This shows that if e² is real, then Im z = n, where n = 0, ±1, ±2, ..., which means that the imaginary part of z can only take the values n
How to determine real numbers?To show that if e² is real, then Im z = n, where n = 0, ±1, ±2, ..., start by assuming that e² is a real number. We can express z in terms of its real and imaginary parts as z = x + iy, where x and y are real numbers.
Using Euler's formula, [tex]e^{(ix)} = cos(x) + i sin(x)[/tex], write e² as:
[tex]e^{2} = (e^{(ix)})^{2}[/tex]
= (cos(x) + i sin(x))²
= cos²(x) + 2i cos(x) sin(x) - sin²(x)
Since e² is real, the imaginary part of e² must be zero. Therefore, the coefficient of the imaginary term, 2i cos(x) sin(x), must be zero:
2i cos(x) sin(x) = 0
For this equation to hold true, either cos(x) = 0 or sin(x) = 0.
If cos(x) = 0, it implies that x is an odd multiple of π/2, i.e., x = (2n + 1)π/2, where n is an integer.
If sin(x) = 0, it implies that x is a multiple of π, i.e., x = nπ, where n is an integer.
Therefore, combining both cases:
x = (2n + 1)π/2 or x = nπ, where n is an integer.
Now let's consider Im z, which is the imaginary part of z:
Im z = y
Since y is the imaginary part of z and z = x + iy, y is directly related to x. From the earlier cases, x can take the values (2n + 1)π/2 or nπ, where n = integer.
For the case x = (2n + 1)π/2, the imaginary part y can be any real number, and therefore Im z can take any value.
For the case x = nπ, the imaginary part y must be zero, otherwise, the imaginary part of e² will not be zero. Therefore, in this case, Im z = 0.
Combining both cases:
Im z = n, where n = 0, ±1, ±2, ...
This shows that if e² is real, then Im z = n, where n = 0, ±1, ±2, ..., which means that the imaginary part of z can only take the values n, where n is an integer.
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a land lady rented out her house for$240,000 for one year. During the year she paid 15%of the rent as income tax. she also paid 25% of the rent property tax and spent $10,000 on repairs. calculate the landlady's total expenses
Answer: 106,000 dollars
Step-by-step explanation:
She paid 36,000 dollars as income tax and 60,000 on the rent property tax and 96,000 dollars + 10,000 dollars is 106,000 dollars.
Use the Central Limit Theorem to find the mean and standard error of the mean of the indicated sampling distribution. The monthly rents for studio apartments in a certain city have a mean of $900 and a standard deviation of $180. If random samples of size 30 are drawn from the population, identify the mean, wx, and standard deviation, 7, of the sampling distribution of sample means with sample size n
Mean of the sampling distribution (wx): $900
Standard error of the mean (σx): Approximately $32.92
To find the mean and standard error of the mean of the sampling distribution, we can use the Central Limit Theorem.
The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution.
In this case, the population mean is μ = $900 and the population standard deviation is σ = $180. We are drawing random samples of size n = 30 from this population.
The mean of the sampling distribution (wx) will be equal to the population mean (μ), which is $900.
The standard deviation of the sampling distribution (σx), also known as the standard error of the mean, can be calculated using the formula:
σx = σ / √n
where σ is the population standard deviation and n is the sample size.
Substituting the given values, we have:
σx = $180 / √30
Calculating this value, we find:
σx ≈ $32.92
Therefore, the mean of the sampling distribution (wx) is $900, and the standard error of the mean (σx) is approximately $32.92.
Please note that the Central Limit Theorem assumes a sufficiently large sample size (typically n ≥ 30) for the approximation to hold.
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fill in the blank to make the statement true.
1.( )+4 506-21 000=1 001
2.698-( )+711=1 388
3.109 006-( )-66 666=23 124
some questions I have already answered only that question make me confused
Consider a drug testing company that provides a test for marijuana usage. Among 308 tested subjects, results from 29 subjects were wrong. (either a false positive or a false negative). Use a 0.05 significance level to test the claim that less than 10 percent of the test results are wrong.
