a) The binomial expansion of (1+12x)^1/2 is given by:
(1+12x)^1/2 = 1^1/2 + (1/2)(1^-1/2)(12x) + (1/2)(1^-1/2)(-1/2)*(12x)^2 + ...
Binomial expression: what is it?
A polynomial with only terms is a binomial. An illustration of a binomial is x + 2, where x and 2 are two distinct terms. Additionally, in this case, x has a coefficient of 1, an exponent of 1, and a constant of 2. As a result, a binomial is a two-term algebraic expression that contains a constant, exponents, a variable, and a coefficient.
The first four terms, in ascending powers of x, are:
1, (1/2)(12x), (1/8)(12x)^2, (1/16)*(12x)^3
b) To use x=1/36 in the expansion, you would substitute x=1/36 into the expansion, and keep only the terms up to x^3 (the fourth term in the expansion). This will give an approximation of the value of (1+12x)^1/2 when x=1/36.
This will be an approximation of √12 because x=1/36 corresponds to 12x = 1 and the initial value of the expansion is 1 + 12x .
We can use this approximation to approximate square root of 12 and also for other square roots by using appropriate x values.
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Jeff wants to purchase a television for his first apartmentThe television costs 750. He can get a zerointerest loan for the television if he pays $100 per month. At the same time, he puts away $200 per month in savings. To figure out when his savings will equal the remaining amount he owes on the television , he starts writing a system of equations :
The correct equation for Jeff after every cost will be y = 750 -100x
Cost of the television = $750
Loan amount paid per month = $100
Amount put in savings = $200
A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.
Since the amount of $200 is being saved, this means that it is not going away or is being spent. Now, as per the question, the initial remaining balance with Jeff is 750. The balance decreases by the loan amount of 100 per month.
This means that the amount is reduced by = 100x.
Therefore, the equation describing the scenario is -
y = 750 -100x
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what are p coordinates?
The coordinates of the point P after the rotation is P' ( 5 , -2 )
What is Rotation?The measure of the amount a figure is rotated about the center of rotation is called the angle of rotation. The angle of rotation is usually measured in degrees. We specify the degree measure and direction of a rotation.
90° clockwise rotation: (x,y) becomes (y,-x)
90° counterclockwise rotation: (x,y) becomes (-y,x)
180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)
270° clockwise rotation: (x,y) becomes (-y,x)
270° counterclockwise rotation: (x,y) becomes (y,-x)
Given data ,
Let the point be represented as P
The coordinates of P = P ( 5 , 2 )
Now , when the point P is rotated by an angle of 270° about the origin ,
On 270° clockwise rotation: (x,y) becomes (-y,x)
And , on 270° counterclockwise rotation: (x,y) becomes (y,-x)
So , assume the rotation is counterclockwise ,
The coordinates of P after rotation is P' ( 5 , -2 )
And , assume the rotation is clockwise ,
The coordinates of P after rotation is P' ( -5 , 2 )
Hence , the coordinates of P after rotation is P' ( 5 , -2 )
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In the last three years, Frederico's basketball team won 50 more games than they lost. If they won 140 games, what was the ratio of wins to losses? Write the ratio in all three forms.
Answer:
Step-by-step explanation:
won 140
lost 140-50=90
Ratio wins to losses: 14:9
Which inequality is true when the value of x is -3? A: -4 - 6 < -3.5 B: -r - 6 > 3.5 C: -x - 6 > -3.5 D: % - 6 > -3.5
The value x = -3 is a solution for the inequality at option C.
-x - 6 > -3.5
Which inequality is true when x is -3?To check this, we just need to evaluate the given inequalities in x = -3, then we can see if the inequality is true or false.
For example, the first one:
-4 - 6 < -3.5
Does not depend on x, so we can ignore that.
The next one is:
-r - 6 > 3.5
Evaluating it on -3 we get:
3 - 6 > 3.5
-3 > 3.5
this is false.
The next one is:
-x - 6 > -3.5
Evaluating we wll get:
3 -6 > -3.5
-3 > -3.5
This is true, so this one is the correct option.
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Trevor loves riding Ferris wheels and roller coasters. While visiting the Putnam County Fair, he first went on the Ferris wheel 1 time and the roller coaster 5 times, using a total of 27 tickets. Then, after taking a break and having a snack, Trevor went on the Ferris wheel 1 time and the roller coaster 1 time, using a total of 7 tickets. How many tickets does it take to ride each attraction?
