Write the equation in POINT-SLOPE FORM of the given graph using the given point.
Help me, please

To prove that EVPI > 0 for the simple decision sapling, we need to understand what EVPI is. EVPI stands for Expected Value of Perfect Information, and it is the difference between the expected value of the decision with perfect information and the expected value of the decision without perfect information.

For the simple decision sapling, we have two options: a sure thing of value r, and a risky gamble with a random payoff G. The expected value of the decision without perfect information is simply the maximum of the two options:
EV = max(r, EG)
The expected value of the decision with perfect information is the maximum of the two options, knowing the outcome of the gamble:
EVPI = max(r, G) - max(r, EG)
Since we know that G is a random variable, the expected value of G is simply the average of all possible outcomes. Therefore, the expected value of the decision with perfect information is simply the maximum of the two options, knowing the average outcome of the gamble:
EVPI = max(r, EG) - max(r, EG) = 0
Therefore, EVPI > 0 for the simple decision sapling.

To prove that the EVPI will be larger if the random payoff of the gamble is replaced by a noisy version of the original gamble, we need to understand how the expected value of the decision changes with the addition of the noise variable Y. The expected value of the decision without perfect information is now:
EV = max(r, EG + EY) = max(r, EG)
Since EY = 0, the expected value of the decision without perfect information does not change. However, the expected value of the decision with perfect information now becomes:
EVPI = max(r, G + Y) - max(r, EG)
Since Y is a random variable with mean 0, the expected value of Y is 0. However, the variance of Y is not necessarily 0, which means that the addition of Y adds variability to the decision. This means that the expected value of the decision with perfect information is now larger, because we have more information about the possible outcomes of the gamble. Therefore, the EVPI will be larger when the random payoff of the gamble is replaced by a noisy version of the original gamble.

To know more about EVPI refer here:

https://brainly.com/question/30198698

#SPJ11

The initial population is 4001 and the maximum population is 4000

The given population growth model is [tex]12000/3+e^{-0.02(t)}.[/tex]

To find the initial population, we need to plug in t=0 into the equation.

[tex]12000/3+e^{-0.02(0)}[/tex]

= [tex]12000/3+1[/tex]

= [tex]4000+1[/tex]

= [tex]4001[/tex]

So the initial population is 4001.

To find the maximum population, we need to find the limit of the equation as t approaches infinity.


= [tex]12000/3+0[/tex]

= [tex]4000[/tex]

So the maximum population is 4000.

In conclusion, the initial population is 4001 and the maximum population is 4000.

See more about initial population at: https://brainly.com/question/29326858

#SPJ11

The region that contains the solution to the system is region B

Given the following system of inequalities

y ≤ x - 2

y ≥ 1/4x - 4

Next, we analyze each of the inequalities expressions

Inequality 1: y ≤ x - 2

Inequality 2: y ≥ 1/4x - 4

Using the above as a guide, we have the following:

This means that the intersection regionn contains the solution

This is represented by the region (b)

Read more about inequality at

https://brainly.com/question/25275758

#SPJ1

Answer:

x=2

Step-by-step explanation:

2(5x+3)=26

expand the bracket first

10x+6=26

take away 6 from both sides

10x=20

divided by 10 on each side

x=2

Answer:

You would use the PEDMAS rule for this equation.

First, distribute 2 inside the parentheses so the equation is 10x + 6 = 26

Then, subtract 6 from both sides 10x = 20

The answer is x = 2.

Step-by-step explanation:

The exponential decay model for this situation is P(t) = 200,000 * (1 - 0.03)^t

Suppose the value of which a thing is expressed in percentage is "a'

Suppose the percent that considered thing is of "a" is b%

Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).

We need to Write the percent as a fraction in simplest form

16.24%

So, we can rewrite it as;

16.24 / 100

16 and 24/100 or 16 6/25

So the percent as a fraction is 16 6/25

We are given that;

Population in 1992=200000

Time=6years

Rate=3%

To write an exponential decay model to represent this situation, we can use the formula:

P(t) = P * (1 - r)ᵗ

where P(t) is the population after t years, P is the initial population, r is the annual rate of decline as a decimal, and t is the number of years. In this case, the initial population P is 200,000, the annual rate of decline r is 0.03 (since the population declines by 3% each year), and t is the number of years from 1992, so t = 0 corresponds to 1992 and t = 6 corresponds to 1998.

