What is an inequality in math?

Answer: The trigonometric identity for the tangent of a sum of angles can be used to simplify the expression:

tan (480° + 300°) = (tan 480° + tan 300°) / (1 - tan 480° tan 300°)

Using the identity for the tangent of half an angle, the tangent of 480° can be expressed as follows:

tan 480° = tan (450° + 30°) = (tan 450° + tan 30°) / (1 - tan 450° tan 30°) = (1 + tan 30°) / (1 - tan 30°) = (1 + √3/3) / (1 - √3/3) = (1 + √3) / (√3 - 1)

Using the identity for the sine and cosine of a multiple of 30°, the tangent of 300° can be expressed as follows:

tan 300° = tan (30° * 10) = tan 30° / (1 - tan 30°) = √3 / (1 - √3) = √3 / (-1 - √3)

Plugging these values back into the expression for tan (480° + 300°), we get:

tan (480° + 300°) = (tan 480° + tan 300°) / (1 - tan 480° tan 300°) = ( (1 + √3) / (√3 - 1) + √3 / (-1 - √3) ) / (1 - (1 + √3) / (√3 - 1) * √3 / (-1 - √3) )

Expanding the denominator and simplifying, we get:

tan (480° + 300°) = ( (1 + √3) / (√3 - 1) + √3 / (-1 - √3) ) / ( (√3 - 1) / (-1 - √3) - (1 + √3) / (-1 - √3) )

Using the identity for the sine and cosine of a sum of angles, the sine and cosine of 480° + 300° can be expressed as follows:

sin (480° + 300°) = sin 480° cos 300° + cos 480° sin 300°

Finally, using the identity for the tangent of an angle in terms of sine and cosine, we get:

tan (480° + 300°) = sin (480° + 300°) / cos (480° + 300°) = sin 780° / cos 780°

The other trigonometric functions in the expression can be simplified using similar techniques, but the final result may be complex. However, it can be verified that the expression is equal to 3/2 by using a calculator or numerical methods.

Step-by-step explanation:

The composition of functions f and g is:

(f o g) =6x^2 + 15x + 2

Here we have two known functions:

f(x) = 3x + 2

g(x) = 2x^2 + 5x

We want to find the composition:

(f o g)

That just means that we need to evaluate function f(x) in x = g(x), we wikk get:

(f o g) = 3*g(x) + 2

Now we can replace g(x) there to get:

(f o g) = 3*(2x^2 + 5x) + 2

          = 6x^2 + 15x + 2

That is the composition.

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Answer:

Yes, dilation does not change the angle measurements.

Step-by-step explanation:

Sally paid $1,043.33 for the leather chair, including tax.

The marked up price of the leather chair is 100% more than the original cost, which means it is twice the original cost.

        $494.47 x 2 = $988.94

       $988.94 x 0.055 = $54.39

        $988.94 + $54.39 = $1,043.33.

Therefore, Sally paid $1,043.33 for the leather chair, including tax.

The calculation provided is applicable for any problem that involves finding the total cost of an item after a percentage markup and the addition of a sales tax. The general formula for calculating the final cost of an item after a markup and sales tax is:

Final cost = (1 + markup percentage) x original cost x (1 + sales tax percentage)

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Answer:

A: 5

Step-by-step explanation:

Using the side-splitter theorem (parallel lines and similar triangle ratios).

[tex]\frac{16}{4x} = \frac{28}{35}[/tex]

Through cross mutliplying, we get 112x = 560

x = 5

Answer:

So, Donna served 9, Alonzo served 19 and kevin served 57 orders.

Step-by-step explanation:

Let orders be :

Donna - x, Alonzo - y, Kevin - z

x = y - 10 - eqn i

z = 3y - eqn ii

Now, By the question,

x + y + z = 85

we know from eqn i and eqn ii,

y - 10 + y + 3y = 85

or, 5y = 95

so, y = 19

Now,

z = 3(19) = 57

x = 19 - 10 = 9

The amount that should be deposited today is given as follows:

$70,663.15.

The amount of money earned, in compound interest, after t years, is given by:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which:

The parameters for this problem are given as follows:

t = 15, A(t) = 115000, r = 0.033, n = 1.

Hence the deposit is obtained as follows:

115000 = P(1.033)^15

P = 115000 x (1.033)^(-15)

P = $70,663.15.

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The following numbers are the result:

22 play trumpet, 11 play trombone, and 8 play saxophone.

Two algebraic expressions having the same value and symbol '=' in between are called an equation.

