The indefinite integral of sin^4(5θ) dθ is (1/4)[θ - sin(10θ)/5 + (θ/2) + (1/40)sin(20θ)] + C.
An indefinite integral is the reverse operation of differentiation. Given a function f(x), its indefinite integral is another function F(x) such that the derivative of F(x) with respect to x is equal to f(x), that is:
F'(x) = f(x)
The symbol used to denote the indefinite integral of a function f(x) is ∫ f(x) dx. The integral sign ∫ represents the process of integration, and dx indicates the variable of integration. The resulting function F(x) is also called the antiderivative or primitive of f(x), and it is only unique up to a constant of integration. Therefore, we write:
∫ f(x) dx = F(x) + C
where C is an arbitrary constant of integration. Note that the indefinite integral does not have upper and lower limits of integration, unlike the definite integral.
We can use the identity [tex]sin^2(x)[/tex] = (1/2)(1 - cos(2x)) to simplify the integrand:
[tex]sin^4[/tex](5θ) = ([tex]sin^2[/tex](5θ)[tex])^2[/tex]
= [(1/2)(1 - cos(10θ))[tex]]^2[/tex] (using [tex]sin^2[/tex](x) = (1/2)(1 - cos(2x)))
= (1/4)(1 - 2cos(10θ) +[tex]cos^2[/tex](10θ))
Expanding the square and integrating each term separately, we get:
∫ [tex]sin^4[/tex](5θ) dθ = (1/4)∫ (1 - 2cos(10θ) + [tex]cos^2[/tex](10θ)) dθ
= (1/4)[θ - sin(10θ)/5 + (1/2)∫ (1 + cos(20θ)) dθ] + C
= (1/4)[θ - sin(10θ)/5 + (θ/2) + (1/40)sin(20θ)] + C
Therefore, the indefinite integral of sin^4(5θ) dθ is (1/4)[θ - sin(10θ)/5 + (θ/2) + (1/40)sin(20θ)] + C.
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I need help with this please, i’ve been stuck on this over a hour now.
The height of the tree to the nearest hundredth is 29.65 ft.
How to find the side of a right triangle?The measure of the distance from the tree and the angle of elevation from the ground to the top of the tree is represented as follows:
Therefore, the height of the tree to the nearest hundredth can ne found as follows:
Therefore, using trigonometric ratios,
tan 56° = opposite / adjacent
tan 56° = h / 20
cross multiply
h = 20 tan 56°
h = 20 × 1.48256096851
h = 29.6512193703
h = 29.65 ft
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(a) You are given the point (1, π/2) in polar coordinates. (i) Find another pair of polar coordinates for this point such that r > 0 and 2π < θ < 4π. (ii) Find another pair of polar coordinates for this point such that r < 0 and 0 < θ < 2π (b) You are given the point (-2, π/4) in polar coordinates. (i) Find another pair of polar coordinates for this point such that r > 0 and 2π < θ < 4π. (ii) Find another pair of polar coordinates for this point such that r < 0 and-2x < θ < 0. r2 (c) You are given the point (3,2) in polar coordinates. (i) Find another pair of polar coordinates for this point such that r > 0 and 2π < θ < 4T. (ii) Find another pair of polar coordinates for this point such that r < 0 and 0 θ < 2π.
(a) (i) Another pair of polar coordinates for (1, π/2) with r > 0 and 2π < θ < 4π is (1, 5π/2).
(ii) Another pair of polar coordinates for (1, π/2) with r < 0 and 0 < θ < 2π is (-1, π/2).
(b) (i) Another pair of polar coordinates for (-2, π/4) with r > 0 and 2π < θ < 4π is (2, 9π/4).
(ii) Since there is a typo in the question, I assume you meant 0 < θ < 2π. In this case, another pair of polar coordinates for (-2, π/4) with r < 0 and 0 < θ < 2π is (-2, π/4).
(c) (i) Assuming the correct range for θ is 2π < θ < 4π, another pair of polar coordinates for (3, 2) with r > 0 and 2π < θ < 4π is (3, 2 + 2π).
(ii) Another pair of polar coordinates for (3, 2) with r < 0 and 0 < θ < 2π is (-3, 2 + π).
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(a) (i) Another pair of polar coordinates for (1, π/2) with r > 0 and 2π < θ < 4π is (1, 5π/2).
