What is the length of x in the diagram below?

A triangle with vertical bisector forms 2 triangles with a right angle. One triangle has a side length of 5 and another angle with a measure 45 degrees. Another triangle has a hypotenuse with length x and another angle with measure 30 degrees.
StartFraction 5 Over StartRoot 3 EndRoot
StartFraction 10 Over StartRoot 3 EndRoot EndFraction
5 StartRoot 3 EndRoot
10

Pοlynοmial f(x) has exactly 3 cοmplex rοοts

The fundamental theοrem οf algebra, alsο knοwn as d'Alembert's theοrem, οr the d'Alembert–Gauss theοrem, states that every nοn-cοnstant single-variable pοlynοmial with cοmplex cοefficients has at least οne cοmplex rοοt. This includes pοlynοmials with real cοefficients, since every real number is a cοmplex number with its imaginary part equal tο zerο.

The fundamental theοrem οf algebra states that any nοn-cοnstant pοlynοmial οf degree n has exactly n cοmplex rοοts (cοunting multiplicities). In this case, the pοlynοmial [tex]f(x) = 8x^3[/tex]+ 216 is a nοn-cοnstant pοlynοmial οf degree 3, sο it has exactly 3 cοmplex rοοts (cοunting multiplicities).

We can alsο use the factοr theοrem tο cοnfirm that f(x) has exactly 3 rοοts. The factοr theοrem states that a pοlynοmial f(x) has a factοr (x - a) if and οnly if f(a) = 0. In this case, we can factοr οut 8 frοm the pοlynοmial tο get:

[tex]f(x) = 8(x^3 + 27)[/tex]

Setting f(x) = 0, we get:

[tex]8(x^3 + 27) = 0[/tex]

This equatiοn is satisfied if and οnly if [tex]x^3 + 27 = 0.[/tex] We can factοr this equatiοn as fοllοws:

[tex]x^3 + 27 = (x + 3)(x^2 - 3x + 9)[/tex]

The quadratic factοr [tex]x^2 - 3x + 9[/tex] has nο real rοοts, since its discriminant is negative [tex](b^2 - 4ac = (-3)^2 - 4(1)(9) = -27)[/tex]. Hοwever, it dοes have twο cοmplex rοοts, which are cοnjugates οf each οther. Therefοre, the pοlynοmial f(x) has exactly 3 cοmplex rοοts (cοunting multiplicities).

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Answer:

3 people are getting candy

Step-by-step explanation:

9x3

We have shοwn that the LHS equals the RHS, and hence, we have prοved the identity: sec²x/2) tan(x) = csc²(x).

Trigοnοmetry is a branch οf mathematics that deals with the study οf relatiοnships between the sides and angles οf triangles. It is a fundamental area οf mathematics that has applicatiοns in many fields, including physics, engineering, and astrοnοmy.

Trigοnοmetry invοlves the study οf six trigοnοmetric functiοns: sine (sin), cοsine (cοs), tangent (tan), cοsecant (csc), secant (sec), and cοtangent (cοt). These functiοns describe the relatiοnships between the angles and sides οf a right-angled triangle.

Trigοnοmetry alsο includes the study οf trigοnοmetric identities, which are equatiοns that invοlve trigοnοmetric functiοns and are true fοr all pοssible values οf the variables.

In the given question,

Starting with the left-hand side (LHS) of the given identity:

sec²(x/2) tan(x)

Using the identity, sec²(x) = 1/cos²(x), we can write:

sec²(x/2) = 1/cos²(x/2)

Substituting this into the LHS:

1/cos²(x/2) * tan(x)

Now, using the identity, tan(x) = sin(x)/cos(x), we can write:

1/cos²(x/2) * sin(x)/cos(x)

Rearranging and simplifying:

sin(x) / cos(x) * 1/cos²(x/2)

Using the identity, csc(x) = 1/sin(x), we can write:

1/sin(x) * 1/cos(x) * 1/cos²(x/2)

Now, using the identity, cos(2x) = 1 - 2sin²(x), we can write:

cos(x) =√(1 - sin²(x/2))

Substituting this into the above equation:

1/sin(x) * 1/√(1 - sin²(x/2)) * 1/cos²(x/2)

Simplifying:

1/sin(x) * 1/√(cos²(x/2)) * 1/cos²(x/2)

Using the identity, csc²(x) = 1/sin²(x) and simplifying:

csc²(x) * cos²(x/2) / cos²(x/2)

The cos²(x/2) terms cancel out, leaving:

csc²(x).

