How do you find product of mixed number and a fraction?.

A commutative property in math is

Commutativity says that the numbers we are dealing with can be shifted or swapped without affecting the answer. This property applies to addition and multiplication, but not to subtraction and division.

a + b = b + a; where a and b are

integers. Example: 3 + 4 = 4 + 3 = 7.

a × b = b × a; where a and b are

non-zero integers. Example :

2×3 = 3×2 = 6

So this rule simply states that you can change the order of numbers in a question by adding or multiplying them without affecting the answer.

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The sum of frequencies for all classes will always equal to the number of element in a data set. Between 0-100 there will be 100 numbers.

So that it's 100 and correct option is b

What is frequencies ?

A frequency is a measurement of how frequently a certain value, occurrence, or feature takes place. It is employed in a wide range of disciplines, including science, engineering, mathematics, and statistics. Frequency is often expressed as a number, percentage, or ratio. For example, a frequency chart might show the number of times a particular product was purchased over a certain time period. Frequency can also refer to the number of times a particular wave pattern occurs in a given time period. In physics and engineering, frequency is used to measure the number of cycles per second of a wave or periodic phenomenon. For example, a wave with a frequency of 440 hertz (Hz) has 440 complete cycles per second.

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According to the given quadratic equation  the sum of their pieces:

A. = 8x² + 2x + 10

B. 7x²- 7x -17

C.  incomplete information.

A quadratic equation is a third polynomial equation in one variable using the formula x = ax2 + bx + c = 0 and a 0. The fundamental theorem of algebra ensures that it has at least one solution because it is a second-order polynomial problem. Real or complex solutions are also possible.

In mathematics, the quadratic issue is any problem that involves multiplying a variable by itself, often known as "squaring." This terminology comes from the fact that a square's area is equal to its side length divided by itself. The Latin term quadrature, which means square, is where the word "quadratic" derives.

A. Lisha

= 3x² + 2x + 8

Makulata = 5x² + 2

Sum of the given equation:

=  (3x² + 2x + 8) + ( 5x² + 2)

=(3x² + 5x² +2x + 10)

= 8x² + 2x + 10

B.

= 4x + 6x² - 11x + 2 + x² - 19

= (6x² + x²) + (4x - 11x) + 2 -19

= 7x²- 7x -17

C. incomplete information.

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Answer:

The number is 6

Step-by-step explanation:

"Think of a number,"    [I'm thinking of x]

"double the number"    [OK, that would be 2x]

"and add 6 "                    [If you insist:  2x + 6]   ["The Answer"]

"The answer is 3 times the number"  [Answer = 3x]

We take the answer, 2x+6 and set it equal to 3x

         2x+6 = 3x

At a local print shop, 20 copies can be made for $8. At this rate,  it would cost $30 to make 75 copies.

What is the equivalent ratio?

Equivalent ratios are the ratios that are the same when we compare them. Two or more ratios can be compared with each other to check whether they are equivalent or not. For example, 1:2 and 2:4 are equivalent ratios.

To solve this problem, we can set up a ratio between the number of copies and the cost:

20 copies / $8 = 75 copies / x

We can cross-multiply to find the value of x:

20 copies * x = 75 copies * $8

20x = 600

x = 30

Therefore, it would cost $30 to make 75 copies.

Here's the completed table of equivalent ratios:

Number of copies                        Cost

           20                                 $8

           75                                   x

We know that these ratios are equivalent, so we can set up a proportion:

20/8 = 75/x

Then we cross-multiply:

20x = 75 * 8

20x = 600

And solve for x:

x = 600/20

x = 30

Therefore, it would cost $30 to make 75 copies.

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The subtraction equation the inequality solve is

-9 + -4 = -13

A number line is a horizontal line with uniformly spaced numerical increments.

The number on the line can are represented depending on the numbers that are provided. A number is used to plot a point depending on the question that goes with it and the number type.

For the problem one unit represent a digit number with intervals of 1.