Given:

There are 41 students in group 2.

Twice as many students play the trumpet as play the trombone,

but 8 students play the saxophone.

Let x be the number of students playing the trombone.

So, according to the question,

x + 2x + 8 = 41

3x = 33

x = 11

Hence, 22 play trumpet, 11 play trombone, and 8 play saxophone.

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On x-axis where |x|>0 is located at point E on the coordinate plane.

What does a math definition of a coordinate plane mean?

A surface with two dimensions known as the coordinate plane is created by two number lines. The x-axis is the name given to one horizontal number line. The y-axis is the name given to the vertical number line that is the other number line. A place known as the origin is where the two axes collide.

                        To graph points, lines, and other things, we can utilize the coordinate plane. The Y-axis and the X-axis cross to create a two-dimensional plane known as a coordinate plane, also referred to as a rectangular coordinate plane grid.

on x-axis except at the origin

or on x-axis where |x|>0

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The derivative of the function y = tan⁻¹ x which is expressed in terms of x will be 1 / (1 + x²).

The rate of change of a function with respect to the variable is called differentiation. It can be increasing or decreasing.

The function is given below.

x = tan y

Get the equation for y, then we have

x = tan y

y = tan⁻¹ x

Differentiate the function with reprect to 'x', then we have

y' = (d/dx) tan⁻¹ x

y' = 1 / (1 + x²)

The derivative of the function y = tan⁻¹ x which is expressed in terms of x will be 1 / (1 + x²).

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Answer:

8

Step-by-step explanation:

46 = 10b - 34

Move 34 to the other side by adding 34 to each side

80 = 10b

Divide 10 on each side to get b on one side

8 = b

Using the Divergence Theorem, we get the flux of F across the closed surface S is π/2 (27√2 - 27).

To apply the Divergence Theorem, we will first compute the divergence of F as follows -

div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

= 1 + 1 + 1

= 3

Now we will calculate the flux of F across the closed surface S that completely surrounds T. By the Divergence Theorem, this is equal to the triple integral of the divergence of F over the region T and is given by -

∫∫S F•n dS = ∭T div(F) dV

We can describe the region T using cylindrical coordinates:

0 ≤ r ≤ 2

-π/2 ≤ θ ≤ π/2

0 ≤ z ≤ √(9 - r sin θ)

The bounds on r and θ come from the fact that the half cylinder is contained within the plane x = 2, and the plane y = ±3. The bounds on z come from the equation of the half cylinder.

Now we can write the triple integral as follows -

∭T div(F) dV = ∫0^2 ∫-π/2^π/2 ∫0^√(9 - r sin θ) 3 r dz dθ dr

Evaluating this integral, we get,

∭T div(F) dV = π/2 (27√2 - 27)

Therefore, the flux of F across the closed surface S is π/2 (27√2 - 27).

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We can show that any field IF is a vector space over itself, with the field addition used as vector addition and the field multiplication used as scalar multiplication.

A field is a set of elements, denoted by IF, that satisfies the following axioms:

Commutativity of addition

Associativity of addition

Existence of additive identity

Existence of additive inverse

Commutativity of multiplication

Associativity of multiplication

Distributivity of multiplication over addition

Existence of multiplicative identity

Existence of multiplicative inverse

Given these axioms, we can show that IF is a vector space over itself.

Closure under addition: For any a, b in IF, their sum a + b is also in IF.

Commutativity of vector addition: For any a, b in IF, a + b = b + a.

Associativity of vector addition: For any a, b, c in IF, (a + b) + c = a + (b + c).

Existence of additive identity: The additive identity 0 in the field IF can also be used as the additive identity in the vector space.

Existence of additive inverse: For any a in IF, its additive inverse -a in the field IF can also be used as the additive inverse in the vector space.

Distributivity of scalar multiplication over vector addition: For any a in IF and any u, v in IF, a * (u + v) = a * u + a * v.

Distributivity of scalar multiplication over field addition: For any a, b in IF and any u in IF, (a + b) * u = a * u + b * u.

Existence of multiplicative identity: The multiplicative identity 1 in the field IF can also be used as the multiplicative identity in the vector space.

Therefore, we have shown that any field IF is a vector space over itself, with the field addition used as vector addition and the field multiplication used as scalar multiplication.

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The p-value (rounded to four decimal places) is 0.2709.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

The formula of probability is defined as the ratio of a number of favorable outcomes to the total number of outcomes.