To ride the Ferris wheel, you need 2 tickets.
And to ride the roller coaster, you need 5 tickets.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
Trevor loves riding Ferris wheels and roller coasters.
Let x represents the ticket of Ferris wheel.
And y represents the ticket of roller coaster.
While visiting the Putnam County Fair, he first went on the Ferris wheel 1 time and the roller coaster 5 times, using a total of 27 tickets.
x + 5y = 27 {equation 1}
Then, after taking a break and having a snack, Trevor went on the Ferris wheel 1 time and the roller coaster 1 time, using a total of 7 tickets.
x + y = 7 {equation 2}
Subtracting equation 2 to equation 1,
we get,
4y = 20
y = 5
x = 2.
Therefore, it takes 2 and 5 tickets to ride each attraction.
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The larger of two numbers is 9 more than twice the smaller. The sum of the numbers is 3 less than five times the smaller. Find the numbers.
By solving a system of equations we will see that the larger number is 21 and the smaller one is 6.
How to find the two numbers?Let's define the variables:
x = smaller number
y = larger number.
We know that the larger of two numbers is 9 more than twice the smaller, so we can write:
y -2x = 9
The sum of the numbers is 3 less than five times the smaller.
x + y = 5x - 3
Then we have a system of equations:
y -2x = 9
x + y = 5x - 3
Now we can isolate y on the first equation to get:
y = 9 + 2x
We can substitute that on the other equation to get:
x + 9 + 2x = 5x - 3
3x + 9 = 5x - 3
9 + 3 = 5x - 3x
12 = 2x
12/2 = x
6 = x
And the value of y is:
y = 9 + 2x
y = 9 + 2*6 = 21
These are the two numbers.
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two runners on an east-west highway are 421.2 feet apart heading east. a hot air balloon is flying ahead of them. the angle of elevation to the balloon of the runner closet to the balloon is 24.5o. if the hot air balloon is 1500 feet directly above a point on the highway east of both runners, what is the angle of elevation to the balloon of the runner the greatest distance from the balloon?
The angle of elevation to the balloon of the runner the greatest distance from the balloon is 83.13 degrees.
The concept used in this problem is Trigonometry. Specifically, the problem uses the tangent function, which is one of the three main trigonometric ratios (sine, cosine, and tangent) used to relate the angles and sides of a right triangle.
To solve this problem, we can use the tangent function of trigonometry, which relates the opposite side (the distance from the runner to the balloon) to the adjacent side (the distance from the runner to the point on the highway) and the hypotenuse (the distance from the point on the highway to the balloon).
Let's call the angle of elevation of the runner closest to the balloon as angle A and the angle of elevation of the runner farthest from the balloon as angle B.
tan(A) = opposite / adjacent = (1500 - 421.2) / 421.2 = 878.8 / 421.2
tan(B) = opposite / adjacent = (1500 + 421.2) / 421.2 = 1921.2 / 421.2
We can use the inverse tangent function (arctan) to find the angle of elevation of the runner the greatest distance from the balloon.
B = arctan(1921.2 / 421.2)
The result is in radians, to convert the angle from radians to degrees, we can use the conversion factor of 180/pi.
B = arctan(1921.2 / 421.2) = arctan(4.5536) = 83.13 degrees (approx)
so, the angle of elevation to the balloon of the runner the greatest distance from the balloon is 83.13 degrees.
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621900 house cost 5.4 mortgage interest rate how  much do you have to pay monthly over 5 years
Over a period of 5 years, the amount that you would have to pay monthly is given as $3,201.37
How to solve for the mortgageTo calculate the monthly mortgage payment for a house that costs $621,900 with an interest rate of 5.4% over 5 years, you can use the following formula:
P = (r(1 + r)^n) / (((1 + r)^n) - 1) * L
Where:
P = monthly payment
r = monthly interest rate (5.4%/12 = 0.0045)
n = total number of payments (5 years x 12 = 60)
L = the loan amount ($621,900)
Plugging in the values, we get:
P = (0.0045(1 + 0.0045)^60) / (((1 + 0.0045)^60) - 1) * 621,900
P = $3,201.37
So you have to pay $3,201.37 monthly over 5 years.