The required answer is based on the given data, we do not have sufficient evidence to support the claim that less than 10 percent of the test results are wrong at a 0.05 significance level.
To test the claim that less than 10 percent of the test results are wrong, use a hypothesis test with a significance level of 0.05. Let's define the null and alternative hypotheses:
Null Hypothesis (H0): The proportion of wrong test results is 10 percent or more.
Alternative Hypothesis (Ha): The proportion of wrong test results is less than 10 percent.
Use the binomial distribution to analyze the data. Let p be the true proportion of wrong test results. Since we want to test that the proportion is less than 10 percent, set p = 0.10 for the null hypothesis.
Given that 29 out of 308 tested subjects had wrong test results, calculate the sample proportion, denoted by p^, as follows:
p^ = 29 / 308 = 0.094
To conduct the hypothesis test, we can use the z-test for proportions. The test statistic is calculated as:
z = (p^ - p) / [tex]\sqrt{}[/tex]((p x (1 - p)) / n)
In this case, since we are testing whether the proportion is less than 10 percent, calculate a one-tailed z-test.
Substituting the values into the formula:
z = (0.094 - 0.10) / [tex]\sqrt{}[/tex]((0.10 x (1 - 0.10)) / 308)
Simplifying the expression:
z = -0.006 /[tex]\sqrt{}[/tex] (0.09 / 308)
z ≈ -0.006 / 0.017
Calculating the z-value:
z ≈ -0.353
To determine the critical value for a one-tailed test at a significance level of 0.05, we can consult the z-table or use statistical software. For a significance level of 0.05, the critical z-value is approximately -1.645 (since we are testing for less than 10 percent, in the left tail).
Since the calculated z-value (-0.353) is greater than the critical z-value (-1.645), we fail to reject the null hypothesis. There is not enough evidence to conclude that the proportion of wrong test results is less than 10 percent.
Therefore, based on the given data, we do not have sufficient evidence to support the claim that less than 10 percent of the test results are wrong at a 0.05 significance level.
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find the missing side z
Answer:
Z=[tex]7\sqrt{2}[/tex] m
Step-by-step explanation:
This is a 45º-45º-90º triangle. That means the side lengths are x, x, and [tex]x\sqrt{2}[/tex].
In this case 14 m is equal to [tex]x\sqrt{2}[/tex]. Now solve for X, which is equal to Z.
The rest of the work is written down in the file attached
Don't forget the units though!
hope I could help
4. Solve the Cauchy-Euler equation: x"y" - 2x*y" - 2xy +8y = 0 (12pts)
the general solution to the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0 is given by y(x) = c₁x² + c₂x⁻¹ + c₃x⁻¹ln(x) where c₁, c₂, and c₃ are constants.
To solve the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0, we'll make the substitution y = [tex]x^r[/tex], where r is a constant.
Let's differentiate y with respect to x:
y' = [tex]rx^{r-1}[/tex]
y" = [tex]r(r-1)x^{r-2}[/tex]
y'" = [tex]r(r-1)(r-2)x^{r-3}[/tex]
Now, substitute these derivatives into the original equation:
[tex]x^3(r(r-1)(r-2)x^{r-3} - 2x^2(r(r-1)x^{r-2}) - 2x(rx^{r-1}) + 8x^r = 0[/tex]
Simplifying, we get:
[tex]r(r-1)(r-2)x^r - 2r(r-1)x^r - 2rx^r + 8x^r = 0[/tex]
Combining like terms, we have:
r(r-1)(r-2) - 2r(r-1) - 2r + 8 = 0
Simplifying further, we get:
r³ - 3r² + 2r - 2r² + 2r + 8 - 2r + 8 = 0
r³ - 3r² + 8 = 0
To solve this cubic equation, we can try to find a rational root using the Rational Root Theorem or use numerical methods to approximate the roots.
By inspection, we find that r = 2 is a root of the equation. This means (r - 2) is a factor of the equation.
