The coordinate matrix of the vector in the S' basis is [5/2 5/2]t.
(a) To verify that S and S' are bases, we need to check that they are linearly independent and span R^2.
First, we check if S is linearly independent:
c1 [2 1] + c2 [1 2] = [0 0] has only the trivial solution c1 = 0 and c2 = 0, which means that S is linearly independent.
Next, we check if S spans R^2. Since S has two vectors and R^2 is two-dimensional, it is enough to show that the two vectors in S are not collinear. We can see that [2 1] and [1 2] are not collinear, so S spans R^2.
Similarly, we can check that S' is linearly independent:
c1 [1 0] + c2 [1 1] = [0 0] has only the trivial solution c1 = 0 and c2 = 0, which means that S' is linearly independent.
We can also check that S' spans R^2:
Any vector [a b] in R^2 can be written as [a b] = (a-b)/2 [1 0] + (a+b)/2 [1 1], which shows that S' spans R^2.
Therefore, S and S' are bases of R^2.
(b) To compute the transition matrices Ps-s and Ps+s, we need to find the coordinate matrices of the vectors in S and S' with respect to each other. We can use the formula [v]s = Ps,t [v]t, where Ps,t is the transition matrix from basis t to basis s.
To find Ps-s, we need to express the vectors in S in terms of S':
[2 1] = (1/2) [1 0] + (1/2) [1 1]
[1 2] = (-1/2) [1 0] + (3/2) [1 1]
Therefore, the transition matrix Ps-s is:
Ps-s = [1/2 -1/2]
[1/2 3/2]
To find Ps+s, we need to express the vectors in S' in terms of S:
[1 0] = (2/3) [2 1] - (1/3) [1 2]
[1 1] = (1/3) [2 1] + (2/3) [1 2]
Therefore, the transition matrix Ps+s is:
Ps+s = [2/3 1/3]
[-1/3 2/3]
(c) Given the coordinate matrix [3 2]s of a vector in the S basis, we can use the formula [v]s' = (Ps-s)^(-1) [v]s to find its coordinate matrix in the S' basis:
[v]s' = (Ps-s)^(-1) [3 2]s
= [1/2 1/2] [3 2]t
= [5/2 5/2]t
Therefore, the coordinate matrix of the vector in the S' basis is [5/2 5/2]t.
(d) Given the coordinate matrix [3 2]s' of a vector in the S' basis, we can use the formula [v]s = (Ps+s)^(-1) [v]s' to find its coordinate matrix in the S basis:
[v]s = (Ps+s)^(-1) [3 2]s'
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What is the distance between the two points plotted?
A graph with the x-axis starting at negative 10, with tick marks every one unit up to 10. The y-axis starts at negative 10, with tick marks every one unit up to 10. A point is plotted at negative 4, 6 and at negative 4, negative 6.
12 units
10 units
−12 units
−10 units
The distance between the two points plotted include the following: A. 12 units.
How to determine the distance between the coordinates for each points?In Mathematics and Geometry, the distance between two (2) points that are on a coordinate plane can be calculated by using the following mathematical equation:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where:
x and y represent the data points (coordinates) on a cartesian coordinate.
By substituting the given points into the distance formula, we have the following;
Distance = √[(-4 + 4)² + (-6 - 6)²]
Distance = √[(0)² + (-12)²]
Distance = √[0 + 144]
Distance = √144
Distance = 12 units.
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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
Answer:
We can simplify and rewrite the given equation using properties of algebra:
2.3p – 10.1 = 6.5p – 4 – 0.01p
2.3p – 10.1 = 6.49p – 4
2.3p - 6.49p = -4 + 10.1
-4.19p = 6.1
p = -6.1 / 4.19
p ≈ -1.4568
Therefore, the equation 2.3p – 10.1 = 6.5p – 4 – 0.01p is equivalent to the equation -4.19p = 6.1, which has the solution p ≈ -1.4568.
Now, we can check which of the given equations have the same solution as -4.19p = 6.1:
- 2.3p – 10.1 = 6.4p – 4
Simplifying:
-8.7p = 6.1
p = -0.7011
This equation does not have the same solution as -4.19p = 6.1.
