What is the equation of a circle with center (-3,-5) and radius 4?
A. (x-3)2 + (y- 5)² = 16
B. (x+3)2 + (y+ 5)² = 16
C. (x-3)2 + (v-5)2 = 4
O D. (x+3)2 + (y + 5)² = 4
SUB

The probability that in the next 1,000 passengers

1 No mishandled bags: 0.0295

2 Four or fewer mishandled bags: 0.3449

3 At least one mishandled bag:  0.9705.

4 At least two mishandled bags:  0.8672

1. To discover the likelihood of no misused sacks within the next 1000 travelers, able to utilize the Poisson dispersion equation:

P(X = 0) =[tex]e^(-λ) * (λ^0)[/tex] / 0!

Where λ is the anticipated number of misused sacks per 1000 travelers, which is rise to 3.52.

P(X = 0) =[tex]e^(-3.52) * (3.52^0)[/tex] / 0!

P(X = 0) = 0.0295

Hence, the likelihood of no misused sacks within the other 1000 passengers is 0.0295.

2. To discover the likelihood of  fewer misused packs within another 1000 travelers, we will utilize the total Poisson dispersion:

P(X ≤ 4) = Σ k=0 to 4 [[tex]e^(-λ) * (λ^k)[/tex]/ k! ]

P(X ≤ 4) = [[tex]e^(-3.52) * (3.52^0) / 0! ] + [ e^(-3.52) * (3.52^1) / 1! ] + [ e^(-3.52) * (3.52^2) / 2! ] + [ e^(-3.52) * (3.52^3) / 3! ] + [ e^(-3.52) * (3.52^4)[/tex]/ 4! ]

P(X ≤ 4) = 0.3449

Subsequently, the likelihood of fewer misused sacks within another 1000 travelers is 0.3449.

3. To discover the likelihood of at least one misused sack within the following 1000 travelers, able to utilize the complementary likelihood:

P(X ≥ 1) = 1 - P(X = 0)

P(X ≥ 1) = 1 - 0.0295

P(X ≥ 1) = 0.9705

Subsequently, the likelihood of at slightest one misused pack within the following 1000 passengers is 0.9705.

4. To discover the likelihood of at slightest two misused sacks within the other 1000 travelers, we can utilize the complementary likelihood once more:

P(X ≥ 2) = 1 - P(X = 0) - P(X = 1)

P(X ≥ 2) = 1 - 0.0295 - [ [tex]e^(-3.52) * (3.52^1)[/tex] / 1! ]

P(X ≥ 2) = 1 - 0.0295 - 0.1033

P(X ≥ 2) = 0.8672

Subsequently, the likelihood of at slightest two misused packs within the following 1000 travelers is 0.8672. 

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The balloon that was farther from the town at the beginning, and which traveled more quickly is option D. Tasha's balloon was farther from the town at the beginning, but Henry's balloon traveled more quickly.  

In order for us to know or to figure out which balloon had a faster journey, we can employ the speed equation:

Speed: Distance divided by time

Note that from the question, Henry's balloon was one that covered a distance of  16 miles within a span of 2 hours resulting in its velocity being 8 miles per hour and  Tasha's balloon was situated y = 5x + 25 miles away from the town.

Theis mean that its distance from the town would be y = 5(2) + 25 = 35 miles, after a duration of 2 hours. So, Tasha's balloon covered a distance of 10 miles within a span of 2 hours, showing a speed of 5 mph.

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See text below

Two hot air balloons are traveling along the same path away from a town, beginning from different locations at the same time. Henry's balloon begins 15 miles from the town and is 31 miles from the town after 2 hours. The distance of Tasha's balloon from the town is represented by the function y = 5x+25.

Which balloon was farther from the town at the beginning, and which traveled more quickly?