P(t) = 200,000 * (1 - 0.03)^t

where t is the number of years from 1992 to 1998.

Therefore, by the given percent answer will be P(t) = 200,000 x (1 - 0.03)^t

Learn more about percent here:

https://brainly.com/question/11549320

#SPJ1

In the congruent geometry ,  length of EH is 14 meters .

What is congruent geometry?

In a congruent geometry, the shapes that are so identical. can be superimposed on themselves.

Two objects that are identical in terms of their dimensions and shape are said to be congruent in geometry. Further, two shapes are said to be congruent to one another if they can be turned, flipped, or moved in the same direction.

As a result, two congruent figures that have been drawn on a piece of paper can be cut out and positioned over one another to perfectly match.

Here, The figure shows the design and its photocopy. The ratio of BC: FG is 1:2.

Since the two figures are congruent

BC/FG= AD/EH

1/2 = 7/EH    (from figure AD = 7 )

EH = 2*7

EH = 14 m

Thus, the required length of EH is 14 meters.

To learn more about congruent geometry refer the below link

https://brainly.com/question/3999145

#SPJ1

The percentage change of the depth from 1846 to 1847 is 28%.

The percentage change of the depth from 1846 to 1848 is 14%.

The percentage is calculated by dividing the required value by the total value and multiplying by 100.

Example:

Required percentage value = a

total value = b

Percentage = a/b x 100

Example:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

20% = 20/100 = 1/5

10% = 10/100 = 1/10

We have,

Percentage change formula:

= [ (Final percentage - Initial percentage) / Initial percentage ] x 100

Percentage change of the depth from 1846 to 1847.

= [ (3.6 - 5) / 5 ] x 100

= 1.4/5 x 100

= 1.4 x 20

= 28

Percentage change of the depth from 1846 to 1848.

= [ (5.1 - 5) / 5 ] x 100

= 0.7/5 x 100

= 0.7 x 20

= 14

Now,

Year                              1847        1848

Depth                          3.6 feet   5.7 feet

Percentage change     28%           14%          

Thus,

The percentage change of the depth from 1846 to 1847 and 1846 to 1848 is 28% and 14%

Learn more about percentages here:

https://brainly.com/question/11403063

The correct answer is b. 95%.

According to the Empirical rule, for a symmetric data set, 68% of the data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations of the mean, and 99.7% falls within 3 standard deviations of the mean.

In this particular case, the mean response time is 26 minutes and the standard deviation is 3 minutes. Therefore, 1 standard deviation from the mean is 23 to 29 minutes, 2 standard deviations from the mean is 20 to 32 minutes, and 3 standard deviations from the mean is 17 to 35 minutes.

Since the question asks for the percentage of response times that fall between 20 to 32 minutes, we are looking for the percentage of data that falls within 2 standard deviations of the mean. Therefore, the correct answer is 95%.

Learn more about Empirical rule

brainly.com/question/30700783

#SPJ11

a) Sin 2x = 2 tan x / 1 + tan^2 x is an identity.

b) Tan x + cot x = 2 csc 2x is an identity.

To verify that each equation is an identity, we will simplify both sides of the equation and show that they are equal.

For part a, we will use the double angle formula for sine and the Pythagorean identity.

sin 2x = 2 sin x cos x

2 tan x / 1 + tan^2 x = 2 sin x / cos x / 1 + sin^2 x / cos^2 x

= 2 sin x / cos x / cos^2 x / cos^2 x

= 2 sin x / cos x / 1 - sin^2 x

= 2 sin x / cos x / cos^2 x

= 2 sin x cos x

Therefore, sin 2x = 2 tan x / 1 + tan^2 x is an identity.

For part b, we will use the definitions of the trigonometric functions and the double angle formula for cosecant.

tan x + cot x = sin x / cos x + cos x / sin x

= (sin^2 x + cos^2 x) / (sin x cos x)

= 1 / (sin x cos x)

= 2 / (2 sin x cos x)

= 2 / sin 2x

= 2 csc 2x

Therefore, tan x + cot x = 2 csc 2x is an identity.

