Show that every element of Sn can be written as a product of transpositions of the form (1, k) for 2sksn. (Assume that n >1 so that you don't have to worry about the philosophical challenges of Si-t0) [Hint: why is it enough to show that this is true for transpositions?]

If the given square is named as ABCD, and BD is the diagonal , we have proved that the angles ∠ABD and ∠ADB are congruent.

Since ABCD is a square, all four angles are right angles (90 degrees).

Let's call the intersection of the diagonals AC and BD point E.

We are given that diagonal BD is between B and D.

Now, let's look at triangle ABD.

Since ABCD is a square, we know that AD and AB are congruent sides of the triangle, and therefore angles ABD and ADB must also be congruent (since they are opposite angles).

Now, we can focus on triangle ADB.

We know that the sum of the angles in any triangle is 180 degrees.

Therefore, we have:

∠ADB + ∠ABD + ∠BAD = 180 degrees

Since we know that ∠ABD and ∠BAD are both right angles (90 degrees), we can substitute these values into the equation above to get:

∠ADB + 90 + 90 = 180 degrees

Simplifying this equation, we get:

∠ADB = 90 degrees

Therefore, we have shown that in the square ABCD, the angles ∠ABD and ∠ADB are congruent.

Hence, we have proved that if diagonal BD is between B and D, then the angles ∠ABD and ∠ADB are congruent.

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Answer:

y = 7x

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

m = 7

The Y-intercept is located at (0,0)

So, the equation is y = 7x

The sum of the convergent geometric series is -81/5. To determine whether the geometric series is convergent or divergent, we need to find the common ratio (r) and analyze it. The series is given by:

Σ (5n - 1) from n=2 to infinity

First, let's find the first two terms of the series:

Term 1 (n=2): 5(2) - 1 = 9
Term 2 (n=3): 5(3) - 1 = 14

Now, we'll find the common ratio (r):

r = Term 2 / Term 1 = 14 / 9

Since the absolute value of the common ratio is less than 1 (|14/9| < 1), the geometric series is convergent.

To find the sum of the convergent series, we'll use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum, a is the first term, and r is the common ratio. In this case, a = 9 and r = 14/9.

S = 9 / (1 - 14/9) = 9 / (-5/9) = 9 * (-9/5) = -81/5

Therefore, the sum of the convergent geometric series is -81/5.

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(a) M = ∬[R] ρ(x, y) dA. (b) x = (1/M) * ∬[R] x * ρ(x, y) dA y = (1/M) * ∬[R] y * ρ(x, y) dA. (c) The moment of inertia (I_x) about the x-axis can be found using the following integral expression: I_x = ∬[R] y^2 * ρ(x, y) dA

To set up integral expressions for the mass, center of mass, and moment of inertia about the x-axis, let's consider an object with density function ρ(x,y) in a region R on the xy-plane.

(a) The mass (M) of the object can be found using the following integral expression:
M = ∬[R] ρ(x, y) dA

(b) To find the center of mass, we need to find the coordinates (x, y) using the following integral expressions:
x = (1/M) * ∬[R] x * ρ(x, y) dA
y = (1/M) * ∬[R] y * ρ(x, y) dA

(c) The moment of inertia (I_x) about the x-axis can be found using the following integral expression:
I_x = ∬[R] y^2 * ρ(x, y) dA

These integral expressions provide a foundation for finding the mass, center of mass, and moment of inertia about the x-axis for a given object with a specified density function ρ(x, y) in the region R. To evaluate these expressions, you'll need to know the density function and region for the specific problem you're working on.

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Step-by-step explanation:

x=3 y=7

-3 -7

x-3=0 y-7=0

r = 5

(x-3)² + (y-7)² = 25

using the rule (a - b)² = a² - 2ab + b²

x² - 6x + 9 + y² - 14y + 49 = 25

x² - 6x + y² - 14y + 58 = 25

-25 -25

x² - 6x + y² - 14y + 33 = 0

Answer:

(x, y) (-1/2, 13/2)

Step-by-step explanation:

since y is both equal to these eqaution we can set the equation eqaul together.

3x+8=5x+9

2x = -1

x = -1/2

y = 3(-1/2) + 8

y = 13/2

y = -7t cos(t) - 7t is the solution of the equation dy = y + 7t² sin(t) as it satisfies the differential equation and the initial condition.

To verify that y = -7t cos(t) - 7t is a solution of the initial-value problem dy/dt = y + 7t² sin(t) with y(π) = 0, we need to check that y satisfies the differential equation and the initial condition.

