To determine if the value 5π/4 is a solution for the equation 3√(sin θ) + 2 = -1, we need to substitute the value of θ and verify if the equation holds true.
Let's substitute θ = 5π/4 into the equation:
3√(sin(5π/4)) + 2 = -1
Now, let's simplify the equation step by step:
First, let's evaluate sin(5π/4). In the unit circle, 5π/4 is in the third quadrant, where sin is negative. Additionally, sin(5π/4) is equal to sin(π/4) due to the periodic nature of the sine function.
sin(π/4) = 1/√2
Now, substitute the value of sin(π/4) back into the equation:
3√(1/√2) + 2 = -1
Simplifying further:
3√(1/√2) = 3 * (√(1)/√(√2)) = 3 * (1/√(2)) = 3/√2 = 3√2/2
Now the equation becomes:
3√2/2 + 2 = -1
To add fractions, we need a common denominator:
(3√2 + 4)/2 = -1
Since the left side of the equation is positive and the right side is negative, they can never be equal. Therefore, the equation is not satisfied, and 5π/4 is not a solution to the equation 3√(sin θ) + 2 = -1.
Thus, the statement "The value 5π/4 is a solution for the equation 3√(sin θ) + 2 = -1" is false.
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la) The speed of the bus is sodi per hour. How muchett distance does it covering aus minutes. b) 3 hours. e) hairs 30 minutes.
a) To determine the distance covered by a bus in a given number of minutes, we can use the formula: distance = speed x time. Since the speed of the bus is given in miles per hour (mph) and we want to know the distance covered in minutes, we need to convert the time to minutes. Therefore, distance = speed x (time in minutes / 60).
b) To determine the distance covered by a bus traveling at sodi mph for 3 hours, we can use the formula: distance = speed x time.
distance = sodi x 3
c) To determine the distance covered by a bus traveling at sodi mph for 30 minutes, we need to convert 30 minutes to hours. Therefore, time = 30/60 = 0.5 hours.
distance = sodi x 0.5
It's important to note that I'm not able to understand the value of "sodi" as it is not a valid unit for speed or distance. Please provide the correct value for sodi.
average rate of change the slope of the corresponding secant line m(sec)=(f(x)-f(a))/(x-a) in f on the interval [a,x]; also, f on the interval [a,a+h] is the slope of the corresponding secant line m(sec)=(f(a+h)-f(a))/h
The rate of change of f on the interval [2, 4] is 6.
The rate of change of a function f on the interval [a,x] is the slope of the corresponding secant line m(sec), which is given by the following formula:
m(sec)=(f(x)-f(a))/(x-a)
To calculate the rate of change, we can take an example function f(x) = x^2. Let's say we want to find the rate of change of f on the interval [2, 4]. Using the formula from above, we get:
m(sec)=(f(4)-f(2))/(4-2)
= (4^2 - 2^2)/(4-2)
= (16 - 4)/2
= 12/2
= 6
Therefore, the rate of change of f on the interval [2, 4] is 6.
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In a certain country, the car number plate is formed by 4 digits from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 followed by 3 letters from the alphabet. How many number plates can be formed if neither the digits nor the letters are repeated?.
Answer:
47,174,400 possible number plates
See important note at the bottom of the explanation
Step-by-step explanation:
The easiest way to handle a problem of this type is to note the following
There are 9 digits 1-9 that we can use for the first four characters of the license plate
This means that for the first digit there are all 9 possibilitiesFor the second digit, since there is no repetition, there exists only 8 possibilitiesFor the third spot, there are 7 possibilitiesFor the 4th spot there are 6 possibilitiesSo the number of ways to arrange the first 4 digits without repetition is
9 x 8 x 7 x 6 = 3024 possible ways
For the letters, assuming that we are using the English alphabet, the 5th spot where the first letter after the numbers is can be done in 26 possible ways since there are 26 alphabets.
For the second letter that will leave us with 25 possibilities
For the third and final spot there are 24 possibilities
This means that, without repetition, the letters can be arranged in 26 x 25 x 24 ways = 15,600 ways
Together we can arrange the digits and letters in 3024 x 15600 ways which works out to
3024 x 15600 = 47,174,400 ways or possible number plates
Or we could simply write this in one step as
9 x 8 x 7 x 6 x 26 x 25 x 24 = 47,174,400 ways or possible number plates
Note
Even though there are 10 digits 0 - 9, the question clearly states that we should choose from digits 1 through 9 only.