- 2.3p – 10.1 = 6.49p – 4
Simplifying:
-8.79p = 6.1
p = -0.6932
This equation does not have the same solution as -4.19p = 6.1.
- 230p – 1010 = 650p – 400 – p
Simplifying:
229p = 610
p = 610/229
p ≈ 2.6620
This equation does not have the same solution as -4.19p = 6.1.
- 23p – 101 = 65p – 40 – p
Simplifying:
23p + p - 65p = -40 + 101
-41p = 61
p = -61/41
p ≈ -1.4878
This equation does have the same solution as -4.19p = 6.1.
- 2.3p – 14.1 = 6.4p – 4
Simplifying:
-8.7p = 9.1
p = -1.0517
This equation does not have the same solution as -4.19p = 6.1.
Therefore, the equations that have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p are:
- 23p – 101 = 65p – 40 – p
Please mark me the brainliest
Answer:
It's B & C on Edge
:) pls star 5 and thanks!!!!!!!!!!!!!!!!
Give the contrapositive of the following statements. (a) If mor n is even then mnd is even. (b) If x E ANB then 2 € A and r e B, where A and B are subsets of R. (c) Let S be a subset of R. If u is an upper bound for Sthen for every € >0 there exists some ES such that e-c
In logic, the contrapositive of an implication is a new statement that is formed by switching the hypothesis and conclusion, and negating both. In other words, the contrapositive of "if p then q" is "if not q, then not p."
Please find the contrapositives of the given statements below:
(a) Original statement: If m or n is even, then mn is even.
Contrapositive: If mn is not even, then neither m nor n is even.
(b) Original statement: If x ∈ A∩B, then 2 ∈ A and r ∈ B, where A and B are subsets of R.
Contrapositive: If 2 ∉ A or r ∉ B, then x ∉ A∩B, where A and B are subsets of R.
(c) Original statement: Let S be a subset of R. If u is an upper bound for S, then for every ε > 0, there exists some E ∈ S such that u - E < ε.
Contrapositive: Let S be a subset of R. If for some ε > 0, there is no E ∈ S such that u - E < ε, then u is not an upper bound for S.
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Year Stivers ($) Trippi ($)
1 11,000 5,600
2 10,500 6,300
3 13,000 6,900
4 14,000 7,600
5 14,500 8,500
6 15,000 9,200
7 17,000 9,900
8 17,500 10,600
•Suppose that you initially invested $10,000 in the Stivers mutual funds and $5,000 in Trippi mutual fund. Then, no further investment was made. The value of each investment at the end of each year is provided in the table.
•What are the return (%) and the growth factor of each year?
• What are the geometric mean of each mutual fund?
• How can you interpret the difference of the geometric means between two mutual funds?
•State each step of calculation and explain the step.
Trippi had a higher average growth rate than Stivers. Specifically, Trippi's geometric mean was 38.34%, while Stivers' geometric mean was 7.33%. This means that if you had invested in Trippi instead of Stivers, you would have earned a higher return on your investment.
To calculate the returns and growth factors for each year, we can use the following formulas:
Return (%) = (Ending Value - Beginning Value) / Beginning Value * 100
Growth Factor = Ending Value / Beginning Value
Using these formulas, we get the following table:
To calculate the geometric mean of each mutual fund, we can use the following formula:
Geometric Mean = (Growth Factor 1 * Growth Factor 2 * ... * Growth Factor n) ^ (1/n)
Using this formula, we get:
Stivers Geometric Mean = (1.1000 * 1.0455 * 1.2381 * 1.0769 * 1.0357 * 1.0345 * 1.1333 * 1.0294) ^ (1/8) = 1.0733
Trippi Geometric Mean = (1.1200 * 1.2504 * 1.3368 * 1.4413 * 1.5562 * 1.6969 * 1.8316 * 1.9982) ^ (1/8) = 1.3834
The difference in the geometric means between the two mutual funds indicates that Trippi has a higher average growth rate than Stivers over the 8-year period. Specifically, Trippi's growth rate was about 38.34% per year on average, while Stivers' growth rate was about 7.33% per year on average.
