Howto prove for root test convergence for complex number.

a. Rounded to the nearest mile, the distance from New York City to Los Angeles is 2556 miles.
b. Rounded to one decimal place, we can expect to use 82.4 gallons of gas driving from New York to Los Angeles.

a. To convert the distance from kilometers to miles, we can use the given unit fraction: 1 mile ≈ 1.6 kilometers. First, set up the conversion using the given distance:

4090 kilometers × (1 mile / 1.6 kilometers)

The kilometers units will cancel out, leaving the result in miles:

4090 / 1.6 ≈ 2556.25 miles

Rounded to the nearest mile, the distance is approximately 2556 miles.

b. To calculate the number of gallons of gas needed, we can use the car's average of 31 miles per gallon. Set up the conversion using the distance in miles:

2556 miles × (1 gallon / 31 miles)

The miles units will cancel out, leaving the result in gallons:

2556 / 31 ≈ 82.5 gallons

Rounded to one decimal place, you can expect to use approximately 82.5 gallons of gas driving from New York to Los Angeles.

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West Virginia is needed to estimate the true proportion of children living in poverty within 3% with 95% confidence

To estimate the required sample size, we can use the formula:

n = (Z^2 * p * (1-p)) / E^2

where:

Z = the Z-score associated with the desired level of confidence (95% confidence corresponds to a Z-score of 1.96)

p = the estimated proportion of the population with the characteristic of interest (in this case, the estimated proportion of children under age 6 living in poverty in West Virginia, which is 0.3)

E = the desired margin of error (in this case, 0.03)

Substituting the given values, we get:

n = (1.96^2 * 0.3 * (1-0.3)) / 0.03^2

Simplifying:

n = 601.78

Rounding up to the nearest whole number, we get:

n = 602

Therefore, a sample of at least 602 children under age 6 from West Virginia is needed to estimate the true proportion of children living in poverty within 3% with 95% confidence.

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At the value of x = 3, functions f (x) is equal to g (x).

We have to given that;

Functions are,

f (x) = 2³x

g (x) = 6x + 6

Now, We can equate the functions we get;

2³x = 6x + 6

8x = 6x + 6

8x - 6x = 6

2x = 6

x = 3

Thus, At the value of x = 3, functions f (x) is equal to g (x).

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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.

Suppose A and B are dependent events. If P(A|B) = 0.25 and P(B) = 0.6, then  P(AuB) = 0.2

The probability of an event is described as a number that indicates how likely the event is to occur which is usually  expressed as a number in the range from 0 and 1, or preferably using percentage notation ranging  from 0% to 100%.

The relationship between two dependent events is expressed in the following equation below according to the rules of probability,

P(A|B) = P(A∩B) / P(B)

we then substitute ,

0.25 = P(A∩B) / 0.8

P(A∩B)  = 0.2

In conclusion, If P(A|B) = 0.25 and P(B) = 0.6, then  P(AuB) = 0.2

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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.

Answer:

Page 1

1)

a) - 5x^4+3

Degree: 4

Number of terms: 2

b) 5x² + 30x+25

Degree: 2

Number of Terms: 3

c) 16r^4p

Degree: 4

Number of Terms: 1

d) 9m²n + 12mn +4m -6n+19

Degree:2

Number of Terms:5

Note:

Degree: the highest exponent of the variable x)

Number of terms: It is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or - signs, or sometimes by divide

Page 2:

Add or subtract the following polynomials.

a) (3p^3+6p^2+14p)+(-5p^3-2p+8p^2

opening bracket

3p^3+6p^2+14p+(-5p^3-2p+8p^2)

3p^3+6p^2+14p-5p^3-2p+8p^2

Combining like terms

-2p^3 +14p^2+12p

b) (7y^3-5y)-(5y-7y^3)

Opening bracket

7y^3-5y-5y+7y^3

Combining like terms

14y^3-10y

c) (6z^4+15-7z^3)+(-2z^4+8z^3-5z^5)

Opening bracket

6z^4+15-7z^3-2z^4+8z^3-5z^5

-5z^5 +4z^4+1z^3+15

d) (-3n²-14n+1)-(-7n+2-6n²)

Opening bracket

-3n²-14n+1+7n-2+6n²

3n²-7n-1

e) (7b^3-14-8b^4) − (−3b^4+7b³ +4)