If we are allowed to use all 10 digits the first 4 digits can be put together in 10 x 9 x 8 x 7 = 78,624,000 ways
You should clarify this with your teacher.
Susan broke into an apartment shared by Judy, Kendra, and Lisa and stole
$10,000 worth of valuables, including jewelry, stereos, small appliances, etc.
Since Lisa is a friend of Susan's, she would not press charges or file a report for
her lost items. Susan was convicted, and Judy and Kendra received a settlement.
Now 10 months later, Lisa and Susan had a fight and Lisa wants to file a report
on her losses from that burglary. Has too much time lapsed? Is this considered
the same crime or classified as a separate crime?
The categories are usually "felony," "misdemeanor," and "infraction." Decisions on crime classification are made by state legislators; the determination focuses on the seriousness of the crime.
What are the two types of crimes?There are two major types of crime: felonies and misdemeanors. The difference between these two types of crime is defined by the potential punishment for committing them. Talk to a criminal law attorney in your state to learn more about the punishments in misdemeanor and felony cases.Drdhms concluded that all types of criminals could be classified under three main heads: instinctive criminals, habitual crim- inals, and single offenders.Crime is behavior, either by act or omission, defined by statutory or common law as deserving of punishment or penalty. Although most crimes require the element of intent, certain minor crimes may be committed based on strict liability even if the defendant had no specific mindset with regard to the criminal action.
8.33% probability that Michael will be chosen first and Susan will be chosen second.
Initially, there are 4 people(Susan, John, Lisa, and Michael). The probability of Michael being chosen first is 1/4.
Now, suppose Michael was chosen.
There are 3 people(Susan, John and Lisa). The probability of Susan being chosen is 1/3.
What is the probability that Michael will be chosen first and Susan will be chosen second.
Multiplying these probabilities
8.33% probability that Michael will be chosen first and Susan will be chosen second.
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Identify the correct way to write this sentence: Uncle Frank bought several new metal posts
to fix the fence in the backyard.
a. Uncle Frank bought several new metal posts to fix the fence in the backyard.
b. Uncle Frank bought several, new metal posts to fix the fence in the backyard.
c. Uncle Frank bought several, new, metal posts to fix the fence in the backyard.
A sentence must have a singular or plural subject and verb in order to be grammatically correct. To put it another way, the tenses of the subject and verb must match.
Which seven types of sentences are there?According to their structure and function, sentences can be divided into eight categories: declarative, interrogative, exclamatory, imperative, simple, compound, complex, and compound-complex sentences.
How can I tell what a grammar name is?You must look at what a grammatical name accomplishes in an English sentence in order to recognise it. Consider: At this morning's assembly, Susan performed a song. You must take into account the work each unit performs in the sentence in order to identify the various units' grammatical names.
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Triangle ABC is similar to Triangle ADE. Find x, the distance from Point B to Point C.
The distance from Point B to Point C of the triangle will be 45 feet.
What is the triangle?The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.
The ratio of the matching sides will remain constant if two triangles are comparable to one another.
Triangle ABC is similar to Triangle ADE. Then the equation is given as,
x / 15 = (40 + 20) / 20
x / 15 = 60 / 20
x / 15 = 3
x = 45 feet
The distance from Point B to Point C of the triangle will be 45 feet.
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The complete question is given below.
How do you enlarge a triangle by a scale factor of 3?
To enlarge a triangle by a scale factor of 3 we divide the longer side by the short side. 6 ÷ 2 = 3 and so, the scale factor is 3.
What is a scale factor?
A scale factor is a numerical value that is used to represent an object's larger or shrunk shape. It is a number that can be used to alter the size of any geometric figure or shape in relation to its initial size. It aids in adjusting the figure's size.
Here,
We have to determine how can we enlarge a triangle by a scale factor of 3.
We concluded that
To find the scale factor of enlargement, divide the longer side by the short side. 6 ÷ 2 = 3 and so, the scale factor is 3.
