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It’s Trigonometry and it’s hard.

The factor of  [tex]x^3 -7x + 6[/tex] is (x - 1) (x +3) (x -2).

Now, According to the question:

The given expression to be factorized is :

[tex]x^3 -7x + 6[/tex]

This can be written in the form:

[tex]x^3 -7x + 6[/tex] = [tex]x^3[/tex] - (1 + 6)x + 6

[tex]x^3 -7x + 6[/tex] = [tex]x^3[/tex] -x - 6x + 6

Take common 'x' from the first two terms and -6 from the last two terms. Then we have,

[tex]x^3 -7x + 6[/tex] = x([tex]x^2 -1[/tex]) -6 (x -1)

                  = x{ [tex](x)^2 - (1)^2[/tex] } - 6(x -1)

                  = x(x +1) (x -1) - 6(x -1)

Finally, Take common (x -1) from the above expression:

[tex]x^3 -7x + 6[/tex] = (x -1) {x(x +1) - 6}

                 = (x -1) [tex](x^{2} +x-6)[/tex]

                 = (x -1)([tex]x^{2}[/tex]+3x -2x -6)

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

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

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The complete question is this:

Which is the factor of [tex]x^3 -7x + 6[/tex]

1. x(x -6) (x -1)

2. [tex](x^{2} -6)[/tex] (x -1)

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

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

The expression that you can use to find the number of ways to choose 5 cards from a 52-card deck is  52!/(47! * 5!).

The expression that you can use to find the number of ways to choose 5 cards from a 52-card deck is 52 choose 5, which is equal to 52!/(47! * 5!), where the exclamation point indicates the factorial function. This is known as the binomial coefficient, and it represents the number of ways to choose a certain number of items from a larger set. In this case, the binomial coefficient is used to find the number of ways to choose 5 cards from a deck of 52 cards.

The order of the cards is irrelevant in card games like poker. A five-card draw of 1,2,7,9, K is equivalent to a draw of K,7,9,1,2. What counts is the fundamental overall group.

Order is irrelevant, therefore we utilize a combination (instead of a permutation)

We will substitute n = 52 and r = 5 into the nCr combination formula, which is n C r = (n!)/(r!*(n-r)!)

so we obtain:

n C r = r!*(n-r)!/(n!)

52 C 5 = (52!)/(5!*(52-5)!)

52 C 5 = (52!)/((52-5)!*5!)

The complete question is:-

Which expression can you use to find the number of ways to choose 5 cards from a 52-card deck?

a) (52-5)! / 52!5!

b) 52! / (52-5)!

c) 52! / (52-5)!5!

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The ratio of trees to flowering plants in Marsha’s backyard is shown on the ratio table.

30 numbers complete the ratio table to make an equivalent ratio.

When we compare two ratios, they are said to be equivalent. To determine if two or more ratios are comparable, they can be compared to one another.

Equivalent ratios are two ratios with the same value. By multiplying or dividing the two amounts by the same number, you may get the corresponding ratio. Finding comparable fractions follows the same procedure.

Here in this case,

for trees: 2: 12    [ 2 to 12, 6 times more ]

for plants: 5: 30  [ 5 to 30, it is also 6 times more]

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Answer:

x=4.04 y=7.66

Step-by-step explanation:

I don't see the picture. Can you send it in the message.

Answer:

x     0    2    4    6

y     3    4    5    6

Step-by-step explanation:

y = x/2 + 3

x = 0

y = 0/2 + 3 = 0 + 3 = 3

x = 2

y = 2/2 + 3 = 1 + 3 = 4

x = 4

y = 4/2 + 3 = 2 + 3 = 5

x = 6

y = 6/2 + 3 = 3 + 3 = 6

With the help of Algebra the correct function is C(t) = 0.10t+25.

What is Algebra?

Algebra is a branch of mathematics that deals with the study of symbols and the rules for manipulating them. It is used to solve a variety of equations and problems by using formulae, equations and graphs. Algebra is primarily used to find unknown values, such as the unknown side of a triangle or the value of a variable in an equation. It can also be used to solve problems that involve more than one variable, such as those involving linear equations and systems of equations. Algebra is a powerful tool for problem-solving and can be used in various scientific fields, such as physics, chemistry and engineering. It is essential for developing a good understanding of mathematics and is an important part of any mathematics curriculum.