Therefore, we have shown that the LHS equals the RHS, and hence, we have proved the identity: sec²x/2) tan(x) = csc²(x).

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We can conclude that 5/8 is between the benchmark fractions 1/2 and 3/4.

A fraction represents a part of a whole or a ratio between two quantities. It consists of a numerator (the top number) and a denominator (the bottom number), separated by a horizontal line. The numerator represents the part or quantity being considered, while the denominator represents the whole or total quantity.

To determine between which two benchmark fractions 5/8 lies, we can compare it to the nearest benchmark fractions, which are 1/2 and 3/4.

To summarize, 1/2 < 5/8 < 3/4.

For example, the fraction 3/4 represents three parts out of a total of four parts, or a ratio of three to four. This can also be expressed as a decimal (0.75) or a percentage (75%).

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Answer: 7-2 i

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let's break down the information provided in the problem:

The man spends 2/3 of his high salary on rent and food.

This means that he has 1/3 of his salary remaining.

He then spends 1/4 of this remaining amount on transportation.

He has 6000 rupees left.

To solve for the amount he pays on transportation, we can use the following steps:

Calculate the amount of money he has left after spending 2/3 of his salary on rent and food:

Remaining amount = (1/3) * high salary

Calculate the amount he spends on transportation using the information that he spends 1/4 of the remaining amount:

Transportation cost = (1/4) * remaining amount

Substitute the value of remaining amount from step 1 into step 2:

Transportation cost = (1/4) * [(1/3) * high salary]

We are given that the man has 6000 rupees left, so we can set the equation above equal to 6000 and solve for high salary:

(1/4) * [(1/3) * high salary] = 6000

Simplifying the equation above:

(1/12) * high salary = 6000

Solving for high salary:

high salary = 6000 * 12 = 72000

Finally, substituting high salary from step 6 into step 3 to find transportation cost:

Transportation cost = (1/4) * [(1/3) * 72000] = 6000

Therefore, the man pays 6000 rupees on transportation.

The amount spent by each friend is 5d + 9.75 = 63.75; d = 10.8 and the mathematics sentence is -10(r + 12.5) = 60.5

In the question, we have

Total = 63.75

Meal = 9.75 each

Dessert = d each

So, the equation is

5d + 9.75 = 63.75

Evaluate the like terms

5d = 54

Divide by 5

d = 10.8

So, the equation and the result are 5d + 9.75 = 63.75; d = 10.8

Here, we have

Negative ten times the sum of a number and 12.5 is 60.5.

Let the number be r

So, we have

-10(r + 12.5) = 60.5

Based on the problem statement, we have

8.94 * 3 = 1/3 * (17.95 + j)

So, we have

80.46 = 17.95 + j

Evaluate

j = 62.51

Hence, the cost of the jalapeno peppers is $62.51

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Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.

In mathematics, inequality is a comparison between two values or expressions using an inequality symbol such as >, <, ≥, or ≤. It is used to compare different values to each other and determine whether one is greater than, less than, or equal to the other. Inequality can be used to express relationships between two or more variables, to solve certain equations, and to graph certain data.

In order to answer these questions, we need to compare the given values. In the first question, Brittany's rope was compared to Sam's rope, and in the second question, Alex's rope was compared to Sam's rope.

For the first question, Sam's rope was 1.5 x 4/5 = 6/5. This value was compared to Brittany's rope which was 1 1/4. Since 1 1/4 is greater than 6/5, we know that Brittany's rope is longer than Sam's rope.

For the second question, Sam's rope was 1.5 x 3/2 = 9/4. This value was compared to Alex's rope which was 5/4. Since 5/4 is less than 9/4, we know that Alex's rope is shorter than Sam's rope.

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Answer:

$0.79

Step-by-step explanation:

$9.48 ÷ 12= 0.79

each pen costs $0.79

The difference quotient for f(x)=x is (x+h-x)/h, which simplifies to 1/h. The average rate of change of f(x)=x on the interval (1,3) is 1/2, since h=2.

