(a) Find all points where the function f(z) = (x^2+y^2-2y)+i(2x-2xy) is differentiable, and compute the derivative at those points.

Answer:

  33/10 and 55/15

Step-by-step explanation:

Possible denominators that are multiples of 5 less than 17 are 5, 10, 15.

Corresponding numerator ranges are [15, 20], [30, 40], and [45, 60]. In only two of these ranges are there any multiples of 11.

  33 in [30, 40]

  55 in [45, 60]

So, there are only two possible fractions meeting your requirement:

  33/10 and 55/15

Answer:

The average rate of change is 2, letter a)

Step-by-step explanation:

Given a function y, the average rate of change S of y=f(x) in an interval  [tex][x_{s}, x_{f}][/tex] will be given by the following equation:

[tex]S = \frac{f(x_{f}) - f(x_s)}{x_{f} - x_{s}}[/tex].

In our problem, we have that:

[tex]f(x) = 2x + 7[/tex]

[tex]x_{s} = -1[/tex]

[tex]x_{f} = 0[/tex]

So:

[tex]f(x_{s}) = f(-1) = 2(-1) + 7 = -2 + 7 = 5[/tex]

[tex]f(x_{f}) = f(0) = 2(0) + 7 = 0 + 7 = 7[/tex]

The average rate of change is:

[tex]S = \frac{f(x_{f}) - f(x_s)}{x_{f} - x_{s}} = \frac{7-5}{0 -(-1)} = \frac{2}{1} = 1[/tex]

The average rate of change is 2, letter a)

Answer:

wala akong alam jun

Step-by-step explanation:

i hate math, mathuloggggggg ka, ayieee ?luh asa ka

The validity of the conclusion (Socrates was a philosopher) does not necessarily follow from the premises because there could be other reasons for Socrates not being a historian.

Let's analyze each inference:

Inference i:

If the moon is made of green cheese, then cows jump over it. The moon is made of green cheese, Therefore, cows jump over the moon.

This inference is an example of a valid logical argument known as modus ponens, a form of deductive reasoning. The structure can be broken down as follows:

Premise 1: If the moon is made of green cheese, then cows jump over it.

Premise 2: The moon is made of green cheese.

Conclusion: Therefore, cows jump over the moon.

Since both premises are assumed to be true, the conclusion logically follows.

Inference ii:

If Socrates was a philosopher then he wasn't a historian. Socrates wasn't a historian. So, Socrates was a philosopher.

This inference is an example of the logical fallacy known as affirming the consequent. Even though the conclusion might be true, the argument's structure is invalid. The structure can be broken down as follows:

Premise 1: If Socrates was a philosopher then he wasn't a historian.

Premise 2: Socrates wasn't a historian.

Conclusion: So, Socrates was a philosopher.

Here, the validity of the conclusion (Socrates was a philosopher) does not necessarily follow from the premises because there could be other reasons for Socrates not being a historian.

Answer:

a)There are [tex]7,2236*10^{26}[/tex] possible different passwords.

b)There are [tex]3,0784*10^{19}[/tex] possible different passwords.

Step-by-step explanation:

a) Each character of the string could be any of the 10 numerals or 21 letters, that means, each character may be any of these 31 different options.

If the password had only one character, there would be 31 different possible passwords, if it were a two characters string, there would be [tex]31^{2}=961[/tex], because in the first place we can place any of the 31 characters, and then in the second place, we can do the same, we are asked of combinations without taking into account order of the characters.

A generalized form, to know how many different passwords we can make of n characters is [tex]31^{n}[/tex].

If we have more than one possible lengths, we add those cases, in our case, we have 14 thru 18 characters, all that could be filled by any of the 31 possible characters.

[tex]\[\sum_{i=14}^{18}31^{i} =31^{14}+31^{15} +31^{16}+31^{17}+31^{18}=7,2236*10^{26}     \\\][/tex].

b) If each password must contain at least 15 numerals, that means the minimum length of a password is 15 now.

We proceed the same as the a) point, but taking into account that for 15 characters, there is not 31 possible characters anymore, but 10.

Doing each individual length:

[tex]Passwords_{length=15} =10^{15}[/tex]

That is because we would multiply 10 possible options by the next possible 10 options and so on 15 times.

[tex]Passwords_{length=16} =10^{15}*31[/tex]

Now, for the sixteenth character, we multiply by the now 31 possible options.