Assuming that :  y = kx + b

plug in : ( 0, 25 ) and  ( 50, 30)

25 = b

30 = 50k + 25

so, k = 0.10 after solving this equation

y =  0.10t+25

Hence the function is C(t)=  0.10t+25

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Answer:

Step-by-step explanation:

x= -3  

y= 3

The three statistical measures of central tendency are mean, median, and mode. While describing a set of data, we point out its primary location. This is referred to as the central tendency measure.

The arithmetic mean of a given data is the sum of all observations divided by the number of observations.

The data set's median is the value in the center. Put all the numbers in ascending or descending order, and then choose the middle number to determine the median. There will only be one middle integer if the number of data points is odd. If the number of data points is even, you must choose the middle two numbers, add them together, and divide by two. Your median will be that value.

A mode is defined as the value that has a higher frequency in a given set of values. It is the value that appears the most number of times.

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The solution is Option C.

The equation f'( 2 ) ≤ 0 is true

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

[tex]f ( x ) = \lim_{n \to \infty} [ \frac{( n^{n} ( x + n ) ( x + \frac{n}{2} ) ....( x + \frac{n}{n} )}{n! (x^{2}+n^{2}) ( x^{2} + \frac{n^{2}}{4} ) ....( x^{2} +\frac{n^{2}}{n^{2} } ) } ]^{\frac{x}{n} }[/tex]

Taking logarithm on both sides of the equation , we get

[tex]ln f ( x ) = \lim_{n \to \infty} \frac{x}{n} [\sum _{r=1}^nln\frac{1}{r/n} + \sum _{r=1}^nln\frac{(x + \frac{1}{r/n} )}{(x^{2} + (\frac{1}{r/n})^{2} ) }[/tex]

On simplifying the derivatives of the equation , we get

[tex]ln f ( x ) = x [\int\limits^1_0 {ln ( 1/t )} \, dt + \int\limits^1_0 {ln \frac{( x+1/t )}{(x^{2} + 1/t^{2} ) } } \, dt][/tex]

Now , substituting the limits in the equation , we get

[tex]ln f ( x ) = x [ \int\limits^1_0 {( - ln t+ ln (tx+1)-lnt-ln(t^{2} x^{2}+1)+lnt^{2}] } \, dt[/tex]

Substitute the value of tx = z in the equation , we get

[tex]ln f ( x ) = \int\limits^1_0 {ln\frac{(1+z)}{1+z^{2} } } \, dz[/tex]

Taking exponential on both sides of the equation , we get

[tex]f ( x ) = e^{\int\limits^x_0 {{ln\frac{(1+z)}{1+z^{2} } } \, dz} \, }[/tex]

Differentiating with respect to x , we get

[tex]\frac{f' (x ) }{f ( x ) } = ln\frac{1+x}{1+x^{2} }[/tex]

Now , on further simplification of the equation , we get

f ( x ) > 0 for x > 0

And , f' ( x ) > 0

[tex]ln\frac{1+x}{1+x^{2} } > 0[/tex]

And ,

x > x²

0 < x < 1

Therefore , f ( x ) is increasing ∀ x ∈ ( 0 , 1 )

Hence , the equation f' ( 2 ) ≤ 0 is true

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Explanation:

[tex] \rm f(x) = \lim \limits_{n \to \infty } \left( \dfrac{ \displaystyle \rm \prod \limits_{r = 1}^n \left(1 + \dfrac{rx}n \right ) }{\displaystyle \rm\prod \limits_{r = 1}^n \left(1 + \bigg(\dfrac{rx}n \bigg)^{2}\right )} \right)^{ \dfrac{x}{n} }[/tex]

[tex] \large \rm = {e}^{ x \int \limits_{0}^{1} ln(1 + xy)dy - ln(1 + (xy {)}^{2} ) dy }[/tex]

[tex]\rm f'(x)=f(x) \ln \bigg( \dfrac{1 + x}{1 + {x}^{2} } \bigg)[/tex]

[tex]\rm For \: x \in(0,1) \: it \: i s \: increasing \: function[/tex]

[tex]\rm f'(2) = f(2) \ln( \frac{3}{5} ) < 0[/tex]

[tex]\rm \dfrac{f'(3)}{f(3)} =ln \bigg( \dfrac{2}{5} \bigg), \dfrac{f'(2)}{f(2)} = ln \bigg( \dfrac{3}5 \bigg) [/tex]

The end behavior of a graph can be determined by looking at the leading term in the equation. If the leading term is a positive or negative power of x, then the end behavior of the graph will be either a horizontal asymptoteor an oblique asymptote. If the leading term is a constant, then the end behavior of the graph will be a horizontal line at that constant value.

