The area of this triangle is 64.8 km². Find the height h.
18 km is the base.
The height of the triangle is?

A box-and-whisker plot that shows the number of olives is shown in the image attached to the answer.

To construct the box-and-whisker plot from the given data we first need to find the first quartile.

To find the 1st quartile we will use the interquartile range and 3rd quartile.

Interquartile range = 3rd quartile - 1st quartile

(substitute the values given in the question)

4 = 19 - 1st quartile

1st quartile = 19 - 4

1st quartile = 15

hence we now have the complete data to construct the box-and-whiskers plot.

minimum = 4

first quartile = 15

median = 16

third quartile = 19

maximum = 20.

The plot is in the attached picture.

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Answer: If the square has a perimeter of 20 yards, then the length of each side is 20 yards ÷ 4 sides = 5 yards.

Step-by-step explanation:

The difference in starting Volume is 175 and after 8 minutes both buckets have same volume.

The momentum of a variable is represented by the rate of change, which is used to mathematically express the percentage change in value over a specified period of time.

Given:

The leakage rate of A

= (2000- 2900)/ (1- 10)

= -900/ (9)

= -100 ml/min

The leakage rate of B

= (2050- 2725)/ (1- 10)

= -675/ (9)

= -75 ml/min

Now, 2900- 100t = 2725 - 75t

25t = 175

t= 7

So, when t= 1 min and after 7 min both buckets have the same volume of water.

So, t= 1+ 7 = 8 mins

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A college finds that 10% of students have taken a distance learning class and that 40% of students are part time students.

a. P(D AND E).= 0.08

b. Find P(E|D).= 0.8

c. Find P(D OR E). = 0.42

A) P (D and E) = 0.4 x 0.2 = 0.08

 explanation: 20% of the part times students are taking distance learning classes (D and E)

B) P (E | D) = P ( D and E) / P (D) = 0.08 / 0.1 = 0.8

C) P (D or E) = 0.4 + 0.1 - 0.08 = 0.42

D and E are not independent, because P (D and E) doesn't equal P(D) x P(E)

D and E are not mutually exclusive

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

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A. the parametric equations of the line of intersection:

x = 2 + 9t

y = 3 + 6t

z = 2 - 15t

B. the angle between the two planes will be between 0 and 90 degrees.

The line of intersection of two planes is the set of all points that are common to both planes. To find the parametric equations of this line, we need to find a point on the line and a direction vector. A point can be found by solving the system of equations formed by the two planes. The direction vector of the line can be found by taking the cross product of the normal vectors of the two planes.

The normal vectors of the planes can be found by taking the coefficients of x, y, and z in each equation and using them as the components of a vector:

Plane 1: normal vector = <3, -2, 1>

Plane 2: normal vector = <2, 1, -3>

The direction vector of the line is given by the cross product of these two normal vectors:

d = normal vector 1 x normal vector 2 = <3, -2, 1> x <2, 1, -3> = <9, 6, -15>

Next, we can find a point on the line by solving the system of equations formed by the two planes:

3x - 2y + z = 1

2x + y - 3z = 3

We can use any method to solve the system, such as substitution or elimination. By substitution, we can find that:

x = 2

y = 3

z = 2

So a point on the line is (2, 3, 2).

The angle between the two planes can be found using the dot product of the normal vectors:

cos(θ) = (normal vector 1 . normal vector 2) / (|normal vector 1| * |normal vector 2|)

where θ is the angle between the two vectors.

cos(θ) = (3 * 2 + (-2) * 1 + 1 * -3) / (sqrt(3^2 + (-2)^2 + 1^2) * sqrt(2^2 + 1^2 + (-3)^2))

cos(θ) = (3 - 2 - 3) / (sqrt(14) * sqrt(14))

cos(θ) = -8 / (2 * sqrt(14))

Therefore, the angle between the two planes is:

θ = acos(cos(θ)) = acos(-8 / (2 *sqrt(14)))

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The inequality the determines the value of t as number of tricks Roxanne might do during a competition is t≤ 28.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Several types of inequalities are represented by a variety of notations.

Given that, she has learned a total of 28 different tricks.

If we suppose t as number of tricks Roxanne might do during a competition.

Then the inequality the determines the value of t is t≤ 28 .

Hence, the inequality the determines the value of t is t≤ 28 .

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Vogel's approximation method is used to find an initial feasible solution to a transportation problem.

