Binomial expansion of the given algebraic expression is:
729x⁶ − 1458x⁵y + 1215x⁴y² - 540x³y³ + 135x²y⁴ - 18xy⁵ + y⁶
How to use the Binomial expansion Theorem?The binomial theorem is one that states the principle for expanding the algebraic expression (x + y)ⁿ and expresses it as a sum of the terms involving individual exponents of variables x and y.
Thus, for example (x + y)³ will be expressed as:
1(x³)(y⁰) + 3(x²)(y¹) + 3(x¹)(y²) + 1(x⁰)(y³)
Similarly, for (3x - y)⁶, using pascals triangle, we have:
1(3x)⁶ + 6(3x)⁵(-y)¹ + 15(3x)⁴(-y)² + 20(3x)³(-y)³ + 15(3x)²(-y)⁴ + 6(3x)¹(-y)⁵ + 1(-y)⁶
= 729x⁶ − 1458x⁵y + 1215x⁴y² - 540x³y³ + 135x²y⁴ - 18xy⁵ + y⁶
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Need help please
Which of the following systems of inequalities matches the word problem? You are on the planning committee for the prom You have an 80 budget for food You decide to cater the prom with Little Cesar's $5 pizzas and McDonald's McChicken's, which cost 2 each. Based on a survey given to students, you will need at least 8 Little Cesar's pizzas to feed everyone that wants pizza Let x represent the number of pizzas bought and y represent the number of McChicken's bought
The system of inequalities that matches the given word problem is:
5x + 2y <= 80
x >= 8
What is system of inequalities?A system of inequalities is a set of two or more inequalities in one or more variables. Systems of inequalities are used when a problem requires a range of solutions, and there is more than one constraint on those solutions.
Equation:Let's break down the problem and identify the key information:
The budget for food is 80 dollars
Little Cesar's pizzas cost 5 dollars each
McDonald's McChicken's cost 2 dollars each
At least 8 Little Cesar's pizzas are needed to feed everyone that wants pizza
Let x represent the number of pizzas bought and y represent the number of McChicken's bought. The total cost of pizzas and McChicken's should be less than or equal to 80 dollars, which gives us the first inequality:
5x + 2y <= 80
Additionally, we know that at least 8 pizzas should be bought, which gives us the second inequality:
x >= 8
Therefore, the system of inequalities that matches the word problem is:
5x + 2y <= 80
x >= 8
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it takes a printer 6 minutes to print 40 pages at this rate how much can they print in 15 minutes
Answer:
100
Step-by-step explanation:
[tex]\frac{40}{6} = \frac{x}{15}[/tex]
x = (40 x 15) / 6
x = 100
70% of what number is 30
Answer: 43
43 x 0.7 = 30.1
Answer:
70 % is salty water and 30% IS DRINKING WATER in which 2.1% comes from glaciers
In a circle of radius 3, an arc has length 5.
What's the central angle of the arc?
Answer: 300/π°, or ~95.49°
Step-by-step explanation:
In order to find the central angle of the arc, we first find the ratio of the arc length to the circumference of the circle, since it is proportional to the ratio of the central angle to the total inside angle, 360°.
Finding the circumference
Circumference is equal to 2πr, and r (radius) in this case is 3, so
C=2π*3, or 6π
The ratio of the arc length to circumference is then 5/6π
Finding the angle
Since the ratio of lengths and ratio of angles are proportional, we can set up this equation:
5/6π=x/360
solving for x gets us:
x=300/π°, or ~95.49°
The total weight of 36 packets of crisps is 864 grams. There are 505 calories in 100 grams of crisps. Work out how many calories are there in one packet of crisps. 1
Answer:
121.2 calories.
Step-by-step explanation:
Weight of one packet of crisps = 864/36
= 24 grams.
So, by proportion, the number of calories in one packet
= 24 * 505/100
= 121.2 calories.
Find 3 ratios that are equivalent to the given ratio. 9 4
18:8, 27:12 and 45:20 are ratios are equivalent to 9:4.
