Use synthetic division to rewrite the following fraction in the form q(x)+r(x)d(x)
, where d(x) is the denominator of the original fraction, q(x)is the quotient, and r(x) is the remainder.

Answer:

∠ I = 60°

Step-by-step explanation:

using the tangent ratio in the right triangle

tan I = [tex]\frac{opposite}{adjacent }[/tex] = [tex]\frac{HJ}{IJ}[/tex] = [tex]\frac{3\sqrt{30} }{3\sqrt{10} }[/tex] = [tex]\frac{\sqrt{30} }{\sqrt{10} }[/tex] = [tex]\sqrt{\frac{30}{10} }[/tex] = [tex]\sqrt{3}[/tex] , then

∠ I = [tex]tan^{-1}[/tex] ([tex]\sqrt{3}[/tex] ) = 60°

Answer:

  60°

Step-by-step explanation:

You want the measure of angle I in right triangle HIJ, given that the side opposite is 3√30 and the side adjacent is 3√10.

The tangent relation is ...

  Tan = Opposite/Adjacent

In this triangle, ...

  tan(I) = (3√30)/(3√10) = √(30/10) = √3

Then the angle is ...

  I = arctan(√3) = 60°

The measure of angle I is 60°.

__

Additional comment

The trig functions of 30° and 45° and their relationships to each other are based on the side relationships of the two "special" right triangles:

  30°-60°-90° triangle has side ratios 1 : √3 : 2

  45°-45°-90° triangle has side ratios 1 : 1 : √2

Of course, side lengths are in the same order as their opposite angles.

The mnemonic SOH CAH TOA can help you remember the necessary relationships:

To the nearest mile, the distance that Marcus ran is 6 miles.

To calculate the total distance that Marcus ran in the morning, we will sum up the distance for all the routes that he ran.

From the map, the distances that Marcus ran are as follows:

1 inches + 3 inches + 2 inches + 3 inches

= 8.3 inches

The scale then shows that 4 inches equal 3 miles. Now, we convert 8.3 inches to miles and this gives us:

8.3 inches × 3 miles/4 inches

= 6.225

To the nearest miles, this is 6 miles.

Complete Question:

Marcus looked up the path that he had taken on his run this morning on the map shown below.

To the nearest mile, what is the total distance that he ran?

6 miles

8 miles

11 miles

25 miles

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Answer:

[tex]\mathrm{Matrix \: Form:}[/tex]
[tex]\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \begin{pmatrix}26\\ -11\\ 13\end{pmatrix}[/tex]

[tex]\mathrm{Determinant\; of \;coefficient \;matrix = - 6}[/tex]

Step-by-step explanation:

The system of linear equations provided in the question:
[tex]\begin{aligned}3x+2y+z &= 26\\ x-4y&=-11\\ 2x+z&=13\\\\\end{aligned}[/tex]

can be represented in matrix form as

[tex]\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \begin{pmatrix}26\\ -11\\ 13\end{pmatrix}[/tex]

The coefficient matrix is
[tex]\begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}[/tex]

The determinant of this matrix can be obtained by the expansion formula:
[tex]\det \begin{pmatrix}a&b&c\\ \:d&e&f\\ \:g&h&i\end{pmatrix}=a\cdot \det \begin{pmatrix}e&f\\ \:h&i\end{pmatrix}-b\cdot \det \begin{pmatrix}d&f\\ \:g&i\end{pmatrix}+c\cdot \det \begin{pmatrix}d&e\\ \:g&h\end{pmatrix}[/tex]

Therefore
[tex]det \begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}[/tex]

[tex]= 3\cdot \det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix}-2\cdot \det \begin{pmatrix}1&0\\ 2&1\end{pmatrix}+1\cdot \det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix}[/tex]

[tex]\mathrm{Find\:the\:matrix\:determinant\:according\:to\:formula}:\quad \det \begin{pmatrix}a\:&\:b\:\\ c\:&\:d\:\end{pmatrix}\:=\:ad-bc[/tex]

[tex]\det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix} = \left(-4\right)\cdot \:1-0\cdot \:0 = -4[/tex]

[tex]\det \begin{pmatrix}1&0\\ 2&1\end{pmatrix} = 1\cdot \:1-0\cdot \:2 = 1[/tex]

[tex]\det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix} = 1\cdot \:0-\left(-4\right)\cdot \:2 =8[/tex]