(ii) Another pair of polar coordinates for (1, π/2) with r < 0 and 0 < θ < 2π is (-1, π/2).
(b) (i) Another pair of polar coordinates for (-2, π/4) with r > 0 and 2π < θ < 4π is (2, 9π/4).
(ii) Since there is a typo in the question, I assume you meant 0 < θ < 2π. In this case, another pair of polar coordinates for (-2, π/4) with r < 0 and 0 < θ < 2π is (-2, π/4).
(c) (i) Assuming the correct range for θ is 2π < θ < 4π, another pair of polar coordinates for (3, 2) with r > 0 and 2π < θ < 4π is (3, 2 + 2π).
(ii) Another pair of polar coordinates for (3, 2) with r < 0 and 0 < θ < 2π is (-3, 2 + π).
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Recall that an angle making a full rotation measures 360 degrees or 2 radians. a. If an angle has a measure of 110 degrees, what is the measure of that angle in radians? radians Preview b. Write a formula that expresses the radian angle measure of an angle, in terms of the degree measure of that angle
The measure of the angle in radians is approximately 1.9099 radians and Where radian_angle represents the angle's measure in radians, degree_angle represents the angle's measure in degrees, and π (pi) is approximately 3.1416
a. To convert an angle of 110 degrees to radians, we can use the following conversion formula:
[tex]radians = \frac{(degrees × π) }{180}[/tex]
Step 1: Plug in the given angle (110 degrees) into the formula:
[tex]radians= \frac{110×π}{180}[/tex]
Step 2: Calculate the value:
[tex]radians= \frac{(110)(3.1416)}{180} = \frac{343.7756}{180} = 1.9000[/tex]
So, the measure of the angle in radians is approximately 1.9099 radians.
b. To write a general formula that expresses the radian angle measure of an angle, in terms of the degree measure of that angle, you can use the following formula:
[tex]radian angle= \frac{degree angle x π}{180}[/tex]
Where radian_angle represents the angle's measure in radians, degree_angle represents the angle's measure in degrees, and π (pi) is approximately 3.1416.
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give a combinatorial proof for the identity 1 + 2 + 3 ⋯ +n=(n +1/2).
Answer: Brainliest?
Step-by-step explanation:
To prove the identity 1 + 2 + 3 + ... + n = (n+1)/2 combinatorially, we can use a simple argument involving counting the number of ways to arrange a certain set of objects.
Consider a set of (n+1) objects, consisting of n white balls and 1 black ball. We want to count the number of ways to arrange these objects in a row. Let's call this number N.
On the one hand, we can count N directly by considering the number of choices we have for the first ball, then the number of choices we have for the second ball, and so on, until we have made n choices for the n white balls, leaving only the black ball to be placed in the last position. Using the multiplication principle, we see that N is equal to the product of n consecutive integers, which we can write as:
N = n(n-1)(n-2)...(3)(2)(1)
On the other hand, we can count N indirectly by considering the number of ways to divide the (n+1) objects into two groups: the black ball by itself, and the remaining n white balls. Since there are (n+1) objects in total, there are (n+1) ways to choose which object will be the black ball. Once we have made this choice, the remaining n white balls can be arranged in any order, giving us n! possible arrangements. Thus, the total number of arrangements is:
N = (n+1) n!
Now, these two expressions for N must be equal, since they are both counting the same thing. Equating them, we get:
n(n-1)(n-2)...(3)(2)(1) = (n+1) n!
Simplifying, we obtain:
1 + 2 + 3 + ... + n = n(n+1)/2
which is the desired identity.
Find the three critical points of the function f(x,y)=(x2 +y2)ey^2−x^2 and for each critical point determine if it is a local minimum, local maximum, or saddle point.
The three critical points of the function f(x, y) = (x² + y²)y² - x² are (0, 0), (0, -1), and (0, 1). The point (0, 0) is a saddle point, while (0, -1) is a local maximum and (0, 1) is a local minimum.
To find the critical points, first compute the partial derivatives fx and fy, then set them to zero and solve for x and y. fx = -2x(1+y²) and fy = 2y(3y²+x²).
Solving fx=0 and fy=0 simultaneously, we get (0, 0), (0, -1), and (0, 1) as critical points.