[tex]Passwords_{length=17} =10^{15}*31^{2} \\Passwords_{length=18} =10^{15}*31^{3}[/tex]

For lengths 17 and 18, we do similar stuff, multiplying by 31 for each new character.

To know all the possible combinations, we add the combinations for each length:

[tex]\sum_{i=15}^{18}Passwords_{lenght=i}=3,0784*10^{19}[/tex]

We can see there is approximately 7 orders of magnitude less with the restriction than there is without it.

Answer:

The expected repair cost is 3.96.

Step-by-step explanation:

Given :A particular sale involves four items randomly selected from  a large lot that is known to contain 10% defectives.

The purchaser of  the items will return the defectives for repair, and the repair cost  is given by[tex]C = 3Y^2 + Y + 2[/tex]

To Find : Find the expected repair cost.

Solution:

We are given that A particular sale involves four items randomly selected from  a large lot that is known to contain 10% defectives.

So, The probability of item being defected = 0.10

Let Y denote  the number of defectives among the four sold.

It follows the binomial distribution.

n = 4 , p =0.10

[tex]E(Y)=np = 4 \times 0.10 =0.4[/tex]

[tex]V(Y)=np(1-p)=0.4(1-0.1)=0.36[/tex]

Now we know that [tex]V(Y)=E(Y^2)-[E(Y)]^2[/tex]

[tex]0.36=E(Y^2)-[0.4]^2[/tex]

[tex]0.36=E(Y^2)-0.16[/tex]

[tex]0.36+0.16=E(Y^2)[/tex]

[tex]0.52=E(Y^2)[/tex]

Now we are given an equation that represents the repair cost

[tex]C = 3Y^2 + Y + 2[/tex]

So, Expected repair cost = [tex]E(C) =E( 3Y^2 + Y + 2)[/tex]

[tex]E(C) =3E(Y^2) +E(Y) + 2[/tex]

[tex]E(C) =3 \times 0.52 +0.4+ 2[/tex]

[tex]E(C) =3.96[/tex]

Hence the expected repair cost is 3.96.

The expected number of defectives is 0.4, and using the formula, we find the expected repair cost to be 3.96 units.

To find the expected repair cost, we first need to determine the expected value of the number of defectives, denoted by Y. Since the probability of a defective item is 10% (or 0.1), Y follows a binomial distribution with parameters n = 4 and p = 0.1. The expected value of a binomial random variable is given by E(Y) = np. Hence, the expected number of defectives is E(Y) = 4 × 0.1 = 0.4.

The repair cost is given by the formula C = 3Y² + Y + 2. To find the expected repair cost, we need to calculate E(C). This involves finding E(3Y² + Y + 2).

Using the linearity of expectation:

We already have E(Y) = 0.4. Next, we need to compute E(Y²). For a binomial random variable, E(Y²) can be found using the formula E(Y²) = Var(Y) + [E(Y)]². The variance of a binomial random variable is given by Var(Y) = np(1-p). Thus, Var(Y) = 4 × 0.1 × 0.9 = 0.36.

Thus, E(Y²) = 0.36 + (0.4)² = 0.36 + 0.16 = 0.52.

Putting it all together:

Therefore, the expected repair cost is 3.96 units.

Answer:

The formula for this quadratic function is x*2 +6x+13

Step-by-step explanation:

If we have the vertex and one point of a parabola it is possible to find the quadratic function by the use of this

y= a (x-h)*2 + K

Quadratic function looks like this

y= ax*2 + bx + c

So let's find the a

y= a (x-h)*2 + K where

y is 13, x is 0, h is -3 and K is 4

13= a (0-(-3))*2 +4

13=9a +4

9=9a

9/9=a

1=a

The quadratic function will be

y= 1(x+3)*2 + 4

Let's get the classic form

(x+3)*2 = (x+3)(x+3)

(x*2+3x+3x+9)

x*2 +6x+13

f(0) = 13

The standard form of the quadratic function whose graph is a parabola with the given vertex and that passes through the given point is;

y = x² + 6x + 13

We are given;

Vertex coordinate; (-3, 4)

A point on the graph; (0, 13)

The vertex form of a quadratic equation is given by;

y = a(x - h)² + k

Where h, k are the coordinates of the vertex.

a is the letter in general form of quadratic equation which is;

y = ax² + bx + c

Thus, at point (0, 13) at the vertex of (-3, 4), we have;

13 = a(0 - (-3))² + 4

⇒ 13 - 4 = 9a

9a = 9  

a = 9/9

a = 1  

Since y = a(x - h)² + k is the vertex form, let us put the vertex values for h and k as well as the value of a to get the quadratic equation;

y = 1(x - (-3))² + 4

y = x² + 6x + 9 + 4

y = x² + 6x + 13

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Answer:

Step-by-step explanation:

a. The five-number summary is made up of the following summary means:

1. Minimum: 25

2. First Quartile: 30.5

3. Medium: 34.5

4. Third quartile: 43

5. Maximum: 78

6. P30: 32.1

b.