The shorter line is as -4 and the longer is at +9

adding bot together will be

= -9 + -4

= -9 - 4

= -13

hence the subtraction equation is -9 - 4 = -13

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The equation g(x) by translating the following function 4 units to the right would be g(x) = (x+4)² - 3(x+4) - 10.

What is a function?

A function is a mathematical relationship between a set of inputs and a set of outputs. More formally, a function is a rule that assigns to each input exactly one output. In mathematical notation, a function is typically represented by an equation of form f(x) = y, where x is the input and y is the output. The symbol f(x) is called the function notation, and x is called the independent variable, and y is called the dependent variable.

To translate a function 4 units to the right, we can add 4 to the input variable (x) inside the function. Therefore, the equation for g(x) would be:

g(x) = (x+4)^2 - 3(x+4) - 10

Hence, the equation g(x) by translating the following function 4 units to the right would be g(x) = (x+4)² - 3(x+4) - 10.

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The equation of the parabola with vertex (-1,-3) and the origin as zero point is given by y = (x + 1)2 - 3.

An equation is a mathematical statement that expresses two expressions as being equal. It consists of two expressions, one on either side of an "equals" sign, and indicates that the expressions have the same value. For example, the equation "2 + 2 = 4" states that the value of the expression "2 + 2" is equal to the value of the expression "4".

This equation can be derived by transforming the standard form of a parabola, which is y = ax2 + bx + c, into a form where the vertex of the parabola lies at the origin of the coordinate system. The standard form of the equation can be written as y = (x - (-1))2 -3. When the terms inside the parentheses are multiplied out, the equation becomes y = (x + 1)2 – 3.

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Answer:

Step-by-step explanation:

42 : x = 12 : 14

x = 42 x 14 : 12

x = 49

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

Answer: 33

Step-by-step explanation:

66 - 33

Answer:

x is 33...........................

Step-by-step explanation:

33 +x =66

66-33=x

Answer: x=56 you can just look it up on

Answer:

[tex]x=56[/tex]

Step-by-step explanation:

[tex]\frac{x}{4}+11=25\\\\[/tex]

Mutiply by 4 every each number you see.

[tex]x+44=100[/tex]

Subtract -44 in both sides.

[tex]x+44-44=100-44[/tex]

So,

[tex]x=56[/tex]

Hope this helps :D

The solution of the integral are:

(a) ∫tanx secx dx = sec(x) + C,

(b) [tex]\int\limits{ \frac{y^3}{(1 - 2y^4)^5} \, dy = -\frac{1}{32(1 - 2y^4)^4}[/tex]

What is integral?

Integrals are used to evaluate such quantities as area, volume, work, and, in general, any quantity that can be interpreted as the area under a curve.

We have given,

∫tanx *secx dx = ?

∫y³/(1-2y⁴)⁵ dy = ?

∫tan(x)sec(x)dx

Rewrite the expression

∫ sin(x)/cos²(x) dx

Use the substitution dx = − 1/sin(x) dt to transform the integral

∫ sin(x)/cos²(x) ×(− 1/sin(x))dt

Simplify

∫− 1/cos²(x)dt

Use the substitution t=cos(x) to transform the integral

∫−t⁻²dt

= −(−t⁻¹)

Calculate

t⁻¹

Substitute back

1/cos(x)

Rewrite the expression

1 × sec(x)

Multiply the terms

sec(x)

Solution

sec(x) + C,  C ∈ R

∫y³/(1-2y⁴)⁵ dy =

[tex]= \int\limits {-\frac{1}{8u^5} } \, du[/tex]

[tex]= \frac{1}{8} \int\limits {-\frac{1}{u^5} } \, du[/tex]

[tex]= \frac{1}{8} (-\frac{1}{4u^4} )[/tex]

by substituting u = (1 - 2y⁴)⁴

[tex]= \frac{1}{8} (-\frac{1}{4(1-2y^4)^4} )[/tex]

[tex]= -\frac{1}{32(1 - 2y^4)^4}[/tex]

Hence, the solution of the integral are:

(a) ∫tanx secx dx = sec(x) + C,

(b) [tex]\int\limits{ \frac{y^3}{(1 - 2y^4)^5} \, dy = -\frac{1}{32(1 - 2y^4)^4}[/tex]

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By setting one side of a quadratic equation to zero, factoring the resulting polynomial, and then using the Zero Product Property, you can obtain the solutions, or roots, of these equations.