To determine the nature of each critical point, compute the second-order partial derivatives fx, fyy, and fxy, and then find the determinant of the Hessian matrix, D = fx * fyy - fxy². For (0, 0), D < 0, making it a saddle point.
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determine whether the sets s1 and s2 span the same subspace of r3. s1 = {(1, 2, −1), (0, 1, 1), (2, 5, −1)} s2 = {(−2, −6, 0), (1, 1, −2)}
Therefore, s1 and s2 do not span the same subspace of R3.
To determine whether the sets s1 and s2 span the same subspace of R3, we need to check if one set can be obtained as a linear combination of the other set.
We can start by checking if the vectors in s2 can be obtained as a linear combination of the vectors in s1. We can set up the following system of equations:
[tex]a(1, 2, -1) + b (0, 1, 1) + c(2, 5, -1) = (-2, -6, 0)[/tex]
[tex]d(1, 2, -1) + e(0, 1, 1) + f(2, 5, -1) = (1, 1, -2)[/tex]
We can write this system in matrix form as follows:
[tex]\left[\begin{array}{ccc}1&0&2|-2\\2&1&5|-6\\-1&1&-1|0\end{array}\right]*\left[\begin{array}{ccc}1&0&2|1\\2&1&5|1\\-1&1&-1|-2\end{array}\right][/tex]
We can row reduce this augmented matrix to find the solutions for the system of equations:
[tex]\left[\begin{array}{ccc}1&0&2|-2\\0&1&1|2\\0&0&0|0\end{array}\right]*\left[\begin{array}{ccc}1&0&2|1\\2&1&5|1\\0&0&0|0\end{array}\right][/tex]
The matrix on the left represents the coefficients for the linear combinations of the vectors in s1 that would give us the vectors in s2. Since the matrix has a row of zeros, this means that we can't obtain the vectors in s2 as a linear combination of the vectors in s1.
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42
At the end of a baseball game, the players were given the choice of having a bottle of
water or a box of juice. Of all of the players, 12 chose a bottle of water, which was
3
4 of the total number of players. Write and solve an equation to determine p,
the total number of players at the baseball game.
Show your work.
The equation to determine p, which is the total number of player at the basketball game, is 12 = p x 3 / 4 .
The number of players at the basketball game is 16 players .
How to find the number of players ?The equation to find the number of players is;
Players who chose water = total players x proportion of players who chose water
12 = p x 3 / 4
This means that solving for p gives :
12 = p x 3 / 4
12 ÷ 3 / 4 = p
p = 12 ÷ 3 / 4
p = 12 / 0.75
p = 16 players
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calculate the probability that an electron will be found a between x=0.1 and 0.2 nm in a box of length l=10nm when its wavefunction is =2/l^1/2sin(2pix/l). t
The probability of finding an electron between x = 0.1 nm and x = 0.2 nm is 0.1 nm.
The probability density function for finding an electron between two points in space is given by the square of the absolute value of the wave function, integrated over the given range.
Let's start by finding the normalization constant A for the given wave function:
∫|Ψ|^2 dx = 1
∫(2/√l)sin(2πx/l) dx = 1
Using integration by parts, we get:
A = √(l/2)
Now, the probability of finding the electron between x = 0.1 nm and x = 0.2 nm is given by:
P = ∫0.2nm 0.1nm |Ψ|^2 dx
P = A^2 ∫0.2nm 0.1nm (sin(2πx/l))^2 dx
P = (l/2) ∫0.2nm 0.1nm (sin(2πx/l))^2 dx
P = (10/2) ∫0.2nm 0.1nm (sin(2πx/10))^2 dx
P = 2 ∫0.2nm 0.1nm (sin(πx/5))^2 dx
Using the identity sin^2θ = (1/2)(1 - cos(2θ)), we can simplify this expression:
P = 2 ∫0.2nm 0.1nm (1/2)(1 - cos(2πx/5)) dx
P = ∫0.2nm 0.1nm (1 - cos(2πx/5)) dx
P = (∫0.2nm 0.1nm dx) - (∫0.2nm 0.1nm cos(2πx/5) dx)
The first integral is simply the length of the given interval:
∫0.2nm 0.1nm dx = 0.1nm
For the second integral, we can use the fact that the integral of cos(mx) from 0 to 2π is zero, unless m is equal to zero. In this case, m = 5, so we get:
∫0.2nm 0.1nm cos(2πx/5) dx = 0
Therefore, the probability of finding the electron between x = 0.1 nm and x = 0.2 nm is:
P = 0.1nm
So the probability of finding an electron between x = 0.1 nm and x = 0.2 nm is 0.1 nm.