The method is based on the observation that if there is a row or a column in the transportation tableau having the same minimum positive difference (or penalty) between the costs of two cells, it is likely that one of these cells will get a positive allocation in the optimal solution.

Calculate the penalty for each cell by subtracting the smaller of the two adjacent cells from the larger.

D1 D2 D3 D4 A B C D

5 1 3 3 34 3 6 4

Penalty 0 2 0 1 31 2 2 -1

3 3 5 4 15 3 4 -1

Penalty 0 0 2 1 12 2 2 2

6 4 4 3 12 4 3 2

Penalty 2 1 1 0 6 1 0 1

4 -1 4 2 19 2 3 2

Penalty 5 5 2 0 17 2 0 0

Identify the row and column having the maximum penalty. In this case, the maximum penalty is 5 in row 1 and column 4.

Allocate as much as possible to the cell in the intersection of the row and column with the maximum penalty. The allocation should not exceed the demand or the supply of that row or column.

D1 D2 D3 D4 A B C D

5 1 3 2 34 3 6 4

Penalty 0 2 0 1 31 2 2 -1

3 3 5 3 15 3 4 -1

Penalty 0 0 2 1 12 2 2 2

6 4 4 3 12 4 3 2

Penalty 2 1 1 0 6 1 0 1

4 -1 4 2 19 2 3 2

Penalty 5 5 2 0 17 2 0 0

Repeat the process until all demands are met or all supplies are exhausted.

D1 D2 D3 D4 A B C D

5 1 2 2 34 3 6 3

Penalty 0 2 0 0 32 2 2 -1

3 3 5 3 15 3 3 0

Penalty 0 0 2 0 12 2 1 2

6 4 4 3 12 4 0 1

Penalty 2 1 1 0 6 1 0 0

4 -1 4 2 19 2 3 1

Penalty 5 5 2 0 17 2 0 0

The above table shows an initial feasible solution to the transportation problem using Vogel's approximation method. The total cost of this solution is 34 + 15 + 12 + 19 = 80, which is the same

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Answer:

-225

Step-by-step explanation:

f = -10 and j= -2

-9(6(-2) +17 -2(-10) )

-9(-12 + 17 + 20)

-9(25)

-225

The area of the circle in discuss as required to be determined is; 44/7.

It follows that the radius of a circle is the distance between its center and any point on the circumference.

In this case, center, = (0, 0) and point in circumference= (-1, -1).

Therefore, the radius is;

r = √( (-1 -0)² + (-1-0)² )

r = √2.

Consequently, since area of a circle is given by;

Area = πr²

A = (22/7) × (√2)²

A = 44/7.

Ultimately, the area of the circle is; 44 / 7.

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The hypothesis tested are given as follows:

[tex]H_0: \mu = 500, H_a: \mu < 500[/tex]

The claim for this problem is given as follows:

"Bags are consistently underweight".

At the null hypothesis, we consider that the claim is false, that is, there is not enough evidence to conclude that the bags are underweight, hence:

[tex]H_0: \mu = 500[/tex]

At the alternative hypothesis, we test if there is enough evidence to conclude if the claim is true, hence:

[tex]H_a: \mu < 500[/tex]

The problem asks for the null and for the alternative hypothesis.

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Answer:

Step-by-step explanation:

a

The ratio of games won to games played is options A and E, 16/20 or 4/5 in fraction form and 0.8 in decimal form.

Ratio is defined as the relationship between two quantities where it tells how much one quantity is contained in the other.

The ratio of a and b is denoted as a : b.

Given that,

For every 16 games the football team won, they lost four.

Games won = 16

Games lost = 4

Total games played = 16 + 4 = 20

Ratio of games won to total games in fractional form = 16 / 20 = 4/5

Ratio of games won to total games in decimal form = 0.8

The correct options are A and E.

Hence the ratio of games won to total games is 16/20 or 4/5 in fraction form and 0.8 in decimal form.

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Your question is incomplete. The complete question is given below.

Manchester Football team participates in a tournament. For every 16 games the football team won, they lost four. Represent the ratio of games won to games played in fraction form and decimal form.

Select TWO correct answers.

A. 4/5, 0.8

B.  1/2, 0.5

C.  2/4, 0.5

D. 1 /4, 0.25

E.  16/20, 0.8

The Exact values for the sine and cosine of the given angles are : sin(120°) = √3/2, cos(120°) = -1/2

Given that the terminal ray of an angle with measure of 120 degrees intersects a unit circle at the point (-1/2, √3/2), we can use the definition of sine and cosine to find the exact values of these trigonometric functions for the given angle.