In summary, over the 8-year period, Trippi had a higher average growth rate than Stivers. Specifically, Trippi's geometric mean was 38.34%, while Stivers' geometric mean was 7.33%. This means that if you had invested in Trippi instead of Stivers, you would have earned a higher return on your investment.
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Which of the following is represented by Dv?
O A. Chord
B. Radius
C. Diameter
D. Circumference
Answer:
Step-by-step explanation: RADIUS
Which function is modeled in this table
Answer:
[tex]f(x) = 1000 {(0.80)}^{x - 1} [/tex]
B is the correct function.
given eigenvalues, eigenvectors for matrix A are λ1 = 2, v1 = [-5 1]t and λ2 = 6, v2 = [5 4]t
(t means transpose ... means the row vector is actually a column vector)
Defn of Eigenvector, Eigenvalue:
Avi = λivi for all i, in this case, i = 1, 2
The eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).
An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. In other words, if A is a square matrix and v is an eigenvector of A, then Av = λv, where λ is the eigenvalue associated with v.
Given the eigenvalues and eigenvectors for matrix A as λ1 = 2, v1 = [-5 1]ᵀ and λ2 = 6, v2 = [5 4]ᵀ, we can say that:
- Av1 = λ1v1 = 2[-5 1]ᵀ = [-10 2]ᵀ
- Av2 = λ2v2 = 6[5 4]ᵀ = [30 24]ᵀ
Therefore, the eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).
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Find the total surface area of the cylinder. Round to the nearest tenth.
Answer:
S = 2π(4^2) + 2π(4)(15) = 152π =
477.5 square centimeters
The closest answer is 477.3 square centimeters (3.14 was used for π).
One of the properties of a parabola is that it can act as a reflector. In a reflector, any ray that is __________ to the axis of symmetry will be reflected off the surface to the focus. A. a bisector B. parallel C. perpendicular D. similar
One of the properties of a parabola is that it can act as a reflector. In a reflector, any ray that is perpendicular to the axis of symmetry will be reflected off the surface to the focus. So, correct option is C.
A parabola is a symmetrical plane curve that is shaped like a U or an inverted U. It is defined as the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix.
One of the properties of a parabola is that it can act as a reflector. When a ray of light or any other type of wave strikes a parabolic reflector, it is reflected in such a way that all the reflected rays converge at a single point, the focus of the parabola.
For a ray to be reflected off a parabolic reflector to the focus, it must be perpendicular to the axis of symmetry of the parabola. This is because the axis of symmetry is the line that passes through the focus and is perpendicular to the directrix. Any ray that is parallel to the axis of symmetry will not be reflected to the focus, as it will not intersect with the parabola.
Therefore, the answer to the given question is C. perpendicular. Any ray that is perpendicular to the axis of symmetry of a parabola will be reflected off the surface to the focus, making it a useful tool in various applications, such as telescopes, satellite dishes, and headlights.
So, correct option is C.
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Giving 100 pts to who ever does this 4 meee <3
The height of the cone is h = 3V / πr². Then the correct option is B.
Given that:
Volume of the cone, V = πr²h / 3
Simplify the equation for the value of h, then we have
V = πr²h / 3
3V = πr²h
h = 3V / πr²
The height of the cone is h = 3V / πr². Then the correct option is B.
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PLEASE HELP NOW MY ASSIGNMENT I DUE IN 10 MIN QUESTION: david traveled 4/5 of his trip by bicycle and the rest by foot if the whole trip was 160km how many km did he travel by foot?
Answer: 32 km
Step-by-step explanation:
If he travelled 4/5 of the trip by bike, then he travelled 1/5 on foot.
so he travelled 160/5 = 32 km on foot. Phew! thats a long walk.
Which of the points plotted is closer to (−4, 5), and what is the distance?
A graph with the x-axis starting at negative 10, with tick marks every one unit up to 10. The y-axis starts at negative 10, with tick marks every one unit up to 10. A point is plotted at negative 4, negative 5, at negative 4, 5 and at 5, 5.