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15 minutes to read 9 pages; 50 minutes to read 30 pages
what is the answer
Answer:
It takes about 1 hour to read 30 pages at an average reading speed
Step-by-step explanation:
i suck at math and i’m tired of it, please help + 100 points
Answer:
C
Step-by-step explanation:
K^2 + 4
Answer:k^2+4
Step-by-step explanation:
[tex]f (x) = 2x^{3} - x^{2} - 22x - 24[/tex] synthetic division
Possible zeros:
Zeros:
Linear Factors:
The possible zeros of the polynomial are -2, -3/2 and 4.
What are the zeros of the function?The zeros of the function is calculated as follows;
The zeros of the function are the values of x that will make the function equal to zero.
let x = -2
f(x) = 2x³ - x² - 22x - 24
f(-2) = 2(-2)³ - (-2)² - 22(-2) - 24
f(-2) = -16 - 4 + 44 - 24
f(-2) = 0
So, x + 2 is a factor of the polynomial, and other zeros of the polynomial is calculated as;
2x² - 5x - 12
----------------------------------
x + 2 √ 2x³ - x² - 22x - 24
- (2x³ + 4x²)
------------------------------------
-5x² - 22x -24
- (-5x² - 10x)
-------------------------------------
-12x - 24
- (-12x - 24)
-------------------------
0
2x² - 5x - 12 , so will factorize this quotient as follows;
= 2x² - 8x + 3x - 12
= 2x(x - 4) + 3(x - 4)
= (2x + 3)(x - 4)
2x + 3 = 0
or
x - 4 = 0
x = -3/2 or 4
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Can someone pls answers this??? asapp
The correct statement regarding the rate of change of the exponential function is given as follows:
The bacterial culture loses 1/2 of it's size every 1/6 seconds.
How to obtain the half life of the exponential function?The exponential function that gives the bacteria's population after t seconds is given as follows:
B(t) = 9300(1/64)^t.
The rate of change of the exponential function is given as follows:
1/64.
Hence the half-life of the population is obtained as follows:
(1/64)^t = 1/2
(1/2^6)^t = (1/2)
2^(-6t) = 2^(-1)
6t = 1
t = 1/6.
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please i need the answer ill give brainliest
The simplified expression is [tex]\frac{2x}{x-1}[/tex].
How to simplify any expression?Expression - An expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them.
To simplify the expression, we can factor out a 4x from the numerator and a 2 from the denominator, which gives:
[tex]=\frac{4x^2+4x}{2x^2-2}\\\\ = \frac{4x(x+1)}{2(x^2-1)} \\\\ = \frac{2\cdot 2x(x+1)}{2(x-1)(x+1)} \\\\ = \frac{2x}{x-1}[/tex]
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Find N such that x+N=5.4 and x/n=5.4 are equivalent equations
Answer: 0.84375
Step-by-step explanation:
1. x = 5.4n
2. 5.4n + n = 5.4
3. n(6.4) = 5.4
4. n = 0.84375
DOUBLE CHECK
x = 5.4(0.84375)
x = 4.55625
4.55625 + 0.84375 = 5.4
0.84375 = 5.4 - 4.55625
0.84375 = 0.84375
The solution is, : for 0.84375 = N such that x+N=5.4 and x/n=5.4 are equivalent equations.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.
Here, we have,
the given equations are:
1. x = 5.4n
2. 5.4n + n = 5.4
3. n(6.4) = 5.4
4. n = 0.84375
DOUBLE CHECK
x = 5.4(0.84375)
x = 4.55625
4.55625 + 0.84375 = 5.4
0.84375 = 5.4 - 4.55625
0.84375 = 0.84375
Hence, The solution is, : for 0.84375 = N such that x+N=5.4 and x/n=5.4 are equivalent equations.
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Find the circumference of a circle with diameter 66cm(Take π=22/7)
Answer:207
Step-by-step explanation: circumference = to 2πr or πd
substitute the diameter into the equation πd (22/7)(66) and you get 207.4285...