Using long division or synthetic division, we can divide r^3 - 3r^2 + 8 by (r - 2):
2 | 1 -3 0 8
| 2 -2 -4
_______________________
1 -1 -2 4
The quotient is r² - r - 2.
Factoring r² - r - 2, we get:
r² - r - 2 = (r - 2)(r + 1)
So the roots of the equation r³ - 3r² + 8 = 0 are: r = 2, r = -1 (repeated root).
Therefore, the general solution to the Cauchy-Euler equation x³y'" - 2x²y" - 2xy' + 8y = 0 is given by:
y(x) = c₁x² + c₂x⁻¹ + c₃x⁻¹ln(x)
where c₁, c₂, and c₃ are constants.
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Given question is incomplete, the complete question is below
Solve the Cauchy-Euler equation:
x³y'" - 2x²y" - 2xy' + 8y = 0
Maya's fish tank has 17 liter of water in it. She plans to add 4 liters per minute until the tank has at least 53 liters. What are possible numbers of minutes Maya could add water? Use t for the number of minutes. Write your answer as an inequality solved for t.
Answer: you know it’s 9mins but Im not sure how I should make the equation probably I think (53= 9T + 4)
Step-by-step explanation:
21. Your local grocery store stocks rolls of bathroom tissue in single packages and in more economical 12-packs. You are trying to decide which to buy. The single package costs 45 cents and the 12-pack costs $5. You consume bathroom tissue at a fairly steady rate of one roll every three months. Your opportunity cost of money is computed assuming an interest rate of 25 percent and a fixed cost of $1 for the additional time it takes you to buy bathroom tissue when you go shopping. (We are assuming that you shop often enough so that you don't require a special trip when you run out.) a. How many single rolls should you be buying in order to minimize the annual holding and setup costs of purchasing bathroom tissue? b. Determine if it is more economical to purchase the bathroom tissue in 12-packs.
To minimize the annual holding and setup costs of purchasing bathroom tissue, you should buy single rolls of bathroom tissue. Purchasing the bathroom tissue in 12-packs is not more economical in this scenario.
To determine the optimal choice between buying single rolls and 12-packs of bathroom tissue, we need to consider the annual holding and setup costs.
a. To minimize the annual holding and setup costs, we need to compare the costs of buying single rolls versus 12-packs.
For single rolls:
Cost per roll = $0.45
Holding cost per roll per year = opportunity cost of money * cost per roll = 0.25 * $0.45 = $0.1125
Setup cost per purchase = $1
Consumption rate = 1 roll every 3 months = 4 rolls per year
For 12-packs:
Cost per pack = $5
Number of rolls in a 12-pack = 12
Holding cost per roll per year = 0.25 * ($5 / 12) = $0.1042
Setup cost per purchase = $1
To minimize the annual holding and setup costs, we need to compare the costs of purchasing the required number of single rolls and the cost of purchasing 12-packs.
b. Comparing the costs, it is more economical to purchase single rolls because the holding cost per roll per year is lower for single rolls ($0.1125) compared to 12-packs ($0.1042). Additionally, the setup cost is the same for both options ($1).
Therefore, to minimize the annual holding and setup costs, you should buy single rolls of bathroom tissue rather than 12-packs.
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pls help! correct gets thanks and brainliest :)
Answer:
[tex] m\angle SYX=76\degree [/tex]
Step-by-step explanation:
[tex] m\angle SYX = m\angle UYV[/tex]
(Vertical angles)
[tex] \because m\angle UYV=76\degree [/tex]
[tex] \therefore m\angle SYX=76\degree [/tex]
Let X and Y be two independent N(0,2) random variable and Z= 7+X+Y, W= 1+ Y. Find cov(Z, W) and p(Z,W).
The correlation coefficient (p(Z, W)) between Z and W is sqrt(2) / 2.
To find the covariance of Z and W and the correlation coefficient (p(Z, W)), we can use the properties of covariance and correlation for independent random variables.
Given that X and Y are independent N(0, 2) random variables, we know that their means are zero and variances are 2 each.
