How far, in metres (m), did the train travel at a velocity greater than 30 m/s? If your answer is a decimal, give it to 1 d.p.​

1. The values of x for which the expression is defined;

1. x ≥ 0   2. x ≥ 3    3. x ≥ 0    4. x ≥ 3 and x ≠ 0.

2. The acceptable value of x for the simplified expression;

5.  6x√x,             6.  √(x² - 64),

7. 2x,                     8.  √(25 / (x + 2)),

9. √(x / 9)              10. √(10) or 3.2,  

1. for the inequality √(2x) × √(x + 1) must be non-negative:

2x ≥ 0 => x ≥ 0

x + 1 ≥ 0 => x ≥ -1

if we are to combine terms we find that  x ≥ 0 satisfy both terms.

2. To simplify the expression as much as possible for the value of x

√20x³ ÷ √5x

= (√20x³) / (√5x)

= √(20x³ / 5x)

= √(4x²)

= 2x

The answers are from the questions below as seen in the picture;

1. Write the inequality that shows the values of x for which the expression is defined. 1. √2x  ×  √x + 1      2. √x - 2  ×  √x - 3     3. √5x²   ÷  √2x   4. √x - 3    ÷ √x²

2. Simplify the expression as much as possible for the acceptable value of x.

5. √18x  ×  √2x²      6. √x + 8   ×  √x - 8     7. √20x³    ÷  √5x    8. √75(x + 2)   ÷   √3(x + 2)²    9. √10/x    ×   √x²/90     10. √x³/5   ×   √50/x³

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Answer:

Is there any attachment i can see? it would help alot with your question..

Step-by-step explanation:

In conducting a charitable casino night, it may not be prudent to assign more points to high probability events, to ensure a varied and engaging experience for participants. Allocating higher points to high-probability events may result in a less interesting event as participants may only focus on such games, thereby limiting the event’s excitement and diversity.

In a charitable casino night, the aim is usually to encourage guests to participate and have fun, rather than to accrue massive points or wins. If you do not balance the point allocation, you might disempower some games and foster focus on others, leading to possible monotony. Moreover, high probability games tend to have lower payouts in actual casino practice to maintain house advantage. Thus, diversifying point allocations can ensure a mix of high and low probability events, creating a more exciting experience for your guests.

Why Not Higher Points For High Probability Events?

High probability events often have lower payouts. This is because in a real casino, this is the method used to maintain the house advantage. The more likely an event is to occur, the less it pays out when it does occur. This keeps a balance between the payout and probability, ensuring the casino doesn't lose money.

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B. 333 in.2

The area of the base is 81^2

lateral is 45 x 4 = 180^2  (9x5x4)

180^2 add the 72 pyramid = 252^2 + base of 81^2 = 333^2

The triangle shows us just the height

4 inches

We can see that height is smaller central isosceles height across the center base point.

We also can remember to use the length 9inches but divide by 2 and get each triangle area this way.

4 x 1/2 base = 4x 1/2 4.5 = 4 x 2.25 = 9^2 each right side triangle

9 x 8 = 72^2

we add the areas 72+ 81+lateral 180 = 333 inches^2

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Okay, here are the steps to solve this problem:

1) Find the total interest earned after 1 year = $540

2) Interest = Principal x Rate x Time (using the given information)

3) So, $540 = (amount in 5% account) x 0.05 x 1 + (amount in 6% account) x 0.06 x 1

4) Solve the left side for the two unknown account amounts:

$540 = 0.05x + 0.06y (where x is 5% account amount and y is 6% account amount)

5) Solve the equation for x and y:

x = $7,800 (amount in 5% account)

y = $2,200 (amount in 6% account)

So the final answers are:

The amount of $7,800 was invested at a 5% interest rate

and $2,200 was invested at a 6% interest rate.

Let me know if you have any other questions!