P(E) = Number of favorable outcomes / total number of outcomes

We are given that;

p = 477/1000 = 0.477

p = 0.48

n = 1000

Now,

The test statistic for a hypothesis test for a proportion is given by:

z = (pi - p) / sqrt(p * (1 - p) / n)

where p is the sample proportion, p is the hypothesized proportion under the null hypothesis, n is the sample size, and sqrt is the square root function.

Substituting these values into the formula, we get:

z = (0.477 - 0.48) / sqrt(0.48 * 0.52 / 1000) ≈ -0.61

We find that the area to the left of z = -0.61 is approximately 0.2709 This means that the probability of getting a test statistic as extreme or more extreme than -0.61, assuming the null hypothesis is true, is 0.2709

Therefore, by probability the answer will be 0.2709

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Answer:

 x = 1/2. 

Step-by-step explanation:

Hope this helps!

It cost $17.82 to buy the papers needed for this reports

An equation is an expression that shows the relationship between two or more numbers and variables using mathematical operations. An equation can be linear, quadratic, cubic and so on, depending on the degree of the variable.

A long year-end status report for work is 141 pages long. You need to print 18 copies for a meeting next week. Therefore:

Amount of paper printed = 141 pages long * 18 copies = 2538 pages

Paper is sold in reams (500 pages) for $3.51 each. Hence:

Amount spent on papers = 2538 pages * ($3.51 / 500 pages) = $17.82

About $17.82 was spent on papers

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Answer:

no it can not so false boiiiiiiiii

Step-by-step explanation:

I think it’s false, maybe

Answer:

y = -110x + 1,600

Step-by-step explanation:

Zachary purchased a computer for $1,600 on a payment plan. Four months after he purchased the computer, his balance was $1,160. Five months after he purchased the computer, his balance was $1,050. What is an equation that models the balance y after x months?

1,160 - 1,050 = $110 change from month 5 to 4

Double check if the rate of decrease is steady over time:

1,600 - ($110 * 4 months) = 1,160

1,600 - ($110 * 5 months) = 1,050

This means, Zachary is paying $110 per month for his computer.

SO:

remaining payment plan balance = initial balance - 110 per month

This can be modeled by the algebraic equation:

y = 1,600 - 110x

rearrange right side:

y = -110x + 1,600

Answer:

[tex]y=-110x+1600[/tex]

Step-by-step explanation:

Given information:

Define the variables:

Therefore:

[tex]\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}[/tex]

Determine if the equation is linear by calculating the slope between each pair of (x, y) points:

[tex]\implies \text{Slope}\;(m)=\dfrac{1050-1160}{5-4}=-110[/tex]

[tex]\implies \text{Slope}\;(m)=\dfrac{1160-1600}{4-0}=-110[/tex]

As the slope is the same, the equation is linear.

[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}[/tex]

As the initial purchase price of the computer was $1,600, the y-intercept is 1600.  We have already calculated the slope. Therefore, substitute the found slope and y-intercept into the slope-intercept formula to create an equation that models the balance y after x months:

let's say segment A(1 , 4) through B(6 , 14) gets partitioned by point C

[tex]\textit{internal division of a line segment using ratios} \\\\\\ A(1,4)\qquad B(6,14)\qquad \qquad \stackrel{\textit{ratio from A to B}}{2:3} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{2}{3}\implies \cfrac{A}{B} = \cfrac{2}{3}\implies 3A=2B\implies 3(1,4)=2(6,14)[/tex]

[tex](\stackrel{x}{3}~~,~~ \stackrel{y}{12})=(\stackrel{x}{12}~~,~~ \stackrel{y}{28}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{3 +12}}{2+3}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{12 +28}}{2+3} \right)} \\\\\\ C=\left( \cfrac{ 15 }{ 5 }~~,~~\cfrac{ 40}{ 5 } \right)\implies C=(3~~,~~8)[/tex]

Answer:

Step-by-step explanation:

The inequality 5x+12y ≤ 30 represents a shaded region in the xy-plane that includes all the points that satisfy the inequality. To graph this inequality, we can first graph the equation 5x+12y = 30, which is the boundary line of the shaded region.

To graph the boundary line, we can find its x- and y-intercepts:

x-intercept: Set y = 0 and solve for x: 5x + 12(0) = 30, which gives x = 6.

y-intercept: Set x = 0 and solve for y: 5(0) + 12y = 30, which gives y = 2.5.

So the boundary line passes through the points (6, 0) and (0, 2.5).