First, we can calculate the derivative of y with respect to t as follows:

dy/dt = d/dt (-7t cos(t) - 7t)
= -7 cos(t) - 7 + (-7t)(-sin(t))
= -7(cos(t) + t sin(t))

Next, we can substitute y and dy/dt into the differential equation and simplify:

dy/dt = y + 7t² sin(t)
-7(cos(t) + t sin(t)) = (-7t cos(t) - 7t) + 7t² sin(t)
-7 cos(t) - 7 + 7t sin(t) = -7t cos(t) - 7t + 7t² sin(t)
-7 cos(t) - 7 = -7t cos(t) - 7t + 7t² sin(t) - 7t sin(t)
-7 cos(t) - 7 = -7t(cos(t) + sin(t)) + 7t² sin(t)

This equation is true for all t, so we have verified that y = -7t cos(t) - 7t is a solution of the differential equation.

Finally, we need to check the initial condition. Since y(π) = -7π cos(π) - 7π = 0, the initial condition is satisfied.

Therefore, we have confirmed that y = -7t cos(t) - 7t is a solution of the initial-value problem dy/dt = y + 7t²sin(t) with y(π) = 0.

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The absolute maximum value of f on the interval [0.2, 20] is 625 and the absolute minimum value of f on the interval [0.2, 20] is 5.04.

To find the absolute maximum and absolute minimum values of f on the given interval, we need to first find the critical points of f and then compare the values of f at these critical points and at the endpoints of the interval.

To find the critical points, we need to find where the derivative of f is equal to zero or undefined. Taking the derivative of f, we get:

f'(x) = 1 + 25 = 0
No solution, so the derivative is never equal to zero.

f'(x) is defined for all x in the interval [0.2, 20]. Therefore, the only critical points are the endpoints of the interval.

To find the value of f at the endpoints, we evaluate f(0.2) and f(20):

f(0.2) = (0.2)^2 + 25(0.2) = 5.04
f(20) = (20)^2 + 25(20) = 625

Comparing the values of f at the critical points and the endpoints, we can conclude that the absolute maximum value of f on the interval [0.2, 20] is 625 and the absolute minimum value of f on the interval [0.2, 20] is 5.04.

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The three infinite lines of charge, with densities of +3 (nC/m), -3 (nC/m), and +3 (nC/m), respectively, are parallel to the z-axis and pass through specific points.

To determine the electric field at a point, we need to use Coulomb's law and integrate over the length of each line of charge.

The direction of the electric field is perpendicular to the line of charge, and the magnitude is proportional to the charge density and inversely proportional to the distance from the point to the line of charge. The final result will be a vector sum of the electric fields due to each line of charge.

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complete question:

Three infinite lines of charge, rhol1 = 3 (nC/m), rhol2 = −3 (nC/m), and rhol3 = 3 (nC/m), are all parallel to the z-axis. If they pass through the respective points determine the nature of electric field.

The answer is option a i.e. A.

Hi! I'm happy to help with your question involving points, closest, and the xz-plane.

To determine which point is closest to the yz-plane, we need to look at the x-coordinate of each point. The yz-plane is where x = 0, so the point with the smallest absolute value of the x-coordinate is closest. Comparing the x-coordinates:
A(-4, 0, -4) -> |-4| = 4
B(3, 4, -3) -> |3| = 3
C(2, 3, 7) -> |2| = 2

C has the smallest absolute value of the x-coordinate, so it is closest to the yz-plane. Therefore, the answer is c. C.

To determine which point lies in the xz-plane, we need to look at the y-coordinate of each point. A point lies in the xz-plane when its y-coordinate is 0. Checking the y-coordinates:
A(-4, 0, -4) -> y = 0
B(3, 4, -3) -> y ≠ 0
C(2, 3, 7) -> y ≠ 0

Only point A has a y-coordinate of 0, so it lies in the xz-plane. Therefore, the answer is a. A.

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Answer:

Step-by-step explanation:

1 is 58

2 is 90

Therefore, the probability mass function of N is:

[tex]P(N=3) = p*(1-p)\\P(N=4) = p*p*(1-p) + (1-p)*(1-p)*p\\P(N=5) = p*p*p*(1-p) + (1-p)*(1-p)*(1-p)*p + 2*p*(1-p)*p*(1-p)\\P(N=6) = p*p*p*p*(1-p) + (1-p)*(1-p)*(1-p)*(1-p) + 3*p*p*(1-p)*(1-p) + \    \ \ \  2*p*(1-p)*p*p*(1-p) + 2*p*p*(1-p)*p*(1-p) \\[/tex]

And so on, for larger values of N.