Covariance:
Cov(Z, W) = Cov(7 + X + Y, 1 + Y)
Since X and Y are independent, the covariance between them is zero:
Cov(X, Y) = 0
Using the properties of covariance, we have:
Cov(Z, W) = Cov(7 + X + Y, 1 + Y)
= Cov(X, Y) + Cov(Y, Y)
= Cov(X, Y) + Var(Y)
Since Cov(X, Y) = 0 and Var(Y) = 2, we can substitute these values:
Cov(Z, W) = 0 + 2
= 2
Therefore, the covariance of Z and W is 2.
Correlation Coefficient:
p(Z, W) = Cov(Z, W) / (sqrt(Var(Z)) * sqrt(Var(W)))
To calculate p(Z, W), we need to find Var(Z) and Var(W):
Var(Z) = Var(7 + X + Y)
= Var(X) + Var(Y) (since X and Y are independent)
= 2 + 2 (since Var(X) = Var(Y) = 2)
= 4
Var(W) = Var(1 + Y)
= Var(Y) (since 1 is a constant and does not affect variance)
= 2
Now we can calculate p(Z, W):
p(Z, W) = Cov(Z, W) / (sqrt(Var(Z)) * sqrt(Var(W)))
= 2 / (sqrt(4) * sqrt(2))
= 2 / (2 * sqrt(2))
= 1 / sqrt(2)
= sqrt(2) / 2
Therefore, the correlation coefficient (p(Z, W)) between Z and W is sqrt(2) / 2.
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Define the linear transformation T: RR by T(v) Av. Find the dimensions of R" and Rm. A = [-2-22] 12 dimension of R" dimension of R
The linear transformation T: R^2 → R^2, defined by T(v) = Av, where A = [[-2, -2], [1, 2]], maps a two-dimensional vector space onto itself. The dimension of R^2 is 2.
In the given linear transformation T: R^2 → R^2, the transformation is defined as T(v) = Av, where A is the transformation matrix. The given matrix A = [[-2, -2], [1, 2]] represents the coefficients of the linear transformation. This means that the transformation T takes a two-dimensional vector v in R^2 and applies the matrix A to it.
The dimension of R^2 is 2, indicating that the vector space R^2 consists of all ordered pairs (x, y) where x and y are real numbers. In this case, the linear transformation T maps a vector in R^2 to another vector in R^2, so both the input and output dimensions are 2.
The dimension of R^n refers to the number of components or variables in a vector in R^n. For example, R^2 consists of vectors with two components, while R^3 consists of vectors with three components. In this case, the dimension of R^2 is 2 because each vector in R^2 has two components.
To summarize, the given linear transformation T: R^2 → R^2, with the matrix A = [[-2, -2], [1, 2]], maps a two-dimensional vector space onto itself. The dimensions of both R^2 and R^2 are 2, representing the number of components in the vectors of each space.
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Jennifer ran 2 miles at the track on Tuesday. If one lap around the track is 1/4 of a mile, how many laps did she run?
Answer:
She would have ran 8 laps.
Step-by-step explanation:
4/4 = 1 mile so therefore she would need 8/8 for it to be 2 miles
sketch the graph of a function that has a local maximum at 6 and is differentiable at 6.
To sketch the graph of a function that has a local maximum at 6 and is differentiable at 6, we can consider a function that approaches a maximum value at 6 and has a smooth, continuous curve around that point.
In the graph, we can depict a curve that gradually increases as we move towards x = 6 from the left side. At x = 6, the graph reaches a peak, representing the local maximum. From there, the curve starts to decrease as we move towards larger x-values.
The important aspect to note is that the function should be differentiable at x = 6, meaning the slope of the curve should exist at that point. This implies that there should be no sharp corners or vertical tangents at x = 6, indicating a smooth and continuous transition in the graph.
By incorporating these characteristics into the graph, we can represent a function with a local maximum at 6 and differentiability at that point.
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What is the equation of the asymptote for the functionf(x) = 0.7(4x-3) - 2?