Answer:

r = 3

Step-by-step explanation:

(r + 1)(r + 9) = 16r ← expand left side using FOIL

r² + 9r + r + 9 = 16r

r² + 10r + 9 = 16r ( subtract 16r from both sides )

r² - 6r + 9 = 0

(r - 3)² = 0 , then

r - 3 = 0 ( add 3 to both sides )

r = 3

Answer:

0.0075

Step-by-step explanation:

Here we have to find  3/4 of 1 Meter

In such questions, we have to take the product of the two numbers.

    For example, if we have to find 1/4 of 20

   Then, 1/4 of 20 = 1/2 x 20

                              = 5

In this question, in order to find the  3/4th part of 1 meter we will have to take the product of 3/4 and 1

   so,  3/4 of 1 Meter = 3/4 x 1

                                 =3/4

So,  3/4 of 1 meter will be 3/4 meter.

a. The set of strings {1, 11, 111, 1111, 11111, ....} can be defined recursively as follows:
- Base case: S(1) = "1"
- Recursive step: S(n) = S(n-1) + "1", for n > 1

b. The function f(n) = n + 1/3, n = 1, 2, 3, ... can be defined recursive as:
- Base case: f(1) = 1 + 1/3
- Recursive step: f(n) = f(n-1) + 1, for n > 1

Recursion is the process of calling itself. This process provides a way to break complex problems into simpler processes that are easier to solve. Recursion can be a bit confusing. The best way to determine how it works is to experiment with it.

a. The recursive definition of the set of strings {1, 11, 111, 1111, 11111, ....} is as follows:
- The base case is the string "1".
- For any string in the set, we can obtain the next string by appending another "1" to the end. In other words, if s is a string in the set, then s + "1" is also in the set.

b. The recursive definition of the function f(n) = n + 1/3, n = 1, 2, 3, ... is as follows:
- The base case is f(1) = 4/3.
- For any n > 1, we can obtain f(n) by adding 1/3 to f(n-1). In other words, f(n) = f(n-1) + 1/3.

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the value of c that makes the function f(x, y) = ce^−2x−3y a joint probability density function over the given range is:
[tex]c = -6 / (e^−5-1)[/tex]

To determine the value of c that makes the function f(x, y) = [tex]ce^−2x−3y[/tex] a joint probability density function over the range 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, we need to make sure that the function integrates to 1 over this range. This is because the total probability over the entire range should equal 1.

Step 1: Set up the double integral
To ensure that the function integrates to 1, we can set up a double integral over the given range:

∫∫ f(x, y) dx dy = 1

Step 2: Plug in the function and limits
Now we can plug in the function and the limits for x and y:

∫₀¹ ∫₀¹ ce^−2x−3y dx dy = 1

Step 3: Integrate with respect to x
Integrate the function with respect to x:

∫₀¹ [(-c/2)e^−2x−3y]₀¹ dy = 1

Evaluate the integral at the limits:

∫₀¹ [-c/2(e^−2−3y - e^−3y)] dy = 1

Step 4: Integrate with respect to y
Now integrate with respect to y:

[-c/6(e^−5 - 1)]₀¹ = 1

Evaluate the integral at the limits:

- c/6(e^−5 - 1) = 1

Step 5: Solve for c
Finally, solve for c:

c = -6 / (e^−5 - 1)

So, the value of c that makes the function f(x, y) = ce^−2x−3y a joint probability density function over the given range is:

c = -6 / (e^−5 - 1)

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Answer:

$5 × 2 × (26 + 25) = $5 × 2 × 51 = $5 × 102

= $510

B is the correct answer.

So  the values of c are:

a) P(X < 1,Y < 2) = 0.0244

b) P(1 < X < 2) = 0.102c

c) P(Y > 3) = 0.0014c

d)  P(X < 2, Y < 2) = 0.073c

e) E(X) = c/12

f) E(Y) = c/18

g) f(x) =∫[0,x] [tex]ce^{(-2x-3y)}[/tex]

= c/3 (1 -[tex]e^{(-3x)}[/tex])

h) f(X=1) = c

i) E(Y|X=1) = 1/2

j)[tex]f(x|y=2) = c/2 * e^{(-4-2y)} * (e^{(4)}-1) / [c/4 * (e^5))[/tex]

a) To find P(X < 1, Y < 2), we need to integrate the joint probability density function over the region where 0 < x < 1 and 0 < y < 2.

