A circular flower garden has an area of 314m². A sprinkler at the center of the garden can cover an area of 12 m. Will the sprinkler water the entire garden?

Answer:

Based on the questionnaire about gap year for matrics, a gap year can be a good idea for some students. The main reason for taking a gap year is to gain life experience, travel, and explore career options before committing to a specific field of study. Additionally, a gap year can provide an opportunity for personal growth and development, as well as the chance to save money and gain work experience.

On the other hand, the main reason against taking a gap year is the potential delay in academic progress and the possibility of losing momentum and motivation towards pursuing further education. Additionally, some students may not have the financial means to afford a gap year and may not have access to job opportunities or travel experiences.

The best gap year options may vary depending on individual preferences and interests, but some popular options include volunteer work, internships, language immersion programs, and cultural exchanges.

As for Zanele, the decision of whether or not to take a gap year should be based on her personal goals and circumstances. If she feels that she needs time to explore her interests and gain experience before continuing with her studies, a gap year may be a good option for her. However, if she is motivated to pursue her academic goals and does not want to delay her progress, then she may choose not to take a gap year.

Step-by-step explanation:

Frankie's dad used 2/3 of the carton of eggs for their breakfast. Two equivalent fractions are 16/24 and 24/36.

Frankie's dad used 8 out of 12 eggs from a full carton. To express this as a fraction, we can write it as 8/12. However, this fraction can be simplified by dividing both the numerator and denominator by their greatest common factor, which in this case is 4. Thus, we get:

8/12 = (8 ÷ 4)/(12 ÷ 4) = 2/3

Alternatively, we can also express this fraction in different forms. One way to do this is by multiplying both the numerator and denominator by the same number. For example, if we multiply both by 2, we get:

8/12 = (8 × 2)/(12 × 2) = 16/24

Similarly, we can multiply by 3 to get:

8/12 = (8 × 3)/(12 × 3) = 24/36

Both of these fractions are equivalent to 2/3, and we can find infinitely many other equivalent fractions by multiplying both numerator and denominator by different numbers.

In summary, when Frankie's dad made scrambled eggs using 8 eggs from a full carton of 12 eggs, he used 2/3 of the carton. This fraction can be expressed in different forms that are equivalent to 2/3 by dividing both numerator and denominator by their greatest common factor or by multiplying both numerator and denominator by the same number.

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Answer:

antonio rolls the 10-sided die from the example. what is the probabilty of rolling a number 10 or less? a number greater than 10

Step-by-step explanation:

The 10-sided die has 10 possible outcomes, which are the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

The probability of rolling a number 10 or less is the probability of getting any of these 10 outcomes. Since each outcome is equally likely, the probability of rolling a number 10 or less is:

P(rolling a number 10 or less) = number of outcomes that are 10 or less / total number of possible outcomes

P(rolling a number 10 or less) = 10/10

P(rolling a number 10 or less) = 1

So the probability of rolling a number 10 or less is 1, which means it is certain that Antonio will roll a number 10 or less.

On the other hand, the probability of rolling a number greater than 10 is 0, since there are no outcomes greater than 10.

The sum of 6 and a sum of 4 when rolling two dice. So, we can say that they are dependent events. Answer: No.

a) To get a sum of 6, there are 5 possible outcomes (1,5), (2,4), (3,3), (4,2), and (5,1). Thus, the probability of getting a sum of 6 when you roll two fair dice is 5/36. Answer: 5/36b) To get a sum of 4, there are three possible outcomes (1,3), (2,2), and (3,1). Thus, the probability of getting a sum of 4 when you roll two fair dice is 3/36 or 1/12. Answer: 1/12c) The sum of 6 or 4 can be obtained from (1,3), (2,2), (3,1), (1,5), (2,4), (3,3), (4,2), and (5,1). Thus, the probability of getting a sum of 6 or 4 is 8/36 or 2/9. Answer: 2/9These outcomes are not mutually exclusive because it is possible to get both a sum of 6 and a sum of 4 when rolling two dice. So, we can say that they are dependent events. Answer: No.

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a. The probability that exactly ten arrive during the hour and all ten have no violations is given by the Poisson distribution, P(X=10) = (e-α*(α10)/(10!) = 0.024

b. The probability that y arrive during the hour, of which ten have no violations is given by the Binomial distribution, P(Y=y) = (y!/(10!*(y-10)!))*(0.510)*(0.5y-10)

c. The probability that ten "no-violation" cars arrive during the next hour is given by summing the probabilities in part (b) from y=10 to [infinity]. This is given by P(Y≥10) = Σy=10 to ∞P(Y=y) = 1 - Σy=0 to 9P(Y=y).