A. Tasha's balloon was farther from the town at the beginning, and it traveled more quickly.

B. Henry's balloon was farther from the town at the beginning, and it traveled more quickly.

C. Henry's balloon was farther from the town at the beginning, but Tasha's balloon traveled more quickly.

D. Tasha's balloon was farther from the town at the beginning, but Henry's balloon traveled more quickly.

The requested probabilities are: A. P(X > 1) ≈ 0.628;                                  B. P(X > 0.36) ≈ 0.844; C. P(X < 0.47) ≈ 0.226;                                              D. P(0.32 < X < 2.46) ≈ 0.524

The probability density function of an exponentially distributed random variable with parameter lambda is given by:

f(x) = lambda * e^(-lambda * x), for x >= 0

The cumulative distribution function (CDF) of X is given by:

F(x) = P(X <= x) = 1 - e^(-lambda * x), for x >= 0

Using the given value of lambda = 0.47, we can solve for each probability as follows:

A. P(X > 1) = 1 - P(X <= 1) = 1 - (1 - e^(-0.47 * 1)) = e^(-0.47) ≈ 0.628

B. P(X > 0.36) = 1 - P(X <= 0.36) = 1 - (1 - e^(-0.47 * 0.36)) = e^(-0.1692) ≈ 0.844

C. P(X < 0.47) = P(X <= 0.47) = 1 - e^(-0.47 * 0.47) ≈ 0.226

D. P(0.32 < X < 2.46) = P(X <= 2.46) - P(X <= 0.32) = (1 - e^(-0.47 * 2.46)) - (1 - e^(-0.47 * 0.32)) ≈ 0.524

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The perimeter of the square based on the dimensions of the diagonal is 145.27 millimeters.

We will begin with calculating the side of square from the diagonal of square. It will form right angled triangle and hence the formula will be represented as -

diagonal² = 2× side²

Keep the value of diagonal

(37✓2)² = 2× side²

Side² = 2638/2

Side² = 1319

Side = ✓1319

Side = 36.32 millimetres

Perimeter of the square = 4 × side

Perimeter = 145.27 millimeters

Thus, the perimeter of the square is 145.27 millimeters.

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The magnitude and direction of the vector that represents the straight path from Aaron's home to the park are approximately 8.5 km and N 34° E, respectively.

We can solve this problem by using vector addition. Let's break down Aaron's path into three vectors:

1. The first vector is 3 km at a bearing of N 30° E, which we can represent as a vector with components <2.598, 1.5>.

2. The second vector is 6 km due east, which we can represent as a vector with components <6, 0>.

3. The third vector is 4 km at a bearing of N 50° E, which we can represent as a vector with components <2.828, 3.053>.

To find the vector that represents the straight path from Aaron's home to the park, we need to add these three vectors together. We can do this by adding their components:

<2.598, 1.5> + <6, 0> + <2.828, 3.053> = <11.426, 4.553>

So the vector that represents the straight path from Aaron's home to the park has a magnitude of √(11.426² + 4.553²) = 12.3 km (rounded to the nearest tenth) and a direction of tan⁻¹(4.553/11.426) = 21° (rounded to the nearest degree) north of east.

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The mean life of the 25 items will be less than 5.9 years (rounded to one decimal place) 8% of the time.

We'll use the concepts of normal distribution, mean, standard deviation, and the z-score.

Step 1: Calculate the standard error of the mean. Standard error = (Standard deviation) / sqrt(Number of items) Standard error = 1.6 / sqrt(25) = 1.6 / 5 = 0.32 years

Step 2: Find the z-score corresponding to the 8% probability. We look for the z-score in a standard normal distribution table, which tells us that 8% of the time (0.08 probability), the z-score is approximately -1.4.

Step 3: Use the z-score formula to find the mean life (x) that corresponds to this probability. Z = (x - Mean) / Standard error -1.4 = (x - 6.3) / 0.32

Step 4: Solve for x. x - 6.3 = -1.4 * 0.32 x = 6.3 - (1.4 * 0.32) x ≈ 5.852

The mean life of the 25 items will be less than 5.9 years (rounded to one decimal place) 8% of the time.

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To find the HPD (Highest Posterior Density) interval of 95% of θ, we need to first calculate the posterior distribution of θ using the Beta prior distribution with parameters α = 1 and β = 1, and the observed data of 17 satisfied and 5 not satisfied.

The posterior distribution of θ is also a Beta distribution with parameters α' = α + number of satisfied and β' = β + number of not satisfied. In this case, α' = 1 + 17 = 18 and β' = 1 + 5 = 6.

So, the posterior distribution of θ is Beta(18,6).

To find the HPD interval, we can use a numerical method such as Markov Chain Monte Carlo (MCMC) simulation. However, since the Beta distribution has a closed-form expression for the quantiles, we can use the following formula to calculate the HPD interval:

HPD interval = [Beta(q1,α',β'), Beta(q2,α',β')]

where q1 and q2 are the quantiles of the posterior distribution that enclose 95% of the area under the curve.