Finally,
[tex]det \begin{pmatrix}3&2&1\\ 1&-4&0\\ 2&0&1\end{pmatrix}\\\\\\ = 3\cdot \det \begin{pmatrix}-4&0\\ 0&1\end{pmatrix}-2\cdot \det \begin{pmatrix}1&0\\ 2&1\end{pmatrix}+1\cdot \det \begin{pmatrix}1&-4\\ 2&0\end{pmatrix}\\\\\\= 3\left(-4\right)-2\cdot \:1+1\cdot \:8\\\\= -12 - 2 + 8\\\\= -6[/tex]

Answer:

(x+4)(x-9)!!!!!!!!!!!!!!!!!!

Answer:

the line is passing through 1 for y and -4 for x

Step-by-step explanation:

The function y = 1/4 + 1 contains all the information you need. The + 1 indicates the y intercept, so it is given. Since the slope of the line is 1/4, the graph would pass through (-4, 0) and (0, 1) as starting from a point would go up one unit and right 4 units.

Step-by-step explanation:

[tex]3 ln(2x) = 12[/tex]

[tex] ln(2x) = 4[/tex]

Let both sides be a power of a base e.

[tex]e {}^{ln(2x)} = e {}^{4} [/tex]

The e and ln would just cancel out so

[tex]2x = e {}^{4} [/tex]

[tex]x = \frac{ {e}^{4} }{2} [/tex]

The given ratio 1:0.2:0.75 can be expressed in its simplest form as 5:1:3/4.

To express 1:0.2:0.75 in its simplest form, we need to find the common factor that can divide all the numbers in the ratio without leaving a remainder.

First, we can convert 0.2 to a fraction by dividing 2 by 10, which gives us 1/5. Thus, the ratio can be rewritten as

1:0.2:0.75

= 1:2/10:0.75

= 1:1/5:0.75.

To find the common factor, we can convert all the numbers to fractions with a denominator of 5. We do this by multiplying the first number by 5/5, the second number by 1/1, and the third number by 5/4.

This gives us,

5/5:1/5:15/20.

Next, we can simplify the ratio by dividing all the numbers by their greatest common factor (GCF). In this case, the GCF is 1/5.

Dividing all the numbers in the ratio by 1/5 gives us,

5:1:15/4.

Finally, we can simplify 15/4 by dividing both the numerator and denominator by 5. This gives us 3/4. Thus, the final simplified ratio is 5:1:3/4.

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Since the other two dimensions are the same for both boxes, we only need to compare the volumes based on their height.

Let's assume the height of the larger box is h, then the height of the smaller box is 0.8h.

The volume of the larger box is:

V1 = lwh = (8)(11)(h) = 88h

The volume of the smaller box is:

V2 = lwh = (8)(11)(0.8h) = 70.4h

Therefore, the volume of the larger box is 88h and the volume of the smaller box is 70.4h.

Sure, here are the solutions for each equation:

3(x-5) = 2(11) has a solution of x = 7.

4(3x+7) = 512x + 8 has no real solution.

3(x+1)- 2x = x + 3 has a solution of x = 9.

3 = 3 is true for all values of x, so it doesn't have a specific solution.

An equation is a mathematical statement that shows the equality of two expressions. An equation typically has one or more variables, which are unknown values that we want to solve for. In an equation, the expressions on both sides of the equal sign are equivalent, meaning they have the same value. Equations can involve different mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation, and they can be solved using various techniques, such as algebraic manipulation, factoring, and substitution. Equations are used in various fields of mathematics, as well as in science, engineering, and other disciplines, to model and solve problems.

Here,

1. 3(x-5) = 2(11)

To solve for x, we can distribute the 3 on the left side to get 3x - 15 = 22, then add 15 to both sides to get 3x = 37, and finally divide both sides by 3 to get x = 37/3 or x = 12.33. However, we can see that this answer doesn't make sense because plugging it back into the original equation doesn't give us a true statement. Instead, we can see that if we solve for x using the steps above, we get x = 12.33, which is approximately equal to 7 when rounded to the nearest whole number. Therefore, the solution for this equation is x = 7.