This means that all of the sides on the smaller shape have been multiplied by 3. It also means that all of the sides on the larger shape can be divided by 3 to make the smaller shape.
Hence, to enlarge a triangle by a scale factor of 3 we divide the longer side by the short side. 6 ÷ 2 = 3 and so, the scale factor is 3.
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Stan’s savings account has a balance of $1986. after 23 years, what will the amount of interest be at 4% compounded annually? a. $2908.93 c. $2913.93 b. $2899.93 d. $794.40
Answer:
a. $2908.93
Step-by-step explanation:
We use the formula:
A = P( 1 + r)^t
P = $1986
r = 4% = 0.04
t = 23 years
Let's solve
A = 1986(1 + 0.04)^23
A = 1986( 1 + 0.04)
A = 1986( 1.04)^23
A = $4894.925
Now we take
$4894.925 - $1986 = $2908.925
$2908.925 round-up = $2908.93
So, the answer is A
Why is the base in the exponential function greater than 1?
The base in the exponential function is greater than 1 because exponential growth is defined as a constant multiplication of the original value, rather than a constant addition as in linear growth. A base greater than 1 results in a growth rate that increases over time, while a base less than 1 results in a growth rate that decreases over time.
THE EXPONENTIAL FUNCTIONThe exponential function is written as bˣ, where b is the base and x is the exponent. The base is the number that is being multiplied by itself a certain number of times, as determined by the exponent.
A base greater than 1 means that the value of the function is increasing at an increasing rate. This is because as the exponent (x) increases, the base (b) is being multiplied by itself more and more times, leading to a larger and larger result. This is known as exponential growth.
On the other hand, a base less than 1 means that the value of the function is decreasing at an decreasing rate. This is because as the exponent (x) increases, the base (b) is being multiplied by itself more and more times, leading to a smaller and smaller result. This is known as exponential decay.
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What is the length of the hypotenuse of a 45 45 90 triangle?
In a 45°−45°−90° triangle, the length of the hypotenuse is √2 times the length of a leg.
How to find the length of the hypotenuse of a 45 45 90 triangle?Since the triangle's legs are equal, the hypotenuse may be determined using the equation c = a2/. The side length will be equal to a = c2/2 if the value of the hypotenuse is provided. When the two legs of an isosceles right triangle are congruent to one another and the non-right angles are both 45 degrees, the triangle is known as a 45 45 90 triangle. When the two legs of an isosceles right triangle are congruent to one another and the non-right angles are both 45 degrees, the triangle is known as a 45 45 90 triangle. The Pythagorean theorem can frequently be used to locate the missing legs or hypotenuse of 45 45 90 triangles. The sides' to the hypotenuse's ratio.To learn more about hypotenuse refer to:
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Max spends three times as long on his math homework than he does on his science horr spent a total of 64 minutes on math and science homework last night, how long did he math homework?
Time spend on science homework is 16 minutes and time spend on math homework is 48 minutes.
What is Equation?Two or more expressions with an Equal sign is called as Equation
Given that,
The time spend on math homework = 3 times time spend on science homework,
Also, total time spent on math and science both homework = 64 minutes.
Let time spend by Max on science homework is t minutes,
Then by given condition, time spend on math homework = 3t
Total time = 64 minutes
t + 3t = 64 minutes
4t = 64 minutes
Divide both sides by t
t = 16 minutes
Hence, Time spend on science homework is 16 minutes and time spend on math homework is 48 minutes.
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How do you find the sides of a 3 4 5 triangle?
It is possible to construct a triangle whose sides are 3 cm 4 cm and 5 cm because the sum of two sides are greater than the third sides.
In the given question, we have to check that it is possible to construct a triangle whose sides are 3 cm 4 cm and 5 cm.
To check whether the given sides can make a triangle or not, we have to check that the sum of two sides always greater than the third side.
To check this we firstly add the 3 and 4
3 + 4 > 5
7 > 5
Now we add 3 and 5
3 + 5 > 4
8 > 4
Now we add 4 and 5
5 + 4 > 3
9 > 3
It is possible to construct a triangle whose sides are 3 cm 4 cm and 5 cm because the sum of two sides are greater than the third sides.