To determine the end behavior of a graph from an equation, we first look at the leading term in the equation. If the leading term is a positive or negative power of x, then the end behavior of the graph will be either a horizontal asymptote (positive or negative infinity) or an oblique asymptote (slanted line). If the leading term is a constant, then the end behavior of the graph will be a horizontal line at that constant value. For example, if the leading term of an equation is 3x2, then the end behavior of the graph will be a horizontal asymptote at positive infinity. If the leading term of the equation is 4, then the end behavior of the graph will be a horizontal line at y=4. To determine the exact shape of the asymptote, we must look at the coefficient of the leading term. A positive coefficient will create a positive asymptote (sloping up), while a negative coefficient will create a negative asymptote (sloping down). Additionally, if the graph oscillates around a single point, then the end behavior of the graph is a limit cycle.

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The size and shape of things are preserved by stiff transformations (reflections, translations, and rotations). As a result, archetypes and visuals are always compatible.

The separation between each pair of the geometric object's points is preserved during a rigorous change of the object. In other words, a rigid transformation maintains the object's size and shape. For instance, the rigid transformation of a triangle retains the angles and lengths of the triangle. Imagine a stiff shape distortion, similar to a cardboard shape. The form remains the same after being pushed left and right, rotated, and flipped. When the mold was created using putty, the opposite was true. The thing can then be twisted and stretched. These conversions, nevertheless, are not exact. Stretching a form alters the distances between each pair of its points.

c + 18 + 10 = 42//

[tex]c = 42 -(18 + 10) cm = 14 cm\\s = 21 cm//[/tex]

Area = [tex]\sqrt{s(s - a)(s - b)(s - c) }[/tex]

and, here s = half the perimeter,

[tex](a + b + c)/2.[/tex]

A = [tex]21\sqrt{11}[/tex]

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The four basic graphs used in statistics include bar, line, histogram and pie charts.

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Applying the derivative to the provided line with respect to "x," using the equation of the given line, The response is 3.

In mathematics, the derivative of a function of a real variable measures the sensitivity of the function's value to changes in its argument. The derivative is the fundamental tool in calculus. a function's rate of evolution in respect to a variable Problems involving calculus and differential equations need the use of derivatives.

The slope or gradient of a line in mathematics is a numerical representation of the steepness and direction of the line. Calculate the percentage by dividing the elevation difference between two places by their separation. slope, then multiply the result by 100. The term "rise" refers to the difference in elevation between two sites. The run is the separation between the points. Percent slope is therefore equal to (rise / run) x 100.

y(x) = 3x +1

derivate y(x) with respect to x. we get,

= 3

slope =3

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Answer:

Miyo can make 30 origami figures in 18 minutes

Step-by-step explanation:

tbh, i just counted up by 5's until i reached 18

Answer: 3

Step-by-step explanation: Because we know Miyo can make 5 origami figures in 3 minutes. and we want to know How many origami figures can Miyo make in 18 minutes? so, we know 5*3 is 15. right? well, then we take 18-15 and get 3 so, Miyo can make 3 origami figures in 18 mins.

In the equation y = mx + b, m is the slope of the line and b is the intercept.

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The height of the isosceles triangle is  h = [tex]\sqrt{a^{2}-\frac{c^{2} }{4} }[/tex]

Given,

Isosceles triangle;

An isosceles triangle has two equal sides as well as two equal angles. The Greek words iso (same) and skelos are the source of the name (leg). An equilateral triangle is one in which all of its sides are equal, whereas a scalene triangle is one in which none of its sides are equal.

Follow these procedures to determine an isosceles triangle's height:

That is,

Height, h = [tex]\sqrt{a^{2}-\frac{c^{2} }{4} }[/tex]

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Answer:

C. [tex]3[/tex]

Step-by-step explanation:

[tex]y=-2x^2 + 8x-5 \\ \\ =-2(x^2 -4x)-5 \\ \\ =-2(x^2-4x+4-4)-5 \\ \\ =-2((x-2)^2 -4)-5 \\ \\ =-2(x-2)^2 +8-5 \\ \\ =-2(x-2)^2 + 3[/tex]

Answer:

  1598.9 ft

Step-by-step explanation:

You want the distance between points A and B when observers at each point find the angle of elevation to a lighthouse beacon to be 15° and 6°, respectively. Point A is 1032 ft horizontal distance from the lighthouse.