According to the Principle of Zero Products, if ab = 0, then either an or b, or both a and b, are 0, or both are 0. The roots of a function (f (x) can be any function) are the answers to y = f (x) when y = 0. These are the locations where an equation's graph crosses the x-axis.

To use factoring to solve a quadratic equation:

1. Convert the equation to standard form with a zero on one side.

2. Factor the non-zero side.

3. Reset each component to zero.

4. Solve each of the ensuing equations.

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The solutions to the equation are x = 2.61 and x = -0.61

To solve the quadratic equation 3x² - 16x + 6 = 0, we'll use the quadratic formula:

x = (-b ± √(b²-4ac)) / 2a

Where a = 3, b = -16 and c = 6.

x = (-(-16) ± √((-16)²-4(3)(6)))/2(3)

x = 16 ± √(256-72) / 6

x = 16 ± √(184) / 6

x = 16 ± √184 / 6

x = (16 ± 4√23) / 6

So the solutions are:

x = (16 + 4√23) / 6 ≈ 2. 61 or x = (16 - 4√23) / 6 ≈ -0.61

Therefore, the solutions to the equation are x = 2.61 and x = -0.61.

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No, the tan inverse is not equal to tan

A right triangle's opposing side to its adjacent side is calculated using the trigonometric function known as the tangent function (tan). It is described as tan(x) = adjacent/opposite. The opposite of the tangent function is called the inverse tangent function (tan-1). The angle whose tangent has the specified value is revealed. The formula for it is tan-1(y) = x, where tan(x) = y. An angle between -90 and 90 degrees is what is returned.

In other words, tan-1 undoes the tangent function because it is the inverse of the tangent function. In other words, the input of the tangent function is the output of the inverse tangent function. In conclusion, tan-1 and tan are inverse functions; they are not equal.

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An increase of 106.63 % in gold price is observed in that particular year.

The original value multiplied by 100 is used to get the percentage change, which is the difference between the original and final values. It is expressed as a percentage and can be used to demonstrate an increase or decrease over time. A higher proportion indicates growth, whereas a lower percentage indicates a decline.

initial price of gold per ounce is $513

final price of gold per ounce is $1060

change in rate = final price of gold - inital price of gold

=> 1060 -513

=> 547

so the change in rate = $547

percentange change =  (change in rate) / (inital rate of gold) *100

=> [tex]\frac{547}{513}[/tex] *100

=> 106.63 %

so the percentage change is 106.63%

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When we say "0" or "infinity," what we're really saying is that the number has no end.

The original value of zero corresponds to the logarithm, and the original value of infinity to the logarithm +. We consider the initial values 0 and infinity similarly when we treat both of the conceivable values and + as a single infinity. Georg Cantor, a mathematician, developed the concept of infinity into The Absolute Infinite. You can think of it as a number that is greater than any other possible or inconceivable amount, whether it be transfinite or finite. Infinity is the unfathomable end of the number line, according to mathematics. No actual number is bigger than infinity since it cannot be believed to exist beyond it.

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The slope and the y intercept of the equation 3x + 2y = 5 is -3/2 and 5/2 respectively.

The term equation of the straight line is y = mx + c here m refers the slope and c refers the value where the line cuts the y-axis that is y intercept.

Here we have given that the equation 3x + 2y = 5.