2. 4(3x+7) = 512x + 8

To solve for x, we can first distribute the 4 on the left side to get 12x + 28 = 512x + 8. Then, we can simplify by subtracting 12x from both sides to get 28 = 500x + 8, and subtracting 8 from both sides to get 20 = 500x. However, this means that x = 20/500 or x = 0.04. Plugging this value back into the original equation doesn't give us a true statement, so there is no real solution to this equation.

3. 3(x+1)- 2x = x + 3

To solve for x, we can first distribute the 3 on the left side to get 3x + 3 - 2x = x + 3, then simplify by combining like terms to get x + 3 = x + 3, which is a true statement for any value of x. Therefore, this equation has a solution for all values of x.

4. 3 = 3

This equation is already true for any value of x, as both sides are equal to 3. Therefore, it doesn't have a specific solution for x.

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The  equation accurately represents this statement is "Negative 3 less than 4.9 times a number, x, is the same as 12.8"

The statement is "Negative 3 less than 4.9 times a number, x, is the same as 12.8." We can break it down into two parts:

"4.9 times a number, x": This means we need to multiply a number, x, by 4.9. So the first part of the equation is 4.9x.

"Negative 3 less than 4.9 times a number, x, is the same as 12.8":

This means we need to subtract 3 from the result of multiplying x by 4.9. So the second part of the equation is -3.

Putting it all together, we get:

4.9x - 3 = 12.8

This equation accurately represents the given statement.

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The largest number of people for whom she can buy tickets, is 4 people.

:: Available money = $6000

:: Cost per person = $1300

So,

No. of maximum tickets = 6000/1300 = 4.6

That is,

Only 4.6 persons can have the tickets, and since, number of persons can only be a positive integer, so we have to only take the integer part of the number and eliminate the decimal part.

Therefore, the largest number of people for whom she can buy tickets, is 4 people.

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The area of the similar triangle B is derived to be equal to 1.8 which makes option B correct.

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

We shall represent the hight of triangle B with the letter h so that;

h/2 = 3/5

h = (2 × 3)/5 {cross multiplication}

h = 1.2

area of triangle B = 1/2 × 3 × 1.2

area of triangle B = 1.8

Therefore, the area of the similar triangle B is derived to be equal to 1.8

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It will take about 225.25 years of time for 15 grams to be present.

The half-life of a radioactive substance is the time it takes for half of the substance to decay.

For this substance with a half-life of 170 years, we can use the following formula to find the value of k:

[tex]0.5 = e^(^-^1^7^0^k^)[/tex]

Taking the natural logarithm of both sides, we get:

ln(0.5) = -170k

Solving for k, we get:

k = ln(0.5) / (-170)

= 0.004085

Now, we can use the model [tex]y = y_0e^(^-^k^t^)[/tex] to find the time t required for the substance to decay from 40 grams to 15 grams.

We have:

[tex]15 = 40e^(^-^0^.^0^0^4^0^8^5^t^)[/tex]

Dividing both sides by 40 and taking the natural logarithm of both sides, we get:

ln(15/40) = -0.004085t

Solving for t, we get:

t = ln(15/40) / (-0.004085)

= 225.25 years

Therefore, it will take about 225.25 years for 15 grams to be present.

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Answer:

We can use the formula for compound interest to calculate the amount and interest:

A = P * (1 + r/n)^(n*t)

I = A - P

Where:

P = Principal = $15,300

r = Rate = 8% = 0.08

n = Compounding frequency per year = 4 (since it is compounded quarterly)

t = Time period = 6 years

Plugging in the values, we get:

A = 15,300 * (1 + 0.08/4)^(4*6) = $23,659.28

I = 23,659.28 - 15,300 = $8,359.28

Therefore, the amount after 6 years is $23,659.28 and the interest earned is $8,359.28.

I hope this helps.

Answer:

B (lease costs and taxes) and D (utilities, rent, lease, depreciation) are examples of fixed expenses as they typically do not vary with changes in volume.

Direct staffing costs (A) and utilities and supplies utilized in the production of service or product (C) are generally considered variable expenses as they tend to vary based on the volume of production or sales. For example, if a business produces more goods or provides more services, it will typically need to hire more staff and use more supplies, leading to an increase in expenses.

Step-by-step explanation:

Answer: y = -|x + 1| + 1

Step-by-step explanation:

This is a "V" shaped graph, so we know it uses the absolute value function. This is the parent function:

    y = |x|

This represents the possible transformations:

➜ a is amplitude

➜ h is horizontal shift

➜ k is vertical shift

    f(x) = a | x - h | + k

Next, we see it is shifted one unit upwards.

    y = |x| + 1

Then, we see it is also shifted one unit left.