The tangent relation for sides in a right triangle is ...

  Tan = Opposite/Adjacent

This can be rearranged to ...

  Opposite = Tan × Adjacent

If d is the distance in feet from A to B, two expressions for the same lighthouse height can be written:

  height = tan(15°)(1032)

  height = tan(6°)(1032 +d)

Equating these values of height, we have ...

  tan(6°)(1032 +d) = tan(15°)(1032)

  d = 1032(tan(15°) -tan(6°))/tan(6°) = 1032(tan(15°)/tan(6°) -1)

  d ≈ 1598.946 . . . . feet

The distance from A to B is about 1598.9 feet.

the exponential function is n(t) =n° (1+r)∧t, where t is the power refers the time in months after opening and n denotes average customer.

when the rate of increases is occurred very rapidly. In this question, the trendy restaurant has earned reputation.

Given, average number of customers after 11 months of opening = n(11) =310

average number of customers after 15 months of opening= n(15) =456

during this 4 month increases of customer =456 -310 =146

increasing rate of customer per month = 146/4 = 36.5

now increasing rate in percentages = 36.5 %

exponential growth rate, r= 0.365

let, n° is the initial number of customer while opening at t=0 month

now, the exponential function n(t) that model's average number of customers, n after t months of being open will be

  n(t) = n° ( 1+r)∧t

we are given that after 11 months of opening average customer n(11) = 310

now, 310 = n° (1+ 0.365)∧11

n°  = 10

hence, the exponential function = n(t) = 10(1+r)∧t

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Thermal energy that comes from the movement of particles.

The energy present in a system that determines its temperature is referred to as thermal energy. Thermal energy flows as heat. Thermodynamics is a whole field of physics that studies how heat is transmitted between various systems and how the procedure is carried out (see the first law of thermodynamics).

The energy that results from a substance's temperature is known as thermal energy. The amount of molecular vibration increases with temperature, increasing thermal energy.

Heat energy is another name for thermal energy. A moving object's kinetic energy is its energy. Thermal energy is a type of kinetic energy because it is produced by moving particles.

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In order to solve a inverse function question we switch the coordinates  of x with that of y and the perform the required  calculations.

The result which we get as is considered as a relation and not as a function.

We an calculate inverse for functions which are one to one .

A function is known as a one to one function only when every second element matches the first value.

The inverse question is a type of function , when we solve this function we get another function which is represented as , ‘g’  and is known as the inverse of function .

y = f(x) then,

x = g(y)

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The probability that Hamid guesses both answers correctly is 10%.

Probability means that it is possible. It is a branch of statistics that deals with the occurrence of a random event. The number is expressed from 0 to 1 where 0 represents an impossible event and 1 represents a sure event.

Probability Formula

The probability formula is defined as the ratio of favorable outcomes to the ratio of total outcomes. For any event (E), this can be shown as

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

Since Hamid is playing a trivia game with multiple choice questions, and each question has 2 correct answers among 5 answer choices, and Hamid has no idea what the answers to a certain question are, so he needs to choose two different answers at random, to determine what is the probability that Hamid guesses both answers correctly the following calculation must be performed:

2/5 x 1/4 = X

0.4 x 0.25 = X

0.1 = X

0.1 x 100 = 10

Therefore, the probability that Hamid guesses both answers correctly is 10%.

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Probability denotes possibility. The occurrence of a random event is the subject of this area of statistics. The number is expressed as a number between 0 and 1, where 0 denotes an improbable event and 1 denotes a certain event.

Statistical Formula

The ratio of favorable outcomes to the ratio of all outcomes is the definition of the probability formula. This can be shown as for any event (E) as

P(E) = Number of positive results / Number of total results

Given that each question in a multiple-choice trivia game has two correct answers out of a possible five, and that Hamid must choose two answers at random because he doesn't know what the answers are, the following calculation must be done to ascertain the likelihood that Hamid will correctly guess both answers:

2/5 x 1/4 = X

0.4 x 0.25 = X

0.1 = X

0.1 x 100 = 10

The equation that makes relationship between width, length and area of a room is Area of room = length of room × width of room and perimeter of room = 2× (length of room + width of room)

The size of a room is either square or rectangle in shape. Whenever a room is rectangle shaped, we need to measure all the side lengths and finally area is found by product of two unequal side length. Besides, we can determine the perimeter of room by adding the two unequal side length and finally 2 is multiplied with it.