Define ratioA ratio is a mathematical comparison between two or more quantities that indicates how many times one quantity is contained within another. Ratios are expressed in the form of a fraction, with the first quantity being the numerator and the second quantity being the denominator.
three ratios that are equivalent to 9:4
18:8 - We multiplied both the numerator and denominator by 2.27:12 - We multiplied both the numerator and denominator by 3.45:20 - We multiplied both the numerator and denominator by 5.All of these ratios are equivalent to 9:4, because they represent the same ratio of 9 parts to 4 parts.
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Which phrase describes the linear relationship between the x and y
values in this table?
X3
4
5
y
9
12
15
OA) X is 3 times y
OB) y is 3 times x
OC) y is 3 less than x
OD) y is 3 more than x
We can conclude that the linear relationship between x and y in this table is described by the phrase "y is 3 more than x."
What is the linear function?
A linear function is defined as a function that has either one or two variables without exponents. It is a function that graphs to a straight line.
The given table shows two variables, x, and y, with corresponding values. A linear relationship between two variables means that a change in one variable causes a proportional change in the other variable.
If we look at the table, we can see that as the value of x increases by 1 unit from 3 to 4, the value of y also increases by 3 units from 9 to 12. Similarly, as the value of x increases by 1 unit from 4 to 5, the value of y increases by 3 units from 12 to 15.
This means that for every 1 unit increase in x, the value of y increases by a constant amount of 3 units.
In other words, the slope of the line connecting these points is constant and equal to 3.
Therefore, we can conclude that the linear relationship between x and y in this table is described by the phrase "y is 3 more than x."
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what is the range of the function graphed?
The correct option is A.
y ≥ -3
what is the range of the grapherd function?The range is defined as the set of possible outputs, so just look at the vertical axis, you can see it goes from x = -3 up, then the range is just the set of all values larger than -3
it is written as:
y ≥ -3
The correct option is a-.
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Helena drops a ball from 25 feet above a lake. The formula t= 1/4√25-h describes the time t in seconds that the ball is h feet above the water. How many feet above the water will the ball be after 1 second?
40 points
Please please help I've been struggling for so long
Identify the equation in the point-slope form that matches the graph
Answer: the answer to the question is A.
h(x) = 3x
If you input a 2 into h(x), what do you get for the output?
h(2) = 3(2) = 6
The output is six
an angle measures 18 degrees less than the measure of its supplementary angle what is the measure of each angle
The measure of the angle is 81 degrees, and the measure of its supplementary angle is 99 degrees.
Let x be the measure of the angle in question, and let y be the measure of its supplementary angle.
By definition, supplementary angles add up to 180 degrees, so we have:
x + y = 180
We also know from the problem statement that:
x = y - 18
We can substitute the second equation into the first equation to eliminate y, giving:
(y - 18) + y = 180
Simplifying this equation, we get:
2y - 18 = 180
Adding 18 to both sides, we get:
2y = 198
Dividing both sides by 2, we get:
y = 99
Finally, we can use the equation x = y - 18 to find the measure of the angle in question:
x = 99 - 18 = 81
Therefore, the measure of the angle is 81 degrees, and the measure of its supplementary angle is 99 degrees.
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Can someone know if I got this right, please?
Therefοre, The similarity statement is ΔBAT ~ ΔDER, and the reasοn fοr the similarity is by SSS similarity since the cοrrespοnding sides are in prοpοrtiοn.
What is the similarity?Twο triangles are similar if their cοrrespοnding angles are cοngruent and their cοrrespοnding sides are prοpοrtiοnal. Sο yοu can find the dimensiοns οf a triangle using anοther triangle. If ABC and XYZ are twο similar triangles, yοu can find the assοciated angles and side lengths using the fοrmulas belοw.
Tο determine whether the twο triangles BAT and DER are similar, we can use the similarity criteria. Twο triangles are similar if their cοrrespοnding angles are cοngruent and their cοrrespοnding sides are in prοpοrtiοn.