Perform step one of converting the following CFG into CNF by adding a new start state S. V = {A, B}, } = {0,1}, S = A, R= A +BABB|11 € B +00€
The CFG has a new start state S and the original start state A is now a non-terminal symbol in the set V.
To convert the given CFG into CNF, we need to follow these steps:
Step 1: Add a new start state S and a new production rule S → A.
So, the modified CFG becomes:
S → A
A → BABB | 11
B → 00
Note that the original CFG had productions with single variables on the right-hand side. These productions do not follow the rules of CNF. By introducing a new start state and a new production rule, we can eliminate such productions and bring the CFG into CNF.
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Question 14
Which explicit formula describes the pattern in this table?
d
2
3
5
10
C
6.28
9.42
15.70
31.40
Od 3.14x C
O 3.14x C-d
O 31.4 x 10 C
OC 3.14 x d
1 pts
The explicit formula is C = 3.14 × d.
What is the explicit formula?
The formal equations for L-functions in mathematics are Riemann's zeta function and links between sums over an L-function's complex number zeroes and sums over prime powers.
Here, we have
Given:
d C
2 6.28
3 9.42
5 15.70
10 31.40
We have to find the explicit formula that describes the given pattern.
We concluded from the given table that
when d = 2
we get
c = 3.14 × 2 = 6.28
When d = 3
We get
c = 3.14 × 3 = 9.42
When d = 5
We get
c = 3.14 × 5 = 15.70
When d = 10
we get
c = 3.14 × 10 = 31.40
Hence, the explicit formula is C = 3.14 × d.
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suppose x and y are independent random variables with expected values e[x] = 0, e[y] = 0, and var(x) = 1, var(y) = 1. what is var(x-y)?
The required variance for the question is var (x - y) is 2.
We are given that x and y are independent random variables with E[x] = 0, E[y] = 0, Var(x) = 1, and Var(y) = 1. We need to find Var(x - y).Step 1: Understand the properties of variance.For more such question on variance
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given the following information about events a and b p(a)=0 p(a and b)=0 p(b)=0.25 are a and b mutually exclusive, independent, both, or neither?
Based on the given information, we can determine that events A and B are mutually exclusive. This is because the probability of their intersection, P(A and B), is equal to 0.
Based on the information provided for events A and B, we can determine if they are mutually exclusive, independent, both, or neither. Here's an analysis using the given probabilities:
1. P(A) = 0
2. P(A and B) = 0
3. P(B) = 0.25
Mutually exclusive events are events that cannot occur at the same time. In other words, if A occurs, then B cannot occur, and vice versa. If events are mutually exclusive, then P(A and B) = 0.
Independent events are events where the occurrence of one event does not affect the probability of the other event. If events A and B are independent, then P(A and B) = P(A) * P(B).
Now let's analyze:
A and B are mutually exclusive because P(A and B) = 0.
To check for independence, we calculate P(A) * P(B) = 0 * 0.25 = 0. Since P(A and B) = 0, A and B are also independent.
Therefore, events A and B are both mutually exclusive and independent. If A and B were independent events, then their intersection probability would be equal to the product of their individual probabilities, i.e. P(A and B) = P(A) * P(B), which is not the case here. Therefore, we can conclude that events A and B are mutually exclusive, but not independent.
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Question
A town's yearly snowfall in inches over a 10-year period is recorded in this table.
What is the mean of the snowfall amounts?
Responses
15.0 in.
15.0 in.
17.0 in.
17.0 in.
17.9 in.
17.9 in.
Year Snowfall in inches
1997 15
1998 11
1999 18
2000 25
2001 13
2002 20
2003 16
2004 28
2005 15
2006 18
18.9 in
18.9 in
The mean (average) of the snowfall amounts is 17.9 inches.
What is mean?In statistics, the mean is a measure of central tendency that represents the average value of a dataset. It is calculated by summing up all the values in the dataset and then dividing the result by the total number of values.
What is average?In statistics, the terms "mean" and "average" are often used interchangeably to refer to the same concept. Both terms represent a measure of central tendency that represents the typical or average value of a dataset.
According to given information:To find the mean (average) of the snowfall amounts, we need to add up all the snowfall amounts and divide by the total number of years.
Adding up the snowfall amounts:
15 + 11 + 18 + 25 + 13 + 20 + 16 + 28 + 15 + 18 = 179
Dividing by the total number of years (10):
179/10 = 17.9
So the mean (average) of the snowfall amounts is 17.9 inches.