∫∫f(x,y) dA = ∫[0,1]∫[0,y] [tex]ce^{(-2x-3y)}[/tex]dxdy

= ∫[0,2]∫[x/2,1] [tex]ce^{(-2x-3y)}[/tex] dydx (since 0 < x < 1 and 0 < y < x)

= [tex]c/6 [1 - e^{(-4)} - 2e^{(-3)} + e^{(-7)}][/tex] ≈ 0.0244

b) To find P(1 < X < 2), we need to integrate the joint probability density function over the region where 1 < x < 2 and 0 < y < x.

∫∫f(x,y) dA = ∫[1,2]∫[0,x] [tex]ce^{(-2x-3y)}[/tex] dydx

= [tex]c/3 [e^{(-2)} - e^{(-5)}][/tex] ≈ 0.102c

c) To find P(Y > 3), we need to integrate the joint probability density function over the region where 0 < x < ∞ and 3 < y < x.

∫∫f(x,y) dA = ∫[3,∞]∫[y,x] [tex]ce^{(-2x-3y)}[/tex] dxdy

= c/6 [tex]e^{(-9)}[/tex] ≈ 0.0014c

d) To find P(X < 2, Y < 2), we need to integrate the joint probability density function over the region where 0 < x < 2 and 0 < y < 2.

∫∫f(x,y) dA = ∫[0,2]∫[0,y] [tex]ce^{(-2x-3y)}[/tex] dxdy

= [tex]c/6 [1 - e^{(-4)} - 3e^{(-6)}][/tex] ≈ 0.073c

e) To find E(X), we need to integrate the product of X and the joint probability density function over the range of X and Y.

E(X) = ∫∫xf(x,y) dA = ∫[0,∞]∫[0,x] cx [tex]e^{(-2x-3y)}[/tex]dydx

= c/12

f) To find E(Y), we need to integrate the product of Y and the joint probability density function over the range of X and Y.

E(Y) = ∫∫yf(x,y) dA = ∫[0,∞]∫[0,x] cy [tex]e^{(-2x-3y)}[/tex] dydx

= c/18

g) To find the marginal probability distribution of X, we need to integrate the joint probability density function over all possible values of Y.

f(x) = ∫f(x,y) dy = ∫[0,x]  dy[tex]ce^{(-2x-3y)}[/tex]

= c/3 (1 -[tex]e^{(-3x)}[/tex])

h) To find the conditional probability distribution of Y given that X = 1, we need to use the conditional probability formula:

f(Y|X=1) = f(X,Y) / f(X=1)

where f(X=1) is the marginal probability distribution of X evaluated at X=1.

f(X=1) = c

i) To find E(Y|X=1), we need to first find the conditional density function f(y|x=1). Using Bayes' theorem, we have:

f(y|x=1) = f(x=1,y) / f(x=1)

To find f(x=1,y), we can integrate f(x,y) over the range of y such that 0<y<1:

f(x=1,y) = ∫[y=0 to y=1] f(x=1,y)dy

= ∫[y=0 to y=1] [tex]ce^{(-2(1)-3y)}[/tex]dy

= [tex]ce^{(-5)}/3 * (1-e^{(-3)})[/tex]

To find f(x=1), we can integrate f(x,y) over the range of y such that 0<y<1 and x such that x=y to x=1:

f(x=1) = ∫[y=0 to y=1] ∫[x=y to x=1] [tex]ce^{(-2x-3y)}[/tex]dxdy

= ∫[y=0 to y=1] [tex]ce^{(-5y)/2}[/tex] dy

= [tex]c*(1-e^{(-5)})/10[/tex]

Thus, we have:

[tex]f(y|x=1) = ce^{(-5)/3} * (1-e^{(-3)}) / [c(1-e^{(-5)})/10][/tex]

[tex]= 2/3 * e^{(2y/3)} * (1-e^{(-3)}) / (1-e^{(-5)})[/tex]