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50 POINTS (and its easy im just kinda dmb)
find the number of possible outcomes
1. A softball player bats six times in a game. Each at-bat results in an out, getting on base or hitting a homerun.
2. A sandwich shop has three types of sandwiches: ham, turkey, and chicken. Each s andwich can be ordered with white bread or multi-grain bread. Customers can add any combination of the six available toppings
Answer: 1.18 possibilities 2. 36 possibilities
Step-by-step explanation:
1. 1o, 1g, 1h, 2o, 2g, 2h, 3o, 3g, 3h, 4o, 4g, 4h, 5o, 5g, 5h, 6o, 6g, 6h
2. hw1 hw2 hw3 hw4 hw5 hw6, hg1 hg2 hg3 hg4 hg5 hg6, tw1 tw2 tw3 tw4 tw5 tw6, tg1 tg2 tg3 tg4 tg5 tg6, cw1 cw2 cw3 cw4 cw5 cw6, cg1 cg2 cg3 cg4 cg5 cg6
if vector c is added to vector b, the result is -9i - 8j. if b is subtracted from c, the result is 5i 4j. what direction is b to the nearest degree
221 degree is the direction of B to the nearest degree.
We have to find the vector B. We will substructure Equations
-9i - 8j - 5i - 4j = -14i -12j
= 2(-7i-6j)
From here you get the value of vector B. That is minus seven i and minus of six j. Now we can see that this is both. The signs are negative so it should lie in 3rd chord event.
And for third quarter went in their direction or we can say Angelo factor B is want A D. Plus theater. So when I. D. Plus and universal marvelous of -6 divided by -7 that provides us one or d. plus 41 degree. That is 221 degree.
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Can anyone explain this please ???
The area of the polygon U is 272 cm².
How to calculate the area of U?The area is the volume contained within the boundary of a 2D shape. It is written as a square and is measured in square metres or centimetres, etc.
The idea of area can be thought of as the area inside a particular shape or space. It refers to the volume of space occupied. As a shape gets bigger, its area and perimeter get bigger as well. Area is distinct from volume and only refers to the area occupied by a flat or 2D object.
As they work with different 2D shapes and calculate each one's area, kids will learn that different shapes require the use of different formulas and methods to find the area.
Step1:
3/12 = s/u =k
s t/s u=f² =(s/u)² =s/16
17/s u=s/16
step2:
s u = 17.16 = 272 cm²
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Fraction A pet store that sells birds has 12 male birds for every day 9 female birds write the ratio of female birds
The ratio of female birds is 3/7.
Explain ratio.One way to express something as a percentage of another is to use a ratio, which is a mathematical concept.
How are ratios determined?By typically dividing two figures, ratios contrast them. If you were comparing one data point (A) to another data point, your formula would be A/B. (B). You are multiplying information A by information B, as indicated by this. For instance, if A is 5 and B is 10, your ratio will be 5/10.
The pet store has 12 male birds for every 9 female birds.
total birds = 12 + 9 = 21 birds.
So, the ratio of female birds = female birds / total birds
= 9/21
= 3/7
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√196
What is the square root?
Answer: √196 = 14
Step-by-step explanation: mere guess
they charge $3.89 per square foot for materials and $122.42 per day of labor. chance spent $2882.48 on the renovation. if the number of square feet is 189 more than the number of days it took for the renovation, how long did the renovation take
Let's call the number of days of labor x, and the number of square feet y.
From the information provided, we can set up the following equation:
What does a math equation mean?
A mathematical statement that shows the equality of two mathematical expressions is the definition of an equation in algebra. As an illustration, the two equations 3x + 5 and 14 are joined by the equal sign to get the equation 3x + 5 = 14.
3.89y + 122.42x = 2882.48
We also know that y = x + 189
We can substitute the second equation into the first equation:
3.89(x + 189) + 122.42x = 2882.48
Expanding and simplifying:
3.89x + 730.91 + 122.42x = 2882.48
x = (2882.48 - 730.91) / (3.89 + 122.42) = 22.98 days
So the renovation took 22.98 days.
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. people are given a blind taste test for bottled water and tap water. they cannot differentiate properly. what are the odds that at least 8 out of 9 people will guess the water type correctly?
The binomial distribution is a discrete probability distribution used in probability theory and statistics to explain the number of successes in a set number of trials, where each trial can only result in success or failure. The probability mass function (PMF) shown below defines the binomial distribution as follows:
P(a) = ⁿСₐ * p^a* (1-p)^(n-a)
Where:
a: the number of successes in n trials
n: the total number of trials
p: the probability of success in one trial
ⁿСₐ : n! / (a! * (n-a)!)
the binomial coefficient, which shows how many different ways there are to select k successes from n trials.