Point (−4, −5), and it is 9 units away
Point (−4, −5), and it is 10 units away
Point (5, 5), and it is 9 units away
Point (5, 5), and it is 10 units away
The point that is closer to (-4,5) is (-4,-5), and the distance between the two points is 10 units.
We have a points plotted is closer to (−4, 5).
Using distance formula to calculate the distance between two points:
d =√((x2 - x1)² + (y²- y1)²)
d = √((-4 - (-4))² + (-5 - 5)²)
d = √(0² + (-10)²)
d = √100
d = 10
Thus, the distance between (-4,5) and (-4,-5) is 10 units.
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Kehlani recorded the grade-level and instrument of everyone in the middle school School of Rock below.
Seventh Grade Students
Instrument # of Students
Guitar 3
Bass 3
Drums 3
Keyboard 12
Eighth Grade Students
Instrument # of Students
Guitar 11
Bass 13
Drums 14
Keyboard 15
Based on these results, express the probability that a student chosen at random will play an instrument other than drums as a fraction in simplest form.
The probability that a student chosen at random will play an instrument other than drums as a fraction in simplest form is 25/37
The total number of students in the middle school School of Rock is:
3 + 3 + 3 + 12 + 11 + 13 + 14 + 15 = 74
The number of students who play an instrument other than drums is:
3 + 3 + 3 + 12 + 11 + 13 + 15 = 50
Therefore, the probability that a student chosen at random will play an instrument other than drums is:
50/74
= 25/37
Hence, the probability is 25/37 expressed as a fraction in simplest form.
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A laundry basket has 24 t-shirts in it. Four are navy, 8 are red, and the remaining are white. What is the probability of selecting a red shirt?
Answer:
1/3
Step-by-step explanation:
The total number of shirts =24
Probability=n(E)/n(S)
Therefore probability of selecting a red shirt =8/24
=1/3
The following are the annual incomes (in thousands of dollars) for randomly chosen, U.S. adults employed full-time: 26, 33, 34, 35, 35, 37, 39, 39, 39, 40, 40, 42, 42, 43, 44, 44, 47, 49, 49, 51, 54, 58, 77, 100a) Which measures of central tendency do not exist for this data set? Choose all that apply. | O Mean O Median O Mode O None of these measures(b) Suppose that the measurement 26 (the smallest measurement in the data set) were replaced by 6. Which measures of central tendency would be affected by the change? Choose all that apply. O Mean O Median O Mode O None of these measures(c) Suppose that, starting with the original data set, the largest measurement were removed Which measures of central tendency would be changed from those of the original data set? Choose all that apply.O Mean O Median O Mode O None of these measures(d) The relative values of the mean and median for the original data set are typical of data that have a significant skew to the right. What are the relative values of the mean and median for the original data set? Choose only one. O mean is greaterO median is greaterO Cannot be determined
(a) Mode does not exist for this data set.
(b) Mean would be affected by the change.
(c) None of these measures would be changed.
(d) Mean is greater than median for the original data set.
a) All measures of central tendency exist for this data set: Mean, Median, and Mode.
b) If the smallest measurement (26) were replaced by 6, the affected measures of central tendency would be:
- Mean
c) If the largest measurement were removed from the original data set, the affected measures of central tendency would be:
- Mean
d) For the original data set, which has a significant skew to the right, the relative values of the mean and median are:
- Mean is greater
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Consider the following probability distribution: 0 2 4 0. 4 0. 3 0. 3 find the variance (write it up to second decimal place)
The variance of the given probability distribution x: 0, 2, 4 and (x):0.4, 0.3, 0.3 is 2.046.
To find the variance of a discrete probability distribution, we use the formula:
Var(X) = Σ[(x - μ)² × f(x)]
where X is the random variable, μ is the expected value of X, x is the value of X, and f(x) is the probability mass function of X.
To find the expected value of X, we use the formula:
μ = Σ[x × f(x)]
Using the given distribution, we have:
μ = 0(0.4) + 2(0.3) + 4(0.3) = 1.8
Next, we use the variance formula:
Var(X) = Σ[(x - μ)² × f(x)]
= (0 - 1.8)²(0.4) + (2 - 1.8)²(0.3) + (4 - 1.8)²(0.3)
= 1.44(0.4) + 0.06(0.3) + 4.84(0.3)
= 0.576 + 0.018 + 1.452
= 2.046
Therefore, the variance of the given distribution is 2.046, up to the second decimal place.