Standard Deviation: 12.72

Rank: 53

Interquartile range: 12.5

C. The interquartile range is 12.5 and 1.5 times the interquartile range is (1.5) (12.5) = 18.75. Third quartile plus 1.5 times the interquartile range is 61.75. The value of 78 exceeds 61.75, then 78 is an outlier.

Answer:

3.5 of the liquid will contain 0.875g of the substance.

Step-by-step explanation:

The problem states that a liquid contains 0.25 mg of a substance per milliliter. And asks how many grams of the substance will 3.5 L contain.

First step: Conversion of 3.5L to ml

Each liter has 1000ml. So:

1L - 1,000mL

3.5L - xmL

x = 1,000*3.5

x = 3,500mL

Second step: How many miligrams are there in 3,500mL?

The problem states that each ml of the liquid contains 0.25mg of a substance. So

1ml - 0.25mg

3,500 mL - xmg

x = 3,500*0.25

x = 875mg

Final step: Conversion of 875mg to g.

Each g has 1000 mg. So

1g - 1000mg

xg - 875mg

1000x = 875

[tex]x = \frac{875}{1000}[/tex]

x = 0.875g

3.5 of the liquid will contain 0.875g of the substance.

To find the mass of the substance in 3.5 L, convert the volume to milliliters and then multiply by the concentration. The mass is 0.875 g.

To find the number of grams of the substance, we need to convert from milliliters to liters and then use the given concentration of 0.25 mg/mL to find the mass.

First, we convert 3.5 L to milliliters by multiplying by 1000: 3.5 L x 1000 mL/L = 3500 mL.

Next, we multiply the volume in milliliters (3500 mL) by the concentration (0.25 mg/mL) to find the mass: 3500 mL x 0.25 mg/mL = 875 mg.

Finally, we convert the mass from milligrams to grams by dividing by 1000: 875 mg / 1000 = 0.875 g.

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Answer:

A truth table should consider all possible truth values of its simple statements. In this case there are two simple statements, where each one of them can take true or false value, so each of the estatements needs two rows for their individual truth values and, therefore, the compound estatement requires 2x2 = 4 rows for its truth table.

Step-by-step explanation:

Answer:

4 Rows  

Step-by-step explanation:

Since, the number of input given to the AND gate in terms of simple statement, the output of the AND gate will give output true if both the statements given to the AND gate will true otherwise it will show output false.

Let's see the truth table for the inputs of two statements

Statement 1                  Statement 2            Output

 False                               False                       False

 False                               True                         False

 True                                 False                       False

 True                                 True                         True

As we can see the number of rows in the truth table of an AND gate having two input, will have 4 rows.

Answer:

134

Step-by-step explanation:

to find how many gifts it can make, you must find how many times 34 can go into 4580. do this by deciding 4580 by 34.

4580/34=134.7

Since you can't make less than a whole gift you must round it down to 134

Jordan can make 134 number of gifts with 34 M & M s in each with 4580 personalized M and M s.

Jordan wants to gift the volunteer so he is making the gifts for volunteers.

For gifting purpose the number of personalized M and M s she ordered is given by = 4580.

The number of personalized M and M s she puts in each volunteer 's gift is given by = 34.

So the number of total gifts she can pack with this number of personalized M and M s given by = 4580/34 = 134.7 = 134 approximately.

So Jordan can make 134 gifts with 34 M and M s in each with 4580 personalized M and M s.

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Answer: a) [tex]N(t) = 10^3\exp(0.046\frac{1}{min}t)[/tex]

b) 1,000,000 bacteria at t = 150 min

Step-by-step explanation:

Hi!!

