The function V(x)=x3−6x2 repreent the volume (in cubic inche) of the box hown. The function W(x)=V(12x) repreent the volume (in cubic inche) when x i meaured in feet. Write a rule for W. Then find W(2)

The value of the function (f /g)(x) would be [tex](\frac{f}{g})(x)=x^2+6[/tex].

Option (A) is correct.

What are composite functions?

A composite function is generally a function that is written inside another function. The composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x). The composite function f [g (x)] is read as “f of g of x”.'

Given:

                [tex]f(x)=7x^3-5x^2+42x-30\\\\g(x)=7x-5[/tex]

Now by using the definition of composite functions we can write

      [tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}\\\\(\frac{f}{g})(x)=\frac{7x^3-5x^2+42x-30}{7x-5}\\\\(\frac{f}{g})(x)=x^2+\frac{42x-30}{7x-5}\\\\(\frac{f}{g})(x)=x^2+6[/tex]

Hence, the value of the function (f - g)(x) would be [tex](\frac{f}{g})(x)=x^2+6[/tex].

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The special product formula for the 'square of a sum' is

(a + b)² = a² + 2ab + b².

(a + b)² = (a +b)×(a + b)

By applying Distributive law in multiplication, we know

(a + b)² = a×(a + b) + b×(a + b)

(a + b)² = a² + ab + ba + b²

(a + b)² = a² + 2ab + b²

For example, (3 + 2)²

We know that (3 + 2)² = 5² = 25

But let us try applying the special product formula

(3 + 2)² = 3² + 2×3×2 + 2²

(3 + 2)² = 9 + 12 + 4

(3 + 2)² = 25

So we can use this formula in cases where we don't know the square of a sum. Many applications can be seen like solving quadratic equations.

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So basically for the first one do 14.0%x 4.6

The triangle with sides 54 m , 78 m and 90 m is not a right angle triangle

What is a Triangle?

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

Given data ,

Let the triangle be represented as ΔABC

Now , the value of the sides of the triangle is

AB = 54 m

BC = 78 m

AC = 90 m

Now , for the triangle to be right angle triangle , it should satisfy the Pythagoras Theorem

So , For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

Substitute the values in the equation , we get

AC² = AB² + BC²

90² = 54² + 78²

8100 = 2916 + 6084

8100 ≠ 9000

Therefore , the triangle ΔABC is not a right angle triangle

Hence ,

The triangle with sides 54 m , 78 m and 90 m is not a right angle triangle

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Answer: C. 1200$

Step-by-step explanation:

$600 times twice is $1200

For a sample population of size, n =5, sx = 20 and sx² = 120, the sum of squared deviation is 40.

Therefore, the answer is 40.

Given a sample population of 5 scores, that is n = 5.

sx = ∑x = 20

sx² = ∑x² = 120

Thus mean can be calculated by

mean, x⁻ = ∑x/ n

x⁻ = 20/ 5

x⁻ = 4

The sum of squared deviations

ss = ∑( x - x⁻)²

ss = ∑( x² -2×x×x⁻ + x⁻²)

ss = ∑x² -2×x⁻×∑x + ∑x⁻²

ss = ∑x² -2×x⁻×∑x + nx⁻²

ss = 120 - 2×4×20 + 5×4²

ss = 40

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The simplification of the given expression is x² - x + 3.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the expression is √x⁴ = x - 3.  The expression will be simplified as,

√x⁴ = x - 3

Solve the radical term first and simplify the expression further,

x² = x - 3

x² - x + 3 = 0

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In the second year, the APR (Annual Percentage Rate) will increase quite substantially.

What is Introductory APR?

a temporary promotional interest rate that is lower than the card's standard APR, occasionally as low as 0% APR. Purchases, balance transfers, or both may be covered. After the introductory period has been over, your balance will be subject to the standard APR.

Any outstanding balances will begin to accrue interest once the promotional period is expired. Not just new charges, but also any amounts you charged or transferred to the credit card within the promotional APR period.

Your APR will transition from your promotional interest rate to a standard variable APR rate decided by your lender once it expires.

So, in this case, Trina's credit card has an introductory APR of 7.99-8.99%, but from the second year onwards the APR will increase.

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Following the conclusion of the promotional period, interest will start to be charged on any unpaid balances. Not only fresh charges, but also any sums you added to the credit card or transferred within the promotional APR period.

When the promotional period is up, your APR will change from the promotional interest rate to a standard variable APR rate chosen by your lender.

Thus, Trina's credit card in this instance has an introductory APR of 7.99–8.99%; but, in the second year, the APR will rise.

The equation of line in slope intercept form is, [tex]y=\frac{3}{4}x+9[/tex].

What is slope point form of the line?