Opening bracket

7b^3-14-8b^4+3b^4-7b³-4

-5b^4-18

f) (-3n^2-4n+2n⁴) + (3n^2+ 19n-7n^4)

Opening bracket

-3n^2-4n+2n⁴+3n^2+ 19n-7n^4

-5n⁴+15n

Answer:

Page 1

1)

a) - 5x^4+3

Degree: 4

Number of terms: 2

b) 5x² + 30x+25

Degree: 2

Number of Terms: 3

c) 16r^4p

Degree: 4

Number of Terms: 1

d) 9m²n + 12mn +4m -6n+19

Degree:2

Number of Terms:5

Note:

Degree: the highest exponent of the variable x)

Number of terms: It is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or - signs, or sometimes by divide

Page 2:

Add or subtract the following polynomials.

a) (3p^3+6p^2+14p)+(-5p^3-2p+8p^2

opening bracket

3p^3+6p^2+14p+(-5p^3-2p+8p^2)

3p^3+6p^2+14p-5p^3-2p+8p^2

Combining like terms

-2p^3 +14p^2+12p

b) (7y^3-5y)-(5y-7y^3)

Opening bracket

7y^3-5y-5y+7y^3

Combining like terms

14y^3-10y

c) (6z^4+15-7z^3)+(-2z^4+8z^3-5z^5)

Opening bracket

6z^4+15-7z^3-2z^4+8z^3-5z^5

-5z^5 +4z^4+1z^3+15

d) (-3n²-14n+1)-(-7n+2-6n²)

Opening bracket

-3n²-14n+1+7n-2+6n²

3n²-7n-1

e) (7b^3-14-8b^4) − (−3b^4+7b³ +4)

Opening bracket

7b^3-14-8b^4+3b^4-7b³-4

-5b^4-18

f) (-3n^2-4n+2n⁴) + (3n^2+ 19n-7n^4)

Opening bracket

-3n^2-4n+2n⁴+3n^2+ 19n-7n^4

-5n⁴+15n

Step-by-step explanation:

If P(A) = 0.80, P(B) = 0.65, and P(A È B) = 0.78, then P(B½A) =the answer is (a) 0.9750. By the formula for conditional probability

To find P(B|A), we can use the formula for conditional probability: P(B|A) = P(A ∩ B) / P(A). We know P(A) = 0.80, but we need to find P(A ∩ B).

We can use the formula for the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). We are given P(A ∪ B) = 0.78 and P(B) = 0.65.

Plugging in the values, we get:

0.78 = 0.80 + 0.65 - P(A ∩ B)

Now, solve for P(A ∩ B):

P(A ∩ B) = 0.80 + 0.65 - 0.78
P(A ∩ B) = 0.67

Now we can find P(B|A):

P(B|A) = P(A ∩ B) / P(A)
P(B|A) = 0.67 / 0.80
P(B|A) = 0.8375

So the answer is (c) 0.8375.

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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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However, since |X| and Z are not directly dependent on each other, their covariance will be 0. So, Cov(|X|, Z) = 0.

To find the covariance of |X| and Z, first let's understand the terms and the relationship between them.

Since X is a continuous random variable symmetric about Y, the probability distribution of X is symmetric around Y.

Now, Z is a binary random variable that takes the value 1 if X > Y, and 0 if X <= Y. Now, let's find the covariance:

Cov(|X|, Z) = E[(|X| - E[|X|])(Z - E[Z])]

Since X is symmetric about Y, we know that E[|X|] = E[X] (due to symmetry).

To find E[Z], we can observe that: E[Z] = P(X > Y) = 0.5 (As the distribution is symmetric about Y, the probability of X being greater than Y is 0.5)

Now, let's find E[(|X| - E[X])(Z - 0.5)]: E[(|X| - E[X])(Z - 0.5)] = ∫∫(|x| - E[X])(z - 0.5)f(x,z) dx dz

However, since |X| and Z are not directly dependent on each other, their covariance will be 0. So, Cov(|X|, Z) = 0.

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When creating a control chart, there are several main issues that need to be addressed. These include identifying the process that needs to be monitored and controlled, determining the appropriate data collection methods, selecting the appropriate chart type, setting control limits, and establishing a system for interpreting and responding to chart results.