Using a Beta distribution calculator or software, we can find that the 0.025 and 0.975 quantiles of Beta(18,6) are approximately 0.633 and 0.898, respectively.

Therefore, the HPD interval of 95% of θ is [0.633, 0.898].

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Answer:

option B.

Step-by-step explanation:

The original price of Casey's favorite pizza was $5.50, but now it costs $6. To find the percentage markup, we can use the formula:

(markup / original price) * 100%

The markup is the difference between the new price and the original price:

$6.00 - $5.50 = $0.50

So the markup is $0.50.

Using the formula above:

(markup / original price) * 100% = ($0.50 / $5.50) * 100% = 9.09%

Therefore, the pizza was marked up by about 9%, which is option B.

There are 59,049 possible 5-digit PINs where adjacent digits cannot be identical, and you are permitted to use digits 0-9.

To determine the number of possible 5-digit PINs where adjacent digits cannot be identical and using the digits 0-9, follow these steps:

Step 1: Consider the first digit. Since there are no restrictions, you have 10 choices (0-9).

Step 2: For the second digit, you can't have it identical to the first digit. Therefore, you have 9 choices left.

Step 3: For the third digit, it can't be identical to the second digit. So, you again have 9 choices.

Step 4: Similarly, for the fourth digit, you have 9 choices.

Step 5: Finally, for the fifth digit, you have 9 choices.

Now, multiply the choices for each digit together: 10 × 9 × 9 × 9 × 9 = 59,049.

So, there are 59,049 possible 5-digit PINs where adjacent digits cannot be identical, and you are permitted to use digits 0-9.

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The depth of the ocean is 650 feets at the bottom of the sunlight zone.

The distance travelled by echo sound is given by the formula -

Speed = 2×distance/time

So, calculating the speed of sound from the formula using distance and time

Speed = 2×25/(1/100)

Speed = 50×1000

Speed of sound = 5000 feet/second

Now, calculating the distance or depth of ocean at the bottom of the sunlight zone -

Distance = (speed×time)/2

Distance = (5000×26/100)/2

Distance = 1300/2

Distance = 650 feets

Hence, the depth of ocean is 650 feets.

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The proof that of the above expression on the condition of a/b = c/d = e/f is given below.

Given: a/b = c/d = e/f

Let e/b = e/c = k

Then, a/b = k and c/d = k, so a = kb and c = kd

Now we have:

√((a⁴ +  c⁴)/  (b⁴ + d⁴))   = √(((k b) ⁴ + ( kd )⁴ )/(b ⁴ + d ⁴)  )

= √ (k ⁴ *  (b⁴ + d⁴ ) / (b⁴ + d⁴))

= k²

Let p = 1 and q = k², then:

(p  a² + q * c²)/(p * b² + q * d²) = (a² + k² * c²)/(b² + k⁴ * d²)

= (k² *   b² + k² *    d ²)/(b ² +   k ⁴ * d ²)

= k ²

Therefore, we have shown that √ ((a ⁴ + c ⁴)/(b ⁴ + d ⁴))   = (p x a ² + q * c ²) / (p * b ² + q * d² )

if a/b =  c/ d =  e/f.

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The volume of the cylinder is 12π units³.

Option A is the correct answer.

We have,

The formula for the volume V of a cylinder with radius r and height h is:

V = πr²h

Now,

Radius = 2 units

Height = 3 units

Now,

Volume.

= πr²h

= π x 2 x 2 x 3

= 12π units³

Thus,

The volume of the cylinder is 12π units³.

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The sphere is tangent to all 12 edges, meaning that it just touches each edge at one point without intersecting it.
First, we need to find the radius of the sphere. Since the sphere is tangent to each edge, it can be thought of as inscribed within the cube.

Drawing a diagonal of the cube creates a right triangle with legs of length 6. Using the Pythagorean theorem, we find that the length of the diagonal is $6\sqrt{3}$.
Since the sphere is inscribed within the cube, its diameter is equal to the diagonal of the cube. Therefore, the radius of the sphere is half of the diagonal, which is $\frac{1}{2}(6\sqrt{3}) = 3\sqrt{3}$.

Now that we have the radius of the sphere, we can use the formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$. Substituting in the value for the radius, we get:

$V = \frac{4}{3}\pi (3\sqrt{3})^3 \approx 113.10$

So the volume of the sphere is approximately 113.10 cubic units.
To find the volume of the sphere tangent to all 12 edges of a cube, we'll first need to determine the sphere's radius.