Let's check if the triangles satisfy the similarity criteria:
Angle BAT is cοngruent tο angle DRE, bοth being right angles.
Angle BTA is cοngruent tο angle EDR, bοth being acute angles.
Angle A is cοngruent tο angle E, bοth being acute angles.
Therefοre, the twο triangles have cοrrespοnding angles that are cοngruent.
Next, we can check if their cοrrespοnding sides are in prοpοrtiοn:
BA/BT = 48/84 = 4/7
ED/DR = 8/14 = 4/7
AT/ER = 72/12 = 6/1
We can see that the ratiοs οf the cοrrespοnding sides are equal, which means that the twο triangles are similar.
Therefοre, The similarity statement is ΔBAT ~ ΔDER, and the reasοn fοr the similarity is by SSS similarity since the cοrrespοnding sides are in prοpοrtiοn.
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Describe what happens to the graph of y=a(x-2)^2+5 as ‘a’ varies
Varying 'a' in the equation og graph [tex]y = a(x-2)^2 + 5[/tex] changes the steepness and orientation of the parabolic curve, while keeping the vertex fixed at (2, 5).
What is graph?A graph is a visual representation of data or information that displays a relationship or pattern between two or more variables.
The graph of [tex]y = a(x-2)^2 + 5[/tex] is a quadratic function that has been shifted 2 units to the right and 5 units up from the standard form of [tex]y = x^2.[/tex] The variable 'a' represents the vertical stretch or compression factor of the parabola.
As 'a' varies, the shape of the parabola remains the same, but the steepness of the curve changes. When 'a' is positive and increases, the parabola becomes narrower, or more steeply curved, and the vertex moves higher up. This is because the larger value of 'a' causes the parabola to stretch vertically. When 'a' is negative and increases, the parabola also becomes narrower and the vertex moves lower down, but the orientation of the parabola is flipped upside down.When 'a' equals zero, the graph becomes a horizontal line at y = 5. This is because the parabola has been flattened completely, so there is no curvature in the graph.In summary, varying 'a' in the equation [tex]y = a(x-2)^2 + 5[/tex] changes the steepness and orientation of the parabolic curve, while keeping the vertex fixed at (2, 5).
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The daily high temperatures, in degrees Fahrenheit (°F), of a town are recorded for one year: The median high temperature is 62°F. The interquartile range of hight temperatures is 32. Which statement is most likely true?
A approximately 25% of the days had a high temperature less than 30•F.D.
B Approximately 25% of the days had a high tenperature greater than 62•F.
C Approximately 50% of the days had a high temperature greater then 62•F.
D Approximately 57% of the days had a high temperature less than 94•F
Using the data of interquartile range and median given, only option C is correct because approximately 50% of the days had a high temperature greater than 62°F.
Which of the following statements is true?We are given that the median high temperature is 62°F and the interquartile range is 32. The interquartile range is the difference between the first and third quartiles, which means that the first quartile is at 62 - 16 = 46°F and the third quartile is at 62 + 16 = 78°F.
Option A: approximately 25% of the days had a high temperature less than 30°F.
Since 30°F is much lower than the first quartile (46°F), it is unlikely that 25% of the days had a high temperature less than 30°F. Therefore, option A is not true.
Option B: approximately 25% of the days had a high temperature greater than 62°F.
This statement cannot be true because the median is already at 62°F, which means that 50% of the days had a high temperature greater than or equal to 62°F. Therefore, option B is not true.
Option C: approximately 50% of the days had a high temperature greater than 62°F.
This statement is true because the median high temperature is 62°F, which means that 50% of the days had a high temperature greater than or equal to 62°F, and the interquartile range is 32, which means that the third quartile is at 78°F. Since the third quartile is above 62°F, approximately 50% of the days had a high temperature greater than 62°F. Therefore, option C is most likely true.
Option D: approximately 57% of the days had a high temperature less than 94°F.