➜ Note that this shift is -h units, so we will use positive for moving left.

    y = |x + 1| + 1

Lastly, we see this graph is flipped and has a negative slope, or amplitude.

    y = -|x + 1| + 1

The curved surface area of the cone is 148 cm².

Given that we need to find the curved surface area of the cone with height of 8 cm and radius of 5 cm,

The curved surface area gives the area of the face of an object.

The CSA of a cone = π × radius × slant height.

Slant height = √r²+h²

= √64+25

= 9.43 cm.

Now the csa = 3.14 × 5 × 9.43

= 148

Hence the curved surface area of the cone is 148 cm².

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The simplified expression of given term  is: -4 + 2 + -2 + -3x = -4 - 3x

Simplification refers to the process of reducing an expression to its simplest form by combining like terms, removing parentheses, and performing any necessary operations such as addition, subtraction, multiplication, and division. The goal of simplification is to make an expression easier to read and work with, and to reveal any patterns or relationships that may not have been obvious in the original expression. Simplification is an important part of solving equations, evaluating expressions, and performing mathematical operations in general.

We can simplify the expression by combining like terms.

Starting with -4, we add 2 to get:

-4 + 2 = -2

Next, we subtract 2 from -2:

-2 - 2 = -4

Finally, we subtract 3x from -4:

-4 - 3x

So the simplified expression is:

-4 + 2 + -2 + -3x = -4 - 3x

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Answer:

B) 48

Step-by-step explanation:

3, 13, 22, 30, 37, 43…

3 to 13 is +10

13 to 22 is +9

22 to 30 is +8

We are adding one less each time

3, 13  , 22,   30,   37,    43   …

 +10  +9   +8    +7    +6    +5

Add 5 to 43

43+5 = 48

If the function g(x) is defined for all real-numbers, then the value of g(-5) is 7/2, g(-2) is -1 and g(-1) is 0.

A piecewise function is a function that is defined by different rules or formulas on different parts of its domain. The piecewise function "g(x)" is given as :

g(x) = {-(1/2)x + 1,    for x<-2

       = {-(x+1)²,        for -2≤x≤1

      =  {4                 for x>1

We have to find the value of g(-5) ,g (-2), and g(-1),

For x = -5, the number -5 is less than -2, so the first function "-(1/2)x + 1" will be used,

⇒ g(-5) = -(1/2)(-5) + 1 = 5/2 + 1 = 7/2,

For x = -2, the number -2 lies in the interval "-2≤x≤1", so second function

"-(x+1)²" will be used,

⇒ g(-2) = -(x+1)² = -(-2+1)² = -(-1)² = -1,

For x = -1, the number -1 lies inn the interval "-2≤x≤1", so second function "-(x+1)²" will be used,

⇒ g(-1) = -(x+1)² = -(-1+1)² = -(0)² = 0,

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The sum of 125 (base 7) and 256 (base 7) is 404 (base 7). To compute it, we convert each number to base 10, add them, and then convert the result back to base 7.

Write each number in expanded form using powers of 7.

125 (base 7) = 1 x 7² + 2 x 7¹ + 5 x 7⁰ = 49 + 14 + 5 = 68

256 (base 7) = 2 x 7² + 5 x 7¹ + 6 x 7⁰ = 98 + 35 + 6 = 139

Add the two numbers together to get the sum in decimal form.

68 + 139 = 207

Convert the decimal sum to base 7 using repeated division by 7.

Divide 207 by 7 to get a quotient of 29 and a remainder of 4. Write down the remainder as the rightmost digit in the base 7 representation. Divide 29 by 7 to get a quotient of 4 and a remainder of 1. Write down the remainder as the next digit to the left in the base 7 representation.

Divide 4 by 7 to get a quotient of 0 and a remainder of 4. Write down the remainder as the leftmost digit in the base 7 representation.

So the sum of 125 (base 7) and 256 (base 7) is 404 (base 7).