1. Consider the cube with edge length 6. Let's focus on one of its vertices.
2. At this vertex, there are 3 edges, each tangent to sphere S.
3. Since the sphere is tangent to all these edges, they form a right-angled triangle inside the sphere, with the edges being its legs and a diameter of the sphere being its hypotenuse.
4. Let r be the radius of sphere S.
5. Using the Pythagorean theorem, we have: (2r)^2 = 6^2 + 6^2 + 6^2
6. Simplifying, we get: 4r^2 = 108
7. Solving for r, we have: r^2 = 27, so r = √27

Now, we can find the volume of the sphere using the formula:

Volume = (4/3)πr^3

8. Substitute the value of r into the formula: Volume = (4/3)π(√27)^3
9. Simplifying, we get: Volume ≈ 36π(√27)

Thus, the volume of sphere S tangent to all 12 edges of the cube with edge length 6 is approximately 36π(√27) cubic units.

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To solve an equation, you need to isolate the variable or get it alone on one side of the equation.

Finding the value of the variable is the fundamental goal when solving an equation. You can achieve this by placing the variable alone on one side of the equation or by isolating it. A number of mathematical procedures must be carried out in order to accomplish this while maintaining the equality of the equation and simplifying the expression containing the variable.

The secret is to alter the equation so that the variable term is on its own by adding, removing, multiplying, or dividing both sides by the same number. By using the necessary mathematical procedures, the solution can be derived after the variable has been isolated.

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Answer:

y = 3/2x-14

Step-by-step explanation:

      The given line is y=-2/3x. So, the slope of the given line is -2/3.

Now, we have to find the perpendicular line to y= -2/3x passing through the point (4,-8).

       The product of two perpendicular lines is -1.

m1.m2 = -1.

-2/3.m2= -1

m2 = 3/2

        Now, we need to find the equation of the line passing through the point (4,-8) with slope 3/2.

The equation of slope-point form is (y-y1) = m(x-x1)

        y-(-8) = 3/2 (x-4)

        y+8 = 3/2x -6

Now, we have to add 6 on both sides.

        y + 8 + 6 = 3/2x - 6 + 6.

        y + 14 = 3/2x

        y = 3/2x - 14.

The likelihood of randomly selecting a terrier from the shelter would be unlikely. That is option B

The formula that can be used to determine the probability of a selected event is given as follows;

Probability = possible event/sample space.

The possible sample space for terriers = 15%

Therefore the remaining sample space goes for Labradors and Golden Retrievers which is = 75%

Therefore, the probability of selecting the terriers at random is unlikely when compared with other dogs.

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The equivalent system of first-order differential equations for the given problem is: 1. dv1/dt = v2 2. dv2/dt = v3 3. dv3/dt = 2v2 - v1 + 1 + t*e ^t

Given differential equation: x''' - 2x'' + x' = 1 + t*e ^t

Step 1: Define new variables.
Let's introduce new variables:
v1 = x'
v2 = v1'
v3 = v2'

Now we have:
v1 = x'
v2 = v1'
v3 = v2'

Step 2: Rewrite the given equation using new variables.
Substitute the new variables into the given differential equation:
v3 - 2v2 + v1 = 1 + t*e ^t

Step 3: Write the equivalent system of first-order differential equations.
Now we have the following equivalent system of first-order differential equations:
dv1/dt = v2
dv2/dt = v3
dv3/dt = 2v2 - v1 + 1 + t*e ^t

So, the equivalent system of first-order differential equations for the given problem is:

1. dv1/dt = v2
2. dv2/dt = v3
3. dv3/dt = 2v2 - v1 + 1 + t*e ^t

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The probability of the compound event of spinning a 5 and flipping heads is 1/12.

Assuming that the spinner is fair and each outcome is equally likely, the probability of spinning a 5 is:

P(spinning 5) = number of ways to get 5 / total number of outcomes

P(spinning 5) = 1 / 6

Now, assuming that the coin is fair and has an equal probability of landing on heads or tails, the probability of flipping heads is:

P(flipping heads) = number of ways to get heads / total number of outcomes

P(flipping heads) = 1 / 2

To find the probability of the compound event of spinning 5 and flipping heads, we multiply the probability of spinning 5 by the probability of flipping heads:

P(spinning 5 and flipping heads) = P(spinning 5) x P(flipping heads)

P(spinning 5 and flipping heads) = (1/6) x (1/2)

P(spinning 5 and flipping heads) = 1/12.