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Dayna erases the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11, and leaves only the integer 12 on the chalkboard, by the concept of multiplicative inverse and by using the extended Euclidean algorithm.

The integers from 1 through 6, as well as their multiplicative inverse [tex]$\mod{13}$[/tex], have been erased from the integers 1 through 12. We need to find the only integer that Dayna did not erase. We can find the multiplicative inverse of an integer a modulo 13 by using the extended Euclidean algorithm.

The integers from 1 through 6 are 1, 2, 3, 4, 5, and 6. We need to find their multiplicative inverses modulo 13.

The multiplicative inverse of 1 modulo 13 is 1, since

[tex]$1 \times 1 \equiv 1 \pmod{13}$[/tex].

The multiplicative inverse of 2 modulo 13 is 7, since

[tex]$2 \times 7 \equiv 1 \pmod{13}$[/tex].

The multiplicative inverse of 3 modulo 13 is 9, since

[tex]$3 \times 9 \equiv 1 \pmod{13}$[/tex].

The multiplicative inverse of 4 modulo 13 is 10, since

[tex]$4 \times 10 \equiv 1 \pmod{13}$[/tex].

The multiplicative inverse of 5 modulo 13 is 8, since

[tex]$5 \times 8 \equiv 1 \pmod{13}$[/tex].

The multiplicative inverse of 6 modulo 13 is 11, since

[tex]$6 \times 11 \equiv 1 \pmod{13}$[/tex].

Therefore, the only integer that Dayna does not erase is 12.

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The solution to the logarithmic equation [tex]2log12(-8x) = 6[/tex] is [tex]x = -9/32[/tex] .

What are logarithmic properties ?

Logarithmic properties are the rules that govern the behavior of logarithmic functions. These properties are important in simplifying logarithmic expressions and solving logarithmic equations. Some of the commonly used logarithmic properties include:

Product property: [tex]logb(xy) = logb(x) + logb(y)[/tex]

This property allows us to simplify the logarithm of a product of two numbers into the sum of logarithms of the individual numbers.

Quotient property: [tex]logb(x/y) = logb(x) - logb(y)[/tex]

This property allows us to simplify the logarithm of a quotient of two numbers into the difference of logarithms of the individual numbers.

Power property:[tex]logb(x^y) = ylogb(x)[/tex]

This property allows us to simplify the logarithm of a power of a number by bringing the exponent outside of the logarithm and multiplying it with the logarithm of the base.

Change of base formula: [tex]logb(x) = logc(x) / logc(b)[/tex]

This property allows us to change the base of a logarithm by dividing the logarithm of the number by the logarithm of the base in a different base.

Solving the given logarithmic equation :

The equation can be solved by using logarithmic properties and basic algebraic manipulation.

We can begin by using the property that states [tex]loga(b^n) = nloga(b)[/tex] for any base a and any positive real number b. Applying this property, we can rewrite the left side of the equation as:

[tex]log12((-8x)^2) = log12(64x^2)[/tex]

Next, we can use the property that states [tex]loga(b) = c[/tex] is equivalent to [tex]a^c = b[/tex]. Applying this property, we can rewrite the equation as:

[tex]12^{2log12(64x^2)} = 12^6[/tex]

Simplifying the left side, we get:

[tex]64x^2 = 12^6 / 12^2[/tex]

[tex]64x^2 = 144[/tex]

Dividing both sides by 64, we get:

[tex]x^2 = 144/64[/tex]

[tex]x^2 = 9/4[/tex]

Taking the square root of both sides, we get:

[tex]x=\pm 3/2[/tex]

However, we need to check the solutions for extraneous roots since the original equation has a logarithm with a negative argument. We can see that the solution x = 3/2 is extraneous since it results in a negative argument for the logarithm. Therefore, the only valid solution is x = -9/32.

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One possible solution to this puzzle is:

A=9, B=8, C=7, D=6, E=5, X=2

So the equation would look like:

58,746 X 4 = 23,494

This means that if you substitute the values I provided for the letters in the original equation, it would result in the equation 58,746 X 4 = 23,494 being true.

To find this solution, I used a process of elimination and logic.