Next, we can determine which side of the boundary line to shade. One way to do this is to pick a test point that is not on the line and plug it into the inequality. For example, the point (0,0) is a convenient test point. Plugging in x=0 and y=0 into the inequality, we get:

5(0) + 12(0) ≤ 30

which simplifies to 0 ≤ 30. Since this is true, we know that the region containing the point (0,0) is part of the shaded region, and therefore the shaded region is the one that contains the origin.

Putting it all together, we can graph the inequality by drawing the line 5x+12y = 30 and shading the region below the line, as shown in the figure below:

    |

 3  |       x

    |      /

 2  |     /

    |    /

 1  |   /  

    |  /  

 0  | /    

    ------------

      0  6

The shaded region includes all the points below the line 5x+12y=30, including the points on the line.

Answer:

The First question's answer is AngleW = 77 degrees

The Second question's answe is x = 2

Step-by-step explanation:

First Question:

AngleW = 7x

AngleY = 6x + 11

Since Opposite Angles in a parallelogram are equal,

AngleW = AngleY

or

7x = 6x + 11

7x - 6x = 11

x = 11

Now that we know the value of x, we can find the value of angleW

AngleW = 7x

AngleW = 7(11)

Angle W = 77

Second Question:

LN = 20

UN = 5x

Since LN and KM are diagonals of the parallelogram, they bisect each other at point of contact, so, UN = 1/2LN

or

5x=1/2 × 20

5x = 10

x = 2

Based on the information given the rate of increase is 55%.

Rate of increase:

Using this formula

Rate of increase=Stock value per share/Number of shares of stock×100

Where:

Stock value per share=$55 per share

Number of shares of stock=100 shares

Let plug in the formula

Rate of increase=$55/100×100

Rate of increase=55%

Inconclusion the rate of increase is 55%

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The area of the trapezoid with base lengths of 16 cm and 13 cm and the height with 10 cm, is 145 cm²

A trapezoid, also known as a trapezium, is a flat-closed shape having 4 straight sides, with one pair of parallel sides.

Given that, a trapezoid has base lengths of 16 centimeters and 13 centimeters, the height of the trapezoid is 10 centimeters, we are asked to find the area of the trapezoid,

Area of the trapezoid = (sum of the bases) × height / 2

Length of bases = 16 cm and 13 cm

Height = 10 cm

Area of the trapezoid = 16+13 × 10/2

= 145

Hence, the area of the trapezoid with base lengths of 16 cm and 13 cm and the height with 10 cm, is 145 cm²

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The linear function that models this scenario is given as follows:

y = -0.25x + 13.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

Bridget has 13 gallons of gasoline to use for her lawnmowing business, hence the intercept b is given as follows:

b = 13.

From the graph, when x increases by 20, y decays by 5, hence the slope m is given as follows:

m = -5/20

m = -0.25.

Hence the function is given as follows:

y = -0.25x + 13.

The problem is given by the image presented at the end of the answer.

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Step-by-step explanation:

The two adjacent vertices of the triangle are A(5; 3) and B(7; 7), and one point on the opposite side is P(-2; 9). To find the coordinates of the third vertex, we need to find the midpoint of AB and extend it to P.

The midpoint of AB can be calculated by finding the average of their x-coordinates and y-coordinates:

C = ((A_x + B_x)/2, (A_y + B_y)/2) = ((5 + 7)/2, (3 + 7)/2) = (6, 5)

Next, we can use the midpoint C and point P to find the slope of the line connecting them:

m = (P_y - C_y) / (P_x - C_x) = (9 - 5) / (-2 - 6) = 4 / -8 = -0.5

Since the slope of a line perpendicular to this one is -2, we can use the midpoint C and the slope to find the equation of the line connecting C and the third vertex D:

y - C_y = -2 (x - C_x)

y - 5 = -2 (x - 6)

Expanding and solving for x, we get:

y = -2x + 17

Substituting the y-coordinate of P (-2) into this equation:

-2 = -2x + 17

Solving for x:

x = 7.5

Finally, we can use the x-coordinate to find the y-coordinate:

y = -2x + 17 = -2 * 7.5 + 17 = 5.5

So the third vertex is D (7.5, 5.5).

The vertices of the triangle are A (5, 3), B (7, 7), and D (7.5, 5.5).

B. Theorem is the term that refer to true statements.

According to the definitions of the terms we have:

A confirmed factual assertion is referred to as a theorem.

• Proposition: A truthful assertion that's less significant but nonetheless intriguing.

• Lemma: A true statement that serves as an example of another true statement.

significant theorem that aids in the validation of other findings).