In conclusion, we have verified that both equations are identities.

Learn about Equations

brainly.com/question/29657992

#SPJ11

The doubling time.

A) The exponential function that describes the amount in the account after time t in years is A(t) = 511969*e^(0.052*t), where A(t) is the amount in the account after time t, e is the base of the natural logarithm, and t is the time in years.

B) The balance after 1 year is A(1) = 511969*e^(0.052*1) = 538,926.41
The balance after 2 years is A(2) = 511969*e^(0.052*2) = 567,639.77
The balance after 5 years is A(5) = 511969*e^(0.052*5) = 661,234.83
The balance after 10 years is A(10) = 511969*e^(0.052*10) = 872,032.74

C) The doubling time is the time it takes for the amount in the account to double. We can find this by setting A(t) = 2*511969 and solving for t.
2*511969 = 511969*e^(0.052*t)
2 = e^(0.052*t)
ln(2) = 0.052*t
t = ln(2)/0.052
t = 13.33 years
So the doubling time is 13.33 years.

Learn more about natural logarithm

brainly.com/question/29154694

#SPJ11

The graph of the piece-wise function is given by the image presented at the end of the answer.

A piece-wise function is a mathematical function that is defined by different formulas or expressions over different intervals or pieces of its domain, that is, the function has different definitions based on it's input.

The intervals for which the function has different definitions are given as follows:

The definitions for each interval are given as follows:

Hence the graph of the function, containing these three definitions, is given by the image presented at the end of the answer.

More can be learned about piece-wise functions at https://brainly.com/question/30348444

#SPJ1

Answer:

D

Step-by-step explanation:

The angle opposite to the side length c is given as follows:

θ = 53º.

The Law of Cosines is a trigonometric formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is also known as the Cosine Rule.

The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation holds true:

c^2 = a^2 + b^2 - 2ab cos(C)

The parameters for this problem are given as follows:

a = 122.5, b = 58.2, c = 164.5.

Hence the measure of angle θ is given as follows:

c^2 = a^2 + b^2 - 2ab cos(θ)

164.5² = 122.5² + 58.2² - 2 x 122.5 x 58.2 x cos(θ)

14259cos(θ) = 8666.76

cos(θ) = 8666.76/14259

θ = arccos(8666.76/14259)

θ = 53º.

The angle is opposite to the side length C.

More can be learned about the law of cosines at https://brainly.com/question/4372174

#SPJ1

a) Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. So the option E is correct.

b) There is insufficient information to draw the conclusion that the variables are connected even though the value is greater than the 0.05 threshold for significance.

c) The difference between these percentage are fairly similar which suggests that sex and risk of lower-extremity Injury in  interscholastic soccer are not related.

a) The null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer is:

Null hypothesis: Sex and lower-extremity injury risk in interscholastic soccer are unrelated factors. Alternative hypothesis: sex and the likelihood of sustaining a lower-extremity injury in collegiate soccer are associated variables.

So the option E is correct.

b) The conclusion is that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries. This conclusion is based on the fact that the chi-square statistic (1.63) and the corresponding p-value (0.202) are both greater than the 1.63 X standard for significance. This indicates that there is not enough evidence to show that the variables are related.

c) Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

Girls = 79/997 × 100 = 7.92%

Boys = 156/1664 × 100 = 9.38%

The difference between the two percentages is consistent with the conclusion that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries because the difference between the two percentages is small.

This indicates that the risk of lower-extremity injuries is relatively similar for both boys and girls, which is consistent with the conclusion that there is no significant relationship between gender and the risk of lower-extremity injuries.

To learn more about null and alternative hypotheses link is here:

brainly.com/question/28331914

#SPJ4

The complete question is:

Researchers studied a random sample of high school students who participated in interscholastic athletics to learn about the risk of lower-extremity injuries (anywhere between hip and toe) for interscholastic athletes. Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

(a) Write null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer.

A. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. Alternative hypothesis: Sex and risk of a lower-extremity Injury in interscholastic soccer are not related variables.

B. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not explanatory variables.

C. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are not explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are explanatory variables.

D. Null hypothesis: There is a weak relationship between sex and risk of a lower-extremity injury in interscholastic soccer. Alternative hypothesis: There is a strong relationship between sex and risk of a lower-extremity injury in interscholastic soccer

E. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables.