To find the probability mass function of N, we need to consider the two cases mentioned in the question.

Case 1: A sequence of events followed by an odd.
For this case, the probability of rolling an even number on the first roll is p. The probability of rolling the same even number on the second roll is also p. The probability of rolling an odd number on the third roll is (1-p) because the even numbers have been exhausted. So, the probability of this specific sequence of rolls occurring is p*p*(1-p).

Case 2: A sequence of odds followed by an even.
For this case, the probability of rolling an odd number on the first roll is 1-p. The probability of rolling the same odd number on the second roll is also 1-p. The probability of rolling an even number on the third roll is p because the odd numbers have been exhausted. So, the probability of this specific sequence of rolls occurring is (1-p)*(1-p)*p.

We can then find N's overall probability mass function by adding the probabilities of all possible sequences that lead to observing both an even and an odd number.

The smallest value in support of N must be 3, since it takes at least 3 rolls to observe both an even and an odd number.

Therefore, the probability mass function of N is:

[tex]P(N=3) = p*(1-p)\\P(N=4) = p*p*(1-p) + (1-p)*(1-p)*p\\P(N=5) = p*p*p*(1-p) + (1-p)*(1-p)*(1-p)*p + 2*p*(1-p)*p*(1-p)\\P(N=6) = p*p*p*p*(1-p) + (1-p)*(1-p)*(1-p)*(1-p) + 3*p*p*(1-p)*(1-p) + \    \ \ \  2*p*(1-p)*p*p*(1-p) + 2*p*p*(1-p)*p*(1-p) \\[/tex]

And so on, for larger values of N.

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If the cost of medical care increases by 40 percent, then, other things the same, the CPI is likely to increase by about:
Your answer: 2.4 Percent

Reason:

The CPI (Consumer Price Index) is a measure of the average change over time in the prices paid by consumers for a basket of goods and services. Medical care is just one component of this basket. If the cost of medical care increases by 40%, it will contribute to the overall increase in the CPI, but the impact will be less than the 40% increase, as other components of the basket will not necessarily increase at the same rate. Based on the given options,

the most likely increase in the CPI is 2.4%.

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The answer to the given composition function is: fog(4) is 2/17

Composition of functions is a mathematical operation that involves applying one function to the output of another function, resulting in a new function.

Given two functions f(x) and g(x), the composition of f and g, denoted as (fog)(x), is defined as:

    (fog)(x) = f(g(x))

Applying this knowledge to the question given, then:

(a) (fog)(4) = f(g(4)) = f(2/(4²+1)) = f(2/17) = |2/17| = 2/17

(b) (gof)(2) = g(f(2)) = g(|2|) = g(2) = 2/(2²+1) = 2/5

(c) (fof)(1) = f(f(1)) = f(|1|) = f(1) = |1| = 1

(d) (gog)(0) = g(g(0)) = g(2/(0²+1)) = g(2) = 2/(2²+1) = 2/5

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The volume of the solid obtained by rotating y=1/x from x=1 to infinity about x-axis is finite, while its surface area is infinite.

To show that the volume of the solid obtained by rotating the portion of y=1/x from x=1 to infinity about the x-axis is finite,

we can use the formula for the volume of a solid of revolution:

V = π∫(b, a) y² dx

where y is the distance from the curve to the axis of rotation, and a and b are the limits of integration.

For the curve y = 1/x, the limits of integration are from 1 to infinity, and the distance from the curve to the x-axis is y, so we have:

Therefore, the volume of the solid is π, which is a finite value.

To show that the surface area of the solid is infinite, we can use the formula for the surface area of a solid of revolution:

S = 2π∫(b, a) y √(1 + (dy/dx)²) dx

For the curve y = 1/x, we have dy/dx = -1/x²,

so we can write:

Making the substitution u = 1/x², we get:

Therefore, the surface area of the solid is infinite.

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The Wronskian of the functions e^x and e^3x is :

2e^4x

The Wronskian is a mathematical concept used in the theory of ordinary differential equations to determine if a set of functions is linearly independent.