The equation of the asymptote for the given function f(x) = 0.7(4x-3) - 2 is y = 2.8x - 4.1.
The equation of an asymptote for a function can be determined by analyzing the behavior of the function as x approaches positive or negative infinity.
For the given function f(x) = 0.7(4x-3) - 2, let's simplify it:
f(x) = 2.8x - 2.1 - 2
f(x) = 2.8x - 4.1
As x approaches positive or negative infinity, the term 2.8x dominates the function. Therefore, the equation of the asymptote can be determined by considering the behavior of the linear term.
The coefficient of x is 2.8, so the slope of the asymptote is 2.8. The y-intercept of the asymptote can be found by setting x to 0 in the equation, resulting in -4.1. Therefore, the equation of the asymptote is y = 2.8x - 4.1.
In conclusion, the equation of the asymptote for the given function f(x) = 0.7(4x-3) - 2 is y = 2.8x - 4.1.
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El resultado de la operación combinada 70-25+9-2x10÷2 corresponde a:
A) 9 B)44 C)80
Answer:
B) 44
Step-by-step explanation:
70 - 25 + 9 - 2 × 10 ÷ 2
70 - 25 + 9 - 20 ÷ 2
70 - 25 + 9 - 10
45 + 9 - 10
54 - 10
44
Please answer quickly!!
Which set of data is represented by the box plot?
Answer:
C?
Step-by-step explanation:
If angle A is 76ᴼ what is its supplementary angle? What is its complementary angle?
Answer:
the complementary angle is 14°
the supplementary angle is 104°
Step-by-step explanation:
76+14=90
76+104=180
What is the solution set for -4m ≥ 96?
Answer:
m [tex]\leq[/tex] 24
Step-by-step explanation:
-4m ≥ 96
(divide both sides by -4)
m [tex]\leq[/tex] 24
(sign flips because dividing by a negative)
Help me quick!!!!!
Donny came by and pimped 4 girls on Tuesday, he then came by again on Saturday and Sunday with 17 more! How many girls did that playa get?
Answer:
I don't know thank answer sorry I just really need points you can report me if you want but I REALLY need some
Line with a slope of -3 and passes through
point (-1,7)
Answer:
y=-3x+5.5
Step-by-step explanation:
The answer is Y=-3x+5.5 because we already know the slope which is -3. Now all you have to do is change the Y-Intercept. As you can tell the Y-Intercept is not a whole number. So you have to change it to a decimal to get the point that you want. You can put the last part of the equation as 5.5 or 5.55. It doesn't matter but 5.5 is more official. If you find any fault in my answer let me know. Thanks. Have a good day!
The diagonal of TV set is 39 inches long. length is 21 inches more than the height. Find the dimensions of the TV set a. The height of the TV set is ___ inches. b. The length of the TV set is ___ inches.
Let's assume the height of the TV set is h inches.
a. The height of the TV set is h inches.
Given that the length is 21 inches more than the height, the length can be represented as h + 21 inches.
b. The length of the TV set is h + 21 inches.
According to the given information, the diagonal of the TV set is 39 inches. We can use the Pythagorean theorem to relate the height, length, and diagonal:
(diagonal)^2 = (height)^2 + (length)^2
Substituting the values, we have:
39^2 = h^2 + (h + 21)^2
Expanding and simplifying:
1521 = h^2 + h^2 + 42h + 441
2h^2 + 42h + 441 - 1521 = 0
2h^2 + 42h - 1080 = 0
Dividing the equation by 2 to simplify:
h^2 + 21h - 540 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives us:
(h - 15)(h + 36) = 0
So h = 15 or h = -36.
Since the height of the TV set cannot be negative, we discard h = -36.
Therefore, the height of the TV set is 15 inches.
Substituting this value back into the length equation, we have:
Length = h + 21 = 15 + 21 = 36 inches.
So, the dimensions of the TV set are:
a. The height of the TV set is 15 inches.
b. The length of the TV set is 36 inches.
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Find the margin of error E. A sample of 51 eggs yields a mean weight of 1.72 ounces. Assuming that o = 0.87 oz, find the margin of error in estimating p at the 97% level of confidence. Round your answer to two decimal places.