(b) For these data, the value of the chi-square statistic is 1. 63, and the p-value for the chi-square test is 0. 202. Based on these results, state a conclusion about the two variables in this situation and explain how you came to this conclusion. The p-value is greater than the 1. 63 X standard for significance, so there is not sufficient evidence to be able to conclude that the variables are related.

(c) For each sex separately, calculate the percent of participants who had a lower-extremity injury. (Round your answers to one decimal place. ) girls 7.9% boys 9.4%. Explain how the difference between these percentages is consistent with the conclusion you stated in part (b). These percentages are fairly similar , which suggests that sex and risk of a lower-extremity injury in interscholastic soccer are not related

Answer:

Step-by-step explanation:

We separate into 2 congruent triangles. Both have a base of 3.5 and a height of 9 inches. Using pythagoran theorum, we will square 9 and 3.5

81 and 12.25

Add them up and it's 93.25

[tex]\sqrt{93.25}[/tex]=9.65660395791

Now since it's isosceles, the other side will also be 9.65660395791.

9.65660395791 + 9.65660395791 + 7 =26.3132079158

You want it rounded to the nearest tenth, so 26.3

airplane A

To compare the rate of speed of Airplane A and Airplane B, we need to calculate the speed of Airplane B first.

The formula to calculate distance is:

distance = rate x time

For Airplane A, we know:

distance = 1662 miles

time = 6 hours

So, we can rearrange the formula to solve for the rate:

rate = distance / time = 1662 / 6 = 277 miles per hour

For Airplane B, we know:

y = 922/3

x = rate of speed over 6 hours

We can use the formula to solve for x:

distance = rate x time

922/3 = x * 6

x = (922/3) / 6 = 153.67 miles per hour

Comparing the rates of the two airplanes, we can see that Airplane A traveled at a faster rate with 277 miles per hour compared to Airplane B's rate of 153.67 miles per hour. Therefore, Airplane A traveled at a faster rate.

The value of x is 18.4.

Right angled triangle are those triangle for which one of the angle is 90 degrees.

Given that,

sin (2x + 7)° = cos (3x - 9)°  [Equation 1]

We have the trigonometric rule that,

cos (θ) = sin(90 - θ)

So, cos (3x - 9)° = sin (90 - (3x - 9))

So from equation 1,

sin (2x + 7)° = sin (90 - (3x - 9))

Equating,

2x + 7 = 90 - (3x - 9)

2x + 7 = 90 - 3x + 9

2x + 7 = 99 - 3x

5x = 92

x = 18.4

Hence the value of x is 18.4.

Learn more about Right Angles Triangles here :

https://brainly.com/question/26698211

#SPJ1

Class B has higher percentage of girls than class A that is 26.6 % more.

A percentage is a number or ratio that can be expressed as a fraction of 100. If you need to calculate the percentage of a number, divide the number by the whole number and multiply by 100. Percentages therefore mean 1 in 100.

Given,

Total number of students in class A = 60

Number of girls student in class A = 28

percentage of girls in class A

= (28/60)×100

= 0.467 × 100

= 46.7%

Total number of students in class B = 60

Number of girls student in class B = 44

percentage of girls in class A

= (44/60)×100

= 0.733 × 100

= 73.3 %

Difference in percentage of girls in class A and class B.

= 73.3% - 46.7%

= 26.6%

Hence, class B has 26.6 percent more girls than class A.

Learn more about percentage here:

https://brainly.com/question/30348137

#SPJ1

The possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

The Rational Root Theorem states that if a polynomial [tex]f(x) = anxn + an-1xn-1 + ... + a1x + a0[/tex] has a rational root, then it must be of the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.

For the given polynomial [tex]f(x) = -x + 14x3 - 18x2 - 4x5 - 26x4 + 8[/tex], the constant term is 8 and the leading coefficient is -4.

The factors of 8 are ±1, ±2, ±4, and ±8. The factors of -4 are ±1, ±2, and ±4.

