The question states that 1/4 of students at International are in the blue house, with 1/5 votes for Adam and 1/4 for Franklin. To analyze the results, calculate the fraction of votes for each candidate and multiply by the total number of students.
Based on the information provided, 1/4 of the students at International are in the blue house. The vote went as follows: 1/5 of the votes were for Adam, and 1/4 of the votes were for Franklin.
To analyze the vote results, we need to calculate the fraction of votes for each candidate.
Let's start with Adam:
- The fraction of votes for Adam is 1/5.
- To find the number of students who voted for Adam, we can multiply this fraction by the total number of students at International.
Next, let's calculate the fraction of votes for Franklin:
- The fraction of votes for Franklin is 1/4.
- Similar to before, we'll multiply this fraction by the total number of students at International to find the number of students who voted for Franklin.
Remember, we are given that 1/4 of the students are in the blue house. So, if we let "x" represent the total number of students at International, then 1/4 of "x" would be the number of students in the blue house.
To summarize:
- The fraction of votes for Adam is 1/5.
- The fraction of votes for Franklin is 1/4.
- 1/4 of the students at International are in the blue house.
Please note that the question is incomplete and doesn't provide the total number of students or any additional information required to calculate the specific number of votes for each candidate.
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Find the general solution to the following differential equations:
16y''-8y'+y=0
y"+y'-2y=0
y"+y'-2y = x^2
The general solution of the given differential equations are:
y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)
y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)
y = c₁e^x + c₂e^(-2x) + (1/2)x
(for y"+y'-2y=x²)
Given differential equations are:
16y''-8y'+y=0
y"+y'-2y=0
y"+y'-2y = x²
To find the general solution to the given differential equations, we will solve these equations one by one.
(i) 16y'' - 8y' + y = 0
The characteristic equation is:
16m² - 8m + 1 = 0
Solving this quadratic equation, we get m = 1/4, 1/4
Hence, the general solution of the given differential equation is:
y = c₁e^(x/4) + c₂xe^(x/4)..................................................(1)
(ii) y" + y' - 2y = 0
The characteristic equation is:
m² + m - 2 = 0
Solving this quadratic equation, we get m = 1, -2
Hence, the general solution of the given differential equation is:
y = c₁e^x + c₂e^(-2x)..................................................(2)
(iii) y" + y' - 2y = x²
The characteristic equation is:
m² + m - 2 = 0
Solving this quadratic equation, we get m = 1, -2.
The complementary function (CF) of this differential equation is:
y = c₁e^x + c₂e^(-2x)..................................................(3)
Now, we will find the particular integral (PI). Let's assume that the PI of the differential equation is of the form:
y = Ax² + Bx + C
Substituting the value of y in the given differential equation, we get:
2A - 4A + 2Ax² + 4Ax - 2Ax² = x²
Equating the coefficients of x², x, and the constant terms on both sides, we get:
2A - 2A = 1,
4A - 4A = 0, and
2A = 0
Solving these equations, we get
A = 1/2,
B = 0, and
C = 0
Hence, the particular integral of the given differential equation is:
y = (1/2)x²..................................................(4)
The general solution of the given differential equation is the sum of CF and PI.
Hence, the general solution is:
y = c₁e^x + c₂e^(-2x) + (1/2)x²..................................................(5)
Conclusion: Therefore, the general solution of the given differential equations are:
y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)
y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)
y = c₁e^x + c₂e^(-2x) + (1/2)x
(for y"+y'-2y=x²)
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The particular solution is: y = -1/2 x². The general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²
The general solution of the given differential equations are:
Given differential equation: 16y'' - 8y' + y = 0
The auxiliary equation is: 16m² - 8m + 1 = 0
On solving the above quadratic equation, we get:
m = 1/4, 1/4
∴ General solution of the given differential equation is:
y = c1 e^(x/4) + c2 x e^(x/4)
Given differential equation: y" + y' - 2y = 0
The auxiliary equation is: m² + m - 2 = 0
On solving the above quadratic equation, we get:
m = -2, 1
∴ General solution of the given differential equation is:
y = c1 e^(-2x) + c2 e^(x)
Given differential equation: y" + y' - 2y = x²
The auxiliary equation is: m² + m - 2 = 0
On solving the above quadratic equation, we get:m = -2, 1
∴ The complementary solution is:y = c1 e^(-2x) + c2 e^(x)
Now we have to find the particular solution, let us assume the particular solution of the given differential equation:
y = ax² + bx + c
We will use the method of undetermined coefficients.