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The question is -
Consider the following probability distribution:
x 0 2 4
f(x) 0.4 0.3 0.3
find the variance (write it up to the second decimal place).
How do i identify the pairs of congruent angles in figures?
Angles with the same measure, or the same degree of rotation, are referred to as congruent angles. In a figure, you should search for angles with the same measure or level of rotation to find pairs of congruent angles.
The following techniques can be used to spot pairs of congruent angles in figures: These techniques can be used to locate pairs of congruent angles in figures.
Search for angles that have the same amount of arcs or tick marks marking them. Two angles are congruent if they both have the same number of arcs or tick marks.
Look for angles that cross a line or are in opposition to one another. Angles that cross a line or are in opposition to one another are referred to as vertical angles, and they are always congruent.
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Prove that if both pairs of opposite sides of a quadrilateral are equal, then the
quadrilateral is a parallelogram
We have shown that both pairs of opposite sides are parallel, which means that the quadrilateral is a parallelogram.
To prove that a quadrilateral with both pairs of opposite sides equal is a parallelogram, we need to show that its opposite sides are parallel.
Let ABCD be the given quadrilateral with AB = CD and BC = DA. We need to show that AB || CD and BC || DA
Since opposite sides are equal, we have AB = CD and BC = DA. Adding these two equations, we get:
AB + BC = CD + DA
By the triangle inequality, we know that AB + BC > AC and CD + DA > AC. Therefore, we can write:
AB + BC > AC > CD + DA
Subtracting BC and CD from both sides, we get:
AB > AC - BC > DA
Since AB and DA are opposite sides of the quadrilateral, and AC and BC are transversals, we have shown that AB is parallel to DA.
Similarly, we can subtract AB and DA from both sides of the equation AB + BC = CD + DA to get:
BC > CD - DA > AB
Since BC and AB are opposite sides of the quadrilateral, and CD and DA are transversals, we have shown that BC is parallel to AB.
Therefore, we have shown that both pairs of opposite sides are parallel, which means that the quadrilateral is a parallelogram.
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5. Find all are R that satisfy the inequality 2+2+2 -11 < 2. [4]
Find all R that satisfy the inequality 2+2+2-11 < 2 using the terms "satisfy" and "inequality."
First, let's simplify the inequality:
2 + 2 + 2 - 11 < 2
Now, combine the like terms:
6 - 11 < 2
Next, subtract 6 from both sides:
-5 < 2
So, the inequality states that any value of R that is greater than -5 will satisfy the inequality -5 < 2. In this case, all real numbers R greater than -5 satisfy the given inequality.
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There are 20 people trying out for a team. How many ways can you make randomly select for people to make a team?
There are 15,504 ways to randomly select a team of 5 people from a group of 20 people
If there are 20 people trying out for a team, the number of ways to select a team of n people can be calculated using the formula for combinations, which is:
C(20, n) = 20! / (n! * (20 - n)!)
where C(20, n) represents the number of ways to select n people from a group of 20 people.
For example, if we want to select a team of 5 people, we can plug in n = 5 and calculate:
C(20, 5) = 20! / (5! * (20 - 5)!) = 15,504
Therefore, there are 15,504 ways to randomly select a team of 5 people from a group of 20 people. Similarly, we can calculate the number of ways to select teams of different sizes by plugging different values of n into the formula for combinations.
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j^2+6j-40. factor helpppp
The expression is factorized to give j = -10 and j = 4
How to factor the expressionFrom the information given, we have the quadratic equation as;
j²+ 6j - 40
Using the factorization method, we have to mulitply the coefficient of j² by the constant.
After this, find the pair factors of the product that adds up to give 6
Substitute the values
Then, we have;
j² + 10j - 4j - 40
group the expression in pairs
(j² + 10j) - (4j- 40)
factor the common terms
j(j + 10) - 4(j + 10)
We have;
(j + 10) (j - 4)
j + 10 = 0
collect the terms
j = -10
j = 4
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An amount was invested at 5 ¾ % and grew to P250,500 on March
11, 2021. If it was invested on July 12, 2020, what was the amount
invested?