The difference quotient for f(x)=x can be found by taking the difference between f(x+h) and f(x) and then dividing by h. In this case, f(x+h) = x + h and f(x) = x, so the difference quotient is (x+h-x)/h, which simplifies to 1/h. To calculate the average rate of change of f(x)=x on the interval (1,3), we need to plug in the values of h and x. Here, x is the first endpoint of the interval, which is 1, and h is the length of the interval, which is 2. Thus, the average rate of change is 1/2.

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Answer:

-3 ≤ 3x ≤ 6

To solve for "x", we can divide each part of the inequality by 3:

-1 ≤ x ≤ 2

Therefore, the number "x" must lie between -1 and 2 in order to satisfy the condition in the sentence.

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

The measurements for LM is 30 units and MN is 15 units.

We know that the scale factor for the dilation is 3, which means that each side of triangle RST was multiplied by 3 to create triangle LMN.

Therefore:

LM = 3(RS)

MN = 3(ST)

To find RS and ST, we can use the fact that TAN R = 5/2.5.

Recall that tangent is the ratio of the opposite side to the adjacent side of a right triangle, so we can set up the following equation:

TAN R = RS/ST = 5/2.5

Cross-multiplying, we get:

RS = 5ST/2.5

Simplifying:

RS = 2ST

Now we can substitute this expression for RS into the equations for LM and MN:

LM = 3(RS) = 3(2ST) = 6ST

MN = 3(ST)

So the measurements for LM and MN are:

LM = 6ST

MN = 3(ST)

From triangle RST,

ST = 5

LM = 6 x 5 = 30 units

MN = 3 x 5 = 15 units

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In response to the question, we may say that  As a result, the angle of trigonometry inclination of the ramp is 4 degrees, to the closest degree.

The area of mathematics called trigonometry examines how triangle side lengths and angles relate to one another. The subject first came to light in the Hellenistic era, about in the third century BC, as a result of the use of geometry in astronomical investigations. The area of mathematics known as exact techniques deals with several trigonometric functions and possible computations using them. There are six common trigonometric functions in trigonometry. These go by the designations sine, cosine, tangent, cotangent, secant, and cosecant, respectively (csc). Trigonometry is the study of triangle characteristics, particularly those of right triangles. Consequently, studying geometry entails learning about the characteristics of all geometric forms.

The ratio of an angle's opposing side (such as the rise of a ramp) to its adjacent side is known as the tangent (the length of the ramp).

Angle's tangent is equal to its opposite and adjacent sides.

In this instance, the ramp's opposite side represents its 3/4-foot rise, and the ramp's adjacent side is its 12-foot length.

Angle's tangent is equal to 3/4 / 12 (or 0.0625).

We may take the inverse tangent (or arctangent) of this number to determine the angle itself.

Amount = arctan (0.0625)

Calculating the answer, we obtain:

Angle is 3.58 9 degrees.

When we convert this to degrees, we get:

a 4 degree angle

As a result, the angle of inclination of the ramp is 4 degrees, to the closest degree.

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The answer of the question based on the angles statements the answer is

always true based on condition.

Range is the difference between highest and lowest values.

Based on the provided image, the statement "The range of a function is always a subset of its codomain" is true.

In mathematics, the codomain of a function is the set of all possible output values, while the range is the set of actual output values produced by the function for a given input. Since the range is a subset of the codomain, the statement is true.

One way to determine the truth of such statements is to understand the definitions of the relevant terms and to reason logically based on those definitions.

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Answer:

3/5 - 13/5 => -6.9 + 4.9

2x - 3 => x - 3 + x

5/3 - x + 1/3 => -3/2x + 2 + 1/2x

2x - 6 => -6 + 2x

3x + 0 => 3x

Step-by-step explanation:

Hope it helps :)

In the function [tex]V(t) = 230(1.12)^t[/tex], the number 230 represents the initial or starting value of the company in millions of dollars at the beginning of the time period in question, which is 2018.

The function models the growth of the company's value over time as an exponential function, with a base of 1.12. The exponent t represents the number of years that have passed since 2018, and the resulting value V(t) is the projected value of the company t years after 2018, given the assumed growth rate of 12% per year.

So, at the start of 2018, the company was worth 230 million dollars, according to the model.

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Answer:

I believe the answer is D

Step-by-step explanation:

if you want an explanation let me know and have a great day and I'm 100% sure that D is the right answer

Yes, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.