The maximum temperature that can be associated with the interquartile range of 32°F is the third quartile, which is at 78°F. Therefore, we cannot say that approximately 57% of the days had a high temperature less than 94°F. Therefore, option D is not true.
Hence, the most likely true statement is option C: approximately 50% of the days had a high temperature greater than 62°F.
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Based on the information in the table and the graph, compare the approximate miles per gallon of car M to car P.
Show your work or explain your answer.
Answer and Step-by-step explanation:
You can start off your answer by comparing each gallon/mile of each car given. Compare the values that both cars have in common.
For Car M, the car drives a distance of 50.4 miles and used 2 gallons of gasoline.
For Car P, the car drives a distance of approximately 65 miles and used 2 gallons of gasoline.
For Car M, the car drives a distance of 80.5 miles and used 3 gallons of gasoline.
For Car P, the car drives a distance of approximately 90 miles and used 3 gallons of gasoline.
For Car M, the car drives a distance of 137.5 miles and used 5 gallons of gasoline.
For Car P, the car drives a distance of approximately 160 miles and used 5 gallons of gasoline.
After looking at the given information, we can conclude that for every gallon of gasoline used, Car P drives a farther distance than Car M.
We can see this at the points 2, 3, and 5 gallons, where Car M drives 50.4 miles, 80.5 miles, and 137.5 miles (respectively), and Car P drives 65 miles, 90 miles, and 160 miles (respectively).
Have a good day!
HELP URGENT !!!A bag contains 16 red marbles, 7 yellow marbles, and 19 green marbles. What is the probability that a marble drawn
from the bag is red or yellow? Write your answer as a simplifed fraction.
Answer: 23/42
Step-by-step explanation:
your messageHELP URGENT !!!A bag contains 16 red marbles, 7 yellow marbles, and 19 green marbles. What is the probability that a marble drawnfrom the bag is red or yellow? Write your answer as a simplifed fraction.Sure, I can help you with that! The probability of drawing a red marble is 16/42 (since there are 16 red marbles out of a total of 42 marbles), and the probability of drawing a yellow marble is 7/42. To find the probability of drawing a red or yellow marble, we simply add those two probabilities: 16/42 + 7/42 = 23/42 So the probability of drawing a red or yellow marble is 23/42, which is already in simplified fraction form.
A construction company transported 60 stones that a total of 858 kilograms how did each stone weigh? write your answer as a mixed number
Answer:
14 3/10
Step-by-step explanation:
Divide 858 by 60 and get 14.3
14.3 in a mixed number form is 14 3/10
Therefore, one stone weighs 14 3/10 kilograms.
Describe the range of the relationship shown here
Outside of this range, the relationship's range may differ, and outliers or other significant points may also have an impact.
what is variable ?A variable in mathematics is a symbol or letter that designates a number or value that is subject to change or variation. In order to solve problems and generate expressions and equations that represent the relationships between different quantities, variables are employed. X and Y are variables in the equation y = 3x + 2, for instance. The variable value of x determines the value of y. In order to express unknown numbers that we want to solve for, we can also utilize variables.
given
The scatterplot's depicted relationship is a positive linear relationship. These are some possible descriptions of the relationship's scope:
As the independent variable (x-axis) rises, so too does the dependent variable (y-axis).
There is not much scatter around the trendline, suggesting that the link is reasonably strong. From the lowest value on the y-axis to the greatest value on the y-axis, the relationship's range is defined.
The scatterplot only depicts the relationship between the two variables within the observed range of the data, it is crucial to remember this. Outside of this range, the relationship's range may differ, and outliers or other significant points may also have an impact.
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i’m confused and would love help.
The final equation of the line is y = (-2/3)x - 3 in the coordinate system.
what is coordinate system?
A coordinate system is a system that uses one or more numbers or coordinates to uniquely determine the position of a point or an object in space. In mathematics, the most common coordinate system is the Cartesian coordinate system
what is equation ?