A line's point slope form equation is y - y1 = m. (x – x1). Consequently,

y - 0 = m(x = 0), or y = mx, is the equation of a line passing through the origin with a slope of m.

Let the line passes through the point (-4, 6) and having slope 3/4.

We know that the equation of line passes through the point [tex](x_1, y_1)[/tex] and having slope m is,

[tex]y-y_1=m(x-x_1)[/tex]

Here, [tex](x_1, y_1) = (-4, 6)[/tex]  and slope [tex]m = \frac{3}{4}[/tex]

Plug the values in the above equation.

⇒

[tex]y-6=\frac{3}{4}(x-(-4))\\ y - 6 = \frac{3}{4}(x + 4)\\ y-6=\frac{3}{4} x + 3\\y = \frac{3}{4}x+3+6\\ y=\frac{3}{4}x+9[/tex]

Hence, the equation of line in slope intercept form is, [tex]y=\frac{3}{4}x+9[/tex].

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The coefficient of x² in (3x² – 5)( 4 +4x²) is -8.

A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6.

First, let's multiply the terms in the given expression by opening the brackets.

⇒   (3x² – 5)( 4 +4x²)

Now, further multiplying we would get,

⇒   3x²(4 + 4x²) -5(4 + 4x²)

⇒    12x² +12x⁴-20-20x²

⇒  12x⁴ -8x²-20

As we can see that the term that is in multiplication with x² is -8. Similarly, the term that is coefficient of  x⁴ is +12.

Hence, the coefficient of x² in the equation  is -8.

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(x-3.5), or (x-7/2) are the factors of the polynomial, f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

Given,

The polynomial, f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

We have to find the factors of the polynomial;

Here,

This polygon's 35-factor factors, plus or minus 1, plus or minus 5, plus or minus 7, are potential zeros. Determine whether or not there is a non-zero remainder in each example using synthetic division. If none of these solutions work, try rational divisors derived from 35 and 6 such as 5/6, 7/6, 1/6 and so on.

Now,

f(x) = 6x⁴ – 21x³ – 4x² + 24x – 35

If used correctly, this method will eventually reveal one or two zeros of this poly. Of course, it would be lot simpler if you looked at the potential solutions provided to you for this issue.

I plotted this function and discovered that it crosses the x-axis at 7/2. Another root exists.

To determine whether or whether 7/2 is a root, use synth. div.

Then,

        ___________________________

7/2   /   6    -21    -4    24    -35

                  21      0   -14     35          

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

          6        0     -4    10       0

Since there is no remainder, the polynomial has a root of 7/2 (or 3.5). Therefore, a factor is (x-3.5) or (x-7/2).

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Answer:

It's A

Step-by-step explanation:

The first part is her purchase before the tax, which should be $32. The second part is her purchase after sales tax, which adds 8% to the shoes and results in the purchase being $34.40

You may use the expression to convert the length of 5 rows of ribbons to inches: length = 5w + (5 - 1)s inches, where w is the width of the ribbons and s is the spacing between the rows.

To convert 5 rows of ribbons to inches, you need to know the width of the ribbons and the spacing between the rows.

Let's say the width of the ribbons is w inches, and the spacing between the rows is s inches. In this case, the total length of the ribbons in 5 rows would be 5w + 4s inches, since there are 5 ribbons and 4 spaces between them.

For example, if the width of the ribbons is 0.5 inches and the spacing between the rows is 0.1 inches, the total length of the ribbons in 5 rows would be :5 × 0.5 inches + 4 × 0.1 inches = 2.5 inches + 0.4 inches = 2.9 inches.

In general, to convert the length of n rows of ribbons to inches, you can use the formula: length = nw + (n - 1)s inches, where w is the width of the ribbons and s is the spacing between the rows.

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Answer:

250 square feet

Step-by-step explanation:

1 1/2 = 3/2 = 1.5

1.5 pound bag = 375 square feet

Take 375 divided by 1.5 = 250 square feet

So, 1 pound of wildflower seeds will cover 250 square feet!

Answer:

Step-by-step explanation:

Using AutoSum in Excel is one quick and simple method of adding numbers. Simply click in a blank cell immediately underneath a column of data. Click AutoSum > Sum next on the Formula tab.

Sum is the outcome of adding two or more numbers or quantities. There may be only two terms, or there may be one hundred, thousand, or a million. There are some summations that have an infinite number of terms.

here

Click AutoSum on the Home tab, choose a cell next to the numbers you wish to add, then hit Enter when you're finished.

When you choose AutoSum, Excel automatically creates a formula (using the SUM function) to add the values.

Simply click in a blank cell immediately underneath a column of data. Click AutoSum > Sum next on the Formula tab.

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1440 combinations ways.