A colony that grows exponentially has a number of bacteria:

[tex]N(t) = N_0 \exp(\lambda t)[/tex]

In this case at time t = 0:

[tex]N(0)=N_0=10^3[/tex]

We need to find the value of λ. We use the data:

[tex]N(t=50\;min)10^4 = 10^3\exp(\lambda \;50\;min)[/tex]

[tex]ln(10)=2.3=\lambda\;50\;min\\\lambda= \frac{0.046}{min}\\N(t) = 10^3\exp(\frac{0.046}{min}t)\\[/tex]

To find when there will be 1,000,000 bacteria:

[tex]10^6=10^3\exp(\frac{0.046}{min}t)[/tex]

[tex]\ln(10^3)=3\ln(10) = \frac{0.046}{min}t[/tex]

[tex]t = 150\;min [/tex]

Answer:

H.''n is not divisible by 6 and n is not divisible by both 2 and 3.

Step-by-step explanation:

We are given that  a statement ''n is divisible by 6 or n is divisible by both 2 and 3.''

We have to write the negation of the  given statement.

Negation: If  a statement p is true then its negations is  p is false.

n is divisible by 6 then negation is n is not divisible by 6.

n is divided by both 2 and 3 then negation is n is not divisible  by both 2 and 3.

Therefore, negation of given statement

''n is not divisible by 6 and n is not divisible by both 2 and 3.

Hence, option H is true.

Answer:

H.''n is not divisible by 6 and n is not divisible by both 2 and 3.

Step-by-step explanation:

We are given that  a statement ''n is divisible by 6 or n is divisible by both 2 and 3.''

We have to write the negation of the  given statement.

Negation: If  a statement p is true then its negations is  p is false.

n is divisible by 6 then negation is n is not divisible by 6.

n is divided by both 2 and 3 then negation is n is not divisible  by both 2 and 3.

Therefore, negation of given statement

''n is not divisible by 6 and n is not divisible by both 2 and 3.

Hence, option H is true.

Step-by-step explanation:

Answer:

Tthe minterm expression for the function X is:

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC

(The character ' refers to the complement of the variable)

Step-by-step explanation:

It is important to know that a minterm refers to a product of all variables that have a result of 1 in the truth table. The variables can be with or without complement. For example, consider the following truth table:

A, B, F

0, 0, 0

0, 1, 0

1, 0, 0

1, 1, 1

According to the previous truth table, the minterm would the row that F equals 1. In this case, when A and B are 1, F is 1. So, the variables are used directly (without complement). The minterm is: F = AB

Now, according to the problem, the truth table is presented below. The idea is to complete the X column which is '0' if the product BA x DC is less than 3 and is '1' if the product BA x DC is greater or equal than 3.

The idea is to get the rows that have '1' in the X column. The minterms are are: 8, 11, 12, 14, 15 and 16. The minterm is composed by the four variables. To indicate a '1' just put the letter. To indicate a '0' put a letter with this character: '. For example, the row 8 would be B'ADC, because B is '0' and the others are '1'. The total minter expression is:

X = 'minterm 8' + 'minterm 11' + 'minterm 12' + 'minterm 14' + 'minterm 15' + 'minterm 16'

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC

Thus, the minterm expression for the function X is:

X = B'ADC + BA'DC' + BA'DC + BAD'C + BADC' + BADC

Answer:

Each inhalation has 750 micrograms of metaproterenol sulfate.

Step-by-step explanation:

First step: How many miligrams are there in each inhalation.This can be solved by a rule of three problem.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

In this step, our measures are:

- The number of inhalations.

- The quantity inhaled.

As the number of inhalations increases, so does the quantity inhaled. This means that this is a direct rule of problem.

300 inhalations contains 225mg of metaproterenol sulfate. How many miligrams are in a inhalation?

300 inhalations - 225mg

1 inhalation - xmg

300x = 225

[tex]x = \frac{225}{300}[/tex]

x = 0.75mg

Each inhalation has 0.75mg of metaproterenol sulfate.

Final step: Conversion of 0.75mg to micrograms.

Each mg has 1000 micrograms. So:

1mg - 1000 micrograms

0.75mg - x micrograms

x = 1000*0.75

x = 750 micrograms

Each inhalation has 750 micrograms of metaproterenol sulfate.

To find out how many micrograms of metaproterenol sulfate are contained in each inhalation, convert the total milligrams to micrograms, then divide by the total number of inhalations. Thus, each inhalation contains 750 μg of metaproterenol sulfate.

The problem requires conversion of milligrams (mg) to micrograms (μg) which will allow for the calculation of the amount of metaproterenol sulfate in each inhalation. It is important to know that 1 mg is equal to 1000 μg. Hence, 225 mg of metaproterenol sulfate is equal to 225,000 μg (225 X 1000). To find out how many micrograms are in each inhalation, divide the total number of micrograms (225,000 μg) by the total number of inhalations (300). Hence, each inhalation contains 750 μg of metaproterenol sulfate.