The margin of error E is approximately 0.31 oz
Margin of error is known to be a statistic expressing the amount of random sampling error in a survey's results. The margin of error informs you how close your survey findings are to the actual population's overall results. It is commonly represented by E.
The formula for margin of error is as follows:
z = critical value
σ = standard deviation
n = sample size
E = margin of error
The formula is, E = zσ/ √n
Here,
Sample size n = 51; Mean = 1.72; Standard deviation σ = 0.87 oz
Level of confidence = 97%
The level of confidence that corresponds to a z-score of 1.88 is 97% (using a standard normal table or calculator).
That is, z = 1.88 (by referring to a standard normal table or calculator)
To calculate the margin of error, we need to substitute the values in the formula
E = zσ/ √n
E = (1.88) (0.87) / √51
E = 0.3081 oz (approx)
Hence, the margin of error is approximately 0.31 oz (rounding the answer to two decimal places).
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Find the surface area of a square pyramid with side length 6 mi and slant height 7 mi.
Answer:
total surface area (TSA) = [tex]2bs[/tex] + [tex]b^{2}[/tex]
length = 6 mi
slant height = 7 mi
∴ TSA = 2 × 6 × 7 + [tex]6^{2}[/tex]
= 84 + 36
= 120
The surface area of the square pyramid is 120 sq. mi.
Given:
side length (s) = 6 mi
slant height (l) = 7 mi
*image of a typical square pyramid is shown in the attachment below.
Recall:
Formula for surface area of square pyramid (SA) = [tex]A + \frac{1}{2} Pl[/tex]
Where,
[tex]A =[/tex] area of the base = [tex]s^2 = 6^2 = 36[/tex] sq mi
[tex]P =[/tex] perimeter of the base = [tex]4(s) = 4\times 6 = 24[/tex] mi
[tex]l =[/tex] slant height = 7 mi
Plug in the values
[tex]SA = 36 + \frac{1}{2}\times 24\times 7\\SA = 36 +84\\SA = 120[/tex]
The surface area(SA) of the square pyramid is 120 sq. mi
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Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.29 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12. 19 and 12.25 ounces.
The probability that the bottle contains between 12.19 and 12.25 ounces is 0.1525.
Given, mean (μ) = 12.29 ounces and standard deviation (σ) = 0.04 ounce.
We need to find the probability that the bottle contains between 12. 19 and 12.25 ounces.
So, let X be the amount of beer filled by the machine. Then, X ~ N(12.29, 0.04²)
Let Z be the standard normal random variable.
Then, Z = `(X - μ)/σ`
Substituting the values, we get,Z = `(X - 12.29)/0.04`
For X = 12.19, `Z = (12.19 - 12.29)/0.04 = -2.5`
For X = 12.25, `Z = (12.25 - 12.29)/0.04 = -1
`Now we need to find the probability of Z being between -2.5 and -1.P(Z lies between -2.5 and -1) = P(-2.5 < Z < -1)
We know that P(Z < -1) = 0.1587 and P(Z < -2.5) = 0.0062
From standard normal distribution table, we get
P(-2.5 < Z < -1)
= P(Z < -1) - P(Z < -2.5)P(-2.5 < Z < -1)
= 0.1587 - 0.0062 = 0.1525
Therefore, the probability that the bottle contains between 12.19 and 12.25 ounces is 0.1525.
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Can someone help me with this problem
Answer:
33
Step-by-step explanation:
well we know you have a right angle and a right angle is equivalent to 90 degrees and you also have angle 57 now we know a triangle is equal to 180 so take 90+57=180 add the 90 and 57 then subtract the anwser from 180
Please help me it’s due soon-
Answer:
B-16
Step-by-step explanation:
jsjjsjdjsjsjxjsjsijsjdjdjjsjzjzijzjsndjidjsjsid
Find the area of the circle. Round your answer to the nearest hundredth. diameter 3in
Area of circle = pi × r squared
A= 3.14....× (1.5×1.5)
A= 7.068583471
A= 7.07in( to nearest hundredth)
a right triangle, with a height of 4m and a width of 1m. he wants to build a rectangular enclosure to protect himself. what is the largest the area of gottfried’s enclosure can be?
