible vectors of length 3.

The standard array of the code is:
0011  0101  0110  1001  1010  1100  1101  1111

To decode the words 0011, 0101, 0110 and 1111, we can find the corresponding codeword in the standard array. The codewords for 0011, 0101, 0110 and 1111 are 0011, 0101, 0110 and 1111, respectively.

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If x is the number of months the fee was charged, then the inequality to determine at most months the fee was charged is x ≥ (y + 46.83)/5.95  .

Let x be the number of months the fee was charged.

Let y be the initial balance before the fee was charged.

So, we can write the inequality as : y - 5.95x ≤ -46.83

where y is = positive number representing the initial balance.

Since the initial balance is "y" , we can determine the value of x that satisfies this inequality.

The inequality is represented as : x ≥ (y + 46.83)/5.95 .

Therefore, the inequality to determine number of months is x ≥ (y + 46.83)/5.95.

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The toy has a volume of 200 cm³. The correct answer is Option d.

This is because when an object is submerged in water, it displaces an amount of water equal to its volume. In this case, the pool toy displaces 200 mL of water, which means it has a volume of 200 cm³ (since 1 mL is equal to 1 cm³). The correct answer is Option d.

It is not correct to say that the toy weighs 200 grams (option a), as the weight of the toy is not related to the amount of water it displaces. Similarly, it is not correct to say that the toy absorbed 200 mL of water (option b), as the toy is simply displacing the water, not absorbing it.

Finally, it is  correct to say that the toy has a surface area of 200 cm² (option c), as the amount of water displaced is related to the toy's volume, not its surface area.

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The measure of angle ABC is 64°.

The angle bisector in geometry is the ray, line, or segment which divides a given angle into two equal parts.

Given that, BM bisects ∠ABC and ∠MBC=32°.

Here, ∠ABC=∠MBC+∠MBA

Since, BM bisects ∠ABC, ∠MBC=∠MBA

∠ABC=∠MBC+∠MBC

∠ABC=2∠MBC

∠ABC=2×32°

∠ABC=64°

Therefore, the measure of angle ABC is 64°.

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We can conclude that -

The exponential function is of the form -

f(x) = eˣ

Given is to write the exponential growth and exponential decay equation.

{ 1 } -

The general exponential growth equation is -

[tex]$f(x)=a(1+r)^{x}[/tex]

{ 2 } -

The general exponential decay equation is -

[tex]$\frac {dN}{dt}= -\lambda N[/tex]

Option {A} and option {E} represent the exponential decay. Option {B}, {E} and {D} represent exponential growth.

Therefore, we can conclude that -

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Answer: x = p/2

Step-by-step explanation:

x = next one
p = previous

in order to solve the next one, you would do :
x (the one you are trying to solve) = p (the previous one : 1/8) divided by 2
x = p/2

The answer of  lines intersect at the point is (1, 2, -1)

To determine whether the lines L1(t) = ⟨3 − t, 8 − 3t, 1 − t⟩ and L2(s) = ⟨−1 − 2s, 3 + s, s⟩ intersect, we need to find a value of t and s that make the x, y, and z components of both lines equal.

First, let's set the x components of both lines equal to each other:

3 − t = −1 − 2s

Next, let's set the y components of both lines equal to each other:

8 − 3t = 3 + s

Finally, let's set the z components of both lines equal to each other:

1 − t = s

Now we can solve for t and s. From the first equation, we get:

t = 4 + 2s

Substituting this value of t into the second equation, we get:

8 − 3(4 + 2s) = 3 + s

Simplifying this equation gives us:

-4 − 6s = 3 + s

Solving for s gives us:

s = -1

Substituting this value of s back into the first equation gives us:

t = 4 + 2(-1) = 2

So the lines intersect at t = 2 and s = -1. To find the point of intersection, we can substitute these values back into either line equation. Substituting into L1(t) gives us:

L1(2) = ⟨3 − 2, 8 − 3(2), 1 − 2⟩ = ⟨1, 2, -1⟩

Therefore, the lines intersect at the point (1, 2, -1).

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The total cost of the services from Happy Paws and Woof Watchers will be equal if the dog is kept for 12 hours.

Finding the answer to a linear equation with one, two, three, or more variables is known as solving a linear equation. A linear equation's solution is, to put it simply, the value or values of the variables that make up the equation. Any coefficient can be deleted using the elimination procedure after being initially equated. Elimination is followed by equations being solved to get the other equation.

Let us suppose the number of hours = h.

Given that, Happy Paws charges $18.00 plus $2.50 per hour.

H = 18 + 2.50h

Woof Watchers charges $12.00 plus $3.75 per hour.

W = 12 + 3.75h

The cost is equal for:

18.00 + ($2.50 x h) = $12.00 + ($3.75 x h)

18 - 12 = 3.75h - 2.50h

h = 12

Hence, the total cost of the services from Happy Paws and Woof Watchers will be equal if the dog is kept for 12 hours.