Therefore, the possible rational roots of f(x) are:

p/q = ±1/1, ±2/1, ±4/1, ±8/1, ±1/2, ±2/2, ±4/2, ±8/2, ±1/4, ±2/4, ±4/4, ±8/4

Simplifying gives us the possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±4/2, ±8/2, ±1/4, ±2/4, ±8/4

Simplifying further gives us the final list of possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±2, ±4, ±1/4, ±1/2, ±2

Removing duplicates, the final list of possible rational roots is:

±1, ±2, ±4, ±8, ±1/2, ±1/4

Therefore, the possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

See more about rational roots at: https://brainly.com/question/29504617

#SPJ11

x and y must have values of 3 and 11, respectively.

What is a Parallelogram?

A parallelogram is a geometric shape with sides that are parallel to one another in two dimensions. It is a form of polygon with four sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length. A parallelogram has adjacent angles that add up to 180 degrees.

The angles in a parallelogram are given in the diagram.

As opposite sides are equal and parallel in a parallelogram, the alternate interior angles must also be the same.

This gives:

5y - x = 52 ...(i)

6y - 18 = 48 ...(ii)

Solving (ii)

6y = 66

y = 11

Substituting in (i)

5(11) - x = 52

x = 3

The values of x and y must be 3 and 11 respectively.

To learn more on Parallelograms, click:

brainly.com/question/29147156

#SPJ1

Therefore , the solution of the given problem of proportionality comes out to be connection is not proportional as a result.

When a relationship consistently has the same number, it is considered to be proportional. For instance, the average amount of apples created by each tree determines how many trees there are in an area and how many apples are harvested. Mathematicians refer to a linear relationship between two numbers or factors as being proportional. The other amount doubles if the original amount does.

Here,

When there is a constant multiple between two variables in a relationship, that relationship is called a proportional relationship.

The expense of the 10 apples, which cost $4 plus a $0.20 surcharge, is not directly related to the quantity because the surcharge is a set price regardless of the quantity of apples.

The quantity being measured is growing by a fixed proportion (in this instance, 100%) every time period, so the bacteria doubling every day is an exponential relationship rather than a proportional one.

Last but not least, because the price per book varies, the $9 cost of 4 volumes is also not proportional. We would anticipate spending $18 if we were to purchase 8 books, which is not twice as much as 4 books. The connection is not proportional as a result.

To know more about proportionality visit:

https://brainly.com/question/29126727

#SPJ1

The length οf οne side οf the square is √17 and the area οf the square is 17 square units.

Cοοrdinate geοmetry is a branch οf mathematics that uses algebraic equatiοns tο describe the relatiοnships between pοints and shapes in a plane. It invοlves using the Cartesian cοοrdinate system tο represent pοints with οrdered pairs οf numbers.

We can start by finding the distance between twο οppοsite vertices οf the square using the distance fοrmula:

d = √[(x2 - x1)² + (y2 - y1)²]

Let's find the distance between the pοints (5, 4) and (1, 5), which are οppοsite vertices οf the square:

d = √[(1 - 5)² + (5 - 4)²] = √[16 + 1] = √17

Sο the length οf οne side οf the square is √17.

Tο find the area οf the square, we can use the fοrmula:

area = side²

Substituting the value οf the side that we fοund abοve, we get:

area = (√17)² = 17

Therefοre, the area οf the square is 17 square units.

To learn more about Coordinate Geometry from the given link

https://brainly.com/question/18269861

#SPJ1

Answer:

The volume of a sphere with a diameter of 8.6 m = 333.0 m^3

Step-by-step explanation:

(a) The probability that X = n is 2^n-1 / 2^N - 1.

(b) ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)

(c) The probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.

(d) This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.

To find the probability mass function of X, we need to find the probability that X = n for each n ∈ {1,...,N}. This is the same as finding the probability that the maximum element of a subset of {1,...,N} is n. There are 2^n-1 subsets of {1,...,n-1}, and each of these subsets can be combined with n to form a subset of {1,...,N} with maximum element n. Therefore, the probability that X = n is 2^n-1 / 2^N - 1.

To find the value of ɸ(t) for any t ∈ R, we need to compute the expected value of 2^tx. Using the formula for expected value, we get:

ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)

To find the joint probability mass function of (X,Y), we need to find the probability that X = n and Y = m for each n,m ∈ {1,...,N}. If n = m, then the only subset of {1,...,N} with maximum element n and minimum element m is {n}, so P(X=n, Y=m) = 1 / 2^N - 1. If n ≠ m, then there are 2^(n-m-1) subsets of {m+1,...,n-1}, and each of these subsets can be combined with m and n to form a subset of {1,...,N} with maximum element n and minimum element m. Therefore, the probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.