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The complete question:

You spin the spinner and flip a coin. Find the probability of the compound event. The probability of spinning number 5 and flipping heads is__.

And spinner sample space is {1, 2, 3, 4, 5, 6}

Answer:

cos A = 45

Step-by-step explanation:

90 - sin A = 45 = cos A

90 Minus whatever the cos value for the angle is = the sin and vice versa.

An estimated regression equation that can be used to predict a residence's average monthly gas bill for last year given its age is [tex]$\hat{y} = 115.14 - 3.167x$[/tex]. The average monthly gas bill for last year increases by $0.2456 on average.

Using age as the predictor variable and average monthly gas bill as the response variable, we can use linear regression to develop an estimated regression equation:

[tex]$\hat{y} = b_0 + b_1 x$[/tex]

where [tex]\hat{y}[/tex] is the predicted average monthly gas bill, x is the age of the residence, b₀ is the intercept and b₁ is the slope.

Using the given data, we can find the values of b₀ and b₁:

[tex]$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = 17.6$[/tex]

[tex]$\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i = 55.906$[/tex]

[tex]$s_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} = 8.564$[/tex]

[tex]$s_y = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n-1}} = 24.193$[/tex]

[tex]$r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n} (y_i - \bar{y})^2}} = -0.577$[/tex]

[tex]$b_1 = r \frac{s_y}{s_x} = -3.167$[/tex]

[tex]$b_0 = \bar{y} - b_1 \bar{x} = 115.14$[/tex]

Therefore, the estimated regression equation is:

[tex]$\hat{y} = 115.14 - 3.167x$[/tex]

where [tex]\hat{y}$[/tex] is the predicted average monthly gas bill and x is the age of the residence.

This equation suggests that as the age of the residence increases by one year, the average monthly gas bill for last year increases by $0.2456 on average.

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Answer:

26 units

Step-by-step explanation:

The distance between two points with coordinates

The distance between the points (1, 2) and (1, -10) is 12 square units

We have to find the distance between (1, 2) and (1, -10)

The length along a line or line segment between two points on the line or line segment.

Distance=√(x₂-x₁)²+(y₂-y₁)²

=√(1-1)²+(-10-2)²

=√-12²

=√144

=12 square units

Hence,  the distance between (1, 2) and (1, -10) is 12 square units

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Answer:

I believe (3, 5) and (10, 1) is the answer

Step-by-step explanation:

Answer:a to b is 21 and b to c is 6 so I think you would need 21+6 divided by 2 i don't know for sure.

Step-by-step explanation:

The area of the composite figure is

The area is calculated by dividing the figure into simpler shapes.

The simple shapes used here include

Area of rectangle = length x width

= 12 x 7

= 84 square ft

Area of triangle = 1/2 base x height

= 1/2 x 12 x 6

= 36 square ft

Total area

= 84 square ft + 36 square ft

=  120 square ft

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The value of the variable x for the arc angle m∠AB and the angle it subtends at the center of the circle C is equal to 73.

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

Thus:

3x - 46 = x + 98

3x - x = 98 + 46 {collect like terms}

2x = 146

x = 146/2 {divide through by 2}

x = 73

Therefore, the value of the variable x for the arc angle m∠AB and the angle it subtends at the center of the circle C is equal to 73.

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(a) The number of ways that the event can occur is 6.

(b) Probabilities are :

1) 1/2, 2) 1/6 and 3) 1/3.

(a) Given a bag of different shapes.

Total number of shapes = 6

So, if we select one shape from random,

total number of ways that the event can occur = 6

(b) Number of squares in the bag = 3

Probability of choosing a square = 3/6 = 1/2

Number of circles in the bag = 1

Probability of choosing a circle = 1/6

Number of stars in the bag = 2

Probability of choosing a star = 2/6 = 1/3

Hence the required probabilities are found.

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The expected monthly utility bill for a 2200 square foot home in this city would be $340.

Given regression equation: U = 0.2F - 200

Where U is the monthly utility bill and F is the square footage of the residence.

Substitute F = 2200 in the equation

U = 0.2(2200) - 200

U = 440 - 200

U = $340

Hence, the expected monthly utility bill for a 2200 square foot home in this city is $340.