First, I knew that the result of the multiplication must start with a 2, since 4 times any single-digit number results in a number between 40 and 36. Therefore, X must be either 2 or 3.

Next, I focused on the fact that the two middle digits of the result are the same, which means that either A and E are the same or B and D are the same.

If A and E are the same, they must be either 4 or 5, since they must be less than X.

However, if A and E are the same, then C must also be 4 or 5, since it cannot be higher than A or E. This means that there would be a repeated digit in the equation, which is not allowed.

Therefore, A and E cannot be the same, and B and D must be the same.

Using this information, I continued to eliminate possibilities until I found a solution that worked.

For example, since B and D must be the same and cannot be 4 or 5, they must be either 6, 7, 8, or 9.

However, if they were 6 or 7, then C would have to be 6 or 7 as well, which is not allowed. If they were 8 or 9, then A and E would have to be 4 or 5, which is also not allowed.

Therefore, B and D must be 8 or 9.

From there, I tried different combinations until I found one that worked. This process involved a lot of trial and error, but by using logic and the constraints of the puzzle, I was able to narrow down the possibilities and eventually arrive at a solution.

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Harris ate x - 111 slices of pizza, So the total number of slices in the pizza is n ,where x is the number of slices each of the 44 friends received, and n is the total number of slices in the pizza, which must be a multiple of 44.

Let x be the number of slices each of the 44 friends received, then the total number of slices in the pizza is 44x. Since the pizza is evenly divided, the number of slices must be a multiple of 44.

Let y be the number of slices Harris received. Then, according to the problem, Harris ate 111 fewer slices than he received, so the number of slices he ate is y - 111.

Since the total number of slices in the pizza is 44x, we have:

y + 43x = 44x

Simplifying this equation, we get:

y = x

Therefore, Harris received the same number of slices as each of the other friends. So the number of slices he ate is:

y - 111 = x - 111

Substituting y = x, we get:

x - 111

Therefore, Harris ate x - 111 slices of pizza.

To find the value of x, we need to know the total number of slices in the pizza, which must be a multiple of 44. We can express this as:

44n = 44x

Dividing both sides by 44, we get:

n = x

So the total number of slices in the pizza is n.

Putting everything together, we get: Harris ate x - 111 slices of pizza, where x is the number of slices each of the 44 friends received, and n is the total number of slices in the pizza, which must be a multiple of 44.

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Answer:

Use the slope-intercept form to find the slope and y-intercept.

Slope: 2y-intercept: (0,3)

Step-by-step explanation:

Answer:

(0,3).

Step-by-step explanation:

for example, y=2x+3 tells us that the slope of the line is 2 and the y-intercept is at (0,3).

When performing a hypothesis test based on a 95% confidence level, the chances of making a type II error are 5%.

The process of hypothesis testing is used to determine whether or not a given statistical hypothesis is valid. The objective of this method is to determine whether the null hypothesis can be accepted or rejected based on the sample data obtained.

Hypothesis testing can be used to evaluate two hypotheses. The null hypothesis is the one that must be accepted or rejected, while the alternative hypothesis is the one that must be supported. In other words, hypothesis testing is a way of determining whether the null hypothesis is reasonable or not.

The Type II error is defined as the error that occurs when the null hypothesis is not rejected even though it is incorrect. In hypothesis testing, this type of error is referred to as a beta error or a false-negative error. The chances of making a Type II error depend on several factors, including the sample size, the level of significance, and the power of the test. When the level of significance is lowered to 0.05, the chances of making a Type II error are 5%.

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The z-score for the 75th percentile of the standard normal distribution is given by 0.67 that is option A.

The most significant continuous probability distribution is the Normal Distribution, often known as the Gaussian Distribution. It is also known as a bell curve. The normal distribution represents a large number of random variables either nearly or exactly.

I found one that shows the following:

Z value Table entry

0.67   0.7486

0.68   0.7517

As a result, the Z value for 0.75 is between 0.67 and 0.68.

Interpolation yields the z value of 0.6745.

If you have a TI-84 calculator, you may calculate the z value as follows:

VARS - 2nd (this will show the DISTR menu)

To select invNorm, press 3.

Enter the value for the area/table (0.75)

If you press enter, it will return the z value.

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Complete question:

what is the z-score for the 75th percentile of the standard normal distribution is:

Answer:

  see attached

Step-by-step explanation:

You want a picture of an isosceles triangle with base angles of 70°.

