The Binomial expansion of given expression, (1+x)⁻¹ is 1 − x + x² - x³+ x⁴ + ...
What is binomial theorem?Binomial theorem primarily helps to find the expanded value of the algebraic expression of the form (x + y)ⁿ
Using the Binomial theorem we can expand the series of ( x + y )ⁿ as
( x + y)ⁿ = ∑ ⁿCₖ xⁿ⁻ᵏ yᵏ
k= 0 to n
= xⁿ + (n(n-1)/2!)x⁽ⁿ⁻¹⁾y + (n(n-1)(n-2)/3!)x⁽ⁿ⁻²⁾y² + .......
where n ≥ 0 and (ⁿCₖ ) is known as a binomial coefficient and they should be positive integers.
We have given that (1+x)⁻¹ comparing it with ( x + y )ⁿ we get, y = 1 ; n = -1
Binomial expansion for this is followed the same way as expanding larger power expansion,
So (1+x)⁻¹ = 1 + (-1)x + (-1)(-1–1)/2! x² + (-1)(-1–1)(-1–2)/3! x³+…….
= 1 -x + x²- x³ +….
but this will result in an infinite series and will converge for values belonging to
|x|<1. So, the required binomial series is
1 -x + x² - x³ +….
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The area model at the right shows a right triangle with side lengths a, b, and c such that a^2 + b^2 = c^2.
If the side lengths a increases by a small amount, but the lengths a increases by a small amount, but the lengths b and c do not change, which of the following statements will be true?
Select all that apply.
The statements that are true in the right triangle are
a² + b² > c² andm∠ACB = 90°What is a right angled triangle?A right angled triangle is a triangle in which one of its angles is 90°.
How to show which statements are true for the right triangle?Since the area model at the right shows a right triangle with side lengths a, b, and c such that a² + b² = c².
If the side lengths a increases by a small amount, but the lengths a increases by a small amount, but the lengths b and c do not change, we need to show which of the following statements will be true.
Now, if the side a increases by a small amount Δa, we have that
a + Δa > a
Squarring both sides, we have that
(a + Δa)² > a²
Adding b² to both sides, we have that
(a + Δa)² + b² > a² + b²
(a + Δa)² + b² > c² since a² + b² = c²
So, a² + b² > c² where a is the new value of a = a + Δa
Also, since a is perpendicular to b, we have that angle m∠ACB = 90°. Increasing a by a small amount Δa does not change a from being perpendicular to b, so we still have that m∠ACB = 90°.
So, the statements that are true in the right triangle are
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a loaded coin has a 0.48 (48%) probability of coming up heads and 0.52 (52%) probability of coming up tails. if it is flipped seven times, what is the probability of getting exactly 4 heads?
The probability of getting exactly 4 heads is 0.26
A loaded coin has 0.48 probability of coming up heads
0.52 probability of coming up tails.
coin is flipped 7 times
let's assume n = 7.
The seven flips of the coin are separate from one another, so what happens on one flip is unaffected by the outcomes of the other tosses.
When calculating the likelihood that two events will occur simultaneously, we must multiply the probabilities of events 1 and 2, presuming that event 1 occurred.
Tosses never affect one another, so the "assuming event 1 happened" part is no longer necessary because it has no effect on the probability. As a result, we just need to multiply the chances for each toss. This supposition is essential. We would have needed to know exactly how tosses are dependent on one another if they were interdependent.
Typically, we would have [tex](n! /(r!)*(n - r)!)*p^r*(1-p)^(n-r)[/tex] is the probability of getting r heads out of n, without taking into account the order.
probability of getting exactly 4 heads r = 4.
out of 7 flipping n =7.
7!/(4!*3!) = 35
(0.48)^4 = 0.05308416
(0.52)^3 = 0.140608
35*0.05308416*0.140608 = 0.26124201492 = 0.26
so the probability of getting exactly 4 heads is 0.26.