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Answer:

p1. If I pay attention to class, therefore I'll take good grades on test next week.

p2. I like cheese.

Step-by-step explanation:

Propositions are statements that declare something from/for someone. It always states something. It may be classified as simple (p1) or combined (p2) depending on the presence or absence of logical connectors p1 (If...therefore). A combined proposition is made up of two simple propositions.

We can and ought work with symbolic operators.

Let p1 be translated into symbolic language,

I pay attention to class = q

I'll take good grades on test next week r

q→r

p2= I like cheese. We can just simply call it p2.

Answer:

[tex]\$21.1[/tex]

Step-by-step explanation:

We know that for principal amount P , time period T and rate of interest [tex]R\%[/tex] , simple interest is given by [tex]S.I. = \frac{P\times R\times T}{100}[/tex] .

Here ,

[tex]P=\$1600\\T=2\,\,months=\frac{2}{12}\,\,years=\frac{1}{6}\,\,years\\S.I=\$56.24[/tex]

To find :  simple interest rate i.e., [tex]R\%[/tex]

On putting values of [tex]P\,,\,T\,,\,S.I[/tex] in formula , we get [tex]S.I. = \frac{P\times R\times T}{100}[/tex]

[tex]56.24 = \frac{1600\times R\times 1}{600}\\R=\frac{56.24\times 600 }{1600}=\frac{703\times 3}{100}=\$21.09[/tex]

Now we need to round off the answer to the nearest tenth .

So, simple interest rate is % = [tex]\$21.09[/tex] = [tex]\$21.1[/tex]

Answer:

The period of the [tex]\sin[/tex] function is [tex]2\pi[/tex]. Then three times this period is [tex]6\pi[/tex].

Step-by-step explanation:

Using Mathematica you can use the command Plot as follows:

```

Plot[Sin[x], {x,0,6Pi}]

```

The output is the graph shown below.

Using Mathematica, you can plot a Sin function that covers a range three times its period by using the Plot[] function and specifying the range of x-values from 0 to 6*Pi. This results in a plot from x=0 to x=6π, a range that covers three full periods of the Sin function.

To begin, once you've installed and opened Mathematica, you'll want to make use of the Plot[] function to indeed plot the function. The Sin function is periodic and has a period of 2π. If you'd like to plot a function that covers a range three times this period, you'd want a range of 6π. The code to achieve this is as follows:

Plot[Sin[x], {x, 0, 6*Pi}]

In this code, 'Sin[x]' is the function you want to plot and '{x, 0, 6*Pi}' defines the range of x-values over which to plot the function. The result is a plot of the Sin function from x=0 to x=6π, which covers three full periods of the Sin function.

As a side note, if you wanted to create a scatter plot or calculate a best-fit line as you might do in a statistics application, you would use different commands within Mathematica or possibly use a different software or calculator that specializes in statistical analysis, such as a TI-83/84 calculator. However, these operations are not needed to simply plot a sin function.

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Answer: (D) none of the these.

Step-by-step explanation:

If the shape of our data set is multimodal, the it will show two or more peaks which represents the number modes in the data.

Since it has no relation with mean or median of the data, there for the correct option will be "none of these".

In a multimodal distribution, the relationship between the mean and the median cannot be determined without more information on the distribution's skewness and the relative size of the peaks. Thus, the answer is that none of the provided options are necessarily correct.

When dealing with a multimodal distribution, which is a distribution with more than one peak or "mode," the relationship between the mean, the median, and the mode is not as predictable as it is in symmetric distributions. Since multimodal distributions can have multiple peaks at different points, it can alter the typical order of the mean, median, and mode based on where these peaks occur relative to each other.

The presence of multiple modes can pull the mean toward the larger values if one peak represents higher values significantly, or it can pull the mean towards lesser values if a peak represents lower values significantly. However, without additional specific information about the skewness or the relative size of these peaks, we cannot definitively say whether the mean will be less than, equal to, or greater than the median. Therefore, the correct answer is:

(D) none of these.

Answer:

Option B is the answer.

Step-by-step explanation:

The area of a surface can be measured in square meter (meter²).

Square meter means its a multiplication of two lengths measured in meter.

Option A.

m.cm = unit of length × unit of length

So we can measure the area by this unit.

Option B.

