PLEASE HELP !
Use the figure below to answer the questions

The scale of the map = 682.5.

We kept a safe distance and followed them. She perceives a separation between her and her brother that wasn't there before. Although they were previously close friends, there was now a great deal of gap between them.

We must calculate the ratio of the distance shown on the map to the real distance between the cities in order to ascertain the scale of the map.

We are aware that there are 210 miles separating the two cities. Let x represent the precise location of this distance on the map. From that, we may establish the ratio:

Actual distance / Map Distance = 210 / x

The distance on the map is indicated as 3 1/4 inches, which is also known as 13/4 inches. When we enter this into the percentage, we obtain:

Actual distance divided by (13/4) = 210 / x

We can cross-multiply and simplify to find x's value:

Actual distance: 682.5 = x * 210 x = 3.25 when 210 * (13/4) Equals x.

Consequently, 3.25 inches on the map represent the actual distance between the cities. We can write: To determine the map's scale:

Actual distance divided by 1 inch on the chart equals 210 miles.

When we replace the values we discovered earlier, we obtain:

1 / 210 = 3.25 / scale

If we solve for the scale, we obtain:

scale = 682.5.

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Step-by-step explanation:

4)

based on similar triangles and the common ratio for all pairs of corresponding sides we know

LE/LM = LD/LK = DE/EM

because E and D are the midpoints of the longer sides, all of these ratios are 1/2.

1/2 = DE/8

8/2 = 4 = DE

5)

same principle as for 4)

BQ/BA = BR/BC = QR/AC

again, Q and R are the midpoints, so all these ratios are 1/2.

1/2 = QR/10

QR = 10/2 = 5

Descriptive statistics are used to summarize and describe data, making them useful for providing a clear understanding of a dataset's important features.

When a researcher wants to report the average cost of college tuition from the 1950s until present time, descriptive statistics are the appropriate method to use. Descriptive statistics are used to summarize and describe data, making them useful for providing a clear understanding of a dataset's important features. By using descriptive statistics, the researcher can calculate measures of central tendency, such as the mean, median, and mode, to determine the typical or average cost of college tuition over time. Additionally, measures of variability, such as the range and standard deviation, can be calculated to understand the spread of the data. Descriptive statistics are commonly used in many fields, including business, economics, psychology, and education, and can provide valuable insights into trends, patterns, and distributions within a dataset.

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The open circle at 3 indicates that 3 is not included in the solution set. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

In mathematics, an expression is a combination of numbers, symbols, and operators that represents a value. Expressions can be as simple as a single number or variable, or they can be complex combinations of mathematical operations. For example, 2 + 3 is a simple expression that represents the value 5, while (2 + 3) x 4 - 1 is a more complex expression that represents the value 19. Expressions can be evaluated or simplified using the rules of arithmetic and algebra.

Here,

1. Simplify:

3(4x-2)+ 7X (2-1) + 4 (6+4)+(-8)

Multiplying inside the parentheses first:

12x - 6 + 7x + 4 + 40 - 8

Combining like terms:

19x + 30

Final answer: 19x + 30

2. Graph:

3 > X

This is a simple inequality in one variable (X). To graph it on a number line, we first draw a dot at 3 (since the inequality is strict), and then shade all values less than 3:

<=========o---

The open circle at 3 indicates that 3 is not included in the solution set.

3. Write the inequality:

X < 5

This is a simple inequality in one variable (X). The inequality sign is "less than," and the number on the right-hand side is 5. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

4. Solve for x:

3x - 7 = 42

Adding 7 to both sides to isolate the variable:

3x = 49

Dividing both sides by 3 to solve for x:

x = 16.33 (rounded to two decimal places)

Final answer: x = 16.3

5. Find 32% of $542.50:

To find 32% of $542.50, we can use the formula:

percent * amount = part

where "percent" is the percentage expressed as a decimal, "amount" is the whole amount, and "part" is the result we're looking for.In this case, we have:

0.32 * $542.50 = part

Multiplying:

$173.60 = part

Final answer: $173.60

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Answer:

1. basement: 1,4,7,10,13

middle : 2,5,8,11,14

top: 3,6,9,12,15

2. you just need to skip count ten  times then we will have 30. so she lives in apartment 30.    

