If f(x) = 8x and g(x)= 3x-2, find (fg)(x).

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°

[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]

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.

Answer:

25/99

Step-by-step explanation:

If the tickets are numbered 1-100, half of the tickets will be even and half of the tickets will be odd

Number of even tickets = 50

Number of odd tickets = 50

Let A be the event =>  even ticket on first draw

Let B be the event => odd ticket on second draw

P(even on first draw) = P(A)
= Number of even tickets/total number of tickets

= 50/100

= 1/2

Once a ticket has been drawn and the second draw is without replacement,

Total number of tickets remaining = 100 - 1 = 99

Total number of odd tickets remaining given first ticket drawn is even
= 50

P(odd ticket second draw | first draw is even) = P(B | A)
= 50/99

P(A and B) = P(A) · P(B | A)

= 1/2 x 50/99

= 50 / (2 x 99)

= 50/198

= 25/99

Probability that it is an even number:

[tex] \tt \: P(E) = \frac{F.O.}{T.O.} [/tex]

[tex] \tt \: P(E) = \frac{50}{100} = \frac{1}{2} [/tex]

Pribability that it is an odd number:

[tex] \tt \: P(E) = \frac{F.O.}{T.O.} [/tex]

[tex] \tt \: P(E) = \frac{50}{99} = 0.505051[/tex]

Answer:

30

Step-by-step explanation:

40/4=10

25%=10

40-10=30

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

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!

(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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The Gauss-Jordan elimination method can be used to solve this system of linear equations. First, we need to create an augmented matrix containing all the coefficients of the system:

\begin{bmatrix}
   7 & 7 & 7 & 0 \\
   2 & 2 & -4 & 6 \\
   1 & 1 & -2 & -1 \\
   0 & 0 & 1 & 0 \\
\end{bmatrix}

We then use a series of row operations to reduce the matrix to row echelon form:

\begin{bmatrix}
   7 & 7 & 7 & 0 \\
   0 & -4 & -5 & 6 \\
   0 & 1 & -3 & -1 \\
   0 & 0 & 1 & 0 \\
\end{bmatrix}

Finally, we solve the system of equations by performing back substitution and we get the following solution:

x1 = 0, x2 = 6, x3 = 1, x4 = -2, x5 = 1

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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

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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Answer: 43.96 inches^2

Step-by-step explanation:

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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We can state this by responding to the provided question B, D, and E are expressions the expressions that have values of 1/6.

Mathematical operations include doubling, dividing, adding, and subtracting. A phrase is constructed as follows: Expression, monetary value, and mathematical operation Numbers, parameters, and functions make up a mathematical expression. It is possible to use words and terms in contrast. Every mathematical statement including variables, numbers, and a mathematical operation between them is called an expression, sometimes referred to as an algebraic expression.

We need to simplify each expression to see which ones are equivalent to 1/6:

6. 1/6 is not equivalent to 6/1.

B. 3÷18 = 1/6

C. 2/3 of 2 equals 6 (not equivalent to 1/6)

D. 1÷6 = 1/6

E. 1/3 ÷ 2 = 1/6

B, D, and E are the expressions that have values of 1/6.

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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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The value of angle XYZ is 0.020 degrees using the cosine function.

Simply put, inverse trigonometric functions are the opposites of the fundamental trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant. The terms arcus functions, antitrigonometric functions, and cyclometric functions are also used to describe them. To find the angle for any trigonometric ratio, apply these inverse trigonometric functions. The fields of engineering, physics, geometry, and navigation all heavily utilise the inverse trigonometry functions.

From the given right triangle we observe that:

cos y = 6 / 15 = adjacent side /hypotenuse

y = arccos(6/15)

y = 0.020 degrees

Hence, the value of angle XYZ is 0.020 degrees using the cosine function.

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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 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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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 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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(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 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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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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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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you have too draw a scatter

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.