We stated earlier that the size of a room is either rectangle or square.

area of square shaped room = (side length× side length)

area of rectangle shaped room = (bigger side length × smaller side length)

perimeter of square shaped room = 4 × side length = 4√(area of room)

perimeter of rectangle shaped room = 2(bigger side length + smaller side length)

hence, the above equations make relationship among the width, length, area and perimeter of a room.

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Over the line y = x, a function and its inverse are reflected. It only takes one step to graph a function and its inverse: first, graph the function, then, to graph the inverse, switch all x and y values at each point.

Given,

Inverse function;

A function that can reverse into another function is known as an inverse function or anti function. In other words, the inverse of a function "f" will take y to x if any function "f" takes x to y. When a function is written as "f" or "F," its inverse is written as "f-1" or "F-1."

Here,

To graph the inverse, you only need to alter the x and y values in each point on the function graph. Just take a look at all the numbers that are being reflected along the line y = x as they move from the f(x) function to its inverse g(x) and back again.

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Answer:

A

Step-by-step explanation:

To compare the initial out-of-pocket cost for the lease and loan options, we need to calculate the total cost for each option.

For the loan option, we can calculate the total cost as follows:

Total cost = MSRP + down payment + interest

= $15,000 + $1,500 + $1,050 (7% of MSRP)

= $17,550

For the lease option, we can calculate the total cost as follows:

Total cost = (MSRP x money factor x 36) + acquisition and documentation fees + security deposit

= ($15,000 x 0.00271 x 36) + $1,250

= $5,925 + $1,250

= $7,175

Since the total cost for the lease option is less than the total cost for the loan option, the initial out-of-pocket cost is less for the lease. Therefore, the correct statement is:

a. The initial out-of-pocket cost is less for the lease.

Answer:

GT: 10

TA: 6

Step-by-step explanation:

use the ratio of GR:RE to find GT:TA

since the ratio of GR:RE is 5:3, the ratio of GT:TA=5:3. The total length of GT+TA=16.

5x+3x=16, 8x=16, x=2.

By plugging in the x value, it is shown that GT=10 and that TA=6

Hence,the company should produce 12 pieces of Model A and 6 pieces of Model B to get the maximum profit i.e. 168000.

Let x be the number of pieces of model A and y be the number of pieces of model B.

Let total profit be Z

Then, Total profit =8000x+12000y

                         Z=8000x+12000y .............. (i)

Now we have the following mathematical model for the given problem 

Max Z=8000x+12000y

Subject to constraints 

9x+12y≤180 .......... (Fabricating constraints)

3x+4y≤60   .......... (ii)

x+3y≤30   (Finishing constraints)........ (iii)

Let us evaluate the objective function Z at each point as shown below

At C(0,10),  Z=8000×0+12000×10=120000

Maximum value of Z is 168000 at B(12,6)

Hence, the company should produce 12 pieces of Model A and 6 pieces of Model B to get the maximum profit i.e. 168000.

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The nature of the roots of the equation 5x² + 3x + 2 = 0 are x = -7.123 and x = 0.878.

The equation 5x² + 3x + 2 = 0 has two solutions, or roots, which can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Where a = 5, b = 3 and c = 2,

x = [-3 ± √(3² - 4(5)(2))] / 2(5)

x = [-3 ± √17] / 10

x = (-3 ± 4.123) / 10

x = (-7.123) / 10,  0.878

The roots of the equation 5x² + 3x + 2 = 0 are x = -7.123 and x = 0.878.

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Examples of perpendicular slope are the sides of a fixed square, a window, and the Red Cross emblem.

Lines that are vertical and horizontal are perpendicular to one another. In this instance, the perpendicular line's slope would be equal to a horizontal line's slope, which would be 0. The slope of the perpendicular line is zero, but the slope of the parallel line is unknown. By taking the negative reciprocal of the slope of the provided line, one can determine the slope of a perpendicular line from that line. We obtain m2 = -1/m1 if the slope of the provided line is m1 and the slope of a perpendicular line is m2.

Given

There are a lot of perpendicular lines that we may see in reality. The corners of the chalkboard, the arms of a clock at specific times of the day, the sides of a fixed square, a window, and the Red Cross emblem are a few examples.

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