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[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &500000\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ t=\textit{elapsed time}\dotfill &25\\ \end{cases} \\\\\\ A = 500000(1 + 0.02)^{25} \implies A=500000(1.02)^{25}\implies A \approx 820303[/tex]

(a) The F statistic for testing that all effects equal 0 is given by:
F = [(R2 / k) / ((1 - R2) / (n - k - 1))]
where R2 is the coefficient of determination, k is the number of predictor variables, and n is the total number of observations.
To derive this expression, we start with the definition of the F statistic:
F = [(SSR / k) / (SSE / (n - k - 1))]
where SSR is the sum of squares for regression and SSE is the sum of squares for error.
We know that R2 = SSR / SST, where SST is the total sum of squares. Therefore, we can rewrite SSR as:
SSR = R2 * SST
Substituting this back into the F statistic, we get:
F = [(R2 * SST / k) / (SSE / (n - k - 1))]
We also know that SST = SSR + SSE, so we can substitute this back into the F statistic:
F = [(R2 * (SSR + SSE) / k) / (SSE / (n - k - 1))]
Simplifying the expression, we get:
F = [(R2 / k) / ((1 - R2) / (n - k - 1))]
This is the expression for the F statistic in terms of the R2 value.

(b) The F statistic for comparing nested models is given by:
F = [(R2_2 - R2_1) / (k_2 - k_1)] / [(1 - R2_2) / (n - k_2 - 1)]
where R2_1 and R2_2 are the R2 values for the smaller and larger models, respectively, and k_1 and k_2 are the number of predictor variables in the smaller and larger models, respectively.
To derive this expression, we start with the definition of the F statistic:
F = [(SSR_2 - SSR_1) / (k_2 - k_1)] / [(SSE_2 / (n - k_2 - 1))]
where SSR_1 and SSR_2 are the sum of squares for regression for the smaller and larger models, respectively, and SSE_2 is the sum of squares for error for the larger model.
We know that R2 = SSR / SST, so we can rewrite SSR_1 and SSR_2 as:
SSR_1 = R2_1 * SST
SSR_2 = R2_2 * SST
Substituting these back into the F statistic, we get:
F = [((R2_2 * SST) - (R2_1 * SST)) / (k_2 - k_1)] / [(SSE_2 / (n - k_2 - 1))]
Simplifying the expression, we get:
F = [(R2_2 - R2_1) / (k_2 - k_1)] / [(1 - R2_2) / (n - k_2 - 1)]

This is the expression for the F statistic for comparing nested models in terms of the R2 values.

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It will take approximately 2.9 years (or 2 years and 11 months) for the $2500 investment to earn $400 interest at a rate of 8% compounded quarterly.

A = [tex]P(1 + r/n)^{nt}[/tex]

Where:

A = the total amount that includes principal and interest, P = the principal that is the initial amount invested, r = the annual interest rate, n = the number of times is the interest is compounded per year, and t = the time

$2900 = [tex]$2500(1 + 0.08/4)^{4t}[/tex]

Simplifying this equation, we get:

1.16 = (1 + 0.02

By calculating the natural logarithm of both sides, we arrive at:

ln(1.16) = 4t ln(1.02)

t ≈ 2.9 years

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The fraction amount of pizza eaten is given by A = ( 5/6 ) of the pizza

The amount of pizza left over B = ( 1/6 ) of the pizza

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

Given data ,

Let the fraction of pizza eaten be represented as A

Now , the value of A is

Let the amount of pizza eaten by Tyler be p = ( 1/2 ) of the pizza

Let the amount of pizza eaten by Dean be q = ( 1/3 ) of the pizza

Now , the fraction of pizza eaten A = p + q

On simplifying , we get

The fraction of pizza eaten A = ( 1/2 ) + ( 13 )

The fraction of pizza eaten A = ( 3 + 2 ) / 6

The fraction of pizza eaten A = ( 5/6 ) of the pizza

And , the amount of pizza left B = 1 - A

On simplifying , we get

The amount of pizza left = 1/6 of the total pizza

Hence , the fractions are solved

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The probability of selling exactly 4 houses in one day is 0.0548, or about 5.48%.

To find the probability of selling exactly 4 houses in one day, use the Poisson distribution formula:

P(X = x) = (λ^x)(e^-λ) / x!

Where:

λ = the average number of events (in this case, the average number of houses sold)

x = the number of events we want to find the probability of (in this case, 4 houses)

e = the base of the natural logarithm (approximately 2.71828)

So, plugging in the values: P(X = 4) = (1.6^4)(e^-1.6) / 4! = (6.5536)(0.2019) / 24 = 0.0548

Therefore, since the data is Poisson distributed, selling exactly 4 houses in one day has a probability of 0.0548, or about 5.48%.

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The required gradient and  y-intercept for the given lines is given by ,

1. gradient = 10 and y-intercept = 20

2.  gradient = -20 and y-intercept = 10

3.  gradient = 0.5 and y-intercept = 2.5

Equation of a line in slope-intercept form = y = mx + c.

where 'm' is the slope of the line

And 'c' is the y-intercept.