Given that, height of the right-angled triangle = 4m. Width of the right-angled triangle = 1m. Let's assume that the rectangular enclosure will be built at the base of the right-angled triangle. The area of the rectangular enclosure can be obtained using the formula, Area of rectangle = length × breadth. Length of the rectangle = height of the right-angled triangle = 4mLet the breadth of the rectangle be x, then the length is the width of the right-angled triangle + x = 1 + x Hence, the area of the rectangular enclosure is given by: Area = Length × Breadth= (1+x)×4= 4x + 4m²Now, the maximum area can be obtained by differentiating the above expression with respect to x and equating it to zero: dA/dx = 4 = 0x = -1. This is not a valid solution since we cannot have a negative breadth, hence we conclude that the area is maximum when the breadth of the rectangular enclosure is equal to the width of the right-angled triangle, i.e., when x = 1m. Thus, Area of the rectangular enclosure = Length × Breadth= (1+1)×4= 8 m². Hence, the largest the area of Gottfried’s enclosure can be is 8 m².
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Let 1 f(z) = z²+1 Determine whether f has an antiderivative on the given domain G. You must prove your claims. (a) G=C\ {i,-i}. (b) G= {z C| Rez >0}.
f(z) = z^2 + 1 has an antiderivative on the domain G = C \ {i, -i}.
(b) Hence, we cannot determine whether f(z) = z^2 + 1 has an antiderivative on the domain G = {z in C | Re(z) > 0} based on the Cauchy-Goursat theorem alone.
(a) To determine whether f(z) = z^2 + 1 has an antiderivative on the domain G = C \ {i, -i}, we can check if f(z) satisfies the Cauchy-Riemann equations on G.
The Cauchy-Riemann equations state that for a function f(z) = u(x, y) + iv(x, y) to have a derivative at a point, its real and imaginary parts must satisfy the partial derivative conditions:
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.
For f(z) = z^2 + 1, we have u(x, y) = x^2 - y^2 + 1 and v(x, y) = 2xy.
Calculating the partial derivatives, we find:
∂u/∂x = 2x, ∂v/∂y = 2x,
∂u/∂y = -2y, ∂v/∂x = 2y.
Since ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x hold for all (x, y) in the domain G, f(z) satisfies the Cauchy-Riemann equations on G. Hence, f(z) has an antiderivative on G = C \ {i, -i}.
(b) Now, let's consider the domain G = {z in C | Re(z) > 0}. To determine if f(z) = z^2 + 1 has an antiderivative on G, we can utilize the Cauchy-Goursat theorem, which states that a function has an antiderivative on a simply connected domain if and only if its line integral around every closed curve in the domain is zero.
For f(z) = z^2 + 1, we can calculate its line integral over a closed curve C in G. However, since G is not simply connected (it has a "hole" at Re(z) = 0), the Cauchy-Goursat theorem does not apply, and we cannot conclude whether f(z) has an antiderivative on G based on this theorem.
To provide a definitive answer, further analysis or techniques such as the residue theorem may be required.
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These tables represent an exponential function, find the average rate of change for the interval from x=9 to x=10.
The average rate of change for the interval from x=9 to x=10 is 39366
Exponential equationThe standard exponential equation is given as y = ab^x
From the values of the average change, you can see that it is increasing geometrically as shown;
2, 6, 18...
In order to , find the average rate of change for the interval from x=9 to x=10, we need to find the 10 term of the sequence using the nth term of the sequence;
Tn = ar^n-1
Given the following
a = 2
r = 3
n = 10
Substitute
T10 = 2(3)^10-1
T10 = 2(3)^9
T10 = 39366
Hence the average rate of change for the interval from x=9 to x=10 is 39366
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Answer:
39,366
Step-by-step explanation:
its right