Such a triangle is shown in the attachment.

__

Additional comment

You can draw one of these yourself using a protractor to measure the angles.

The side lengths have the approximate ratio of 13:19:19.

Probability that X = 5:Since, Joe selects only 4 bags of lettuce. X can't be 5.P(X=5) = 0Hence, the probability that X = 0 is 0.3164 and the probability that X = 5 is 0.

The given problem can be solved using the concept of binomial distribution.

In the given question, there are 48 bags of lettuce out of which 12 bags are contaminated with listeria.

Joe selects 4 bags of lettuce. X is the random variable which represents the number of contaminated bags of lettuce selected by Joe. X can take values from 0 to 4. (as Joe selects only 4 bags).

Part A)Number of ways to select 4 bags of lettuce out of 48:This can be solved using the concept of combinations. The formula to calculate the number of combinations is[tex]:nCr = n! / r!(n-r)![/tex]Here, n = 48 and r = 4.

Number of ways = 48C4 = 194,580

Part B)Probability that X = 0:This can be calculated using the formula for the binomial distribution :

[tex]P(X = r) = nCr * p^r * q^(n-r)[/tex]

Here, p = probability of selecting contaminated bag = 12/48 = 0.25q = probability of selecting non-contaminated bag = 1-0.25 = 0.75Also, n = 4 and r = [tex]0P(X=0) = 4C0 * 0.25^0 * 0.75^4= 0.3164[/tex]

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a) The profit or loss for the private label product is $223,714 and the profit or loss for the branded product is $125,143 using plantwide allocation.

b) The profit or loss for the private label product is $159,114 and the profit or loss for the branded product is $132,171 using department allocation.

a) Using plantwide allocation, the total manufacturing overhead of $432,000 is allocated based on the total direct labor cost of $336,000 (sum of direct labor costs for both products). The overhead rate is calculated as $432,000 / $336,000 = $1.2857 per dollar of direct labor cost. The overhead cost for each product is then calculated by multiplying the overhead rate by the direct labor cost for that product.

Private Label Branded Total

Sales revenue   $768,000       $480,000         $1,248,000

Direct material    216,000          156,000              372,000

Direct labor         144,000           96,000              240,000

Manufacturing    184,286            122,857              307,143

Total cost            544,286        354,857                899,143

Profit (loss)          $223,714         $125,143            $348,857

The profit or loss for the private label product is $223,714 and the profit or loss for the branded product is $125,143 using plantwide allocation.

b) Using department allocation, the manufacturing overhead of $432,000 is allocated based on the direct labor costs for each department. The overhead rate for department A-101 is $201,600 / $144,000 = $1.4014 per dollar of direct labor cost, and the overhead rate for department A-102 is $230,400 / $96,000 = $2.4000 per dollar of direct labor cost. The overhead cost for each product is then calculated by multiplying the overhead rate by the direct labor cost for that product in each department.

Private Label Branded Total

Sales revenue                   $768,000        $480,000           $1,248,000

Direct material                    216,000           156,000              372,000

Direct labor                         144,000             96,000              240,000

Manufacturing overhead

Department A-101              $201,600           $64,400            $266,000

Department A-102              46,286             31,429                  77,714

Total overhead                  $247,886          $95,829             $343,715

Total cost                         608,886               347,829             956,715

Profit (loss)                          $159,114             $132,171            $291,285

The profit or loss for the private label product is $159,114 and the profit or loss for the branded product is $132,171 using department allocation.

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Answer:

49.5

Step-by-step explanation:

Given:

5× ______ = - 35

• -35/5

• -7

5 x -7 = -35

Answer:

The answer is 7

Step-by-step explanation:

Divide each term in 5x=-35 by 5 and simplify.x=−7

Answer:

y = 3x - 3

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (3, 6 ) and (x₂, y₂ ) = (- 1, - 6 )

m = [tex]\frac{-6-6}{-1-3}[/tex] = [tex]\frac{-12}{-4}[/tex] = 3 , then

y = 3x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (3, 6 )

6 = 3(3) + c = 9 + c ( subtract 9 from both sides )

- 3 = c

y = 3x - 3 ← equation of line

If all other factors are held constant, decreasing the significance level results in an increase in the probability of a type II error. This is true. we can say that the probability of making a type II error increases when the significance level is lowered.