In order to find the slope and the y intercept we have to rewrite the given equation as follows,

Here we need to find the value of y so we have to move the other terms to the next side then we get,

=> 2y = 5 - 3x

Then it can be rewritten as,

=> y = (5 - 3x)/2

Therefore, the resulting equation is written as,

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

So, the slope of the equation is -3/2 and the y intercept is 5/2.

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Answer: $9.50 per hour

Hope this helps :)

Step-by-step explanation:

Find the rate of change

76/8 = 9.5

8.50 cents per hour because 8.50 x 9 is close to 76

To be a function, each x-value (or input) can only be used once.  Each x-value (or input) can only be paired with a single y-value (or output).

If any x-value (or input) is paired with more than one y-value (or output), it's not a function.

Top Left:

Not a function

"rock" and "leaf" are paired with two outputs.

Top Right:

Not a function

The one input goes to three outputs.

Bottom Left:

Function

Each x-value/input is used only once.

Bottom Right:

Not a function

The x-value of 4 is used twice and paired with two different y-values.

Answer:[tex]\frac{8!}{2!*2!*2!}[/tex] = 5040.

Step-by-step explanation:

Since CALCULUS has 8 letters with 2 indistinguishable C's,

2 indisnguishable L's and 2 indistinguishable U's, the answer is

[tex]\frac{8!}{2!*2!*2!}[/tex] = 5040.

When the rectangular garden has a length of 20 yards and a width of 42 yards the diagonal path will be [tex]\sqrt{2164}[/tex] .

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

Since we have a right-angled triangle, we can use the Pythagorean Theorem to find our answer. The Pythagorean Theorem states the following:

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

In this formula a and b represent the legs of the triangle and c represents the length of the hypotenuse.

Let's substitute our data from the question.

[tex]42^{2} +20^{2} =c^{2}[/tex]

Square.

1764 + 400 = [tex]c^{2}[/tex]

And finally take the root of both sides. We only need to use the positive solution, since the length of the hypotenuse can't be negative.

[tex]c = \sqrt{2164}[/tex]

When the rectangular garden has a length of 20 yards and a width of 42 yards the diagonal path will be [tex]\sqrt{2164}[/tex] .

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Answer: 53

Step-by-step explanation:

The length and width of the garden represent the legs of a right triangle, and the diagonal path represents the hypotenuse.

The Pythagorean Theorem states that a squared plus b squared equals c squared, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse.

Let p equal the length of the diagonal path in yards.

Applying the Pythagorean Theorem gives us p squared equals forty-five squared plus twenty-eight squared.

Now square forty-five and twenty-eight to get p squared equals two thousand twenty-five plus seven hundred eighty-four.

Next, find the sum of two thousand twenty-five and seven hundred eighty-four, which is two thousand eight hundred nine.

Finally, calculate the square root of two thousand eight hundred nine to find that p equals fifty-three.

So the length of the diagonal path is fifty-three yards.

We have proved that BC² ×AD = AC² ×BD using AAA congruence rule.

Triangles that are identical in size and shape are said to be congruent. This implies that the corresponding sides and angles are equal.

Without comparing each of the triangles' sides and angles, we can determine whether two triangles are congruent. The four rules that demonstrate triangle congruence will be discussed in this lesson. They are referred to as the SSS rule, SAS rule, ASA rule, and AAS rule.

We'll discuss the Hypotenuse Leg rule, a right triangle proof, in a later lesson. It is sufficient to demonstrate that the two triangles are congruent if only one of the rules is accurate.

Consider △ACD&△ABC

∠CAD=∠CAB

∠CDA=∠ACB

∴△ACD∼△ABC,{ By AA similarity criterion}

⟹ AC/AB = AD/AC

∴ AC ² =AB×AD  

Similarly, △BCD∼△BAC { by AA similarity criterion}

⟹ BC/BA =BD​/BC

∴BC² =BA×BD

∴ BC²/AC² = AB×BD/AB×AD

∴BC² ×AD = AC² ×BD

Thus, We have proved that BC² × AD = AC² × BD using AAA congruence rule.