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y=-1/2X
thank you for your time
Is 9 hundredths greater than 6/10
Step-by-step explanation:
9 hundredths = 9/100
is 9/100 > 6/10 ?
let's bring both to 1/100
9/100 > 6×10/100
9/100 > 60/100
no, this is wrong. 9/100 is not greater than 60/100 (or 6/10).
9/100 would be greater than 6/100. but not greater than 6/10.
DOUBLE POINTS!! HELP PLEASE
The surface area of the pyramid is [tex]3060mm^{2}[/tex].
What is the surface area of a pyramid?If “P” is the base perimeter, “l” is the slant height, and “B” is the base area, then the total surface area of the regular pyramid formula is given by [tex]\frac{1}{2} PI+B[/tex] Square units.
Given that the base is a square, therefore the area of the base is 30×30mm= 900mm².
So, B=900.
The perimeter of the base is equal to 4×30mm=120mm.
So, P=120.
The slant height is equal to 36mm.
So, I=36.
Now we will substitute the values in the surface area formula of the pyramid.
The surface area of the pyramid is equal to
[tex]\frac{1}{2}*120*36+900=3060mm^{2}[/tex]
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1. Stephen has discovered a blow-out sale at the music store in the mall. The cost of 3 sets of guitar
strings, 2 guitars, and 2 amps is $472. The cost of 2 sets of guitar strings and 1 guitar is $212. The cost
of 4 sets of strings and 1 guitar is $322. If x represents sets of guitar strings, y represents guitars, and z
represents amps, write a system of equations that can be used to determine the sale price of the sets
of strings, the guitars, and the amps? What is the cost for each amp?
Each amp costs_____?
The system of equations that can be used to determine the sale price of the sets of strings, the guitars, and the amps.
3x + 2y + 2z = 472
2x + y = 212
4x + y = 322
The cost of each amp is $51.
What is an equation?An equation is a mathematical statement made up of two expressions connected by an equal sign.
We have,
3x + 2y + 2z = 472 ____(1)
2x + y = 212 ____(2)
4x + y = 322 ____(3)
From (2) and (3) we get,
Multiply 2 on (2).
4x + 2y = 424
4x + y = 322
(-) (-) (-)
y = 102
Now,
2x + y = 212
2x = 212 - y
2x = 212 - 102
x = 110/2
x = 55
Now,
x = 55
y = 102
From (1) we get,
3x + 2y + 2z = 472
3 x 55 + 2 x 102 + 2z = 471
165 + 204 + 2z = 471
2z = 102
z = 51
Thus,
The system of equations that can be used to determine the sale price:
3x + 2y + 2z = 472
2x + y = 212
4x + y = 322
The cost of each amp is $51.
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a tank contains 1,000 l of brine with 16 kg of dissolved salt. pure water enters the tank at a rate of 10 l/min. the solution is kept thoroughly mixed and drains from the tank at the same rate. (a) how much salt is in the tank after t minutes?
The salt that is left after t minutes is y = 16e^[-t/100]
We have to mix problems we use:
dy / dt = (rate in) - (rate out)
The water entering the tank at a speed of 10 L / min but it has no salt, therefore
rate in = 0
The tank has 1000 liters of brine with 16 kg of salt initially, therefore the concentration of salt at time "t" is:
y (t) / 1000
then rate out would be:
rate out = y (t) / 1000 * 10 = y (t) / 100
The difference equation would then be:
dy / dt = 0 - y (t) / 100
1 / y (t) * dy = 1/100 * dt
We integrate from both sides and we have:
ln y = - (1/100) * t + C1
y = e ^ [- (1/100) * t + C1]
y = e ^ [- (1/100) * t] * e ^ [C1]
We assume that C = e ^ [C1]
Thus:
y = C * e ^ [- (1/100) * t]
now y = 16 to t = 0, replacing:
16 = C * e ^ [- (1/100) * 0]
16 = C, therefore we would have:
y = 16*e ^ [- t / 100]
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Valeria has an action figure collection of 500 action figures. She keeps 235 of
the action figures on her wall. What percentage of Valeria's action figure
collection does she keep on her wall?