What is a type II error? In hypothesis testing, a type II error occurs when a false null hypothesis is not rejected. When there is a real effect and the null hypothesis is false, this happens. It's a mistake that occurs when a researcher fails to reject a false null hypothesis.

A false negative is another term for a type II error. The power of the test, the size of the sample, the confidence level, and the effect size are all factors that influence the probability of making a type II error. Only if we decrease the significance level can the probability of a type II error be increased.

What is the significance level? The significance level is also known as alpha. It is the probability of rejecting a null hypothesis when it is true. It is represented by α. It is usually set at 0.05 or 0.01 in most studies. When the significance level is lowered, the probability of making a type I error decreases, but the probability of making a type II error increases. Therefore, we can say that the probability of making a type II error increases when the significance level is lowered.

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a) The new principal (principal + capitalized interest) when Rich begins making loan payments will be $9,425.10.

b) The total interest that Rich will pay over the life of the loan is $4,914.20.

Simple interest (S.I.) is a method of calculating the amount of interest on a specific principal amount at a certain rate. For example, when one takes out a loan of Rs. 5000, at the rate of 10 p.A. Two years The person's two years of interest will be the money borrowed by SI.

Interest on student loans is based on simple interest. If the student does not pay interest, the only interest capitalized is that for the period during which the payment was not made.

Some students choose to pay interest during the non-repayment period to reduce the capitalized interest and principal during the repayment period.

Capitalization refers to the addition of interest to the principal amount.

Period of student loan = 10 years

The amount of the federal unsubsidized student loan = $7,900

Interest rate = 4.29%

Capitalized interest during the non-payment time = $1,525.10 ($7,900 x 4.29% x 4.5)

New principal when Rich begins making loan payments = $9,425.10 ($7,900 + $1,525.10)

Total interest for loan = $4,914.20 ($7,900 x 4.29% x 14.5)

Complete Question:

Rich is attending a 4-year college. As a freshman, he was approved for a 10-year, federal unsubsidized student loan in the amount of $7,900 at 4.29%. He knows he

has the option of beginning repayment of the loan in 4.5 years. He also knows that during this non-payment time, interest will accrue at 4.29%.

If Rich decides to make no payments during the 4.5 years, the interest will be capitalized at the end of that period.

a. What will the new principal be when he begins making loan payments?

b. How much interest will he pay over the life of the loan?

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The valid symbolized arguments is HS-valid. (option e).

Now let's focus on the symbolized arguments presented in this question. To determine their validity, we need to identify the form of each argument. In some cases, we may need to rewrite the argument using double negation or commutativity before assigning a name to its form.

The argument form known as Hypothetical Syllogism (HS) is represented as follows:

P⊃Q

Q⊃R

P⊃R

The given argument can be rewritten using double negation as follows:

H⊃~M

M⊃H

H⊃~H

Then, we can see that the argument follows the form of HS, where if H implies not M, and not M implies not H, then we can conclude that H implies not H. This conclusion is always true because it is a contradiction. Therefore, the argument is valid.

Hence the correct option is (e).

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Complete Question:

Determine whether the following symbolized arguments are valid of invalid by identifying the form of each. In some cases the argument must be rewritten using double negation or commutativity before it has a named for. Those arguments without a specific name are valid.

H⊃~M

M

______

~H

a. DA-Invalid.

b. MP-valid.

c. AC-invalid.

d. MT-valid.

e. HS-valid.

Answer:

? ≈ 5.0

Step-by-step explanation:

using the sine ratio in the right triangle

sin24.3° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{2.06}{?}[/tex] ( multiply both sides by ? )

? × sin24.3° = 2.06 ( divide both sides by sin24.3° )

? = [tex]\frac{2.06}{sin24.3}[/tex] ≈ 5.0 ( to 1 decimal place )

An expression is a combination of numbers, variables, and operations that can be evaluated to produce a value. Expressions can be simple or complex and can involve arithmetic, logical, or comparison operations.