The sine of an angle is defined as the y-coordinate of the point on the unit circle that the terminal ray of that angle intersects. So, for this angle, the sine is:

sin(120°) = √3/2

The cosine of an angle is defined as the x-coordinate of the point on the unit circle that the terminal ray of that angle intersects. So, for this angle, the cosine is:

cos(120°) = -1/2

So, the exact values for the sine and cosine of the angle with measure of 120 degrees are:

sin(120°) = √3/2

cos(120°) = -1/2

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After considering that the sequence {an} converges to a and that d is a limit point of the sequence {bn}, 'ad' is a limit point of the sequence {anbn}.

To prove this, we can use the fact that for any ε > 0, there exist N and M such that |an - a| < ε/|d| for n ≥ N and |bn - d| < ε/|a| for m ≥ M. Then, we have:

Using the bounds we obtained for |an - a| and |bn - d|, we can simplify this inequality to:

|anbn - ad| ≤ ε + |d|ε/|a| for n ≥ N and m ≥ M

This shows that for any ε > 0, there exists an index k such that |anbn - ad| < ε for k ≥ max(N, M), which means that ad is a limit point of the sequence {anbn}.

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A function is a rule that for every input it assigns exactly one output. The Option 1 is correct.

A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain.

To be able to define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule). The key thing that makes rule actually a function is there is exactly one output for each input. That is, it is important that the rule be a good rule.

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Answer:

No solution

Step-by-step explanation:

Given the simultaenous equations:

[tex]\displaystyle{\begin{cases} 3.25x - 1.5y = 1.25 \\ 13x-6y=10 \end{cases}}[/tex]

The first equation can be rewritten as:

[tex]\displaystyle{\dfrac{325}{100} x - \dfrac{15}{10}y = \dfrac{125}{100}}[/tex]

Clear the denominators by multiplying both sides with 100:

[tex]\displaystyle{\dfrac{325}{100} x \cdot 100 - \dfrac{15}{10}y \cdot 100= \dfrac{125}{100} \cdot 100}\\\\\displaystyle{325x - 150y = 125}[/tex]

The whole terms are multiple of 25, so we can simplify even lower by dividing both sides by 25:

[tex]\displaystyle{\dfrac{325x}{25} - \dfrac{150y}{25} = \dfrac{125}{25}}\\\\\displaystyle{13x-6y=5}[/tex]

So now, we have:

[tex]\displaystyle{\begin{cases}13x-6y=5\\ 13x-6y=10 \end{cases}}[/tex]

There are various ways to solve simultaenous equations but I'll use the elimination method. Multiply negative in either first or second equation but I'll choose first:

[tex]\displaystyle{\begin{cases}-13x+6y=-5\\ 13x-6y=10 \end{cases}}[/tex]

Add both equations with like terms:

[tex]\displaystyle{0=5}[/tex]

Since this equation is false. Therefore, there is no solution to the simultaenous equation.

Answer:

Step-by-step explanation:

You want to know what to do first with the system of equations ...

You are taught a couple of strategies for solving systems of linear equations algebraically. The "substitution" strategy requires you use one of the equations to write an expression that can be substituted into the other equation. The purpose of this is to reduce the number of variables in the remaining equation. Usually, that means solving for one of the variables to obtain the expression to substitute.

Another strategy you are taught is the "elimination" strategy. It is also called the "addition" or "combination" strategy. It is executed by adding (or subtracting) some multiple of one of the equations from the other equation, or some multiple of it. The purpose of this is to make the coefficient of one of the variables be zero in the combined equation.

The substitution strategy is easiest to execute if you already have one or both equations in "y=" or "x=" form. It is nearly as easy to execute if the coefficient of one of the variables is +1 or -1, or if that can be easily made to be the case. So, this is what you look for to see if the substitution strategy is an appropriate choice.

The elimination strategy is easiest to execute if the coefficients of one of the variables are the same or opposites. If they are the same, that variable can be eliminated by subtracting one equation from the other. If they are opposites, the variable can be eliminated by adding one equation to the other. So, this relation between coefficients is one of the next things you look for when deciding what your strategy will be.

The elimination strategy can also be effectively used if the coefficients of one of the variables are a nice (integer) multiple of one another. In this problem, we notice that the coefficients of y are -1.5 and -6, which are related by a factor of 4. (It is helpful to be very familiar with multiplication facts.) As it happens, the coefficients of x have the same relation: 13 is 4 times 3.25.