The percentage of Valeria's action figure collection is 47% which she keeps on her wall.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
To find the percentage of Valeria's action figure collection that she keeps on her wall, we need to divide the number of action figures she keeps on her wall by the total number of action figures in her collection and then multiply by 100%.
Since Valeria has a collection of 500 action figures and she keeps 235 of them on her wall, the percentage of her collection that she keeps on her wall is :
⇒ 235 / 500 × 100% = 47%.
Thus, the percentage of Valeria's action figure collection is 47%
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the length, x centimeters, of eels in a river may be assumed to be normally distributed with mean 53 and standard deviation 7. an angler catches an eel from the river. determine the probability that the length of the eel is less than 62 centimeters. round your answer to four decimal places.
The probability that the length of the eel is less than 62 centimeters is 0.0985.
In the given question, the length, x centimeters, of eels in a river may be assumed to be normally distributed with mean 53 and standard deviation 7.
An angler catches an eel from the river.
We have to determine the probability that the length of the eel is less than 62 centimeters.
Eels in a river may be assumed to be normally distributed with mean = μ = 53
Eels in a river may be assumed to be normally distributed with standard deviation = σ = 7
Now the probability that the length of the eel is less than 62 centimeters is
P(x > 62) = 1 - P(x < 62)
P(x > 62) = 1 - P((x - μ)/σ < (62 - 53)/7)
P(x > 62) = 1 - P(z < 1.29)
P(x > 62) = 1 - 0.9015
P(x > 62) = 0.0985
Hence, the probability that the length of the eel is less than 62 centimeters is 0.0985.
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Two planes are flying at altitude of 15,000 feet. The angle of depression from one plane to the airport is 70°, the angle of depression from one plane to the airport is 60°. Which plane is closer to the airport and by how much?
The plane with an angle of depression of 70° is closer to the airport.
The distance between the plane and the airport is 15957 feet.
What are trigonometric identities?There are three commonly used trigonometric identities.
Sin x = 1/ cosec x
Cos x = 1/ sec x
Tan x = 1/ cot x or sin x / cos x
Cot x = cos x / sin x
We have,
Two planes are flying at an altitude of 15,000 feet.
The angle of depression from one plane to the airport is 70°.
This means,
The distance of the plane from the airport.
Sin 70 = 15,000 / Distance
Distance = 15,000/ sin 70 = 15,000 / 0.94 = 15957 feet _____(1)
The angle of depression from one plane to the airport is 60°.
The distance of the plane from the airport.
Sin 60 = 15,000 / Distance
Distance = 15,000/ sin 60 = 15,000 / 0.87 = 17241 feet _____(2)
Thus,
The plane with an angle of depression of 70° is closer to the airport.
The distance between the plane and the airport is 15957 feet.
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how do i simplify the expression?
Answer: ≈1.71
Step-by-step explanation:
[tex]\displaystyle\\(5^\frac{2}{3}) ^\frac{1}{2} =\\\\5^{\frac{2}{3}(\frac{1}{2})} =\\\\5^\frac{2(1)}{3(2)} =\\\\5^\frac{1}{3} =\\\\\sqrt[3]{5} }=\\\\\approx1.71[/tex]
in the distribution system, the water main sizes most commonly in use in newer systems are what sizes?
In the distribution system, the water main sizes most commonly in use up to 3.65m to tiny 12.7mm.
Water pipes come in a variety of sizes, from enormous mains with a diameter of up to 3.65 m to tiny 12.7 mm pipes used to supply individual outlets within a building.
Any pipe or tube intended to deliver drinking water to consumers is referred to as a water pipe. Depending on the situation, water may be treated either before distribution or at the point of use (POU). Water is typically treated before distribution in well-thought-out water distribution networks, and occasionally it is also chlorinated, to prevent recontamination on the way to the end user.
Polyvinyl chloride (PVC), cast iron, copper, steel, and, in older systems, concrete or fired clay are materials frequently used to build water pipes. Long runs of water pipe can be joined together using flange, nipple, compression, or soldered joints (SCOTT 2011).