[tex]\frac{ft^{3} }{m^{2} }[/tex]

Since ft³ is a unit of volume and m² of area. When we divide these units we get the unit of length.

Therefore, we can not measure the area by this unit.

Option C.

Inch² = unit of length × unit of length

So we can measure the area by this unit.

Option D.

m.ft = Unit of length × unit of length

Which shows its a unit of area.

Option E.

[tex]\frac{ft^{3} }{m}[/tex] = [tex]\frac{\text{Unit of volume}}{\text{Unit of length}}[/tex]

    = unit of area

Therefore, we can use this unit to measure the area.

Option B. is the answer.

Answer:  The required inverse of the given matrix is

[tex]P^{-1}=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]

Step-by-step explanation:  We are given to find the inverse of the following orthogonal matrix :

[tex]P=\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right] .[/tex]

We know that

if M is an orthogonal matrix, then the inverse matrix of M is the transpose of M.

That is, [tex]M^{-1}=M^T.[/tex]

The transpose of the given matrix P is given by

[tex]P^T=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]

Therefore, according to the definition of an orthogonal matrix, the inverse of matrix P is given by

[tex]P^{-1}=P^T=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]

Thus, the required inverse of the given matrix is

[tex]P^{-1}=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]

Answer: There are 1820 sets of four marbles other than lavender.

Step-by-step explanation:

Since we have given that

Number of red marbles = 3

Number of green marbles = 5

Number of lavender marbles = 1

Number of yellow marbles = 2

Number of orange marbles = 6

So, Total number of marbles = 3 + 5+ 1 + 2 + 6 = 17

We need to find the sets of 4 marbles other than lavender.

so, Number of total marbles other than lavender  becomes = 17 -1 =16

Number of marbles in a set = 4

So, Number of ways becomes

[tex]^{16}C_4\\\\=1820[/tex]

Hence, there are 1820 sets of four marbles other than lavender.

Answer:

y=2 e^{(5x +5)} + e^{(-x - 1)}

Step-by-step explanation:

Here we have an ODE, matched to zero, so it is an  

homogeneous equation. The typical aproach here is to propose a solution to y and then find the constants that fullfit the equation.

We propose [tex]y=e^{rx} \\\\So \frac{dy}{dx} = re^{rx}\\And \\\frac{dy^{2} }{dx^{2} } = r^{2} e^{rx}Replacing this in the original equation, we getr^{2} e^{rx} - 4re^{rx} -5e^{rx} = 0\\[/tex]

Taking the exponential as a factor, we obtain:

[tex]e^{rx}(r^{2}  - 4r -5) = 0\\[/tex]

An exponential function is always greater than zero, so the only way of matching the equation is to find two "r" that reduce the second term to zero(you can factorize or use the Quadratic formula (see imagen below).

[tex](r^{2}  - 4r -5) = 0\\ r= 5 and r=-1\\\\[/tex]

So, this gives us the two parts of our solution:

[tex]y= C e^{5x} + D e^{-x}[/tex] , with C and D being real numbers.

In order to find C and D, we will use the initial values given in the question.

[tex]y = C e^{5x} + De^{-x}\\\\\frac{dy}{dx} = 5C e^{5x} - D e^{-x} \\y(-1) = 3= C e^{-5} + De^{1}\\\frac{dy}{dx} = 9 = 5C e^{-5} - D e^{1} \\[/tex]

That is a linear equations system of two equations and two unknowns, which is resolveable :

[tex]\left \{ {{C e^{-5} + De = 3} \atop {5C e^{-5} - De= 9}} \right. \\[/tex]

To make it more clear, we will make a change of variables:

[tex]C e^{-5} = A\\De = B\\\\So\\\left \{ {{A + B= 3 } \atop {5A - B=9}} \right. \\\\[/tex]

Clearing B:

[tex]B = 3 - A\\\\5A - 3 + A = 9\\6A = 12, A = 2[/tex]

For B, we go back to (1)

[tex]B=3-A  (1)\\B = 3-2, B=1[/tex]

Now, we undo the change of variable :

[tex]A= C e^{-5} = 2 \\C= \frac{2}{e^{-5} } \\\\B = De= 1\\D=\frac{1}{e}[/tex]

Finally, we just replace C and D in y and then work a bit with it to have a more aesthetic response:

[tex]y=C e^{5x} + De^{-x}\\y=  \frac{2}{e^{-5} } e^{5x}  + \frac{1}{e} e^{-x}\\y = 2 e^{5x - (-5)} + e^{(-x - 1)}\\y=2 e^{(5x +5)} + e^{(-x - 1)}[/tex]

To solve the given initial value problem, we can use the characteristic equation and the known initial conditions to find the solution. The solution to the initial value problem is y(x) = 3.440e^(-x) - 6.440e^(5x).