The Wronskian of the functions e^x and e^3x is given by the determinant of a matrix formed using these functions and their derivatives. Here's the calculation:

Wronskian(W) = | e^x    e^3x  |
                      | (d/dx)e^x (d/dx)e^3x |

Wronskian(W) = | e^x    e^3x  |
     | e^x    3e^3x |

Wronskian(W) = (e^x)(3e^3x) - (e^3x)(e^x) = 2e^4x

So, the Wronskian of the functions e^x and e^3x is 2e^4x.

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To find the least squares solution of the system Ax = b, we first need to find the pseudoinverse of A (denoted as A+). Then, we can use the formula x = A+ b to find the least squares solution.

To find the pseudoinverse of A, we can use the Moore-Penrose inverse formula:

A+ = (A^T A)^-1 A^T

where A^T is the transpose of A.

Using this formula, we get:

A^T A =
1 0 3 0
0 10 1 4
3 1 2 2
0 4 2 2
0 0 0 6
1 -1 0 0

Taking the inverse of A^T A, we get:

(A^T A)^-1 =
0.0447 -0.0206 0.0358 -0.0323 -0.0171 0.0478
-0.0206 0.0111 -0.0115 0.0074 0.0035 -0.0155
0.0358 -0.0115 0.0505 -0.0395 -0.0125 0.0383
-0.0323 0.0074 -0.0395 0.0356 0.0082 -0.0295
-0.0171 0.0035 -0.0125 0.0082 0.0068 -0.0099
0.0478 -0.0155 0.0383 -0.0295 -0.0099 0.0451

Multiplying A^T and b, we get:

A^T b =
1
1
-1
-1
1
-2

Using the formula x = A+ b, we get:

x =
0.2
0.1
-0.6

Therefore, the least squares solution of the system Ax = b is:

x = (0.2, 0.1, -0.6)

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The absolute maximum of f(x) on [−1, 3] is (−1, 11), and the absolute minimum is (2, −5)

To find the absolute maximum and minimum of a function on a closed interval [a, b], we need to evaluate the function at its critical points (where the derivative is zero or undefined) and at the endpoints of the interval, and then compare the values.

First, we find the derivative of f(x):

f'(x) = 3x^2 - 4x - 4

Setting f'(x) = 0 to find the critical points:

3x^2 - 4x - 4 = 0

Using the quadratic formula, we get:

x = (-(-4) ± sqrt((-4)^2 - 4(3)(-4)))/(2(3))

x = (-(-4) ± sqrt(64))/6

x = (-(-4) ± 8)/6

x = -2/3 or x = 2

Next, we evaluate f(x) at the critical points and the endpoints of the interval:

f(-1) = 11

f(3) = 10

f(-2/3) = 22/27

f(2) = -5

Therefore, the absolute maximum of f(x) on [−1, 3] is (−1, 11), and the absolute minimum is (2, −5)

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The absolute maximum of f(x) on [−1, 3] is (−1, 11), and the absolute minimum is (2, −5)

To find the absolute maximum and minimum of a function on a closed interval [a, b], we need to evaluate the function at its critical points (where the derivative is zero or undefined) and at the endpoints of the interval, and then compare the values.

First, we find the derivative of f(x):

f'(x) = 3x^2 - 4x - 4

Setting f'(x) = 0 to find the critical points:

3x^2 - 4x - 4 = 0

Using the quadratic formula, we get:

x = (-(-4) ± sqrt((-4)^2 - 4(3)(-4)))/(2(3))

x = (-(-4) ± sqrt(64))/6

x = (-(-4) ± 8)/6

x = -2/3 or x = 2

Next, we evaluate f(x) at the critical points and the endpoints of the interval:

f(-1) = 11

f(3) = 10

f(-2/3) = 22/27

f(2) = -5

Therefore, the absolute maximum of f(x) on [−1, 3] is (−1, 11), and the absolute minimum is (2, −5)

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x = 4 and x = -1/2 are real zeros of the polynomial

To find all the real zeros of the polynomial p(x) = 2x³ - 15x² + 24x + 16, we can follow these steps:

Step 1: Identify potential rational zeros using the Rational Root Theorem.
The Rational Root Theorem states that any potential rational zeros will be of the form ±p/q, where p is a factor of the constant term (16) and q is a factor of the leading coefficient (2). In this case, the possible rational zeros are ±1, ±2, ±4, ±8, ±1/2, ±2/2 (±1), and ±4/2 (±2).

Step 2: Test each potential rational zero using synthetic division.
We can use synthetic division to test each potential rational zero. If the remainder is 0, the potential rational zero is a real zero of the polynomial.