Co-interior angles, also known as consecutive interior angles, are those between two lines that are split by a third line (transversal), and are located on the same face of the transversal.

The majority of the time, it comes from the latin word "com-," which often means "along with." Co-interior angles are located on the same face of a transversal as well as between two lines. The significant correlations angles in each diagram are referred to as co-interior angles. Co-interior angles are supplementary if the two lines remain parallel since they add to 180 degrees.

In the given diagram:

= 75 + 105 (co-interior angles)

= 180° (supplementary angles)

So,

line 1 || line 2

Thus, the line 1 and line 2 are parallel lines as the sum of both given angles is 180°.

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The value of n is 3, n is a positive integer

We know that:

1 + 2 + 3 + 4 + 5 + 6 = 21

Therefore:

(1 + 2 + 3 + 4 + 5 + 6)² = 21² = 441

Now, let's look at the sum of cubes:

1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²

We already know that 1 + 2 + ... + 6 = 21, so we can rewrite the equation as:

1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²

1³ + 2³ + ... + n³ = (1 + 2 + 3 + 4 + 5 + 6 + ... + n)²

We want to find the value of n that makes this equation true. We know that the sum of the first n positive integers is:

1 + 2 + 3 + ... + n = n(n+1)/2

So we can rewrite the equation as:

1³ + 2³ + ... + n³ = [n(n+1)/2]²

Now we substitute the value we know for 1 + 2 + 3 + 4 + 5 + 6:

441 = [6(7)/2]²

441 = 21²

So n(n+1)/2 = 7, which means:

n(n+1) = 14

The only positive integer solution for n in this case is 3, because:

n(n+1) = 14

n² + n - 14 = 0

(n-3)(n+4) = 0

The positive integer solution is n = 3, which means:

1³ + 2³ + 3³ = [3(4)/2]² = 36² = 441

So the value of n is 3.

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Answer: 4

Step-by-step explanation:

The florist can make 23 bunches of flowers with the lilies, carnations, and roses they have for the wedding.

What is the ratio?

A ratio is a comparison of two or more quantities that indicates their relative sizes or amounts. It expresses the relationship between two or more numbers or values by dividing one by the other.

Let's start by finding out how many flowers in each bunch are lilies.

Since 60 percent of the flowers are lilies, we can calculate that 0.6 x 40 = 24 of the flowers in each bunch are lilies.

That means the remaining 16 flowers in each bunch are split between carnations and roses.

Let's use the ratio of 3:5 to split these flowers between carnations and roses.

First, we need to find the total number of parts in the ratio:

3 + 5 = 8

That means for every 8 flowers in the bunch, 3 are carnations and 5 are roses.

Now we can set up an equation to find out how many bunches of flowers the florist can make:

140 (number of carnations) ÷ 3 (number of carnations in each 8-flower group) = 46.67

So the florist can make 46 bunches of flowers with the carnations they have.

But we need to make sure that there are enough roses to go with these carnations.

If 3 of every 8 flowers in each bunch are carnations, that means 5 of every 8 flowers are roses.

So for each bunch, there are 5/8 x 16 = 10 roses.

To make 46 bunches of flowers, the florist will need 46 x 16 = 736 flowers in total.

If 24 of every 40 flowers in each bunch are lilies, that means 16 of every 40 flowers are carnations and roses combined.

So for 736 flowers, there will be 368 carnations and roses.

Since each bunch contains 16 flowers, the florist can make 368 ÷ 16 = 23 bunches of flowers with the carnations and roses they have.

Therefore, the florist can make 23 bunches of flowers for the wedding.

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The probability that Harry, Jason, and Sarah will all be accepted into their desired schools is;P(Harry, Jason, Sarah) = P(Harry) * P(Jason) * P(Sarah) = 0.1949 * 0.2413 * 0.1492 = 0.007 Possible answer: 0.007 (or 0.7%)

The solution to the question is given below; The mean score of the GMAT exam is μ = 550 and the standard deviation is σ = 115.There are three friends - Harry, Jason, and Sarah - who want to be admitted to business schools. Harry requires a score of 650 or above, while Jason requires a score of 630 or higher and Sarah requires a score of 670 or above. Because the admission procedure is independent for each of the friends, the joint probability can be calculated using the multiplication rule.