To solve the initial value problem y'' - 4y' - 5y = 0, y(-1) = 3, y'(-1) = 9, we can use the characteristic equation. The characteristic equation for this differential equation is r^2 - 4r - 5 = 0. Solving this equation, we find the roots r = -1 and r = 5.

Therefore, the general solution to the differential equation is y(x) = c1e^(-x) + c2e^(5x), where c1 and c2 are constants.

Using the initial conditions, y(-1) = 3 and y'(-1) = 9, we can solve for c1 and c2. Substituting the values of y and y' into the general solution and rearranging the equations, we get c1e + c2e^(-5) = 3 and -c1e - 5c2e^(-5) = 9.

Solving these two equations simultaneously, we find c1 ≈ 3.440 and c2 ≈ -6.440. Therefore, the solution to the initial value problem is y(x) = 3.440e^(-x) - 6.440e^(5x).

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Answer:

h =  v/lw .

Step-by-step explanation:

v=lwh

Divide both sides by lw:

v  / lw = h.

Answer:  The required age of Nate is 4 years.

Step-by-step explanation:  Given that Alice is 3 years old and Nate is half of the sum of Alice's age and 5.

We are to find the age of Nate.

Let x and y represents the ages of Alice and Nate respectively in years.

Then, according to the given information, we have

[tex]x=3.[/tex]

and also,

[tex]y=\dfrac{1}{2}(x+5)\\\\\\\Rightarrow y=\dfrac{1}{2}(3+5)\\\\\\\Rightarrow y=\dfrac{1}{2}\times8\\\\\Rightarrow y=4.[/tex]

Thus, the required age of Nate is 4 years.

Answer:

electric energy = 980 J

Step-by-step explanation:

Given,

Diameter of wind turbine = 100 m

Speed of wind,v = 7 m/s

Density of air = 1.25 kg/m

Hence, the total mass of air enters into turbine =Speed of wind x Diameter of wind turbine

                                                                             = 1.25 x 100

                                                                              = 125 kg

Hence, total mechanical energy of turbine can be given by,

[tex]E\ =\ \dfrac{1}{2}.m.v^2[/tex]

    [tex]=\ \dfrac{1}{2}\times 125\times 7^2[/tex]

    = 3062.5 J

Efficiency of turbine = 32%

Hence,

total electric energy = 32% of mechanical energy

                                  [tex]=\dfrac{32}{100}\times 3062.5\ J[/tex]

                                  = 980 J

So, the total electric energy produced by the turbine will be 980 J.

Final answer:

The electricity produced by a 100m diameter wind turbine exposed to 7 m/s wind and with a density of 1.25 kg/m^3, assuming a 32% efficiency, would be approximately 5.39 megawatts.

Explanation:

To calculate the electricity produced by a 100-meter diameter wind turbine exposed to a wind speed of 7 m/s, we initially need to find out the power generated by the turbine before efficiency losses. We use the formula for the power captured by the wind turbine:

P = ½ ρ A v³

Where ρ (rho) is the air density, A is the swept area of the turbine blades, and v is the wind velocity.

First, let's calculate the swept area (A):
A = πr²
A = π(½ × 100 m)²
A = π(× 50 m)²
A = π × 2500 m²
A = 7853.98 m²

Next, plug in the values into the power equation:
P = 0.5 × 1.25 kg/m³ × 7853.98 m² × (7 m/s)³
P = 0.5 × 1.25 × 7853.98 × 343
P = 16839589.89 watts or 16.84 MW before efficiency losses.

To obtain the electricity produced considering the efficiency of the turbine, we multiply the initial power by the efficiency factor:

Electricity produced = P × efficiency
Electricity produced = 16.84 MW × 0.32
Electricity produced = 5.39 MW

Thus, the wind turbine would produce approximately 5.39 megawatts of electricity under the given conditions.