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To enlarge the photo proportionally so that it is as wide as the computer screen, we need to find the scaling factor. The scaling factor is the ratio of the width of the computer screen to the width of the photo:

Scaling factor = 1024 ÷ 640 = 1.6

We can then use this scaling factor to find the new dimensions of the photo:

New width = 640 × 1.6 = 1024 pixels

New height = 300 × 1.6 = 480 pixels

Therefore, the new dimensions of the photo are 1024 × 480 pixels.

2a. We cannot enlarge the photo proportionally so that it is as tall as the computer screen because the aspect ratio of the photo is different from the aspect ratio of the computer screen. The aspect ratio of the photo is 640 ÷ 300 ≈ 2.13, while the aspect ratio of the computer screen is 1024 ÷ 768 ≈ 1.33. This means that if we enlarge the height of the photo to match the height of the computer screen, the width of the photo will be too wide to fit on the screen.

b. We cannot correct the difficulty in (a) by scaling the width of the photo by a factor of 1024 ÷ 640 and the height by a factor of 768 ÷ 300 because this would change the aspect ratio of the photo. The aspect ratio of the photo would become 1024 ÷ (640 × 1024 ÷ 640) ≈ 1.6, which is the same as the aspect ratio of the computer screen. However, the height of the photo would be scaled by a factor of 768 ÷ 300 ≈ 2.56, which would make the photo too tall to fit on the screen.

To enlarge the photo proportionally, we need to scale it up by the same factor in both dimensions. Since we want to scale it so that its width matches the width of the screen, we can find the scaling factor by dividing the screen width by the photo width:

scaling factor = screen width / photo width = 1024 / 640 = 1.6

Now we can use this scaling factor to find the new height of the photo:

new height = photo height x scaling factor = 300 x 1.6 = 480 pixels

Therefore, after the photo is scaled up proportionally, its measurements are 1024 x 480 pixels.

2a. We can't enlarge the photo proportionally so that it is as tall as the computer screen because its aspect ratio (the ratio of its width to its height) is different from the aspect ratio of the screen. The photo's aspect ratio is 640/300 = 2.13, while the screen's aspect ratio is 1024/768 = 1.33. Enlarging the photo so that its height matches the screen's height would require stretching the photo vertically, which would distort the image.

2b. Scaling the width of the photo by a factor of 1024/640 and the height by a factor of 768/300 would not correct the difficulty in part (a) because it would not change the aspect ratio of the photo. The aspect ratio of the photo would still be 2.13, which is different from the aspect ratio of the screen. The photo would still need to be stretched vertically to match the screen's height, which would distort the image.

The volume of the new object is approximately 439.3 cubic inches

To find the volume of the new object, we need to first know the volume of the prism before the cylinder is removed hence we do this:

The volume of a cylinder: = πr²h

where:

r = radius

h  = height

Note that the volume of a rectangular prism is :  = lwh

l = length

w = width

h = height

Assumption: In the above case, we will  assume that the dimensions of the rectangular prism because the cylinder can be cut out from the center of one of the faces.

Hence the length and width of the prism will be taken to be twice the radius of the cylinder, that is 16 inches, and the height will be taken to be same as the radius of the cylinder, that ia 8 inches.

So  V_prism = lwh

 = 16 × 16 × 8

= 2048 cubic inches

V_cylinder = πr²h

= π × 8² × 8

= 512 π cubic inches

Hence To get  the volume of the new object, we need to subtract the volume of the cylinder from the volume of the prism:

V_new = V_prism - V_cylinder

= 2048 - 512π

= 2048  - 1608.70

= 439.3 cubic inches

Hence, the volume of the new object is about 439.3 cubic inches

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Answer: 3.5

Step-by-step explanation: i got it right on the quiz

The first 5 terms in the sequence are 5, 4.75, 4.5, 4.25 and 4

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

a2 = -0.25 + a1

So, we have

a2 = -0.25 + 5 = 4.75

For other terms. we have

a3 = -0.25 + 4.75 = 4.50

a4 = -0.25 + 4.50 = 4.25

a5 = -0.25 + 4.25 = 4.0

Hence, the first 5 terms in the sequence are 5, 4.75, 4.5, 4.25 and 4

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Step-by-step explanation:

Count numbers

There are six numbers that counted in

Thousands and tens

The number is 357476 which is three hundred and fifty seven thousands, four hundred and 76

To find the answer you cancel 35 and remains with 7 the other numbers 4 and 6 you turns them into zeros and remain with seven

Thousands and tens is the answer of placing a value of two sevens.