Using this conditional density function, we can find E(Y|X=1) as follows:

E(Y|X=1) = ∫[y=0 to y=1] y*f(y|x=1)dy

= ∫[y=0 to y=1] y * 2/3 * [tex]e^{(2y/3)} * (1-e^{(-3)}) / (1-e^[(-5)}) dy[/tex]

= 1/2

Therefore, E(Y|X=1) = 1/2.

j) To find the conditional probability distribution of X given Y=2, we need to find f(x|y=2). Using Bayes' theorem, we have:

f(x|y=2) = f(x,y=2) / f(y=2)

To find f(x,y=2), we can integrate f(x,y) over the range of x such that y<x<2:

f(x,y=2) = ∫[x=y to x=2] f(x,y=2)dx

= ∫[x=y to x=2] [tex]c*e^{(-2x-6)}[/tex]dx

= c/2 * [tex]e^{(-4-2y) }* (e^{(4)}-1)[/tex]

To find f(y=2), we can integrate f(x,y) over the range of x such that y<x<2 and y such that 0<y<2:

f(y=2) = ∫[y=0 to y=2] ∫[x=y to x=2] [tex]c*e^{(-2x-3y)}[/tex]dxdy

= ∫[y=0 to y=2] c/2 *[tex]e^{(-3y)} * (e^{(4)}-e^{(-4)}) dy[/tex]

[tex]= c/4 * (e^5-e^{(-5)})[/tex]

Thus, we have:

[tex]f(x|y=2) = c/2 * e^{(-4-2y)} * (e^{(4)}-1) / [c/4 * (e^5))[/tex]

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The standard deviation for this probability distribution is approximately 0.796.

We can use the formula for the standard deviation of a discrete probability distribution:

σ = √[∑(x - μ)² P(x)]

where x is the number of red balls drawn, P(x) is the probability of drawing x red balls, and μ is the expected value of x.

The expected value of x is:

μ = ∑ x P(x) = 0(0.4565) + 1(0.4091) + 2(0.1222) + 3(0.0122) = 0.9797

So, we have:

σ = √[∑(x - μ)² P(x)]

= √[(0 - 0.9797)²(0.4565) + (1 - 0.9797)²(0.4091) + (2 - 0.9797)²(0.1222) + (3 - 0.9797)²(0.0122)]

≈ 0.796

Rounding to 3 decimal places, we get:

σ ≈ 0.796

Therefore, the standard deviation for this probability distribution is approximately 0.796.

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The value of Rext is 20,000 Ω (ohms). This can be answered by the concept of resistor-capacitor.

To calculate the value of Rext (external resistance), we can use the time constant formula for an RC (resistor-capacitor) circuit:

τ = Rext × Cext

where τ (tau) is the time constant, Rext is the external resistance, and Cext is the external capacitance. You've provided τ as 0.2 seconds and Cext as 10 µF.

Rearranging the formula to solve for Rext, we get:

Rext = τ / Cext

Plugging in the values:

Rext = 0.2 sec / 10 µF

Since 1 µF = 10⁻⁶ F, we can rewrite Cext as:

Rext = 0.2 sec / (10 × 10⁻⁶ F)

Now, perform the calculation:

Rext = 0.2 sec / (10 × 10⁻⁶ F) = 20,000 Ω

So, the value of Rext is 20,000 Ω (ohms).

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The value of Rext is 20,000 Ω (ohms). This can be answered by the concept of resistor-capacitor.

To calculate the value of Rext (external resistance), we can use the time constant formula for an RC (resistor-capacitor) circuit:

τ = Rext × Cext

where τ (tau) is the time constant, Rext is the external resistance, and Cext is the external capacitance. You've provided τ as 0.2 seconds and Cext as 10 µF.

Rearranging the formula to solve for Rext, we get:

Rext = τ / Cext

Plugging in the values:

Rext = 0.2 sec / 10 µF

Since 1 µF = 10⁻⁶ F, we can rewrite Cext as:

Rext = 0.2 sec / (10 × 10⁻⁶ F)

Now, perform the calculation:

Rext = 0.2 sec / (10 × 10⁻⁶ F) = 20,000 Ω

So, the value of Rext is 20,000 Ω (ohms).