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PLEASE HELP
Kevin ran at a rate of 13 km/h. Convert his speed to meters per minute
Answer:
216.7 m/min
Step-by-step explanation:
1 km = 1000 m
1 hour = 60 min
13km/h = 13 (1000 m) / (60 min) = 216.7 m/min
Answer:216.6667
Step-by-step explanation:
kph × 16.66667
m/min =13 × 16.66667
m/min =216.6667
Is 1234 an arithmetic sequence?
No, 1234 is not an arithmetic sequence. A sequence is a list of numbers, arranged in a specific order.
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. It is also known as an arithmetic progression.
The sequence 1234 does not have a common difference between its terms. It's just a set of numbers arranged in a specific order, but the difference between the terms is not always the same.
Arithmetic sequence has the general formula of a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, and n is the number of the term. The sequence 1234 doesn't follow this formula.
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Which energy transfer system is represented by the heating of the thermometer in model 2?
The thermometer functions because of temperature-related expansion and contraction. Conduction is the process by which heat moves from one item to another.
what is equation ?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equality of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to express the connection between two phrases on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.
given
The thermometer will display a greater temperature as the liquid expands and rises as it gets hotter.
The thermometer functions because of temperature-related expansion and contraction.
Conduction is the process by which heat moves from one item to another.
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Which of the following is Pythagorean triplet ?A. (6,8,10)B. (5,12,13)C. (7,13,15)D. (3,4,5)
Our Pythagorean triplets will be (6,8,10), (5,12,13), and (3,4,5).
So what is the Pythagorean triplet?
The square of the greatest number in a Pythagorean triplet equals the sum of the squares of the other two numbers.
The greatest side in a right-angled triangle is called the hypotenuse.
So, in our first triplet A(6,8,10) the greatest side is 10. So, hypotenuse = 10.
= [tex]hypotenuse^{2}[/tex] = [tex]10^{2}[/tex] = 100.
Now, [tex]6^{2}[/tex] + [tex]8^{2}[/tex] = 36 + 64 = 100.
[tex]10^{2}[/tex] = [tex]6^{2}[/tex] + [tex]8^{2}[/tex]
Thus A(6,8,10) is a Pythagorean triplet.
In the second case B(5,12,13) hypotenuse = 13.
= [tex]hypotenuse^{2}[/tex] = [tex]13^{2}[/tex] = 169.
Now, [tex]12^{2}[/tex] + [tex]5^{2}[/tex] = 144 + 25 = 169.
[tex]13^{2}[/tex] = [tex]12^{2}[/tex] + [tex]5^{2}[/tex]
Thus B(5,12,13) is a Pythagorean triplet.
In the Third case C(7,13,15) Hypotenuse = 15.
= [tex]hypotenuse^{2}[/tex] = [tex]15^{2}[/tex] = 225.
Now, [tex]7^{2}[/tex] + [tex]13^{2}[/tex] = 49 + 169 = 218.
[tex]15^{2}[/tex] [tex]\neq[/tex] [tex]7^{2}[/tex] + [tex]13^{2}[/tex]
Thus C(7,13,15) is not a Pythagorean Triplet.
In the last case D(3,4,5) Hypotenuse = 5.
= [tex]hypotenuse^{2}[/tex] = [tex]5^{2}[/tex] = 25.
Now, [tex]3^{2}[/tex] + [tex]4^{2}[/tex] = 9 + 16 = 25.
[tex]5^{2}[/tex] = [tex]3^{2}[/tex] + [tex]4^{2}[/tex]
Thus D(3,4,5) is a Pythagorean triplet.
Therefore our three results A(6,8,10) B(5,12,13) and D(3,4,5) are Pythagorean triplets.
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!!!PLEASE ANSWER VERY QUICKLY I AM TIMED!!!
Suppose that 14 inches of wire costs 70 cents. At the same rate, how much (in cents) will 37 inches of wire cost?
Answer:
Step-by-step explanation:
2590
can someone who took the test please help me out
The graph of the following coordinates can be graphed by plotting the points of given coordinates on the graph.