• Corollary: A true statement that can be inferred directly from a premise or theorem.

• Proof: The justification for why a claim is accurate.

• Conjecture: An assertion that is deemed true but for which there is no supporting evidence. (An assertion that is being put forth as being true).

• Axiom: A fundamental presumption regarding a mathematical scenario. (a premise we take to be true)

Hence, B. Theorem is the term that refer to true statements.

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The number of hours of television watched and the number of hours of exercise, correct to three decimal places is 3.940.

A decimal is any number expressed in base-10 notation, which includes a decimal point and the numbers 0 through 9. Decimals are used to represent fractions, numbers that are not whole numbers, and numbers that are exceptionally large or tiny.

To calculate the correlation coefficient between hours of television watched and hours of exercise, we will use the formula

r = (∑xy − (∑x)(∑y) / n) /[tex]\sqrt{((2x-x/n)(2y-2/n))}[/tex]

where x is the number of hours of television watched and y is the number of hours of exercise.

Given the data provided, we have

x = 3, 14, 21, 28, 43

y = 38, 27, 36, 15, 30

Therefore,

∑x = 3 + 14 + 21 + 28 + 43 = 109

∑y = 38 + 27 + 36 + 15 + 30 = 156

∑xy = (3)(38) + (14)(27) + (21)(36) + (28)(15) + (43)(30) = 1291

∑x2 = 32 + 142 + 212 + 282 + 432 = 4552

∑y2 = 382 + 272 + 362 + 152 + 302 = 4644

n = 5 (number of members surveyed)

Substituting these values into the formula, we get

r = (1291 − (109)(156) / 5) /[tex]\sqrt{(4552-(109)2/5)(4644-(156)2/5)))}[/tex]

r = (1291 − 17784 / 5) /[tex]\sqrt{(4552-10902/5)(4644-24336/5)}[/tex]

r = (-16493 / 5) / [tex]\sqrt{(3456)(2036)}[/tex]

r = -3298.6 / [tex]\sqrt{695936}[/tex]

r = -3298.6 / 835.2

r = -3.94

Rounded to the nearest thousandth, r = -3.940.

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According to the information, it can be inferred that Rebecca spent 31 minutes and 25 seconds on her phone from Monday to Thursday

To calculate the time Rebecca spent on her phone from Monday through Thursday we must add the number of minutes (and seconds) she used her phone during this period. For this we must rely on the information in the table:

According to the table, Rebecca made two calls each day, and each call was delayed:

Monday

call 1: 2 minutes 15 seconds

call 2: 4 minutes 23 seconds

Tuesday

call 1: 5 minutes 53 seconds

call 2: 1 minute 20 seconds

Wednesday:

call 1: 6 minutes 40 seconds

call 2: 3 minutes

Thursday:

call 1: 4 minutes 12 seconds

call 2: 3 minutes 42 seconds

Based on the above, we need to add the time in minutes and seconds and then calculate how much time in total she used her phone:

So finally we must divide the number of seconds into 60 (number of seconds that a minute has) and calculate how many minutes they are:

205 / 60 = 3.4

This equates to 3 minutes and 25 seconds. In total, Rebeca spent 31 minutes and 25 seconds on her phone from Monday to Thursday.

Note: This information is incomplete. Here is the complete information:

Monday

call 1: 2 minutes 15 seconds

call 2: 4 minutes 23 seconds

Tuesday

call 1: 5 minutes 53 seconds

call 2: 1 minute 20 seconds

Wednesday:

call 1: 6 minutes 40 seconds

call 2: 3 minutes

Thursday:

call 1: 4 minutes 12 seconds

call 2: 3 minutes 42 seconds

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The cube root can be estimated in the range:

3 < ∛29 < 4

And the value is ∛29 = 3.1

First we need to find two perfect cubes that bound our number.

We know that:

3*3*3 = 3^3 = 27

4*4*4 = 64

Then:

27 < 29 < 64

∛27 < ∛29 < ∛64

3 < ∛29 < 4

So we have an estimation of the cube root, and it will be closer to 3 than to 4, so we can estimate this as:

∛29 = 3.1

3.1 = 29.7

So it is a good estimation.

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Considering the expression c^n = d, for n > 1, the number c is the nth root of number d.

The expression for this problem is defined as follows:

c^n = d

To obtain the number that is the nth root to the other, an expression representing the nth root in the problem must appear.

The expression representing the nth root can appear isolating the variable c, as follows:

[tex]c = \sqrt[n]{d}[/tex]

Hence the number c is the nth root of number d.

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