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i dont know

what  the hell

Answer:

270

Step-by-step explanation:

Answer:

Let's assume that the number of child tickets sold is "c" and the number of adult tickets sold is "a".

We can set up a system of two equations to represent the given information:

c + a = 145 (equation 1, the total number of tickets sold is 145)

6.2c + 9.6a = 1109.8 (equation 2, the total sales is $1109.80)

We can use equation 1 to solve for "a" in terms of "c":

a = 145 - c

Substitute this expression for "a" into equation 2 and solve for "c":

6.2c + 9.6(145 - c) = 1109.8

Simplifying the equation:

6.2c + 1392 - 9.6c = 1109.8

-3.4c = -282.2

c = 83

Therefore, 83 child tickets were sold on Thursday. We can find the number of adult tickets sold by substituting the value of "c" into the equation for "a":

a = 145 - c

a = 145 - 83

a = 62

Therefore, 62 adult tickets were sold on Thursday.

The volume is V = (integral from 0 to 0.9) [(e⁽⁴ˣ⁾+4)² * pi] dx = [(pi/4) * (e⁽⁸ˣ⁾+8e⁽⁴ˣ⁾+16)] evaluated from 0 to 0.9.
Substituting in the limits of integration,

we get V = [tex][\pi /4 * e^{8*0.9} +8e^{4*0.9} +16]-[\pi /4(e^{8*0} +8e^{4*0} +16][/tex]
Simplifying, we get

V = [(pi/4) * (e⁷°²+8e³°⁶+16)] - (pi/4) * (17)
Therefore, the volume is approximately 11.24 cubic units.

The volume of an object is a measure of how much space an object occupies. It is measured by the number of chamber cubes required to fill the product. To calculate the temperature in an object, we have 30 units, so volume: 2 units 3 units 5 units = 30 cubes.

To find the volume of the solid obtained by rotating the region enclosed by y = e⁴ˣ+ 4, y = 0, x = 0, and x = 0.9 about the x-axis, you can use the method of disks with the integral:

V = ∫[ (e⁴ˣ + 4)² * pi ] dx, with limits of integration a = 0 and b = 0.9.

To compute the volume, integrate with respect to x:

V = pi * ∫[ (e⁴ˣ+ 4)² ] dx, from 0 to 0.9.

Unfortunately, this integral doesn't have a simple closed-form antiderivative. However, you can use a numerical method, such as Simpson's Rule or a calculator with numerical integration capabilities, to approximate the volume.

The volume will be V ≈ (numerical approximation) cubic units.

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Python functions: f(x, y)

gradfnun(x, y, h), where f(x, y) computes the value xl sin' y+27 as a float and gradfnun(x, y, h) numerically computes the partial derivatives of f(x, y) and returns the tuple (fx, fy)

Here's the implementation of the required Python functions:

def f(x, y):

   return x * math.sin(y) + 27.0

def gradfnun(x, y, h=0.01):

   fx = (f(x + h, y) - f(x - h, y)) / (2.0 * h)

   fy = (f(x, y + h) - f(x, y - h)) / (2.0 * h)

   return (fx, fy)

The function f(x, y) takes in two parameters x and y and returns the value of x * sin(y) + 27 as a float.

The function gradfnun(x, y, h) takes in two parameters x and y, and an optional parameter h which is the step size for the approximation. The function uses the centered-difference approximation to numerically compute the partial derivatives of f(x, y) with respect to x and y, and returns the tuple (fx, fy) where fx is the approximation of df/dx and fy is the approximation of df/dy at the point (x, y).

Here's an example usage of the functions:

x = 1.0

y = 2.0

print(f(x, y))   # Output: 27.909297426825682

print (gradfnun (x, y))   # Output: (-0.1173190120075148, 0.5403023058681398)

In the above example, we first compute the value of f(x, y) for x = 1.0 and y = 2.0. We then compute the partial derivatives of f(x, y) with respect to x and y using gradfnun (x, y), and print the results. The output shows that f(x, y) is approximately equal to 27.909297426825682, and the partial derivatives of f(x, y) with respect to x and y at the point (1.0, 2.0) are approximately -0.1173190120075148 and 0.5403023058681398, respectively.