The fact that both sets of coefficients are related by the same factor raises a red flag regarding these equations. It means they are either dependent (have infinite solutions) or are inconsistent (have no solution).

The equations are dependent if one equation is a multiple of the other. Here, we can check that by multiplying the first equation by 4:

  4 × (3.25x -1.5y) = 4 × 1.25

  13x -6y = 5

We notice the other equation is ...

  13x -6y = 10

Values of x and y that make 13x-6y=5 cannot also make that same sum be 10. These are called "inconsistent" equations, and they have No Solution.

Hypothetical: If the first equation were 3.25x -1.5y = 2.5, then multiplying it by 4 would give 13x -6y = 10, the same as the second equation. In this case, the equations would be called "dependent," and any of the infinite number of solutions to the first equation would also be a solution to the second equation.

After you look at the equations to determine if any of the coefficients are 1, or have nice relations with the coefficients of the other equation, you can formulate a strategy for elimination or substitution. As we saw above, it can be useful to eliminate any fractions to start with. Sometimes, it is also useful to factor out any common factors. For example, 2x + 6y = 8 can be reduced to x +3y = 4 by factoring out 2 from every term.

Then, the variable that you solve for first will be the one that is left after you have done your substitution or elimination.

As we saw above, the given equations can be rewritten as ...

We already know the same coefficients and different constants mean these equations are inconsistent and have No Solution. If we need further convincing we can subtract one from the other. Here, too, we can plan ahead a little bit: subtracting the first from the second will leave a positive constant:

  (13x -6y) -(13x -6y) = (10) -(5)

  0 = 5 . . . . . . . simplify (false)

There are no values of x and y that will make this false statement true, hence no solution.

Darrel's total earning for the week he sold 225 items is $2,805.

We are given that Darrel has a weekly salary of $430.

This means that no matter how many items he sells, he will always earn at least $430 for the week.

However, Darrel also earns an additional $19 for every item he sells in excess of 100 items.

This means that for the first 100 items he sells, he will not earn any additional money beyond his $430 weekly salary.

But for every item he sells beyond 100, he will earn an additional $19.

Now, for selling 225 items, Darrel sold 125 items in excess of the 100 item baseline.

Thus, the additional amount he earned from selling 125 items is:

= 125 items ×  $19 per item

= $2,375

Therefore, his total earnings for the week would be:

$430 (weekly salary) + $2,375 (amount earned from selling items in excess of 100)

= $2,805

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Step-by-step explanation: A stem and leaf plot is a way to plot data where the data is split into stems (the largest digit) and leaves (the smallest digits). They were widely used before the advent of the personal computer, as they were a fast way to sketch data distributions by hand.

Answer:

See below

Step-by-step explanation:

-1 => not a real number √-1 is a complex number it does not exist

0 => 7, √0 is just 0

9 => √9 + 7 => 3 + 7 => 10

81 => √81 + 7 => 9 + 7 => 16

(a) A regression problem because CEO salary is a continuous variable.

(b) A classification problem because the response variable is categorical (success or failure)

(c) A regression problem because the response variable is continuous

(a) This scenario involves collecting data on the top 500 firms in the US and recording profit, number of employees, industry, and CEO's salary. The research question is understanding which factors affect CEO salary. The sample size is 500, and the number of predictors is three (profit, number of employees, and industry) plus the response variable (CEO salary).

(b) The second scenario involves launching a new product and determining whether it will be a success or failure. Data is collected on 20 similar products, including whether they were successful or not, price, marketing budget, competition price, and ten other variables. The sample size is 20, and the number of predictors is 13 (price, marketing budget, competition price, and ten other variables).

(c) The third scenario involves predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Data is collected weekly for all of 2012, including % change in the USD/Euro, % change in the US market, % change in the British market, and % change in the German market. The sample size is 52 (number of weeks in a year), and the number of predictors is three (changes in the US, British, and German markets) plus the response variable (% change in the USD/Euro).