Substituting y in the differential equation:y" + y' - 2y = x²a(2) + 2a + b - 2ax² - 2bx - 2c = x²
Comparing the coefficients of x² on both sides, we get:-2a = 1
∴ a = -1/2
Comparing the coefficients of x on both sides, we get:-2b = 0 ∴ b = 0
Comparing the constant terms on both sides, we get:2c = 0 ∴ c = 0
Thus, the particular solution is: y = -1/2 x²
Now, the general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²
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a. Find the measure of each interior angle of the regular hendecagon that appears on the face of a Susan B. Anthony one-dollar coin.
The regular hendecagon is an 11 sided polygon. A regular polygon is a polygon that has all its sides and angles equal. Anthony one-dollar coin has 11 interior angles each with a measure of approximately 147.27 degrees.
Anthony one-dollar coin. The sum of the interior angles of an n-sided polygon is given by:
[tex](n-2) × 180°[/tex]
The formula for the measure of each interior angle of a regular polygon is given by:
measure of each interior angle =
[tex][(n - 2) × 180°] / n[/tex]
In this case, n = 11 since we are dealing with a regular hendecagon. Substituting n = 11 into the formula above, we get: measure of each interior angle
=[tex][(11 - 2) × 180°] / 11= (9 × 180°) / 11= 1620° / 11[/tex]
The measure of each interior angle of the regular hendecagon that appears on the face of a Susan B. Anthony one-dollar coin is[tex]1620°/11 ≈ 147.27°[/tex]. This implies that the Susan B.
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The measure of each interior angle of a regular hendecagon, which is an 11-sided polygon, can be found by using the formula:
Interior angle = (n-2) * 180 / n,
where n represents the number of sides of the polygon.
In this case, the regular hendecagon appears on the face of a Susan B. Anthony one-dollar coin. The Susan B. Anthony one-dollar coin is a regular hendecagon because it has 11 equal sides and 11 equal angles.
Applying the formula, we have:
Interior angle = (11-2) * 180 / 11 = 9 * 180 / 11.
Simplifying this expression gives us the measure of each interior angle of the regular hendecagon on the coin.
The measure of each interior angle of the regular hendecagon on the face of a Susan B. Anthony one-dollar coin is approximately 147.27 degrees.
To find the measure of each interior angle of a regular hendecagon, we use the formula: (n-2) * 180 / n, where n represents the number of sides of the polygon. For the Susan B. Anthony one-dollar coin, the regular hendecagon has 11 sides, so the formula becomes: (11-2) * 180 / 11. Simplifying this expression gives us the measure of each interior angle of the regular hendecagon on the coin. Therefore, the measure of each interior angle of the regular hendecagon on the face of a Susan B. Anthony one-dollar coin is approximately 147.27 degrees. This means that each angle within the hendecagon on the coin is approximately 147.27 degrees. This information is helpful for understanding the geometry and symmetry of the Susan B. Anthony one-dollar coin.
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Calculate the eigenvalues of this matrix: [Note-you'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues. You can use the web version at xFunctions. If you select the "integral curves utility" from the main menu, will also be able to plot the integral curves of the associated diffential equations. ] A=[ 22
120
12
4
] smaller eigenvalue = associated eigenvector =( larger eigenvalue =
The matrix A = [[22, 12], [120, 4]] does not have any real eigenvalues.
To calculate the eigenvalues of the matrix A = [[22, 12], [120, 4]], we need to find the values of λ that satisfy the equation (A - λI)v = 0, where λ is an eigenvalue, I is the identity matrix, and v is the corresponding eigenvector.
First, we form the matrix A - λI:
A - λI = [[22 - λ, 12], [120, 4 - λ]].
Next, we find the determinant of A - λI and set it equal to zero:
det(A - λI) = (22 - λ)(4 - λ) - 12 * 120 = λ^2 - 26λ + 428 = 0.
Now, we solve this quadratic equation for λ using a graphing calculator or other methods. The roots of the equation represent the eigenvalues of the matrix.
Using the quadratic formula, we have:
λ = (-(-26) ± sqrt((-26)^2 - 4 * 1 * 428)) / (2 * 1) = (26 ± sqrt(676 - 1712)) / 2 = (26 ± sqrt(-1036)) / 2.
Since the square root of a negative number is not a real number, we conclude that the matrix A has no real eigenvalues.
In summary, the matrix A = [[22, 12], [120, 4]] does not have any real eigenvalues.
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