In mathematics, an equation is a statement that shows the equality between two expressions. It contains an equals sign (=) between the two expressions that are being compared. Equations can be used to describe relationships between variables, to solve problems, or to represent real-world situations.
In the given question,
To draw the line going through (-6,1) with a slope of -2/3, we can follow these steps:
Plot the point (-6,1) on a coordinate system. This will be one point on the line.
From that point, use the slope to find another point on the line. A slope of -2/3 means that for every 3 units to the right we move, we must move 2 units down. So starting from (-6,1), we can move 3 units to the right and 2 units down to get to the point (-3,-1). This is another point on the line.
Draw a straight line through the two points to represent the line going through (-6,1) with a slope of -2/3.
To write the equation of the line, we can use the point-slope form of the equation:
y - y1 = m(x - x1)
where (x1,y1) is one point on the line, m is the slope, and (x,y) are the coordinates of any other point on the line. Let's choose (-6,1) as our point:
y - 1 = (-2/3)(x - (-6))
Simplifying, we get:
y - 1 = (-2/3)x - 4
Adding 1 to both sides, we get the final equation of the line:
y = (-2/3)x - 3
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What’s is the perimeter and area of 14.3 yds and 12 yds
Answer:
Step-by-step explanation:
The perimeter of a rectangle is given by the formula:
perimeter = 2(length + width)
In this case, the length is 14.3 yards and the width is 12 yards. Substituting these values into the formula, we get:
perimeter = 2(14.3 + 12)
perimeter = 2(26.3)
perimeter = 52.6 yards
Therefore, the perimeter of the rectangle is 52.6 yards.
The area of a rectangle is given by the formula:
area = length x width
Substituting the length and width values, we get:
area = 14.3 x 12
area = 171.6 square yards
Therefore, the area of the rectangle is 171.6 square yards.
A spinner contains six equal sections: 3 red, 1 yellow, 1 blue, and 1 purple. What color do we expect the spinner to land on most frenquently?
A. Red
B. Yellow
C. Blue
D. Purple
Answer:
A) Red
Step-by-step explanation:
We can find the probability of the spinner landing on each color by dividing the number of sections for that color by the total number of sections.
Probability of landing on red: 3/6 = 1/2
Probability of landing on yellow: 1/6
Probability of landing on blue: 1/6
Probability of landing on purple: 1/6
The spinner is most likely to land on red because it has the highest probability, 1/2 or 50%. Therefore, the answer is A. Red.
A, red
It would frequently land on 3 because 3 takes up the bigger portion of the spinner
I need help with number 24.
A principal was interested in the number of hours slept per night by 14-year-olds in a particular town. She gathered data from a random sample of ten 14-year-olds from a local youth group in the town and wanted to create an appropriate graphical representation for the data.
Which graphical representation would be best for her data?
Stem-and-leaf plot
Histogram
Circle graph
Bar graph
The best graphical representation for the given data would be a histogram. A histogram is a graph that displays the distribution of continuous numerical data. It consists of a series of bars, where each bar represents a range of values and the height of the bar represents the frequency or proportion of data points that fall within that range. Since the variable of interest in this case is the number of hours slept per night, which is a continuous numerical variable, a histogram would be an appropriate way to display the distribution of the data. A stem-and-leaf plot is another way to display numerical data, but it is better suited for smaller data sets. Circle graphs and bar graphs are used to display categorical data.
HELP ME QUICK
what is 6:12 in the simplest form
Answer:
1 : 2
Step-by-step explanation:
First, find the GCF of 6 and 12, which is 6.
So divide both sides by 6:
6 / 6, 12 / 6
= 1 : 2
Write the equation of a line perpendicular to y = 1/3 x + 5 and passing through the point (2,6).
1. y = 3x +18
2. y = 3x
3. y = -3x +12
4. y = -3x
The equation of the line perpendicular to y = 1/3 x + 5 and passing through (2,6) is y = -3x + 12, which is option 3.