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Complete question:

Triangle ABC is right angled at C and CD is perpendicular to AB. Prove that BC² ×AD = AC² ×BD

The track rises 31.88 feet over a horizontal distance.

One of the six fundamental functions in trigonometry is the tangent function. As stated, Tan A = Opposite Side/Adjacent Side is the Tangent Formula. Tangent may be modeled using sine and cosine as follows: Tan A = Sin A / Cos A.

Given:

A train track slopes upward at an angle of 12°.

The track rise over a horizontal distance of 150 feet.

From the attached image:

tan(12)° = AB/OA

AB = 150 x tan12°

AB = 31.88 feet.

Therefore, the required distance is 31.88 feet.

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The component form of the vector u and v are < -60.11, 9.89 >

To find the component form of the sum of u and v, we first need to express each vector in terms of its components in the x and y directions. To do this, we use the trigonometric relationships between the magnitude and direction angle of a vector and its x and y components.

The x and y components of u can be found using the following formulas:

x_u = u * cos(u)

y_u = u * sin(u)

where u is the magnitude of u and u is the direction angle of u measured in radians.

So, x_u = 14 * cos(45°) = 9.89

and y_u = 14 * sin(45°) = 9.89

The x and y components of v can be found using the following formulas:

x_v = v * cos(v)

y_v = v * sin(v)

where v is the magnitude of v and v is the direction angle of v measured in radians.

So, x_v = 70 * cos(180°) = -70

and y_v = 70 * sin(180°) = 0

Now, we can find the x and y components of the vector sum by adding the corresponding components of u and v.

x_sum = x_u + x_v = 9.89 + (-70) = -60.11

y_sum = y_u + y_v = 9.89 + 0 = 9.89

Therefore, the component form of the sum of u and v is given by the vector < -60.11, 9.89 >

Note: the direction angle is measured in radians and the angle given in the question was in degrees, so it's important to convert the angle from degrees to radians before use it.

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Gross profit is the profit that a business makes after the manufacturing and selling costs are subtracted from the total sales.

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The value of x in the supplementary angle pair is obtained as x = 14°.

What is a supplementary angle?

The definition of "supplementary" in mathematics relates to angles that combine to form a straight angle. It indicates that when two angles sum up to 180 degrees, they are said to be supplementary angles.

Two angles are supplementary, if -

One of its angles is an acute angle and another angle is an obtuse angle.

Both of the angles are right angles.

The given angles are -

Angle 1, ∠1 = 156°

Angle 2, ∠2 = (x + 10)°

Since, both angles lie on same plane which a straight line it forms a supplementary angle.

The equation for supplementary angle pair is -

∠1 + ∠2 = 180°

Substituting the values in the equation -

∠1 + ∠2 = 180°

156° + (x + 10)° = 180°

166° + x = 180°

x = 180° - 166°

x = 14°

Therefore, the value of x = 14°.

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The height of this hill is given as :  51.88 feet.

How to solve for the height of the hill

We can use trigonometry to find the height of the hill. Let's call the height of the hill "h".

From the information given, we know that:

tan(27°) = h / (103/2)

h = (103/2) * tan(27°)

Using a calculator, we would have to go ahead to get the value of the variable h or height.

So  that h ≈ 51.88 feet.

So, the height of the hill is approximately 51.88 feet.

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The solution to the system of equations y = 1.5x - 4 and y = -x  is (1.6,-1.6)

The coordinates of the ordered pair (s) that satisfy all of the system's equations make up the solution set to an equational system. This means that the equations will be true for those x and y values. The intersection of all of the graphs, when a system of equations is graphed, is the solution.

These are the equations:

y = 1.5x - 4

y = -x

Put y = -x in place of y in the first equation.

-x = 1.5x - 4

Assemble similar terms

-1.5x - x = -4

This results

-2.5x = -4

Add -2.5 to both sides.

x = 1.6

Change y = -x to x = 1.6.

y = -1.6

As a result, the system of equations solution is (1.6,-1.6).

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