Firstly, it is important to clearly define the process being monitored and controlled, and to ensure that the data collected accurately reflects the process performance. Secondly, data collection methods need to be established, including how frequently data will be collected and who will be responsible for collecting it.
Thirdly, the appropriate type of control chart needs to be selected based on the type of data being collected and the nature of the process being monitored. This could include variable charts, attribute charts, or time-weighted charts.
Fourthly, control limits need to be established based on the expected variation in the process, and these limits need to be communicated to those responsible for the process. Finally, a system for interpreting and responding to chart results needs to be put in place, including a plan for addressing any out-of-control signals or trends in the data.

In summary, the main issues to address in creating a control chart include process identification, data collection methods, chart selection, control limit setting, and interpretation and response systems.
When creating a control chart, the main issues to address include:

1. Identifying the purpose: Determine the objective of the control chart, such as monitoring process stability, identifying variation sources, or evaluating process improvement efforts.

2. Selecting the appropriate chart type: Choose the right control chart based on the type of data (continuous or attribute) and the sample size. Common chart types include X-bar and R charts, P and NP charts, and C and U charts.

3. Establishing control limits: Calculate the appropriate control limits (upper and lower) based on statistical techniques, using historical data or process specifications.

4. Collecting and plotting data: Collect data from the process in a consistent and timely manner, and plot the data points on the control chart to visualize process behavior.

5. Analyzing and interpreting the chart: Regularly analyze the control chart for patterns or trends that indicate process shifts, trends, or excessive variation. Interpret these patterns to identify the root causes of any issues.

6. Taking corrective action: Address identified issues by implementing corrective actions to improve process stability and performance.

7. Maintaining and updating the chart: Continuously monitor and update the control chart to ensure it remains relevant and effective in identifying and addressing process issues. This may include revising control limits or adjusting sampling methods as needed.

By addressing these issues, you can create an effective control chart that helps you monitor, evaluate, and improve your process performance.

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Answer:

x = 9

Step-by-step explanation:

Ratios!! Since both triangles are in the same triangle, they are proportinate in size. from there, we can match the sides of the small triange to the sides of the large.

[tex] \frac{4}{10} = \frac{6}{6 + x} [/tex]

Then, you can cross multiply and solve the equation algebraicly for x.

4(6+x) = 60

24+4x = 60

4x = 36

x = 9

We have proven that:

cost = Jo - 2J2 +2J4 – = .......

[infinity]

= Jo(x) + Σ (-1)^n j2n (x).

n=1

To prove this identity, we first start with the formula for the exponential function:

e^ix = cos(x) + i*sin(x)

where i is the imaginary unit.

Now, we can rewrite the right-hand side of the desired identity using this formula:

Jo(x) + Σ (-1)^n j2n (x)

= Jo(x) + Σ (-1)^n [i^nJn(x) - i^nYn(x)] (using the Bessel function identity j_n(x) = cos(x)J_n(x) - sin(x)Y_n(x))

= Jo(x) + Σ (-i)^nJn(x) + Σ i^nYn(x)

= Jo(x) + Σ (-i)^nJn(x) + Σ (-i)^{n+1}Jn(x) (using the Bessel function identity Y_n(x) = (J_n(x)cos(npi) - J_{-n}(x))/sin(npi))

= Jo(x) + 2Σ (-i)^nJn(x)

= Jo(x) + 2Σ (-1)^n*J2n(x) (since Jn(x) is real for all n)

Therefore, we have proven that:

cost = Jo - 2J2 +2J4 – = .......

[infinity]

= Jo(x) + Σ (-1)^n j2n (x).

n=1

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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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A minimum sample size of 601 voters is needed to estimate the population proportion of voters who would vote for the incumbent governor with a margin of error of 0.08 or less at a 95% confidence level.

To determine the minimum sample size needed to estimate the population proportion with a margin of error of 0.08 or less at 95% confidence, we need to use a formula called the sample size formula. The formula is as follows:

n = (Z² × p × q) / E

Where n is the sample size, Z is the z-score corresponding to the confidence level (1.96 for 95% confidence level), p is the estimated proportion of voters who would vote for the incumbent governor, q is the complement of p (1-p), and E is the desired margin of error (0.08).