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the system below shows a periodic function and an absolute value function. how many solutions does the system have?
The system that contains periodic function and an absolute value function has two solution
In the given system there are two function, those are periodic function and absolute value function
The function is defined as the statement that shows the relationship between one dependent variable and other independent variables. The function consist of different types of variables numbers and the mathematical operators
The periodic function is a function that repeats its value at a regular interval of time
The absolute value function is a function that contains the mathematical expression with absolute value symbols
The system contains periodic and absolute value function and it will contains two solution
Therefore, the system has two solution
I have solved the question in general, as the given question is incomplete
The complete question is:
The system below shows a periodic function and an absolute value function. how many solutions does the system have?
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A Birr 1,000 deposits is made at a bank that pays 12% interest compounded monthly. How much will be in the account at the end of 10 years?
Answer:
$15400
Step-by-step explanation:
1 year = 12 month
10 years = 120 months
12% = month
1440% = 120 months
1440% = 14.4
Take 1000 times 14.4 = $14400
So, we add the two numbers
1000 + 14400 = $15400
So, $15400 will be in the account at the end of 10 years.
X over 100 = 100
What is X?
Answer:
x is 1000
Step-by-step explanation:
hope this helps
Please help it’s due today
The figures below show the length of a new pencil and its length after seema used it for a day. New pencil - 18 cm After seema used it - 15 cm
By what fraction of the original length of the pencil reduce that day?
A. 1/4
B. 1/5
C. 1/6
D. 1/18
The pencil was reduced by 1/6 of its original length. Hence, the correct option is Option C.
To calculate the fraction, you must divide the remaining length (15 cm) by the original length (18 cm). In order to get a fraction, you have to divide it by their highest common factor, which in this case is 3.
= \frac{15}{18}
= \frac{5}{6}
This is the length of the pencil left at the end of the day. To find the fraction used, you have to subtract this fraction from its whole.
= 1 - \frac{5}{6}
= \frac{5}{6} - \frac{5}{6}
= \frac{6-5}{6}
= \frac{1}{6}
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Which linear function represents the line given by
the point-slope equation y - 8 = 1/2(x - 4)?
O f(x) = =x + 4
O f(o) = %x + 6
O 1(x) = X - 10
O 10x) = x - 12
The line given by the point slope equation y - 8 = 1/2 (x - 4) represents the linear function f(x) = [tex]\frac{1}{2}[/tex] x + 6.
What is Linear Function?Linear functions are those functions whose graph drawn is a straight line. It consists of one dependent variable and one independent variable.
Linear functions are usually represented as f(x) = y = p + qx.
The given equation of the line in point slope form is,
y - 8 = 1/2 (x - 4)
⇒ y - 8 = (1/2 × x) - (1/2 × 4)
⇒ y - 8 = 1/2x - 2
⇒ y = 1/2x -2 + 8
⇒ y = 1/2x + 6
When x = 0, y = f(0) = (1/2 × 0) + 6 ⇒ f(0) = 6
Hence linear function f(x) = [tex]\frac{1}{2}[/tex] x + 6 represents the line given by the point slope equation y - 8 = 1/2 (x - 4).
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A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°
Answer:
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°
Step-by-step explanation:
m∠5 + m∠6 are linear pairs, so their sum has to be 180°.
m∠2 + m∠3 = m∠6 because of the exterior angle theorem, which states that the measure of an exterior angle is equal to the sum of the two opposite interior angles. (using this logic, m∠1 equals the sum of m∠5 and m∠3, and m∠4 equals the sum of m∠2 and m∠2).
m∠2 + m∠3 + m∠5 = 180° because the triangle sum property says that the sum of a triangle's interior angles will equal 180°.
g an epidemic of measles has started at what rate is the modeled by the funtion p epidemic spreading after 3 days
The modeled epidemic rate cannot be determined from the information provided.