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a).north component = 44.0 km/h * sin(40.0°) ≈ 28.31 km/h

East component = 44.0 km/h * cos(40.0°) ≈ 33.71 km/h

The difference between the northern components of the fishing boat and the speedboat is:

North difference = 28.31 km/h - 0 km/h = 28.31 km/h

So the time it takes for the speedboat to intercept the fishing boat is:

Time = North distance / North difference = 6.21 km / (28.31 km/h) = 0.219 hours

Converting to minutes:

Time = 0.219 hours * 60 minutes/hour ≈ 13.1 minutes

Therefore, it takes the speedboat about 13.1 minutes to intercept the fishing boat.

b) tan θ = east component / north component

θ = tan⁻¹(east component / north component)

θ ≈ 51.3°

So the direction in which the speedboat travels is 51.3° east of north. Therefore, the compass bearing with respect to due north is:

Bearing = 90° - 51.3° ≈ 38.7° east of north.

In mathematics, a function is a relationship between two sets of values, such that each input value (also known as the argument or independent variable) is associated with exactly one output value (also known as the value or dependent variable). A function is usually denoted by a symbol such as "f(x)" or "y" and is defined by a rule or formula that specifies how the input value is transformed into an output value.

For example, consider the function f(x) = 2x. This function takes an input value x, multiplies it by 2, and returns the result as the output value. So, for example, when x is 3, the output value is 6. When x is 5, the output value is 10.

Functions can be represented graphically as well. The graph of a function is a set of points in a two-dimensional coordinate system, where the x-coordinate is the input value and the y-coordinate is the output value. For example, the graph of the function f(x) = x^2 is a parabola.

Functions are widely used in many areas of mathematics, science, engineering, economics, and more. They provide a powerful tool for modeling real-world situations, making predictions, and analyzing data.

(a) To find how much time it takes for the speedboat to intercept the fishing boat, we first need to find the position of the fishing boat at the moment the speedboat launches.

From the lighthouse's perspective, the fishing boat is located at a bearing of 15.0° east of north and a distance of 24.0 km. Using trigonometry, we can find the north and east components of the fishing boat's position:

North component = 24.0 km * sin(15.0°) ≈ 6.21 km

East component = 24.0 km * cos(15.0°) ≈ 22.76 km

Now we can use the relative velocity between the fishing boat and the speedboat to find the time it takes for the speedboat to intercept the fishing boat. The speedboat's velocity can be broken down into north and east components:

North component = 44.0 km/h * sin(40.0°) ≈ 28.31 km/h

East component = 44.0 km/h * cos(40.0°) ≈ 33.71 km/h

The difference between the northern components of the fishing boat and the speedboat is:

North difference = 28.31 km/h - 0 km/h = 28.31 km/h

So the time it takes for the speedboat to intercept the fishing boat is:

Time = North distance / North difference = 6.21 km / (28.31 km/h) = 0.219 hours

Converting to minutes:

Time = 0.219 hours * 60 minutes/hour ≈ 13.1 minutes

Therefore, it takes the speedboat about 13.1 minutes to intercept the fishing boat.

(b) To find the direction in which the speedboat travels, we can use trigonometry to find the angle between the speedboat's velocity vector and the north direction.

The north component of the speedboat's velocity is 28.31 km/h, and the east component is 33.71 km/h. Using the tangent function, we can find the angle:

tan θ = east component / north component

θ = tan⁻¹(east component / north component)

θ ≈ 51.3°

So the direction in which the speedboat travels is 51.3° east of north. Therefore, the compass bearing with respect to due north is:

Bearing = 90° - 51.3° ≈ 38.7° east of north.

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The people may easily accept the conclusion of the study described because of a common misunderstanding of correlational studies.

The Correlational Studies are used to examine the relationship between two variables, and a positive correlation suggests that the two variables are related in a certain way .

In this study, the Researchers found a strong positive correlation between sugary snacks and hyperactivity, suggesting that as the amount of sugary snacks offered increased, the number of times children left their seats also increased.

Therefore , people easily accept the conclusion of the study that sugar causes hyperactivity due to a misunderstanding of correlational studies and a lack of knowledge about the need for experimental designs to establish causality.

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The given question is incomplete , the complete question is

Researchers conducted a naturalistic study of children between the ages of 5 and 7 years. The researchers visited classrooms during class party celebrations. As a measure of hyperactivity, they recorded the number of times children left their seats. The researchers found a strong positive correlation between sugary snacks offered at the parties and hyperactivity.

Based on these findings, the researchers concluded that sugar causes hyperactivity.

Explain why people may easily accept the conclusion of the study described above? Include in your explanation a misunderstanding of correlational studies

The equation that can be used to find the year y in which Arizonia became a state is y = x+ 96.

Two expressions that are set equal to one another in a mathematical statement is the definition of an algebraic equation. A variable, coefficients, and constants are the typical components of an algebraic equation.