For the line Y = 10x + 20

Compare with standard form we get,

Slope of the line is 10 .

And the y-intercept is 20.

This implies,

gradient is 10 and the y-intercept is (0, 20).

For the line  Y=10-20x

Compare with standard form we get,

Slope of the line is -20

And the y-intercept is 10.

This implies,

the gradient = -20

And the y-intercept = (0, 10).

For the line  Y=-2. 5+0. 5x

Compare with standard form we get,

Slope of the line is 0.5

And the y-intercept is -2.5

This implies,

the gradient = 0.5

And the y-intercept = (0, -2.5)

Therefore, the gradient and the y-intercept for each line is equal to,

Y=10x+20 , gradient = 10 and y-intercept = 20

Y=10 -20x  , gradient = -20 and y-intercept = 10

Y=-2. 5+0. 5x , gradient = 0.5 and y-intercept = 2.5

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70 hours will Audrey spend on her English assignments.

2.9 days will it take Audrey to complete her assignments.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Mr. Dalton has told her she will have 28 assignments this semester.

She needs to spend 2.5 hours on each assignment.

So, she will spend on Assignments

= 28 x 2.5

= 70 hours

Now, In one day = 24 hours

So, to complete the assignments it will take

= 70/ 24

= 2.9 days

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In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞). In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

To find the domain of the given rational expression, we need to determine the values of x that make the denominator equal to zero. This is because division by zero is undefined and therefore, these values of x cannot be included in the domain.

The denominator of the given expression is x^(2)-25. We can set this equal to zero and solve for x:

x^(2)-25=0

(x+5)(x-5)=0

x=-5 or x=5

These values of x make the denominator equal to zero, so they cannot be included in the domain. Therefore, the domain of the given rational expression is all real numbers except -5 and 5.

In interval notation, the domain can be written as: (-∞, -5) ∪ (-5, 5) ∪ (5, ∞)

In set notation, the domain can be written as: {x ∈ ℝ | x ≠ -5 and x ≠ 5}

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The answer choice which represents the solution to the inequality given as required is; b > -90.

Quotient refers to a number resulting from the division of one number by another.

As evident in the task content; the given inequality is;

Eight more than the quotient of a number b and 45 is greater than 6.

The symbols of inequalities are;

Greater than >

Less than <

Equal to =

Greater than or equal to ≥

Less than or equal to ≤

Therefore, the algebraic expression is;

b/45 + 8 > 6

Therefore; b/45 > -2

b > -90.

Ultimately, the required inequality is; b/45 + 8 > 6 and it's solution is; b > -90.

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The length of line BC is 10cm ,The area of the circle is 78.5 cm² and The circumstance of Circle  is  31.4 cm.  

Part-1

To Find the length of BC, We have the lenghts of CD=8cm and DE=12cm

To find out length. We have to make a perpendicular line construction joins AE.

Triangle AED forms a right angled triangle.

As we all know pythagoras theorem.

Pythagoras formula c²=a²+b²

To findout radius of circle

Let AE=AC=r  

      AD=CD+AC

      AD=8+r

                               AD²=AE²+DE²

Submit all values in the above equation.

                               (8+r)²=r²+12²

(a+b)²=a²+b²+2ab use the formula

                             64+r²+16r=r²+144

Cancel r² each other.

                             64+16r=144

                                    16r=144-64

                                     16r=80

                                         r=5

Radius of the circle    r=5cm

Therefore , the value of BC=2AC

                                       BC=2r

                                        BC=2*5

                                        BC=10cm

The length of line BC is 10cm

Part-2

To find out area of the circle

We have the formula for area of circle.

                                       Area=πr²

                                         Area=(22/7)*5*5

                                         Area=78.5 cm²

The area of the circle is 78.5 cm².

To find out the circumstance of Circle

We have the formula for circumstance of circle.  2πr

                              circumstance=2*(22/7)*5

                              circumstance=31.4 cm

The circumstance of Circle  is  31.4 cm.  

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The area of the yellow part of the pattern is found as: 72 cm²

The dimension of the square: 12 x 12

Area of yellow part = Area of triangle with base height equals side of square.

Then,

Area of yellow part = 1/2 * B * H

Area of yellow part = 1/2 *12 *12

Area of yellow part = 72 cm²

Thus, the area of the yellow part of the pattern is found as: 72 cm².

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The correct question is-

Work out the area of the yellow part of the pattern. This is a pattern that is being used to make a patch quilt. The pattern is a square, measuring 12cm

by 12cm square.

Answer:

990 applicants

Step-by-step explanation:

We know

A university law school accepts 4 out of every 11 applicants.

If the school accepted 360 ​students, find how many applications they received.

To get from 4 to 360, we time 90

We take 11 times 90 = 990 applicants

So, they received 990 applicants.