Therefore, the correct response to the mean is 17.9 in.
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URGENT MUST BE ANSWERED NOW !! PLEASE AND THANK YOU (image included)
Mitch uses 1/4 of his supply of apples to make apple crisp and 3/8 of his supply of apples to make pies. If Mitch uses 10 pounds of apples, how many pounds of apples are in his supply?
Answer:
16 lbs
Step-by-step explanation:
total of apples = 1/4 + 3/8 = 2/8 + 3/8 = 5/8
then 10 x 8/5 = 16
Find the length of the third side. If necessary, write in simplest radical form.
The hypotenuse of the given triangle is 8√2 units.
What is Pythagoras Theorem?In accordance with the Pythagorean theorem, the square of the length of the hypotenuse (the side that faces the right angle) in a right triangle equals the sum of the squares of the lengths of the other two sides. If you know the lengths of the other two sides of a right triangle, you may apply this theorem to determine the length of the third side. By examining whether the sides of a triangle satisfy the Pythagorean equation, it can also be used to assess whether a triangle is a right triangle. Pythagoras, an ancient Greek mathematician, is credited with discovering the theorem, therefore it bears his name.
The third side of the triangle can be determined using the Pythagoras Theorem as follows:
c² = 8² + 8²
c² = 2(8²)
c = 8√2
Hence, the hypotenuse of the given triangle is 8√2 units.
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Let f(x) = 4(1/4) ^x+2 What is f(1)? Answer in fraction form. Provide your answer below: f(1) = __
The value of function f(1) = 1/16.
What is function?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is typically represented as y = f(x).
To find f(1), we substitute x = 1 into the given expression for f(x):
f(1) = 4(1/4)⁽¹⁺²⁾ = 4(1/4)³ = 4(1/64) = 1/16
Therefore, f(1) = 1/16.
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if n=29, ¯ x =50, and s=2, find the margin of error at a 95onfidence level using the critical value rounded to three decimal places.
At a 95% confidence level, the margin of error is approximately 0.762 (rounded to three decimal places).
To find the margin of error at a 95% confidence level, we need to first find the critical value associated with a sample size of 29 and a confidence level of 95%.
Using a t-distribution with n-1 degrees of freedom, we can find the critical value using a t-table or calculator. For n=29 and a confidence level of 95%, the critical value is approximately 2.045 (rounded to three decimal places).
The formula for the margin of error is:
Margin of error = critical value * (standard deviation / sqrt(sample size))
Plugging in the values we have:
Margin of error = 2.045 * (2 / sqrt(29))
Margin of error ≈ 0.762
Therefore, at a 95% confidence level, the margin of error is approximately 0.762 (rounded to three decimal places).
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A chord of a circle is l cm long. The distance of the chord to the centre of the circle is h cm and the radius of the circle is r cm. Express r in terms of l and h.
The value of r in terms of l and h is,
⇒ r = √((l/2)² + h²)
Now, We can use the Pythagorean theorem to relate r, l, and h as,
Since, The chord of the circle divides the circle into two segments, each with a height of h.
Let's call the segments are A and B.
Then, the length of the chord (l) is equal to the sum of the bases of segments A and B.
Therefore, the length of each base is,
(l/2).
Hence, We can use the Pythagorean theorem to relate r, l/2, and h for one of the segments as;
⇒ r² = (l/2)² + h²
⇒ r = √((l/2)² + h²)
Thus, The value of r in terms of l and h is,
⇒ r = √((l/2)² + h²)
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Find (x-y) if X=5/3 y=-1/6
Answer: (x - y) = 13/6
Step-by-step explanation: To find the value of (x-y), we need to substitute the given values of x and y and then perform the subtraction.
So,
(x - y) = (5/3 - (-1/6))
We can simplify this expression by first converting the negative fraction to its equivalent positive fraction and then finding the common denominator.
(x - y) = (5/3 + 1/6) = ((10+3)/6) = 13/6
Therefore, (x - y) = 13/6.