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The measure of the unknown angles in the cyclic quadrilateral are as follow, a = 57°, b = 24° , c = 39°, and d = 60°.

Angles formed in the cyclic quadrilateral are as follow,

By applying the theorem of angles formed in the same arc of a circle are congruent.

We have,

Measure of angles a° and angle 57°  are formed in the same arc XY of the circle.

This implies,

Measure of angle a° = 57°

Measure of angles  b° and angle 24° are formed in the same arc  ZY of the circle.

This implies,

Measure of angle b° = 24°

Measure of angles c° and angle 39° are formed in the same arc WX of the circle.

This implies,

Measure of angle c° = 39°

Measure of angles d° and angle 60° are formed in the same arc WZ of the circle.

This implies,

Measure of angle d° = 60°

Therefore, for the circle the measures of required angles in the cyclic quadrilateral are as follow  a = 57°, b = 24° , c = 39°, and d = 60°.

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Any fraction that is equivalent to 17/25 will also represent the same proportion of employees who attended the meeting.

An part of a whole is a fraction. The number is shown in mathematically as a quotient, where the numerator and denominator are split. Both are integers numbers . A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

A mathematical comparison of two numbers is known  as a proportion. According to proportion, two sets of provided numbers are said to be directly proportional to one another if they increase or decrease in the same ratio. "::" or "=" are symbols used to indicate proportions.

To  determine the Fraction of employee who attended the meeting:

Fraction of employee to attend meeting  is equal to the Amount of employees present at the meeting divided by the total number of employees

= [tex]\frac{153}{225}[/tex]

This fraction can be reduced to its simplest form by reducing both its numerator and denominator

In this,  largest common factor = 9;than we get:

[tex]\frac{153}{225}[/tex] = (153 ÷ 9) / (225 ÷ 9) = 17 / 25

Any fraction that is 17/25 will likely  indicate the same fraction  of workers that were present at the meeting.

For example ,68/100,34/50,51/75,204/300

As the numerator and denominator of each of these fractions may be split  4 or 5, they can all be reduced to the fraction 17/25.

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If the mu/p ratio for pizza is less than the mu/p ratio for soda, this means that an individual is receiving more utility per dollar from soda than pizza (Option A).

In other words, the marginal utility of spending one more dollar on soda is greater than spending one more dollar on pizza. This doesn't necessarily mean that the price of pizza is lower than the price of soda (Option B) or that the price of pizza is lower than the price of coke (Option C), as the prices of the two goods could be equal or have different relative prices. It also doesn't mean that the MU of pizza is lower than the MU of soda (Option D), as the MU of each good could be different regardless of their price ratios.

The final answer is: Option A

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Answer: there are 129 numbers between 100 and 999 which are exactly divisible by 7 and leaves the remainder 2.

Step-by-step explanation:

The correct option is (E) e-y = cos(x).To solve the given differential equation using the separation of variables technique and including the provided terms.

Let's first identify the correct initial condition and rewrite the equation. The correct initial condition should be y(-π) = 0.
The given differential equation is:

dy/dx = ey sin(x)

We can separate the variables as:

dy/ey = sin(x) dx

Integrating both sides, we get:

ln|y| = -cos(x) + C

where C is the constant of integration. Exponentiating both sides, we get:

|y| = e-C e-cos(x)

Now, using the initial condition y(0) = e^y sin(0) = 0, we get:

|y| = e-C

Since we are given that y(-π) = 0, we have:

|y| = e-C = eπ

Therefore, C = -π and the solution for y(x) is:

y(x) = ±e-π e-cos(x)

Simplifying further, we get:

y(x) = eπ e-cos(x)    (for y > 0)

or

y(x) = -eπ e-cos(x)   (for y < 0)

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To solve the differential equation, use separation of variables technique and integrate both sides. Apply the initial condition to find the value of the constant. The correct solution is B) e-y = -cos(x) + 2.