A) To factor [tex]p^2-x^2+6x-9[/tex] first notice that the expression inside the parentheses is the difference of two squares, so we can use the formula:

[tex]p^2 - x^2 = (p + x)(p - x)[/tex]

Therefore, we can write the expression as:

[tex](p + x + 3)(p - x + 3)[/tex]

To check this, you can use FOIL to expand the expression:

[tex](p + x + 3)(p - x + 3) = p^2 - px + 3p + px - x^2 + 3x + 3p - 3x + 9 = p^2 - x^2 + 6x + 9[/tex]

B) To factor b^2-c^2-10(b-c), notice that the first two terms form a difference of squares, so we can use the formula again:

[tex]b^2 - c^2 = (b + c)(b - c)[/tex]

The expression can then be written as:

[tex](b + c)(b - c) - 10(b - c) = (b - c)(b + c - 10)[/tex]

To check this, we can use FOIL again:

[tex](b - c)(b + c - 10) = b^2 - bc - 10b - cb + 10c = b^2 - c^2 - 10(b - c)[/tex]

Therefore, the factored form of the expression is [tex](b - c)(b + c - 10)[/tex].

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Answer:

12

Step-by-step explanation:

i think its right

The difference in time between the two trains is speed of 0.5 hours, or 30 minutes.

Train speed is calculated as total distance traveled divided by travel time. The time it takes for two trains to cross each other is defined as (a+b) / (x+y) if their lengths, let's say a and b, are known and they are going at speeds of x and y, respectively.

We can use the following calculation to determine how long the two trains are apart in time:

Time = Speed / Distance

Manchester to London is approximately 163 miles away on Train A. (we can assume a direct route). 110 mph is listed as the average speed. Hence, the duration of A train is:

time A is calculated as follows: 163 miles at 110 mph, which equals 1.4827 hours.

With the detour, the distance for Train B is roughly 200 miles from Manchester to London. Although we don't know the speed of Train B, we do know that it arrived in London 30 minutes later than Train A did. As a result, Train B's travel time is:

Time B equals Time A plus 0.5 hours, or 1.9827 hours.

The two trains' respective departure times are:

time diff is equal to 1.9827 hours minus 1.4827 hours, which is 0.5 hours.

Thus, there is a 0.5-hour (30-minute) gap in time between the two trains.

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Answer: 1.2 miles

Step-by-step explanation:

1 hour = 60 min

60 x 0.02 = 1.2         1 minute and 20 seconds has elapsed

60 miles/ every 60 minutes or 1 mile a minute

1 x 1.2 = 1.2

she has traveled 1.2 miles

Answer: The answer is 120 mph (miles per hour)

Step-by-step explanation:

The one thing you need to do is to figure out how many mph did Ella drive for 2 hours.

So, you need to do 60 x 2, and you will get the answer 120.

And there's your answer!

Let a function f be analytic everywhere in a domain D. Prove that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D. Suppose a function f(z) is analytic and real-valued throughout a domain D. Assume that there are two different points in D, say w and z, so that f(z) ≠ f(w). Therefore, we have a closed path consisting of the line segment from z to w and the line segment from w to z. Now let γ be any closed path in D that encloses the path from z to w.  Since f(z) ≠ f(w), f(z) - f(w) ≠ 0, so f(z) - f(w) / z - w ≠ 0. This makes the function 1 / (f(z) - f(w)) analytic throughout D. Thus, by Cauchy's theorem, $\oint_{\gamma } \frac{1}{f(z)-f(w)}dz = 0$.where gamma is a function.

frac stands for fraction. This leads to $\int_{w}^{z} \{1}/{f(z)-f(w)}dz + \int_{z}^{w} \{1}/{f(z)-f(w)}dz = 0$. But by substituting f(w) = f(z) and reversing the direction of integration in the second integral, we get, $\int_{w}^{z} \frac{1}{f(z)-f(w)}dz - \int_{w}^{z} \frac{1}{f(w)-f(z)}dz = 0$ . This results in the integral$\int_{w}^{z} \frac{f'(w)}{f(z)-f(w)}dz - \int_{w}^{z} \frac{f'(z)}{f(w)-f(z)}dz = 0$. Now it is time to simplify and evaluate each of the integrals. The first integral has value $\ln|f(z)-f(w)|$ and the second integral has value $\ln|f(z)-f(w)|$. Therefore, $f'(z)/f(z)-f(w) = f'(w)/(f(w)-f(z))$. This equation, however, is not possible unless $f(z) = f(w)$. Hence, we conclude that the function f(z) is constant throughout D.

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Answer:

50

Step-by-step explanation:

Let the number = x.

20% of the number is 20% of x = 0.2x.

Since the number exceeds 20% of itself by 40, then teh number minus 20% of itself is 40.  

x - 0.2x = 40

0.8x = 40

x = 40/0.8

x = 50