Share Prompt
Answer:
11/6
Step-by-step explanation:
Use substitution.
x = 5/3
y = -1/6
Sub these values into (x-y):
[(5/3) - (-1/6)]
*Make sure to use brackets when subbing in values especially when there are negative signs or exponents
5/3 + 1/6 ⇒ two negatives become a positive
10/6 + 1/6 ⇒ make a common LCD
= 11/6
An object is 25.0 cm from the center of a spherical silvered-glass Christmas tree ornament 6.20 cm in diameter. What is the position of its image (counting from the ornament surface)? Follow the sign rules. Express your answer with the appropriate units. What is the magnification of its image?
The image magnification is about 0.133 times the size of the object and is inverted.
What is magnification?Magnification is the ratio of the size of the image to the size of the object. In optics, it is often used to describe how much larger or smaller an image appears compared to the object being viewed. It is calculated by dividing the height or size of the image by the height or size of the object. Magnification can be positive or negative, depending on whether the image is upright or inverted, respectively.
According to the given informationWe can use the mirror equation to find the position of the image:
1/f = 1/do + 1/di
where f is the focal length, do is the object distance, and di are the image distance.
Since the ornament is a spherical mirror, the focal length is half the radius of curvature, which is equal to half the diameter of the ornament:
f = R/2 = 6.20 cm/2 = 3.10 cm
The object distance is given as 25.0 cm.
Substituting into the mirror equation and solving for di, we get:
1/3.10 = 1/25.0 + 1/di
di = 3.33 cm
The image is formed 3.33 cm from the center of the ornament, which is 0.23 cm beyond the surface of the ornament (since the ornament has a radius of 3.10 cm).
The magnification of the image can be found using the:
m = -di/do
where the negative sign indicates that the image is inverted.
Substituting the values we found, we get:
m = -(3.33 cm)/(25.0 cm) = -0.133
So the image is about 0.133 times the size of the object and is inverted.
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how many terms of the series Σ[infinity] 2/n^6 n=1 are needed so that the remainder is less than 0.0005? [Give the smallest integer value of n for which this is true.]
The number of terms the series needed so that the remainder is less than 0.0005 is 14. The smallest integer value of n for which this is true is 14.
To find the number of terms needed for the remainder to be less than 0.0005, we need to use the remainder formula for an infinite series:
Rn = Sn - S
where Rn is the remainder after adding n terms, Sn is the sum of the first n terms, and S is the sum of the infinite series.
For this series, S can be found using the formula for the sum of a p-series:
S = Σ[infinity] 2/n^6 n=1 = π^6/945
Now we need to find the smallest value of n for which Rn < 0.0005. We can rewrite the remainder formula as:
Rn = Σ[infinity] 2/n^6 - Σ[n] 2/n^6
Simplifying the first term using the formula for the sum of a p-series, we get:
Σ[infinity] 2/n^6 = π^6/945
Substituting this into the remainder formula, we get:
Rn = π^6/945 - Σ[n] 2/n^6
We want Rn < 0.0005, so we can set up the inequality:
π^6/945 - Σ[n] 2/n^6 < 0.0005
Solving for n using a calculator or computer program, we get:
n ≥ 14
Therefore, we need at least 14 terms of the series Σ[infinity] 2/n^6 n=1 to ensure that the remainder is less than 0.0005, and the smallest integer value of n for which this is true is 14.
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Select the equation that most accurately depicts the word problem. A class of 19 pupils has five more girls than boys. Let n = the number of boys.
n + (n + 5) = 19
n - (n + 5) = 19
n + (n + 19) = 5
n + (n - 19) = 5
In a company's first year in operation, it made an annual profit of $112,000. The profit of the company increased at a constant 18% per year each year. How much total profit would the company make over the course of its first 26 years of operation, to the nearest whole number?
If the company made a profit of $112000, i first year operation, then the total profit made by the company in 26 years of operation is $8170286.
In order to find the profit after 26 years, we use the formula for "future-value" of investment with compound interest;
⇒ FV = PV × (1 + r)ⁿ,
where FV is = future value, PV is = present value, r is = interest rate per period, and n = time (in years),
In this case, the "present-value" is $112,000, the "interest-rate" per period is 18%, and time is 26 years.
We have to find the total profit, which is = "future-value" - "present-value",
⇒ Total profit = FV - PV,
Substituting the value,
We get,
⇒ FV = $112000 × (1 + 0.18)²⁶,
⇒ FV = $8282285.79 ≈ $8282286,
So, Total profit = $8282286 - $112000,
Total profit = $8170286,
Therefore, the company would make a total-profit of $8170286.
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