EquationsTo find the equation of a line perpendicular to y = 1/3 x + 5 and passing through the point (2,6), we first need to determine the slope of the original line and the slope of y = 1/3 x + 5 is 1/3.
To find the slope of a line perpendicular to this, we can use the fact that the product of the slopes of two perpendicular lines is -1 and therefore, the slope of the line we want is the negative reciprocal of 1/3, which is -3.
Now that we have the slope, we can use the point-slope form of the equation of a line to write the equation of the line passing through (2,6) with slope -3:
y - 6 = -3(x - 2)
y = -3x + 12
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5) ||2x - 3| - X+1| = 4x − 1
Answer:
X= -2x - 1,2x + 3
Step-by-step explanation:
1)Break down the problem into these 2 equations.
(2x-3) - X+ 1 = 4x − 1
-(-(2x-3) - X + 1) = 4x − 1
2)Solve the 1st equation: (2x-3) - X + 1 = 4x − 1.
X=-2x 1
3)Solve the 2nd equation: -(-(2x-3) − X + 1) = 4x – 1.
X = 2x + 3
4)Collect all solutions.
X= -2x - 1,2x + 3
Prove that X^n + Y^n is divisible by X + Y for any odd natural number n, where n ≥ 1.
Step-by-step explanation:
We can prove that X^n + Y^n is divisible by X + Y for any odd natural number n, where n ≥ 1, using mathematical induction.
Base case: n = 1 When n = 1, we have X^1 + Y^1 = X + Y, which is clearly divisible by X + Y.
Inductive step: Assume that X^k + Y^k is divisible by X + Y for some odd natural number k ≥ 1. We will show that X^(k+2) + Y^(k+2) is divisible by X + Y.
We can expand X^(k+2) + Y^(k+2) using the binomial theorem: X^(k+2) + Y^(k+2) = (X^2)(X^k) + (Y^2)(Y^k)
We can factor out X + Y from the first term and Y + X from the second term, since X + Y = Y + X: X^(k+2) + Y^(k+2) = (X + Y)(X^k + XY^k) + (X + Y)(Y^k + XY^k)
We can then simplify this expression by factoring out XY^k from the first term and X^kY from the second term: X^(k+2) + Y^(k+2) = (X + Y)XY^k + (X + Y)X^kY X^(k+2) + Y^(k+2) = (X + Y)XY^k + (X + Y)X^kY
We can then factor out (X + Y) from this expression to get: X^(k+2) + Y^(k+2) = (X + Y)(XY^k + X^kY)
Since X + Y is a common factor of X^(k+2) + Y^(k+2), we have shown that X^(k+2) + Y^(k+2) is divisible by X + Y.
Therefore, by mathematical induction, we have proven that X^n + Y^n is divisible by X + Y for any odd natural number n, where n ≥ 1.
the relative frequency of a class is computed by: group of answer choices dividing the frequency of the class by the number of classes dividing the frequency of the class by the class width dividing the frequency of the class by the total number of observations in the data set subtracting the lower limit of the class from the upper limit and multiplying the difference by the number of classes adding the lower limit of the class to the upper limit and multiplying the sum by the number of classes
The relative frequency of a class is computed by dividing the frequency of the class by the total number of observations in the data set. This statement (C) is true.
What is the relative frequency?
The relative frequency is a statistical term that represents the percentage of times a certain event occurs in an experiment or study. It is calculated by dividing the number of times an event occurred by the total number of trials in the experiment or study.
The class width is the difference between the upper limit and lower limit of a class. Class width is used to construct frequency histograms, cumulative frequency curves, and frequency polygons.The frequency of a class is the number of times the data falls within a certain class.
So, to compute the relative frequency of a class, divide the frequency of the class by the total number of observations in the data set. Therefore, option c, "dividing the frequency of the class by the total number of observations in the data set," is correct.
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how do you write 2350 million in standard form
Answer:
WE CAN WRITE IN STANDARD FORM IN THE FOLLOWING WAY..
LMX0