Both sides have equal weight, therefore it is balanced. Make sure that every modification made to one side of the equation is reflected on the other side to prevent a mistake from throwing the equation out of balance.

Let us suppose the year Indiana became a state = x.

Given that, Arizonia became a state 96 years later than Indiana.

This can be written algebraically as follows:

y = x + 96

Hence, the equation that can be used to find the year y in which Arizonia became a state is y = x+ 96.

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A Congruency Proof is a method of proving two figures are congruent by showing that their corresponding sides and angles are equal.

Congruency Proof is important in Math because it provides a rigorous and systematic way to establish that two geometric figures have the same shape and size.

There are three types of congruency proofs.

a) Congruence of side-angle-side (SAS).

Two congruent sets of sides, and the included angle between them is congruent.

b) SSS congruence occurs when two triangles have three pairs of congruent sides.

c) Angle-side-angle congruence (ASA):

The two triangles have two sets of congruent angles and a congruent included side.

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The correct responses are:

If you have SSA, the triangle is determined by the third side. It does not form a triangle if it is too short to intersect the other side. A triangle can also be formed if the third side is the ideal length to connect to the other two sides. Last but not least, the third side may be long enough to intersect the opposite side twice, forming two triangles.

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complete question:

When using the law of sines, why can the SSA case result in zero, one, or two triangles?

The required solution of the expression to put in the equation is x = 3.

To cross multiply two fractions, multiply the first fraction's numerator by the second's denominator and the second fraction's numerator by the first fraction's denominator.

To solve the equation 2.5/x = 10/12 for x, we can cross-multiply to eliminate the fractions:

2.5/x = 10/12

12(2.5) = 10x

30 = 10x

x = 3

Therefore, the solution to the equation is x = 3. To check, we can substitute x = 3 back into the original equation:

2.5/3 = 10/12

0.8333 = 0.8333

This confirms that x = 3 is indeed the solution to the equation.

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Answer:

Part A: length = 4x - 5

Part B: See explanation below

Step-by-step explanation:

Part A

Area of a rectangle = length x width

Given area and width we can find the length as

[tex]length = \dfrac{area}{width}[/tex]

[tex]Area = 12x^2 - 15x\\\\Width = 3x\\\\Length = \dfrac{12x^2 - 15x}{3x}\\\\= \dfrac{12x^2}{3x} - \dfrac{15x}{3x}\\\\= 4x - 5\\\\[/tex]

Answer to Part A

Part B

[tex]length = 4x- 5\; (from part A)}[/tex]

[tex]width = 3x \;(given)}[/tex]

[tex]area = length \times width[/tex]

[tex]=(4x - 5)(3x)\\\\= 4x(3x) - 5(3x)\\\\= 12x^2 - 15x\\\\[/tex]

Hence verified

Both sides of the equation simplify to a, and we can conclude that min(a, min(b, c)) = min(min(a, b), c) is true for all real numbers a, b, and c where a is the smallest.

One way to do this is through mathematical induction, which involves proving a statement for a specific set of values and then showing that it holds true for all other values. In this exercise, we will apply this method to prove two statements involving integers and real numbers.

Statement involving integers:

The given statement is n² + 1 ≥ 2n, where n is an integer in the range [1, 4]. In order to prove this statement, we need to verify it for all values of n in this range. We start with n = 1, which gives us 1² + 1 ≥ 2(1), or 2 ≥ 2. This is true, so we move on to n = 2, which gives us 2² + 1 ≥ 2(2), or 5 ≥ 4. This is also true. Continuing in this manner, we can verify that the statement is true for all values of n in the given range. Therefore, we can conclude that n² + 1 ≥ 2n is true for all integers in the range [1, 4].

Statement involving real numbers:

The given statement is min(a, min(b, c)) = min(min(a, b), c), where a, b, and c are real numbers and we assume that a is the smallest of the three. To prove this statement, we start with the left-hand side and simplify it using the assumption that a is the smallest real number:

min(a, min(b, c)) = min(a, b) if a ≤ b, otherwise min(a, b) = a

= min(min(a, b), c) if a ≤ min(a, b), otherwise min(min(a, b), c) = a

Next, we move on to the right-hand side of the equation and simplify it:

min(min(a, b), c) = min(a, c) if a ≤ c, otherwise min(a, c) = a

Since we assumed that a is the smallest real number, we know that a ≤ b and a ≤ c.

To know more about integer here.

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