A variety of phenomena, including coin tossing, die rolling, and other tests with a set number of trials and only two possible outcomes, can be usefully modelled using the binomial distribution.
In a blind tasting test, if participants are unable to distinguish between bottled and tap water, we can assume that their guesses are random. The probability of accurately predicting the type of water in this situation is 50%.
The likelihood that 8 out of 9 people will properly predict their water type is
(1/2)^8 * (1/2)^1 = (1/2)^9 = 1/512
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How do you solve for x?
Step-by-step explanation:
please attached for details
The cafeteria sells each apple at one price and each banana at another price. For 5 apples and 3 bananas Dan pays $5.70. For 3 apples and 5 bananas Chris pays $4.70. The price of one apple is how much more then the price of one banana, in cents?
In this question, why do you multiply by 5 to the top equation and 3 to the bottom equation?
Answer: In this question, we are trying to find the difference in price between one apple and one banana. We can set up a system of two equations with two variables to represent the information given in the problem. Let x be the price of one apple in cents and y be the price of one banana in cents.
The first equation represents the information that Dan paid $5.70 for 5 apples and 3 bananas:
5x + 3y = 570
The second equation represents the information that Chris paid $4.70 for 3 apples and 5 bananas:
3x + 5y = 470
We can use these equations to solve for the value of x (the price of one apple) in terms of y (the price of one banana).
We can use the first equation and solve for x:
5x = 570 - 3y
x = (570 - 3y)/5
We can then substitute this expression of x into the second equation:
3((570 - 3y)/5) + 5y = 470
By multiplying both sides of the equation by 5 and 3 respectively, we can get the equation in a form that we can solve for y and then substitute back for x.
We can solve for y by subtracting 3((570 - 3y)/5) from both sides:
5y = 470 - 3((570 - 3y)/5)
Expanding and simplifying the right side of the equation:
5y = 470 - 3(114 - y)
5y = 470 - 342 + 3y
y = (470 - 342)/8
We can substitute this value of y back into the first equation and solve for x
x = (570 - 3y)/5
x = (570 - 3(470 - 342)/8)/5
x = (570 - 441)/5
Therefore, the price of one apple is (570 - 441)/5 = 129 cents more than the price of one banana.
Step-by-step explanation:
In 1957 an earthquake in Alaska measured about 9.0 on the Richter scale. An earthquake with magnitude 6.0 occurred in Japan in early 2007.
How many times more intense was the earthquake in Alaska than the one in Japan?
The number of times more intense that the earthquake in Alaska was than the one in Japan was 1, 000 times more intense.
How does the Richter Scale work ?The Richter magnitude is determined by measuring the height of the greatest wave that has ever been seen at a given distance from the seismic source.
Each order of magnitude is 10 times more intense than the previous one since the Richter scale is a base-10 logarithmic scale. To put it another way, a two is ten times as intense as a one, and a three is one hundred times stronger.
This means that the number of times more intense that the earthquake in Alaska was than the one in Japan was:
= 10 x 10 x 10
= 1, 000 times
Note : Alaska and Japan earthquake difference:
= 9 - 6
= 3
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Yusuf drove 156 miles in 3 hours. If he continued at the same rate, how far would he travel in 2 hours?
Answer:104 miles in 2 hours.
Step-by-step explanation:
so first since we need to know how much he drove in 2 hours we just divide 156 by 3 to get 52 and then multiply 52 by 2 since we need to know how many miles he traveled in 2 hours then you get your final answer of
104
Which of these systems of linear equations has no solution?
2x+8y =15
4x+10y =30
2x-y=18
4x+ 2y = 38
4x+7y= 17
8x-14y - 36
4x-3y = 16
8x-by = 34
The system of linear equations: 2x-y=18 & 4x-3y = 16 has no solution.
The system of linear equations has no solution if the equations are inconsistent, meaning that there is no set of values for the variables that satisfies all of the equations simultaneously. One way to check for inconsistency is to try to find the solution by solving for one of the variables in one equation and then substituting it into another equation to see if both equations are satisfied. If the equations yield different values for the same variable, then the system has no solution.