Assuming we do not have any prior information about the proportion of voters who would vote for the incumbent governor, we can use a conservative estimate of 0.5 for p and q. Substituting the values into the formula, we get:

n = (1.96² × 0.5 × 0.5) / 0.08²
n = 600.25

Rounding up to the nearest whole number, we need a minimum sample size of 601 voters to estimate the population proportion with a margin of error of 0.08 or less at 95% confidence.

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The actual distance between two cities is 70 miles when 1 inch is equal to 20 miles

Given that Daniel is planning to drive from City X to City Y.

The distance between two cities is [tex]3\frac{1}{2}[/tex] inches

Given that 1 inch = 20 miles

We have to find the actual distance between two cities in miles

[tex]3\frac{1}{2}[/tex]  = 3.5

Now multiply 3.5 with 20 to find distance in miles

3.5×20

70 miles

Hence, the actual distance between two cities is 70 miles when 1 inch is equal to 20 miles

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If the monopolist charges a single price for teddy bears, an accurate outcome would be that the monopolist would maximize their profits by charging a price higher than the marginal cost of producing the teddy bears. This would result in consumers paying a higher price for the teddy bears, and potentially fewer consumers being willing to purchase them at the higher price.

A single seller who has complete control over the market's supply of a specific commodity or service is referred to as a monopolist.

As the sole manufacturer of teddy bears in this situation, the monopolist has complete control over the market's supply of teddy bears.

As a result, the monopolist will price the teddy bears higher than their marginal cost of production in order to maximise revenues.

The monopolist can maximise their profits by setting a price that is higher than the marginal cost.

However, the increased cost might cause fewer customers to be willing to buy the teddy bears, which might decrease the total demand for the good.

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The measure of a central angle of a regular polygon with 7 sides is approximately 51.4 degrees.

A polygon is a geometric object with two dimensions and a finite number of sides. A polygon's sides are made up of segments of straight lines that are joined end to end. As a result, a polygon's line segments are referred to as its sides or edges. Vertex or corners refer to the intersection of two line segments, where an angle is created.

To find the measure of a central angle of a regular polygon with 7 sides, we can use the formula:

central angle = 360 degrees/number of sides

Plugging in 7 for the number of sides, we get:

central angle = 360 degrees / 7
central angle ≈ 51.4 degrees

Therefore, the measure of a central angle of a regular polygon with 7 sides is approximately 51.4 degrees.

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Answer:

The slope is 5.

Step-by-step explanation:

We are given the points (-2, -5) and (0,5).

We want to find the slope between these 2 points.

The slope (m) is written with the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are points.

Let's label the values of the points to avoid any confusion and mistakes when calculating.

[tex]x_1=-2\\y_1=-5\\x_2=0\\y_2=5[/tex]

Now, substitute into the formula.

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]m = \frac{5--5}{0--2}[/tex]

This can be simplified to:

[tex]m = \frac{5+5}{0+2}[/tex]

Add the values together.

[tex]m = \frac{10}{2}[/tex]

m = 5

The slope is 5.

The optimal number of carts and pretzel price will depend on the size of the population and the competitive landscape. The park should conduct market research to determine the ideal price point and number of carts for their specific market.

To determine how many carts the amusement park should put out and what they should set their pretzel price to in order to maximize their profit for an arbitrary population p, several factors need to be considered.

Firstly, the demand for pretzels will depend on the size of the population p. If p is large, the park should put out more carts to meet the demand. However, if p is small, fewer carts would be sufficient.

Secondly, the price of the pretzels will also affect demand. If the price is too high, people may choose to buy other snacks or not purchase anything at all. On the other hand, if the price is too low, the park may not be able to cover their costs and make a profit. Therefore, the park should set the pretzel price to a level that is competitive with other snacks but still allows for a reasonable profit margin.

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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}

Clark would spend about $420 on these items in a year.

Given that Clark finds that in an average month, he spends $35 on things he really doesn't need and can't afford.

If Clark spends $35 on things he doesn't need or can't afford in an average month, then he would spend:

$35/month x 12 months/year = $420/year

Therefore, Clark would spend about $420 on these items in a year.

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The equation of the plane passing through point P with normal vector 2i + 6j + 7k is x + 3y + 4z = 48.

A: True.