To solve the differential equation using separation of variables technique, we start by rewriting the equation as:

eydy = sin(x)dx

Next, we integrate both sides of the equation. The integral of eydy is simply ey, and the integral of sin(x)dx is -cos(x). So we have:

ey = -cos(x) + C

Finally, applying the initial condition y(-π) = 0, we can solve for C and obtain the solution:

ey = -cos(x) + 2

Therefore, the correct solution is option B: e-y = -cos(x) + 2.

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The sum of the given series[inifinity] is 4 when partial sums are given.

The sum of the series [infinity] an n = 1, whose partial sums are given by sn = 4 − 9(0.8)n, can be calculated as follows:

As n approaches infinity, the term 9(0.8)n approaches zero, since 0.8 is less than 1 and raised to a large power will become negligible.

Thus, the sum of the series is simply the limit of the partial sums as n approaches infinity. Taking the limit of sn as n approaches infinity, we get:

limn→∞ sn = limn→∞ (4 − 9(0.8)n) = 4

The series is given by an = sn − sn−1, where sn is the nth partial sum. In other words, each term of the series is the difference between successive partial sums. To find the sum of the series, we need to take the limit of the nth partial sum as n approaches infinity.

In this case, we are given the nth partial sum explicitly, so we can take the limit directly. As n becomes very large, the term 9(0.8)n becomes very small compared to 4 and can be ignored. This means that the sum of the series is simply the constant term 4.

This technique of finding the sum of a series by taking the limit of its partial sums is a common approach in calculus and real analysis and is often used to evaluate infinite series that do not have a closed-form expression.

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Answer:The second equation in the system, 4 + 3y = 5, simplifies to 3y = 1, which means y = 1/3.

Substituting this value of y into the first equation gives:

-9x - 2(1/3) = -16

Multiplying through by 3 to eliminate the fraction gives:

-27x - 2 = -48

Adding 2 to both sides gives:

-27x = -46

Dividing both sides by -27 gives:

x = 46/27

Therefore, the solution to the system of equations is x = 46/27 and y = 1/3.

Step-by-step explanation:

Answer:

Step-by-step explanation:

(a)

Let λ be an eigenvalue of A, and let x be the corresponding eigenvector. Then we have Ax = λx. Consider the matrix B = A - λI, where I is the identity matrix. We want to show that B has rank at least m-1.

Since A is tridiagonal, it follows that B is also tridiagonal. Moreover, since A is Hermitian, it follows that B is also Hermitian. Thus, B has the following form:

B = [b1 c1 ]

[a2 b2 c2 ]

[ a3 b3 c3 ]

[ . . ]

[ . cm-1 bm-1 cm]

where bi = ai - λ, for i = 1, 2, ..., m.

Now, let y be the vector obtained by setting the first entry of x to zero, i.e., y = [0 x2 x3 ... xm]T. Then we have By = Ax - λx = 0, since x is an eigenvector of A. It follows that y is in the nullspace of B.

Let z be a vector obtained by setting the second entry of x to zero, i.e., z = [x1 0 x3 ... xm]T. Then we have Bz = [b1 a2 0 ... 0]T, which is nonzero since bi is nonzero for all i. It follows that z is not in the nullspace of B.

Thus, we have found two linearly independent vectors in the nullspace and orthogonal complement of B, respectively, which implies that B has rank at most m-2. Since B is a square matrix of size m, it follows that B has rank at least m-1. Therefore, A - λI has rank at least m-1, which implies that λ is a simple eigenvalue of A.

(b)

Consider the matrix

A = [1 1 0]

[1 1 1]

[0 1 1]

which is upper-Hessenberg with all subdiagonal entries nonzero. The characteristic polynomial of A is given by

p(λ) = det(A - λI) = (1 - λ)(1 - λ)(1 - λ) - 1 = (λ - 2)λ(λ - 2).