In this case, the system of linear equations:
2x-y=18
4x-3y = 16
Has no solution. We can solve for x in the first equation and then substitute it into the second equation. If the equation yields a different value for y, then the system has no solution.
2x-y=18
2x=y+18
x=y/2 + 9
Now, substituting x into the second equation:
4x-3y=16
4(y/2 + 9) - 3y = 16
2y + 36 - 3y = 16
-y = -20
This result implies that y is not a real number, so the system of equations has no solution.
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please help ! Identify the special angle pair created by angles 1 and 2 in each diagram.
The angle relationship between ∠1 and ∠2 are as follows:
Same side interior anglesalternate interior angles corresponding angles.How to find angle relationships in parallel lines?When parallel lines are cross by a transversal line, angle relationships are formed such as corresponding angles, alternate interior angles, alternate exterior angles, linear angles, same interior angles, vertically opposite angles etc.
Therefore. let's find the relationship of the angle 1 and 2 in the parallel lines cut by the transversal.
Hence, the relationship are:
1. Same side interior angles
2. alternate interior angles
3. corresponding angles.
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WILL MAKE BRAINIEST!!
Identify a 30 year mortgage
O. A mortgage that will be paid off in 15 years?
O. A mortgage that has the same interest rate and same monthly payment for 30 years.
O. A mortgage that fluctuates interest rates
O.None of the above
Because a 30-year mortgage has a longer term, your monthly payments will be lower and your interest rate on the loan will be higher.
What is the term for a 30-year mortgage?A 30-year fixed-rate mortgage is commonly used to refer to conventional loans. Conventional loans are not backed by the government; however, a 30-year fixed FHA, USDA, or VA loan that is insured by the government is available. Rocket Mortgage® does not currently provide USDA loans.
A 30-year fixed-rate mortgage is a house loan with a 30-year repayment period and an interest rate that remains constant during the loan’s duration. When you take for a 30-year fixed-rate home loan, the amount you make each month remains the same until the loan is paid off.
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After you have constructed a 60° angle, what other appropriately used construction will get you a 30° angle?
Answer:
Construct an angel bisector
suppose that professor rodgers needs one small group of students from his class to participate in a focus group. there are 12 12 students in the class. how many different combinations of four selected students can professor rodgers create?
Professor Rodgers can select the group of 4 students in 11880 ways.
Here the professor has to select 4 students from a group of 12. hence we see that
to select the first student, the professor can do so in 12 ways.
To select the second student, the professor now has 11 students left. Hence he can do in 11 ways.
To select the third student, he is left with 10 students, hence the thirst student can be selected in 10 ways
The fourth student can be selected in 9 ways.
Hence, Professor can select the group in 12 X 11 X 10 X 9 ways
= 11880 ways.
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What property is 4 x (X •10) = (4•x) • 10
Answer:
Your answer is Identitive property
HELPPPPPPPPPPPPPPPPPPPPPP!!!!!!!!!!!!!!!!!!!
UR AMAZING
Answer:
1/
Step-by-step explanation:
1/3+1/2=5/6
1-5/6=1/6
1/
I really need help on this!
# 1 Use the relationships and patterns in the tabel to identify the missing
value from the ordered pair. Type the missing value in the answer space
provided.
X
y
2
25
(2,25)
(3,__)
(4, 100)
(5, 125)
3
4
100
5
125
The missing value in the space provided is given as follows:
75.
How to model the proportional relationship?A proportional relationship is modeled as follows:
y = kx.
In which k is the constant of proportionality.
The constant for the relationship in this problem is given as follows:
k = 125/5 = 100/4 = 25/1 = 25.
Hence the relationship is given as follows:
y = 25x.
The missing value is the output when x = 3, hence:
y = 25(3)
y = 75.
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Find veda and kalbes climming speeds
How many miles is the diagonal path from Jim house to the mall
The distance from Jim's house to the mall is given as follows:
11.25 miles.
How to obtain the distance between two points?Suppose that we have two points with coordinates given as follows:
[tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex]
Then the equation for the distance between these two points is given as follows:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
The coordinates are given as follows:
Jim's house: (17, 13).Mall: (8,1).Hence the distance is given as follows:
[tex]D = \sqrt{(17 - 8)^2+(13 - 1)^2}[/tex]
D = 15 units.
Each unit represents 0.75 miles, hence the distance in miles is given as follows:
15 x 0.75 = 11.25 miles.
Missing InformationThe problem is given by the image presented at the end of the answer.
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