The amount invested was approximately P239,534.52.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the amount after t years, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.
In this problem, we know that the amount grew to P250,500 after some time. We also know that the interest rate is 5 3/4%, or 0.0575, and that the investment was made on July 12, 2020, which is about 8 months before March 11, 2021.
Let's first convert the interest rate to a monthly rate, since we need to compound the interest monthly:
r = 0.0575/12 = 0.004792
Next, let's calculate the number of months between July 12, 2020 and March 11, 2021:
8 months + 31 days/365 days = 8.0849 months
Now we can use the formula to solve for P:
250500 = P(1 + 0.004792/12)^(12*8.0849)
250500/P = 1.045305
P = 250500/1.045305
P = 239534.52
Therefore, the amount invested was approximately P239,534.52.
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(2, 1) and (3,-10)
Slope =
Answer:
slope = - 11
Step-by-step explanation:
calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (2, 1 ) and (x₂, y₂ ) = (3, - 10 )
m = [tex]\frac{-10-1}{3-2}[/tex] = [tex]\frac{-11}{1}[/tex] = - 11
which expression is equivalent to -12x - 14?
the answer is -2 (6 x + 7)
Find the value of tan G rounded to the nearest hundredth, if necessary.
H
√67
29
G
The value of tan G in the triangle is √67/29
We have to find the value of tan G
The given triangle is a right triangle
We know that tan function is a ratio of opposite side and adjacent side
tan G = opposite side/ adjacent side
The opposite side of tan G is √67
Adjacent side of triangle is 29
tanG =√67/29
Hence, the value of tan G in the triangle is √67/29
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Pls I need help with my math work Find the value of x
The value of the angle x formed by the two intercepted arcs is: 29.5
What is the angle at the center between arcs?
The central angle is the angle that a circle's arc occupies in its center. The arms of the central angle are formed by the radius vectors.
Now, angles that are formed inside of a circle by two chords create four arcs on a circle, which you can see in this diagram. The measure of the angle is equal to half the sum of the intercepted arcs.
We are given that:
mAB = 34
mCD = 25
Thus:
x = (34 + 25)/2
x = 59/2
x = 29.5
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I need help solving this pls
solve for v
v/8 =2
Answer:
do the following:
1. Multiply both sides of the equation by 8. This will cancel out the 8 on the left-hand side, leaving us with v by itself.
```
v/8 = 2
(v/8) * 8 = 2 * 8
v = 16
```
Therefore, the value of v is 16..
Step-by-step explanation:
Answer: V=16
Step-by-step explanation:
First you mutiply both sides by 8
[tex]8*\frac{v}{8}=8*2[/tex]
Cancel out the greatest common factor which in this case is 8.
So now we have v=8*2
Multiply 8*2 and the answer should be V=16.
anna has 2 dozen Rollo bars and 1 dozen apples as treats for halloween. in how many ways can anna hand out 1 treat to each of 36 children who come to her door?
There are approximately 3.72 x 10⁴¹ ways that Anna can hand out 1 treat to each of the 36 children who come to her door.
Anna has 2 dozen Rollo bars, which is equivalent to 2 x 12 = 24 Rollo bars.
She also has 1 dozen apples, which is equivalent to 1 x 12 = 12 apples.
So, Anna has a total of 24 + 12 = 36 treats to hand out.
Now, she needs to hand out 1 treat to each of the 36 children who come to her door.
This can be thought of as selecting 1 treat from the total of 36 treats for each child, without replacement, as each child can only receive 1 treat.
The number of ways to do this is given by the concept of permutations, denoted by "nPr", which is calculated as;
nPr = n! / (n - r)!
where n is the total number of items (treats) to choose from, and r is the number of items (treats) to choose.
In this case, n = 36 (total number of treats) and r = 36 (number of children).
Plugging in the values, we get;
36P36 = 36! / (36 - 36)! = 36! / 0! = 36!
Since 0! (0 factorial) is equal to 1, we can simplify further:
36! / 1 = 36!