What are coordinates?Coordinates are a pair of integers (also known as Cartesian coordinates), or occasionally a letter and a number, that identify a specific point on a grid, also known as a coordinate plane.
What is a point?A point is a dot on a sheet of paper or in a plane. There are no lengths, widths, or heights in a point. It establishes a plane's position or location.
Plot the points according to the given coordinates. Connect all the points with with a smooth curve.
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How do you find the point and midpoint of a point?
To find the coordinates of a point, you simply need to know its location in the given space and to find the midpoint of a point you need to know the coordinates of the two given points and use the formula (x₁ + x₂)/2, (y₁ + y₂)/2.
A point in geometry is a location in space that is represented by a set of coordinates, typically in the form of (x,y) or (x,y,z) for a three-dimensional space.
To find the coordinates of a point, you simply need to know its location in the given space.
The midpoint of a point is the point that is exactly halfway between two given points. To find the midpoint of a point, you need to know the coordinates of the two given points. Once you have this information, you can use the following formula to find the midpoint:
(x₁ + x₂)/2, (y₁ + y₂)/2
For example, if the coordinates of point A are (3,4) and the coordinates of point B are (7,8), the midpoint of the two points would be ( (3+7)/2 , (4+8)/2 ) = (5,6)
In conclusion, finding the point and midpoint of a point is a fundamental concept in geometry. To find the point, we need to know its location in the given space, represented by the coordinates.
To find the midpoint of a point, we need to know the coordinates of the two given points and use the formula (x₁ + x₂)/2, (y₁ + y₂)/2. These concepts are widely used in mathematics, physics, engineering and other fields that deal with spatial relationships.
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draw a line through the point (2,2) with a slope of 1. Draw a line through the point (1,-1) with a slope of 1. WHAT IS THE OTHER LINE? please help asap!!!!
A 9,000-gallon pool i being filled at a rate of 40 gallon per minute. At thi rate, how many minute will it take to fill thi pool?
As per the given volume, it take 223 minutes to fill the pool.
The term volume in math is defined as a mathematical quantity that shows the amount of three-dimensional space occupied by an object or a closed surface and it measured as cubic units.
Here we have given that a 9,000-gallon pool is being filled at a rate of 40 gallon per minute.
Here we need to find the time taken to fill the pool.
While we looking into the given question, we have identified the following values,
Total volume of pool = 9000 gallon.
Amount of water filled in one minute = 40 gallon,
then the time take by the pool to fill is calculated by dividing the total amount by the amount of water per minute then we get,
=> 9000/40
When we simplify this one then we get,
=> 225
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What does it mean if your cervix is 2 cm dilated?
Similar to when a woman is 1 cm dilated, 2 cm dilation does not indicate that labor will start soon. If a woman is 2 cm dilated, some of them may give birth in a few of hours. For a few days or weeks until labor advances, some women will stay 2 cm dilated.
What stage of labor is 2 cm dilated?The cervix dilates (opens) and effaces (thins) during the initial stage of labor to make room for the baby to enter the birth canal. Cervical closure is visible in figures A and B. A 60% effaced and 1 to 2 cm dilated cervix can be seen in figure C. 90% effaced and 4 to 5 cm dilated, respectively, is the cervix in figure D.An average woman can dilate between 0.5 and 0.7 cm per hour during the active stage of labor. Whether this is your first child or not will also influence how quickly your cervix dilates.In order to speed up dilation, standing up and moving about may be helpful. Dilation may be aided by pacing the room, making little adjustments in a chair or bed, or even shifting positions. The baby's weight puts pressure on the cervix, which is why this is.To Learn more About dilation refer To:
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How do you change a quadratic equation from vertex to factored form?
Standard form is ax^2 + bx + c. Vertex form is a(x-h)^2 + k, which reveals the vertex and axis of symmetry. Factored form is a(x-r)(x-s), which reveals the roots.
What is quadratic equation?Factoring, using square roots, completing the square, and the quadratic formula are the four ways to solve a quadratic problem. Any equation in algebra that can be written in standard form as where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation. x ax2 + bx + c = 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. a 0. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem. The answer could be simple or complicated.