We know that the total arrangements will be

6!= 720 because its a 6letter word

So if we take MI to be a one letter word

Same for IM

which will Also be 6!= 720

SO TOTAL possible combination will be 720+720= 1440

Combinations are a way to calculate the total outcomes of an event where order of the outcomes does not matter. To calculate combinations, we will use the formula nCr = n! / r! * (n - r)!, where n represents the total number of items, and r represents the number of items being chosen at a time.

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Answer:

A. [tex]\frac{1}{5}[/tex]

Step-by-step explanation:

⭐ What is the multiplicative rate?

To find the multiplicative rate of this exponential function, we need to make an equation to see what each y-value gets multiplied by to get the next y-value.

I am picking the y-values 2 and 2/5, but you can choose any two consecutive y-values.

Equation (let x= the multiplicative rate of change):

[tex]2x = \frac{2}{5}[/tex]

Solve for x:

[tex]\frac{2x}{2} = \frac{2}{5}(2)[/tex]

[tex]x = \frac{2}{10}[/tex]

Simplify:

[tex]x = \frac{1}{5}[/tex]

∴ The multiplicative rate of change of the function is 1/5.

The three steps for graphing linear inequalities are consists of solving the variables, plotting the points on graph and bound the region.

while solving the inequalities of two variables the inequal sign are changed by equal sign for a certain time. After determining the unknown value, we plot the points on the XY axis.

while graphing the inequal variables the region that satisfy the inequality is bounded by solid line or dashed line depends on the inequal sign.

hence, we need to plot the linear inequalities on XY axis in three steps.

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Answer:

Step-by-step explanation:

If the derivative of g(x) with respect to x (denoted d/dx g(x)) is equal to 0, then the derivative of the product g(x)sin(x) with respect to x can be found using the product rule for derivatives.

The product rule states that if f(x) and g(x) are two functions, then the derivative of the product f(x)g(x) with respect to x is given by:

d/dx (f(x)g(x)) = f(x) * d/dx g(x) + g(x) * d/dx f(x)

In our case, f(x) is g(x) and g(x) is sin(x). Substituting these values into the equation above, we get:

d/dx (g(x)sin(x)) = g(x) * d/dx sin(x) + sin(x) * d/dx g(x)

Since the derivative of sin(x) with respect to x is cos(x) and the derivative of g(x) with respect to x is 0, we can simplify this expression to:

d/dx (g(x)sin(x)) = 0 * cos(x) + sin(x) * 0

Which simplifies to:

d/dx (g(x)sin(x)) = 0

Therefore, if the derivative of g(x) with respect to x is equal to 0, then the derivative of the product g(x)sin(x) with respect to x is also equal to 0.

The values of x for the given quadratic equation 2x² = 4x - 7

are given by:

[tex]x = \frac{2 +i\sqrt{10} }{2}[/tex]

[tex]x = \frac{2 -i\sqrt{10} }{2}[/tex]

What is the solution of a quadratic equation?
The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations.

According to the given question:

For this case we have the following quadratic equation:

2x² = 4x - 7

Rewriting the equation we have:

2x² - 4x + 7 = 0

From here, we have:

a = 2b = -4c = 7

Substituting values in the quadratic equation we have:

x = (-b ± √b² - 4ac)/2a

x = (-(-4) ± √(-4) - 4(2)(7)/2(2)

Rewriting the equation we have:

x = (4 ± √16 - 56)/4

x = (4 ± √-40)/4

x = (2 ± i√10)/2

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The commutative, associative, and distributive property are the properties of addition and multiplication.

Commutative Property - The Commutative Property is derived from the word commute, which means to move. It states that the product or sum of two numbers do not change even if the places of numbers are changed or if the numbers move from one place to another. The commutative property of addition states that a + b = b + a and the commutative property of multiplication states that a × b = b ×  a.

Associative Property - It is related with the grouping of numbers. It states that the product or sum of three numbers do not change even if the grouping of numbers are changed. The associative property of addition states that (a + b) + c = a + (b + c) and the associative property of multiplication states that: (a × b) × c = a × (b ×  c).

Distributive Property - It is related with the distribution of numbers. It states that the product of three numbers do not change even if the distribution of numbers are changed. The distributive property of multiplication states that: a × (b + c) = (a × b) + (a ×  c).

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Answer:

ƒ(6) = − 50

Step-by-step explanation:

ƒ(x) = − x² – 14, Find f(6)

ƒ(6) = − 6² – 14

ƒ(6) = − 36 – 14

ƒ(6) = − 50

Repeat the procedure of dividing, multiplying, subtracting, and bringing the next digit down until you have solved the division problem's quotient.