Step 3: Check for any irrational zeros using the quadratic formula.
If we find a quadratic factor during synthetic division, we can use the quadratic formula to find any remaining irrational zeros.

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12 = 2 x 2 x 3

18 = 3 x 3 x 2

32 = 2 x 2 x 2 x 2 x 2

100 = 2 x 2 x 5 x 5

140 = 2 x 2 x 5 x 7

76 = 2 x 2 x 19

75 = 3 x 5 x 5

45 = 3 x 3 x 5

42 = 2 x 3 x 7

110 = 2 x 5 x 11

The system of recurrence relations is:

[tex]$a_n = 4a_{n-1}$[/tex] for [tex]$n \geq 2$[/tex], with initial conditions [tex]$a_1 = 0$[/tex] (there are no 2's in a 1-digit sequence) and [tex]$a_2 = 1$[/tex] (the only 2-digit sequence that satisfies the conditions is 23).

Let[tex]$a_n$[/tex] be the number of n-digit quaternary sequences that contain an even number of 2's and an odd number of 3's. We can find a recurrence relation for [tex]$a_n$[/tex] as follows:

Case 1: The last digit is 0, 1, or 3. In this case, the parity of the number of 2's and 3's in the sequence remains the same. Therefore, the number of (n-1)-digit sequences that satisfy the conditions is [tex]$a_{n-1}$[/tex].

Case 2: The last digit is 2. In this case, the parity of the number of 2's changes from even to odd, and the parity of the number of 3's remains odd. Therefore, the number of (n-1)-digit sequences that end in 0, 1, or 3 and satisfy the conditions is [tex]$3a_{n-1}$[/tex], and the number of (n-1)-digit sequences that end in 2 and have an even number of 2's and an even number of 3's is $a_{n-1}$. Therefore, the number of n-digit sequences that end in 2 and satisfy the conditions is [tex]$a_n = 3a_{n-1} + a_{n-1} = 4a_{n-1}$[/tex].

Therefore, the system of recurrence relations is:

[tex]$a_n = 4a_{n-1}$[/tex] for [tex]$n \geq 2$[/tex], with initial conditions [tex]$a_1 = 0$[/tex] (there are no 2's in a 1-digit sequence) and [tex]$a_2 = 1$[/tex] (the only 2-digit sequence that satisfies the conditions is 23).

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Option B is correct. The volume of the solid created is approximately 2.356 cubic units.

We can use the disk method.

First, we need to find the equation of the line passing through the points (1,0) and (1,1), which is simply x=1.

Next, we can find the equation of the line passing through the points (3,1) and (1,1) using the slope-intercept form: y - 1 = (1-1)/(3-1)(x-3) => y = -x/2 + 2

Now, we can find the points of intersection of the two lines:

x = 1, y = -x/2 + 2 => (1, 3/2)

Using the disk method, we can find the volume of the solid as follows:

V = ∫[1,3] πy² dx

= ∫[1,3] π(-x/2 + 2)² dx

= π∫[1,3] (x²- 4x + 4)dx/4

= π[(x³/3 - 2x² + 4x)] [1,3]/4

= π(3/4)

= 0.75π

Hence, volume of the solid created is 2.356 cubic units. Answer is closest to option B.

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The complete sentences are

Draw a segment 6 centimeters long.

Then from one endpoint, draw a 30° angle.

To construct a triangle, we need to know the length of three sides, the length of two sides and the measure of the angle between them, or the length of one side and the measure of the two adjacent angles.

In the given problem we know the length of the sides hence, we can follow the given steps to draw the required triangle

Here we have

Equal length of the side of the triangle = 6 cm

Since the sides are equal the resultant triangle will be an equilateral triangle

To draw a triangle with all side lengths equal to 6 centimeters, we need to follow these steps:

Once all three sides are drawn, you can verify that the triangle is equilateral by measuring the length of each side.

Therefore,

The complete sentences are

Draw a segment 6 centimeters long.

Then from one endpoint, draw a 30° angle.

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In a case whereby Ken and Hamid run around a track where it take Ken 80 seconds to complete a lap It take Hamid 60 seconds to complete a lap. the number of  laps they will each have run when they next meet on the start line is that Ken will have run 3 laps and Hamid will have run 4.

The LCM of 80 nd 60 seconnds can be written as 240,  however when 240 seconds go then they will both be at the start line.