Answer:

a) 56x = 16800 - 14p  

b) $1200

c) 300 pounds

Step-by-step explanation:

Given:

At p₁ = $900 ; x₁ = 75 pounds

at p₂ = $956 ; x₂ = 75 - 14 = 61 pounds

Now,

from the standard equation of line, we have

[tex](x - x_1)=\frac{(x_2-x_1)}{(p_2-p_1)}\times(p-p_1)[/tex]

on substituting the respective values, we get

[tex](x - 75)=\frac{(61-75)}{(956-900)}\times(p-900)[/tex]

or

( x - 75 ) × 56 = -14p + 12600

or

56x - 4200 = -14p + 12600

or

56x = 16800 - 14p        (relation between the unit price p and demand x)

b) For no consumers x = 0

thus, substituting in the relation we get

56 × 0 = 16800 - 14p

or

14p = 16800

or

p = $1200

c) For free , p = $0

on substituting in the above relation derived, we get

56x = 16800 - ( 14 × 0 )

or

x = 300 pounds

Answer:

Angle A = 100°

Angle B = 80°

Step-by-step explanation:

Supplementary angles sum 180°

Angle A to Angle B ratio is 5:4, you can write that ratio using a factor.

Angle A = 5x

Angle B = 4x

Since the sum is 180°

Angle A + Angle B = 180°

5x + 4x = 180°

9x = 180°

x = 20°

Replacing x = 20°

Angle A = 5(20°) = 100°

Angle B = 4(20°) 80°

Answer with Step-by-step explanation:

Let A and B be non- empty bounded subset of R

a.We have to find why [tex]sup(A\cup B)[/tex]exist

If A and B are bounded set

Then there exist  constant  such that

[tex]a\leq A\leq b[/tex] and [tex]c\leq B\leq d[/tex]

Then , sup of A =b and sup of B=d

When  a set is bounded then all elements lie in the set are lie between the constants s and t.

All elements are less than or equal to t then t is supremum of set.

Because both set are bounded and sup of both set A and B are exist.All elements A union B are less than or equal to sup A or sup B.

[tex]sup(A\cup B)=max(sup A, sup B)[/tex]

Then, [tex]sup (A\cup B)[/tex] exist.

b.We have to prove that

[tex] sup (A\cup B)=max(sup A,sup B)[/tex]

Suppose ,A =(1,2) and B=(2,3)

Sup A=2 , sup B=3

[tex](A\cup B)=(1,2)\cup (2,3)[/tex]

Upper bound of [tex]A\cup B)=3[/tex]

Hence, [tex]Sup (A\cup B)=3[/tex]

If A=(4,5),B=(2,3)

Sup A=5,Sup B=3

[tex]A\cup B=(4,5)\cup (2,3)[/tex]

[tex]Sup(A\cup B)=5[/tex]

Hence, [tex]Sup(A\cup B)=5[/tex]

Hence, we can say that [tex]sup(A\cup B)=max(sup A,sup B)[/tex].

Answer:

90% of people marry there 7th grade love. since u have read this, u will be told good news tonight. if u don't pass this on nine comments your worst week starts now this isn't fake. apparently if u copy and paste this on ten comments in the next ten minutes you will have the best day of your life tomorrow. you will either get kissed or asked out in the next 53 minutes someone will say i love you

Step-by-step explanation:

Answer:   [tex](182.356,\ 197.644)[/tex]

Step-by-step explanation:

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Sample size : n=85

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

Sample mean : [tex]\overline{x}=190[/tex]

Standard deviation : [tex]\sigma=11.7[/tex]

The confidence interval for population mean is given by :-

[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\\\\=190\pm(1.96)\dfrac{11.7}{\sqrt{9}}\\\\=190\pm7.644\\\\=(190-7.644,\ 190+7.644)=(182.356,\ 197.644)[/tex]

Thus, the 95% confidence interval for the true mean cholesterol content, μ, of all such eggs = [tex](182.356,\ 197.644)[/tex]

Hence, we conclude that the true population mean of amount of cholesterol lies between 182.356 and 197.644.

Answer:

The equation is an differential equation of second order.

The dependent variable is x, while t is the independent variable.

Step-by-step explanation:

The order of the equation depends on the greatest grade of the derivative, in this case it's the second derivative (x'')

Since x is a function of t, we would have that t is the independent variable while x is the dependent variable.

Final answer:

The given equation is mx'' + bx' + kx = 8t^6. The order of the equation is 2 and the independent and dependent variables are t and x, respectively. The parameters in the equation are m, b, and k.

Explanation:

The given equation is mx'' + bx' + kx = 8t^6, where x is a function of t. Let's break down the equation:

Therefore, the order of the equation is 2, as it contains the second derivative of x.