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The monthly utility bill would be expected for a 2200 square foot home in this city is  $240

The given regression line represents a relationship between the monthly utility bill (U) and the square footage of a residence (F) in a certain city. The equation U = 0.2F - 200 is in the form of a linear equation, where the coefficient 0.2 represents the rate of change in the utility bill for every one unit increase in square footage.

To find the expected monthly utility bill for a 2200 square foot home, we substitute F = 2200 into the equation. By plugging in this value, we can calculate the corresponding value of U, which represents the expected utility bill for that particular square footage.

Substituting F = 2200 into the equation U = 0.2F - 200, we get:

U = 0.2(2200) - 200

Calculating the expression within the parentheses gives us:

U = 440 - 200

Simplifying further:

U = 240

Therefore, the expected monthly utility bill for a 2200 square foot home in this city is $240. This means that, based on the given regression line, on average, residents with a 2200 square foot home can expect a monthly utility bill of $240 in this city.

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The complex fourth roots of 3−33i are: [tex]z_0[/tex] = 3.062[tex]e^{(-21.603)}[/tex], [tex]z_1[/tex] = 1.513[tex]e^{(22.247)}[/tex], [tex]z_2[/tex] = 0.3826[tex]e^{(22.247)}[/tex] and [tex]z_3[/tex] = 1.198[tex]e^{(76.247)}[/tex].

To find the complex fourth roots of 3-33i, we can use the polar form of the complex number:

3-33i = 33∠(-86.41)

Then, the nth roots of this complex number are given by:

[tex]z_k[/tex] = [tex]33^{(1/n)}[/tex] × ∠((-86.41 + 360k)/n) for k = 0, 1, 2, ..., n-1

For n = 4, we have:

[tex]z_0[/tex] = [tex]33^{(1/4)}[/tex] × ∠(-86.41/4) ≈ 3.062∠(-21.603°)

[tex]z_1[/tex] = [tex]33^{(1/4)}[/tex] × ∠(88.99/4) ≈ 1.513∠(22.247°)

[tex]z_2[/tex] = [tex]33^{(1/4)}[/tex] × ∠(196.99/4) ≈ 0.3826∠(49.247°)

[tex]z_3[/tex] = [tex]33^{(1/4)}[/tex] × ∠(304.99/4) ≈ 1.198∠(76.247)

So the complex fourth roots of 3-33i are approximate:

[tex]z_0[/tex] = 3.062[tex]e^{(-21.603)}[/tex]

[tex]z_1[/tex] = 1.513[tex]e^{(22.247)}[/tex]

[tex]z_2[/tex] = 0.3826[tex]e^{(22.247)}[/tex]

[tex]z_3[/tex] = 1.198[tex]e^{(76.247)}[/tex]

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The general solution of the given differential equation is:
y = (r^(1/2)) * (1 + Ce^(2r^(1/2))) or y = (r^(1/2)) * (-1 + Ce^(2r^(1/2)))
where C is the constant of integration.

To find the general solution of r = y^2 - 1 dr, we need to separate the variables and integrate both sides. We can start by rearranging the equation as:
dr/(y^2 - 1) = dy/r

Now, we can integrate both sides. On the left side, we can use partial fractions to make the integration easier. We can write:
dr/(y^2 - 1) = [1/(2*(y-1))] - [1/(2*(y+1))] dy

Integrating both sides, we get:
1/2 * ln|y-1| - 1/2 * ln|y+1| = ln|r| + C
where C is the constant of integration.

We can simplify this as:
ln|(y-1)/sqrt(r)| - ln|(y+1)/sqrt(r)| = 2C

Using logarithmic properties, we can simplify further as:
ln|[(y-1)/sqrt(r)] / [(y+1)/sqrt(r)]| = 2C
ln|[(y-1)/(y+1)]| = 2C

Exponentiating both sides, we get:
|[(y-1)/(y+1)]| = e^(2C)

Taking the positive and negative cases separately, we get:
(y-1)/(y+1) = e^(2C)

or
(y-1)/(y+1) = -e^(2C)

Solving for y in each case, we get the general solution as:

y = (r^(1/2)) * (1 + Ce^(2r^(1/2))) or y = (r^(1/2)) * (-1 + Ce^(2r^(1/2)))

where C is the constant of integration. This is the general solution of the given differential equation.

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