So the lap that  Ken will covered in 240s = 240/80 = 3laps

So the lap that  Hamid will covered in 240s = 240/60 = 4laps

Therefore, we can come into conclusion that Ken will have to run   3laps  where Hamid will have run  4Laps.

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Miss kito wins the 45 games and Mr. Fishman wins the 36 games.

(a) The setup of the equations is:

3.5%x + 5.75% ($ 780,000 - x) = $33,600

(b) The farmer invested $500,000 at 3.5% and $280,000 at 5.75%

Miss Kito and Mr. Fishman played 81 games of their favorite 2-player game, 7 Wonders Duel.

We have to find the how many games did they each win?

Let's Miss Kito wins 'x' games

So, the equation will be:

x + (x - 9) = 81

2x - 9 = 81

2x = 90

x = 45

And, Mr. Fishman = 45 - 9 = 36

Miss kito wins the 45 games and Mr. Fishman wins the 36 games.

(a) A farmer bought a scratch ticket and found out later that he won $1,200,000. After 35% was deducted for income taxes he invested the rest; some at 3.5% and some at 5.75% .

$1,200,000 × (1 - 3.5%)= $780,000

Suppose that he invested x at 35%

and ($ 780,000 - x) at 5.75%

3.5%x + 5.75% ($ 780,000 - x) = $33,600

(b) 3.5% + 5.75%($ 780,000 - x) = $33,600

3.5%x - 5.75% + 44,850 = 33,600

2.25%x = $11,250

x = $500,000

=> $780,000 - $500,000

= $280,000

So, the farmer invested $500,000 at 3.5% and $280,000 at 5.75%

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The given question is incomplete, complete question is :

Miss Kito and Mr. Fishman played 81 games of their favorite 2-player game, 7 Wonders Duel. Miss KIto ultimately won 9 more games than Mr. Fish did. How many games did they each win?

A farmer bought a scratch ticket and found out later that he won $1,200,000. After 35% was deducted for income taxes he invested the rest; some at 3.5% and some at 5.75% . If the annual interest earned from his investments is $33,600 find the amount he invest at each rate.

a. Define variables to represent the unknowns and setup the necessary equations to answer the question.

b. [4 points] Algebraically solve the equation you created and express your final answer using a complete sentence and appropriate units. (You will not receive full credit if a trial and error method is used in place of an algebraic method.)

The limit of the given function as t approaches negative infinity is 1.

To evaluate the limit of the given function as t approaches negative infinity, we need to determine the behavior of the function as t becomes increasingly negative.

First, note that as t approaches negative infinity, both the numerator and denominator of the fraction become increasingly negative.

To simplify the expression, we can divide both the numerator and denominator by the highest power of t that appears in the denominator, which is t³

[tex]\lim_{t \to- \infty}[/tex] (3t² + t)/(t³ - 7t + 1)

= [tex]\lim_{t \to- \infty}[/tex] (3/t - 1/t²)/(1 - 7/t² + 1/t³)

As t approaches negative infinity, the dominant term in the denominator is -7/t², which becomes increasingly negative. Therefore, the limit of the denominator as t approaches negative infinity is negative infinity.

Now let's look at the numerator. As t approaches negative infinity, the dominant term in the numerator is 3/t, which becomes increasingly negative. Therefore, the limit of the numerator as t approaches negative infinity is negative infinity.

Using the quotient rule for limits, we can conclude that:

[tex]\lim_{t \to- \infty}[/tex] (3t² + t)/(t³ - 7t + 1) = [tex]\lim_{t \to- \infty}[/tex] (3/t - 1/t²)/(1 - 7/t² + 1/t³) = -[infinity]/-[infinity] = 1

Thus, the limit t approaches negative infinity is 1.

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The homogeneous solution is y_H(x) = C1*cos(2x) + C2*sin(2x)

Given the differential equation y" + 4y = cos(2x), you want to find the homogeneous solution y_H.

To find the homogeneous solution y_H, we need to solve the homogeneous differential equation y" + 4y = 0.

Step 1: Identify the characteristic equation.
The characteristic equation is given by r^2 + 4 = 0, where r represents the roots.

Step 2: Solve the characteristic equation.
To solve the equation r^2 + 4 = 0, we get r^2 = -4. Taking the square root of both sides, we obtain r = ±2i.

Step 3: Write the general solution for the homogeneous equation.
Since we have complex conjugate roots, the general homogeneous solution y_H can be written as:

y_H(x) = C1*cos(2x) + C2*sin(2x)
Here, C1 and C2 are constants determined by the initial conditions.

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