The probabilities that Harry, Jason, and Sarah will achieve the desired scores are calculated separately using the z-score equation, which is as follows;For Harry:Z = (650 - 550) / 115 = 0.87 Probability of Harry getting admitted = P(z > 0.87) = 0.1949 (using z-table)For Jason:Z = (630 - 550) / 115 = 0.7 Probability of Jason getting admitted = P(z > 0.7) = 0.2413 (using z-table)For Sarah:Z = (670 - 550) / 115 = 1.04 Probability of Sarah getting admitted = P(z > 1.04) = 0.1492 (using z-table)The probability that Harry, Jason, and Sarah will all be accepted into their desired schools is;P(Harry, Jason, Sarah) = P(Harry) * P(Jason) * P(Sarah) = 0.1949 * 0.2413 * 0.1492 = 0.007 Possible answer: 0.007 (or 0.7%)

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a) The theoretical probability of a fair coin landing on heads is 1/2

b) The experimental probability of landing on heads is 40%. Note that the experimental probability is lower than the theoretical probability.

Part A: The theoretical probability of a fair coin landing on heads is 1/2 or 0.5. Thus, it is possible for either of the teams to get the ball first.

Part B: The frequency of each outcome after flipping the coin 10 times may vary, but for example:

Heads: 4

Tails: 6

The experimental probability of landing on heads can be calculated by dividing the frequency of heads by the total number of flips: 4/10 = 0.4 or 40%.

Comparing the experimental probability of 0.4 to the theoretical probability of 0.5, we can see that the experimental probability is lower than the theoretical probability.

This difference may be due to random chance or factors such as the way the coin is flipped, the surface it lands on, or the wind. As more flips are made, the experimental probability should approach the theoretical probability.

Note that Theoretical probability is the likelihood of an event occurring based on reasoning or calculation, without actually performing the event.

Experimental probability is the probability of an event occurring based on actual repeated trials or experiments. It is determined by dividing the frequency of the event by the total number of trials.

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Answer: About 75 years

Step-by-step explanation:

2.5x30=75

The angle m∠P is equal to 50 degrees.

Since triangle OPQ is isosceles with PQ congruent to QO, we know that angle OPQ is congruent to angle OQP. Let's call this angle x. Then, we can set up an equation based on the fact that the angles in a triangle add up to 180 degrees: x + x + 50 = 180

Simplifying the equation, we get

2x + 50 = 180

Subtracting 50 from both sides, we get

2x = 130

Dividing by 2, we get:

x = 65

Therefore, angle OPQ and angle OQP are both equal to 65 degrees. Since angle OPQ and angle P are supplementary (they add up to 180 degrees), we can find angle P as:

m∠P = 180 - m∠OPQ

=> 180 - 2x

=> 180 - 2(65)

=> 50 degrees.

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Answer:

18

Step-by-step explanation:

In this you have to use means and extremes method, when you multiply 72 and 16 you’ll get 1152. Then divide it with 64, you’ll get 18.

[tex]\huge\text{Hey there!}[/tex]

[tex]\large\textsf{Original equation/problem }\\\\\large\boxed{\mathsf{72:64}}\\\\\large\textsf{Both numbers (72 \& 64) are factors of 4, so we will DIVIDE them by 4}\\\\\large\boxed{\mathsf{\rightarrow 72\div4: 64\div4}}\\\\\large\textsf{Which results in}\downarrow\\\\\large\boxed{\mathsf{\rightarrow 18:16}}\\\\\\\huge\text{Therefore your answer should be:}\\\huge\boxed{\mathsf{\bold{18}:16}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

Answer:

252 square

Step-by-step explanation:

Considering a recipe of the dessert for 16 people, the needed amounts of butter, sugar, flour and cheese are given as follows:

The amounts are obtained applying the proportions in the context of the problem, as we are given the amount needed for 8 people, hence we must obtain the ratio between the number of people and 8, and then multiply the amounts by this ratio.

For 8 people, the amounts of the ingredients are given from the problem as follows:

The ratio between 16 people and 8 people is given as follows:

16/8 = 2.

Hence the amount of each ingredient will double, thus the needed amounts are given as follows:

Gina is preparing a recipe for 8 people, and the amounts are given as follows:

The problem asks for the necessary amounts for 16 people.

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