To find the probability mass function of W = X - Y, we need to find the probability that W = k for each k ∈ {0,...,N-1}. This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.

Learn more about Probability

brainly.com/question/11234923

#SPJ11

The solutions to the equation z^2 = -361 are:

z = ±19i and z = ±19i*(-1)^k, k = 0 or 1.

To find all n complex solutions of the equation x^n = k, where k is a complex number, we can use the polar form of complex numbers. If we write k in polar form as k = re^(iθ), where r = |k| is the modulus of k and θ = arg(k) is the argument of k, then we can write x in polar form as x = ρe^(iφ), where ρ = |x| is the modulus of x and φ = arg(x) is the argument of x.

Substituting these polar forms into the equation x^n = k, we get:

(ρe^(iφ))^n = re^(iθ)

Taking the modulus of both sides, we get:

|ρ^n| = |r|

Since r is a complex number, its modulus is a positive real number, so we can write r = Re^(i0) in polar form, where R = |r|.

Substituting this into the equation above, we get:

|ρ^n| = R

Taking the argument of both sides, we get:

nφ = 0 + 2πk

where k is an integer. Solving for φ, we get:

φ = (2π*k) / n

Substituting this expression for φ back into the polar form of x, we get:

x = ρe^(i(2π*k)/n)

where ρ is a positive real number. This gives us n complex solutions for x, which are:

x = ρe^(i(2π*k)/n), k = 0, 1, 2, ..., n-1.

Now let's apply this method to solve the equation z^2 = -361.

Writing -361 in polar form, we have:

-361 = 361e^(iπ)

Taking the square root of both sides, we get:

z = ±sqrt(361)e^(iπ/2 + iπk)

where k = 0 or 1. Simplifying, we get:

z = ±19e^(i(π/2 + π*k))

So the solutions to the equation z^2 = -361 are:

z = ±19i and z = ±19i*(-1)^k, k = 0 or 1.

Learn more about complex

brainly.com/question/17027861

#SPJ11

Answer: (0, 2)

Step-by-step explanation:

The solution of a system of equations is where the lines intersect at.

We will use substitution to solve the system.

Equations:

y = 1/2x + 2

y = -1/5x + 2

Set both equations to equal each other:

1/2x + 2 = -1/5x + 2

Simplify:

7/10x = 0

x = 0

Plug 0 back in:

y = 1/2(0) + 2

y = 0 + 2

y = 2

The solution is (0, 2)

(This can also be seen by looking at the graph)

Hope this helps!

Assuming this is a simple interest situation:

The 8.75% account will earn (0.0875)(5000) each year.
     $437.5 earned per year

The 9.5 % account will earn (0.095)(7000) each year

     $665 earned per year

Together, these accounts earn $1102.5 per year.

If x is the number of years until the desired interest is earned, then you need to solve 1102.5x = 8820.  Dividing by 1102.5 on both sides, you find x=8.

Again, assuming this is simple interest and not compound interest, it will take 8 years.

The Complete value for the Table is

x               f(x)                   g(x)

1                8                         1.1

5               20                  1.61051

10              35                  2.5937

20             65                   6.7274

The function f(x) grows with faster rate.

A function is an expression, rule, or law in mathematics that describes a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given:

We have f(x) = 3x+ 5 and g(x) = [tex](1.1)^x[/tex]

The Complete value for the Table is

x                   f(x)                                           g(x)

1               3x+ 5 = 3(1) +5= 8                         1.1

5              3x+ 5 = 3(5) +5= 20                      [tex](1.1)^5[/tex]= 1.61051

10             3x+ 5 = 3(10) +5= 35                    [tex](1.1)^{10}[/tex]= 2.5937

20            3x+ 5 = 3(20) +5= 65                    [tex](1.1)^{20}[/tex]= 6.7274

So, the function f(x) grows with faster rate.

Learn more about Function here:

https://brainly.com/question/12431044

#SPJ1