The rate of an epidemic spreading is a complex process that is influenced by many factors such as the population density, the number of people who are vaccinated, and the infectiousness of the virus. Therefore, it is not possible to accurately model the rate of an epidemic spreading after 3 days based on the information provided. To accurately model the rate of an epidemic spreading, additional information such as the population density, the number of people who are vaccinated, and the infectiousness of the virus would be needed.
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Which expression is equivalent to (ab^7)^4? ab^28 ab^11 a^5b^11 a^4b^28
The expression that is equivalent to (ab⁷)⁴ is (a⁴b²⁸)
What is the integer exponent property?
Exponents that are integers are known as integer exponents. Exponents with positive integer values tell us how many times to multiply the base by itself. We are instructed by negative integer exponents to first flip the numerator and denominator before multiplying the result by the specified number of times.
Here, we have
Given expression: (ab⁷)⁴
We have to find the expression that is equivalent to this expression.
Using integer exponent property:
= (ab⁷)⁴
= (a⁴b²⁸)
Hence, the expression that is equivalent to (ab⁷)⁴ is (a⁴b²⁸).
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Graph the function g(x)=2f(x-1)
x f(x)
0 0
-2 -4
4 -2
You need to draw y = 2 f (x-1)
So you need to choose x so that x-1 = 0, -2, 4 then you can use the above values.
New function
x-1 = 0, x = 1, y = 2 f (x-1) = 2 f(0) = 2 x 0 = 0
x-1 = -2 x = -1 y = 2 f (x-1) = 2 f(-2) = 2 x -4 = -8
x-1 = 4 x = 5 y = 2 f (x-1) = 2 f (4) = 2 x -2 = -4
So 3 points on the new graph are (1, 0) (-1, -8) and (5, -4)
Basically going from y = f(x) to y = f(x-a) where a is a constant positive number you translate (move) y = f(x) a units to the right.
y = f(x) to y = 2 f(x), you dilate f(x) with center (0,0) factor 2, so (a,b) > (2a, 2b)
So basically here you are doing both things.
So if (a,b) is on y = f(x) then after the translation (a,b) > (a+1, b) then after the dilation (a+1,b) > (2a+2, 2b).
what is graph?
A diagram or pictorial depiction of facts or values that is arranged might be referred to as a graph. The relationships between two or more items are frequently represented by the points on a graph.
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elect the correct answer. Which system of equations is satisfied by the solution shown in the graph? A coordinate plane linear graph on inequalities in which a line intersects Y-axis at 6 and another line intersects y-axis at 10. Both lines intersect X-axis at minus 2 and Y-axis at 8. A. x + 2y = 6 and x − y = 10 B. x + y = 6 and x − 2y = 10 C. x + 2y = 10 and x − y = 6 D. x + y = 6 and x − y = -10
We can factor this polynomial with 4 terms without grouping.
What is factor?Factor is a thing that influence decision or the situation.
How to factor polynomials with 4 term without groupingLet P(x) be a polynomial with four terms.
To factor P(x) without grouping, substitute
x = -1, 1, -2, 2, -3, 3.......
P(-1) = 0 ----> (x + 1) is a factor of P(x)
P(1) = 0 ----> (x - 1) is a factor of P(x)
P(-2) = 0 ----> (x + 2) is a factor of P(x)
P(2) = 0 ----> (x - 2) is a factor of P(x)
If P(-1) ≠ 0, then (x + 1) is not a factor of P(x).
Then, try x = 1, x = -2, x = 2 and so on.
Once one of the linear factors of P(x) is found, the other factors can bound easily (the rest of the process has been explained in the following examples).
Factor the following polynomials without grouping :
d as (x2 + ax + b).
Then,
(x + 1)((x2 + ax + b) = x3 - 2x2 - x + 2
Comparing the coefficients of x and constants,
b + a = -1 ----(1)
b = 2
Substitute b = 2 in (1).
2 + a = -1
a = -3
x2 + ax + b = x2 - 3x + 2
Factors of (x2 - 3x + 2) are (x - 1) and (x - 2).
Therefore,
x3 - 2x2 - x + 2 = (x + 1)(x - 1)(x - 2)
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Determine a series of transformations that would map polygon ABCD onto polygon A'B'C'D'?