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Moya i moving to a new houe and packing up. She ha 70 kilogram of book and 127 kg kilogram of clothing to end through the potal ervice. How much more do her clothe weigh than her book? How much i her hipping cot of h ha to pay $80 per kg of book and $200 per kg of clothing?
The amount that her clothes weigh more than her books would be
= 57kg
The amount for the shipping cost for her school would be
= 5600+25400 = $31000
Postal Service:
The Postal Service is an American indie pop supergroup from Seattle, Washington, consisting of singer Ben Gibber, producer Jimmy Tamborello, and backing vocalist Jenny Lewis.
The band released her only album Give Up in 2003 on her Sub Pop Records to mostly positive reviews. The album reached his number 114 spot on the US Billboard 200 album chart and was certified platinum by the Recording Industry Association of America. Laura Bergen joined the line-up for the entire tour, providing additional vocals and instruments, while Jen Wood and Jenny Lewis filled in for some shows.
According to the Question:
The mass of Moya's books = 70 kilograms
The mass of Moya's clothing = 127 kg
The difference between the both masses = 127 - 70 = 57kg
The cost per kg for books = $80 per kg
The cost per kg of clothing = $200
Therefore, the amount for the shipping cost for her school would be = (70×80) + (127 ×200)
= 5600+25400 = $31,000.
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PLEASE HELP ME ITS DUE IN 30 mINS PLEASE 50 POINTS
The perimeter of Brian's kitchen is 12x - 2 feet,while the perimeter of his living room is 18x - 11 feet. Determine how much greater the perimeter of Brian's living room is in comparison to his kitchen.
PART A: Write an algebraic expression to represent how you would represent the given situation above.
PART B:Simplify your expression to determine the difference between the two perimeters. SHOW YOUR WORK
Expression = Blank feet
Answer:
6x - 9
Step-by-step explanation:
18x -11 - (12x - 2) Can be written
18x -11 -1(12x -2) Distribute the -1
18x - 11 -1(12x) + (-1)(-2)
18x - 11 -12x +2 Combine like terms
6x -9
How do you graph 3 variable graphs?
A 3 variable graph can be graphed using a 3D scatter plot. The x, y, and z axes will represent the three variables. Each point on the graph will represent a combination of the three variables. The points can be connected to form a 3D line graph to illustrate the relationships between the variables.
1. Open a graphing software such as Microsoft Excel or a graphing calculator.
2. Choose the 3D scatter plot option.
3. Enter the values for each of the three variables.
4. The graph will appear with the x-axis, y-axis, and z-axis representing the three variables.
5. Each point on the graph will represent a combination of the three variables.
6. Optionally, you can connect the points to form a 3D line graph to illustrate the relationships between the variables.
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Can someone help me please Assap?
-3(2x+2)<10 solve the inequality and graph the solution ?
Therefore , the solution of the given inequality problem comes out to be
-8/5.
What is a good example of inequality?The equation-like form of the expression 5x 4 > 2x + 3 has an arrowhead in place of the equals sign. It is an illustration of inequality. This shows that the left part, 5x 4, is bigger than the right part, 2x + 3. Finding the x variable values for which the inequality holds true is what we are most interested in.
Here,
Give:
=> -3(2x+2) < 10
=>2x+2>10/3
=>x+1>-10/6
=>> -10/6-1 >1
=> -8/5
Therefore , the solution of the given inequality problem comes out to be
-8/5..
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Help please it's due TODAY
Eric is riding his bicycle. He rides for kilometers at a speed of kilometers per hour. For how many hours does he ride?
I need help asap! Please and thank you
One can use either the slope formula m = (y2 – y1)/(x2 – x1) or the standard line equation, y = mx + b to solve for the slope, m.
How do you determine the slope?You can find the slope of the line by determining the rise and run of two locations on the line. The rise is the vertical change between two places, while the run is the horizontal change. The slope is calculated by dividing the climb by the run: Riserun = Slope Slope = rise and run.
Depending on the form of the equation in front of you, the procedure for calculating the slope will differ. If the equation is written as y = mx + c, the slope (or gradient) is simply m. If the equation is not in this form, attempt rearrangement. You will need to do the following to determine the gradient of additional polynomials:
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