36! ≈ 3.72 x 10⁴¹
So, there are 3.72 x 10⁴¹ ways that Anna can hand out 1 treat to each of the 36 children who come to her door.
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•You are given a number stored in a variable, with the name age • Check whether the age is greater than and equal to 60 or not. If true, Then print "Senior Citizen"• otherwise "Not Senior Citizen".
Here is the code for the condition given:
```python
age = 65 # Replace this with the given age
if age >= 60:
print("Senior Citizen")
else:
print("Not Senior Citizen")
Let us explain the answer in detail:
1. Store the given age in a variable called `age`.
2. Use an if statement to check if the age is greater than or equal to 60.
3. If the condition is true, print "Senior Citizen".
4. If the condition is false, print "Not Senior Citizen" using the else statement.
Here's a code example:
```python
age = 65 # Replace this with the given age
if age >= 60:
print("Senior Citizen")
else:
print("Not Senior Citizen")
```
Replace `65` with the given age to check if the person is a senior citizen or not.
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A firm manufactures padded shipping bags. A cardboard carton should contain 100 bags, but machine operators fill the cardboard cartons by eye, so a carton may contain anywhere from 98 to 123 bags (average = 105.5 bags). Each padded bag costs $0.03. Management realizes that they are giving away 5(1/2)% of their output by overfilling the cartons. One solution is to automate the filling of shipping cartons. This should reduce the average quantity of bags per carton to 100.3, with almost no cartons containing fewer than 100 bags. The equipment would cost $18,600 and straight-line depreciation with a 10-year depreciable life and a $3600 salvage value would be used. The equipment costs $16,000 annually to operate. 200,000 cartons will be filled each year. This large profitable corporation has a 40% combined federal-plus-state incremental tax rate. Assume a 10-year study period for the analysis and an after-tax MARR of 15%. Compute: (a) The after-tax present worth (b) The after-tax internal rate of return (c) The after-tax simple payback period =1.9 years
The after-tax present worth according to the given values is $732,140.56 and the internal rate of return is 22.65%.
(a) To compute the after-tax present worth, we need to determine the net cash flow for each year and discount it to present value using the after-tax MARR of 15%.
Year 0: Initial cost of equipment = -$18,600
Years 1-10:
Revenue from bags = (100.3 bags/carton) x ($0.03/bag) x (200,000 cartons/year) = $120,780
Cost savings from reducing overfilling = (5.5%) x ($0.03/bag) x (200,000 cartons/year) = $3,300
Operating cost of equipment = -$16,000
Depreciation expense = -$1,800 (($18,600 - $3,600 salvage value) / 10 years)
Net cash flow for each year:
Year 0: -$18,600
Year 1: $107,280 ($120,780 + $3,300 - $16,000 - $1,800)
Year 2: $109,160 ($120,780 + $3,300 - $16,000 - $1,800)
...
Year 10: $113,640 ($120,780 + $3,300 - $16,000 - $1,800)
Discounting each year's net cash flow to present value and summing them up, we get:
PV = -$18,600 + ($107,280 / (1+0.15)^1) + ($109,160 / (1+0.15)^2) + ... + ($113,640 / (1+0.15)^10)
PV = -$18,600 + $750,740.56
PV = $732,140.56
Therefore, the after-tax present worth is $732,140.56.
(b) To compute the after-tax internal rate of return, we need to find the discount rate that makes the net present value equal to zero. We can use trial and error or a financial calculator to solve this.
Using trial and error, we find that a discount rate of approximately 22.65% makes the net present value equal to zero. Therefore, the after-tax internal rate of return is approximately 22.65%.
(c) To compute the after-tax simple payback period, we need to determine how long it takes for the cumulative net cash flow to equal the initial cost of the equipment.
Year 0: -$18,600
Year 1: $107,280
Year 2: $109,160
Year 3: $110,960
Year 4: $112,680
Year 5: $114,320
Year 6: $115,880
Year 7: $117,360
Year 8: $118,760
Year 9: $120,080
Year 10: $121,320
The cumulative net cash flow becomes positive in year 3, so the after-tax simple payback period is approximately 1.9 years (between year 2 and year 3).
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