Therefore , Due to a sequence of transformations, polygon ABCD will therefore overlap polygon A'B'C'D'.
What is polygon?A polygon is a closed, two-dimensional, flat or planar structure that is circumscribed by straight sides. There are no curves on its sides. The edges of a polygon are another name for its sides. A polygon's vertices (or corners) are the places where two sides converge.
Here,
a 4 unit translation to the left after reflecting polygon ABCD across the x-axis.
The coordinates for point A in the polygon ABCD are (1, -5).
ABCD's polygon should be mapped onto A'B'C'D',
Reflect the ABCD polygon across the x-axis in accordance with the rule,
A(x, y) → A'(x, -y) (x, -y)
Following this criterion, A's coordinates will be,
A(1, -5) → A'(1, 5) (1, 5)
ABCD is also moved four units to the left.
The rule for the translation is,
A'(x, y)=> A"(x - 4, y) (x - 4, y)
A'(1, 5) => A"(1 - 4, 5) (1 - 4, 5)
=> A"(-3, 5) (-3, 5)
Due to a sequence of transformations, polygon ABCD will therefore overlap polygon A'B'C'D'.
a 4 unit translation to the left after reflecting polygon ABCD across the x-axis.
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6. The people at the party ate 16 hot chickin wings and 26 mild chickine wings. What percent of the chickin wings ere mild?
28.1%
O 22.5%
O 61.5%
O 61.9%
Answer: 61.9%
Step-by-step explanation:
First, add 16 and 26 together to get the total of chicken = 42
Then take 26 divided by 42, then times 100% = 61.9%
for a research study with 2 levels of factor a, 3 levels of factor b, and 5 in each treatment condition, what are the df values for the f-ratio evaluating the main effect for factor a?
The degrees of freedom for the f-ratio evaluating the main effect of factor A would be (1, 8).
A factorial simple is what?A factorial is a mathematical operation denoted by the symbol (!) that multiplies a number (n) by each integer that comes before it. To put it another way, the factorial function instructs you to multiply all the whole numbers starting at the selected number and decreasing by one. The factorial of a number (n!) is equal to n in more mathematical terminology (n-1)
How do you calculate factorial?The factorial of a number (n!) is equal to n in more precise mathematical terminology (n-1). For instance, if you wanted to compute the factorial for four, you would write: 4! = 4 x 3 x 2 x 1 = 24.
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Luke has blue and red ballsEvery day he wins 2 blue balls and loses 3 red onesAfter 5 days he has the same amount of blue and red ballsAfter 9 days, he has twice as many red balls as blue ballsHow many red balls did he have at the beginning?
The number of red balls Luke has at the start is 47.
blue ball per day = 2
red ball per day = -3 ('-' ve sign as he is loosing them)
let the number of red balls be represented by r
let the number of blue balls be represented by b
after 5 days
b+ 10 = r -15 -(i)
After 9 days
b + 18 =2(r-27)-(ii)
subtracting (i) from (ii)
b + 18 - b - 10 = 2r - r - 54 +15
8 = r - 39
r = 39 + 8
r = 47
The number of red balls Luke has at the start is 47
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what is the x intercept of the equation 3x-3y=21
Find domain and range please! Also, due tmrw!
The Domain of the Function is {-2, -1, 0,1 ,2, 3, 4, 5} and the Range of the Function is {4, 3, 2, 1, 0, -1}
What are Functions?
A function is an expression, rule, or law in mathematics that describes a connection between one variable (the independent variable) and another variable (the dependent variable).
Solution:
We neeed to find the Domain and Range of the Function
Domain refers to all the values of x = {-2, -1, 0,1 ,2, 3, 4, 5}
Range refers to all the values of y = {4, 3, 2, 1, 0, -1}
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JRB has been looking for a longboard for a while now. She found one for 80% of the original price. She paid $720 for it. What was the original price?
Answer:900
Step-by-step explanation:
720=.80x
720/.80=900
to check it
900*.20=720