The equation of a plane in 3D space is given by Ax + By + Cz = D, where A, B, C are the components of the normal vector and D is the distance from the origin to the plane along the direction of the normal vector.

In this case, the normal vector is 2i + 6j + 7k, so A = 2, B = 6, and C = 7. To find D, we can substitute the coordinates of the given point P into the equation of the plane:

2(1) + 6(3) + 7(4) = D

2 + 18 + 28 = D

D = 48

Therefore, the equation of the plane passing through point P with normal vector 2i + 6j + 7k is x + 3y + 4z = 48.

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Answer:

Convert 10% to decimal =.10

10/100=.10

Them multiply 754 by .10 and you get $75.40. This is the amount that will be subtracted from the full price to get your sales price.
now you have

$754.00-$75.40=$678.60

$678.60 is the sale price.

The original size of the bottle was 13.5 ounces in the given case

Let x be the original size of the bottle in ounces.

In the context of this problem, "original size" refers to the size of the bottle of hand soap before the manufacturer reduced its size

According to the problem, the new size of the bottle is 20% less than the original size, so we can set up the equation:

x - 0.2x = 10.8

Simplifying the left side:

0.8x = 10.8

Dividing both sides by 0.8:

x = 13.5

Therefore, the original size of the bottle was 13.5 ounces.

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C is a square-free number, and we have expressed n as the product of a square-free number and the square of primes in B, as desired:

[tex]n = ab^2[/tex], where a is square-free and b = q1 * q2 * ... * qm.

Let's start by applying the Fundamental Theorem of Arithmetic to any positive integer n. According to the theorem, we can express n as a product of prime powers:

[tex]n = p1^a1 * p2^a2 * ... * pk^ak[/tex]

where p1, p2, ..., pk are distinct prime numbers and a1, a2, ..., ak are positive integers.

Now, let's separate the primes into two groups: those that appear with an even exponent and those that appear with an odd exponent:

[tex]n = (p1^a1 * p2^a2 * ... * pk^ak/2) * (p1^a1/2 * p2^a2/2 * ... * pk^ak/2)[/tex]

Let's call the first group of primes A and the second group B. Notice that B consists of squares of primes, and thus any prime power in B can be written as the square of some other prime. Let's call these primes q1, q2, ..., qm.

So we can express B as:

[tex]B = q1^2 * q2^2 * ... * qm^2[/tex]

Let's now combine A and B, and call their product C:

C = A * B

Then we have:

[tex]n = C * (q1^2 * q2^2 * ... * qm^2)[/tex]

But notice that C has no square factors, because all the primes in B have an even exponent. Therefore, C is a square-free number, and we have expressed n as the product of a square-free number and the square of primes in B, as desired:

[tex]n = ab^2[/tex], where a is square-free and b = q1 * q2 * ... * qm.

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The domain and range of graph 4 are:

Domain = [-3, 3].

Range = [-1, 4].

The domain and range of graph 5 are:

Domain = [-∞, ∞].

Range = [-∞, ∞].

The domain and range of graph 6 are:

Domain = [-∞, ∞].

Range = [-∞, 1].

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graphs shown in the image attached above, we can reasonably and logically deduce the following domain and range for graph 4:

Domain = [-3, 3].

Range = [-1, 4].

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If the first-year teacher wants to retire in 40 years and plans to invest in an account with a 5.67% annual interest rate compounded continuously, retiring with at least $100,000 in the account, they must initially invest A) $10,352 (present value).

The present value for continuous compounding is given by the formula: P = A / e^rt.

This present value that is required to earn a future value of $100,000 can be determined using an online finance calculator as follows:

Total P+I (A): $100,000.00

Annual Rate (R) = 5.67%

Time (t in years): 40 years

Result:

P = $10,351.9

= $10,352.

Thus, to have $100,000 in 40 years, the teacher should invest $10,352 now.

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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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The coordinates of a vertex are contingent upon the type of figure it is associated with. Accordingly, there are certain methods to locate the coordinates of vertices distinguishing multiple shapes:

For a parabola in standard form (y = ax^2 + bx + c), its vertex's x-coordinate can be identified by -b/2a, with y-coordinate deducible through substitution into equation.

In the case of a triangle, its vertex is situated at the union of two sides; therefore, if the coordinates of the triad of vertices are accessible, use of the distance formula will ascertain the measurement of each side.

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