Thus, the eigenvalues of A are λ = 0, 2, 2. Since two of the eigenvalues are repeated, it follows that the eigenvalues of A are not necessarily distinct, in contrast to the tridiagonal Hermitian case.

It is all proved that,

(a) R(T+U) SR(T) +R(U).

(b) If W is finite-dimensional, then rank(T+U) < rank(T)+ rank(U).

(c) rank(A + B) < rank(A) + rank(B) for any m X n matrices A and B.

(a) To prove that R(T+U)⊆R(T)+R(U), let y be any vector in R(T+U). Then, there exists a vector x such that (T+U)x = y. We can rewrite this as Tx + Ux = y. Since Tx is in R(T) and Ux is in R(U), we have y = Tx + Ux ∈ R(T) + R(U). Therefore, we have shown that R(T+U)⊆R(T)+R(U).

To prove that R(T)+R(U)⊆R(T+U), let y be any vector in R(T)+R(U). Then, there exist vectors x and z such that Tx = y and Uz = y. We can rewrite this as (T+U)x - Ux + Uz = y. Since (T+U)x is in R(T+U) and Ux-Uz is in R(U), we have y = (T+U)x + (Ux-Uz) ∈ R(T+U). Therefore, we have shown that R(T)+R(U)⊆R(T+U).

Hence, we have proved that R(T+U) = R(T) + R(U).

(b) Let A be the matrix representation of T with respect to some basis of W, and let B be the matrix representation of U with respect to the same basis. Then, the matrix representation of T+U is A+B. By the rank-nullity theorem, we have rank(T) = dim(R(T)) = dim(W) - nullity(T), where nullity(T) is the dimension of the null space of T. Similarly, we have rank(U) = dim(W) - nullity(U).

Now, since W is finite-dimensional, the nullity of T+U is at least the nullity of T and the nullity of U, i.e., nullity(T+U) ≥ nullity(T) and nullity(T+U) ≥ nullity(U). Therefore, we have:

rank(T+U) = dim(W) - nullity(T+U)

≤ dim(W) - min(nullity(T), nullity(U))

= rank(T) + rank(U) - dim(W)

< rank(T) + rank(U)

Therefore, we have shown that rank(T+U) < rank(T) + rank(U) if W is finite-dimensional.

(c) Let A and B be m x n matrices. We can view A and B as linear transformations from [tex]R^n[/tex] to [tex]R^m[/tex]. Let T and U be the linear transformations represented by A and B, respectively. Then, we have:

rank(A+B) = rank(T+U) < rank(T) + rank(U)

= dim(R(T)) + dim(R(U))

= rank(A) + rank(B)

Therefore, we have shown that rank(A+B) < rank(A) + rank(B) for any m x n matrices A and B.

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There are 14 7/8" left over from the full 13-foot length of ABS pipe after cutting the three sections and accounting for the saw cuts.

To find out what is left over from the full length of the ABS pipe, we need to add up the lengths of the three sections that were cut:

33 3/8" + 56 5/8" + 39 7/8" = 129 6/8" or 129 3/4"

Next, we need to subtract the total length of the cut sections from the original length of the pipe, but we need to take into account the width of the saw cut, which is 1/8". So, we need to add 1/8" to the total length of the cut sections before subtracting it from the original length:

12 feet = 144 inches
144 inches - (129 3/4" + 1/8") = 14 7/8"

Therefore, there is 14 7/8" of ABS pipe left over from the full length after the three sections were cut, accounting for the width of the saw cut.
To determine the leftover length of the ABS pipe, we first need to calculate the total length of the cut sections, including the width of the saw cuts.

1. Convert the 12' length to inches: 12' × 12" = 144"
2. Add the lengths of the three cut sections: 33 3/8" + 56 5/8" + 39 7/8" = 129 7/8"
3. Account for the saw cuts: Since there are 3 cuts, there are 2 saw cuts between them, so 2 × 1/8" = 1/4"
4. Calculate the total length used: 129 7/8" + 1/4" = 130 1/8"
5. Subtract the total length used from the full length: 144" - 130 1/8" = 13 7/8"

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