For example, consider the long-division problem:

4567 ÷  123

In this problem, we need to repeat the process of dividing, multiplying, subtracting, and bringing the next digit down three times, since the divisor has three digits.

However, in some cases, you may need to repeat the process more or fewer times than the number of digits in the divisor.

For example, if the dividend has fewer digits than the divisor, you may need to repeat the process fewer times. Or, if you need to carry a digit over to the next step, you may need to repeat the process more times.

Thus, in general, you should repeat the process of dividing, multiplying, subtracting, and bringing the next digit down until you have found the quotient of the division problem. This will typically be the number of digits in the divisor, minus one, but it may be more or fewer in some cases.

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The coefficient of yA² for the three given expressions are 2, 10 and 5 respectively.

It is a mathematical statement that defines a mathematical concept as well as the system of a mathematical operation express in a symbolic form.

Mathematical expression is made up of at least two numerical terms along with a sign or at least two terms along with a sign. The expression having two or more numerical terms is called numerical expression.

As example 4 + 3 or 100÷4

The expression which has both variables and numbers is called algebraic expression.

As example X+6 or 25+5y

In an expression, the numbers which are accompanied by variables is called coefficient. As example, 24 + 3y, where 3 is the coefficient of y

but 24 is a constant number.

We are given three expressions, these are 2xA²y, 10xyA² and 5yA²

for the 1st expression, 2 is the coefficient of A²y

for the 2nd expression, 10 is the coefficient

for the last 5 is the coefficient.

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Step-by-step explanation:

The general quadratic equation [tex]ax^2+2hxy+by^2+2gx+2fy+c=0[/tex] represents pair of straight lines if its discriminant [tex]\Delta=0[/tex] and [tex]h^2-ab\geq0.[/tex] The discriminant of this equation is given by the determinant,

[tex]\Delta=\left|\begin{array}{ccc}a&f&g\\f&b&h\\g&h&c\end{array}\right|[/tex]

Here the given equation is,

[tex]\longrightarrow x^2+2xy+2y^2+2x+2y+1=0[/tex]

So,

[tex]a=1[/tex]

[tex]b=2[/tex]

[tex]c=1[/tex]

[tex]f=1[/tex]

[tex]g=1[/tex]

[tex]h=1[/tex]

Thus,

[tex]\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\1&1&1\end{array}\right|[/tex]

Performing the operation [tex]R_3\to R_3-R_1[/tex] over this determinant,

[tex]\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\0&0&0\end{array}\right|[/tex]

Now the 3rd row is completely so the discriminant is equal to zero.

[tex]\longrightarrow\Delta=0[/tex]

So our equation represents degenerate conics, which are,

pair of distinct straight lines intersecting each other at any point if [tex]h^2-ab > 0.[/tex]

pair of straight lines, either coincident or distinct but parallel to each other, if [tex]h^2-ab=0.[/tex]

a single point if [tex]h^2-ab < 0.[/tex]

We see that,

[tex]\longrightarrow h^2-ab=1^2-1\times 2=-1[/tex]

[tex]\longrightarrow h^2-ab < 0[/tex]

So our equation represents a single point.

Hence the equation does not represent pair of straight lines.

The pigeonhole principle can be used to demonstrate that there must be at least one duplicate in the set nums. Each possible number that can appear in nums is a "pigeonhole" in this context, and each number within nums is a "pigeon."

Method or evidence to confirm that Nums must contain multiple duplicate number: Simple Strategy: Utilizing nested loops, it is intended to determine whether or not each element appears in the array more than once for each element.

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When given two values in algebra, it is simple to find the third value. The x value is 5.

x Multiplicand Equals Product

X = product / multiplicand

* x = 10

Product

The x value is 5.

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The relationship between the volume of a cone and a pyramid:

the volume of cone = volume of pyramid

if and only if the pyramid has circular base and both have equal radii and vertical heights

In this question we need to find the relationship between the volume of a cone and a pyramid.

We know that a pyramid is three dimensional shape with polygon base and trigular lateral faces which meet at vertex.

The formula for the volume of pyramid is,

V = 1/3 * base area * height

As we know a cone is nothing but a pyramid with circular base.

Consider a cone with radius r and vertical height h.

The formula for the volume of cone,

V = 1/3 × π × r² × h

V = 1/3 * area of circle * height

Therefore, the volume of a cone and a pyramid with circular base are equal.

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Isosceles triangle has three sides, three angles, a base, a height, and an area in which two sides are equal to each other .

the sum of triangle is 180, sum of all the interior angles is equal to 180.

so if you know two angles then apply sum of triangles and find out the value of the third triangle.

isosceles triangle

An isosceles triangle is a type of triangle that has any two sides equal in length. The two angles of an isosceles triangle, opposite to equal sides, are equal in measure.

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