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let B be the event that the selected coin is biased, and F be the event that the selected coin is fair. Let H be the event that the coin toss shows a head.

We want to find P(B|H), the probability that the selected coin is biased given that the coin toss shows a head. By Bayes' theorem, we have:

P(B|H) = P(H|B) * P(B) / P(H)

We know that P(H|B) = 1/4 (since the biased coin has a probability of 1/4 of showing a head), and that P(B) = 1/2 (since there are two coins, one of which is biased).

To find P(H), we can use the law of total probability:

P(H) = P(H|B) * P(B) + P(H|F) * P(F)

P(H) = (1/4) * (1/2) + (1/2) * (1/2)

P(H) = 3/8

Putting it all together:

P(B|H) = P(H|B) * P(B) / P(H)

P(B|H) = (1/4) * (1/2) / (3/8)

P(B|H) = 1/3

Therefore, the probability that the selected coin is biased given that the coin toss shows a head is 1/3.

The statement " all of the following are things you can do when faced with any ethical dilemma as a managerial accountant except: always contact the board of directors first" is false because contacting the board of directors is not always necessary or appropriate when faced with an ethical dilemma as a managerial accountant

Contacting the board of directors is not always necessary or appropriate when faced with an ethical dilemma as a managerial accountant. While the board of directors may need to be informed in some situations, there are other steps that can be taken depending on the specific circumstances. For example, seeking guidance from a supervisor, consulting with a professional organization, or seeking legal advice may be more appropriate in some cases. The appropriate course of action will depend on the specific situation and ethical principles involved.

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Using binomial theorem, the generating function is G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256 while the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

[tex](x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)[/tex]

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

[tex](x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

Therefore, the generating function of (x+4)^4 is:

[tex]G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]

The associated sequence can be read off by finding the coefficients of each power of x:

To find the generating function of (x+4)^4, we expand it using the binomial theorem:

(x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)

where C(n,k) denotes the binomial coefficient "n choose k".

Simplifying the terms, we get:

(x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256

Therefore, the generating function of (x+4)^4 is:

G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256

The associated sequence can be read off by finding the coefficients of each power of x:

The coefficient of x^k is the k-th term of the sequence.

In this case, the sequence is given by the coefficients of G(x):

a₀ = 256

a₁ = 256

a₂ = 96

a₃ = 16

a₄ = 1

Therefore, the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.

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If we round up, we get x = 12, which means that 12 students took history and 36 students took mathematics.

A Venn diagram is a graphical representation of sets, often used to show relationships between different sets of data. It consists of one or more circles, where each circle represents a set. The circles are usually overlapping to show the relationships between the sets. The region where the circles overlap represents the elements that belong to both sets. The non-overlapping parts of each circle represent the elements that belong to only one of the sets.

Here,

A. To illustrate the data on a Venn diagram, we need to draw two overlapping circles, one for history and one for mathematics. We can label the regions as follows:

The region where the circles overlap represents the students who took both history and mathematics.

The region outside both circles represents the students who did not take either history or mathematics.

The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics.

The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history.

In this diagram, let's assume that x students took history and 3x students took mathematics. Then the total number of students who took the examination is x + 3x = 4x. We can label the regions on the diagram accordingly:

The region where the circles overlap represents the students who took both history and mathematics. Since there are 3x students who took mathematics, and all of them are included in the overlap region, the number of students who took both is 3x.

The region outside both circles represents the students who did not take either history or mathematics. Since there are 46 students in total, and 4x of them took the examination, the number of students who did not take either is 46 - 4x.

The region within the history circle but outside the mathematics circle represents the students who took history but not mathematics. Since x students took history, and 3x took mathematics, the number of students who took history but not mathematics is x - 3x = -2x. However, since we cannot have a negative number of students, we can assume that this region is empty.

The region within the mathematics circle but outside the history circle represents the students who took mathematics but not history. Since there are 3x students who took mathematics, and some of them also took history, the number of students who took mathematics but not history is 3x - 3x = 0. Therefore, this region is also empty.

B. We know that the total number of students who took the examination is 46. Therefore, we have:

x + 3x = 46

4x = 46

x = 11.5

However, since x represents the number of students who took history, it must be a whole number. Therefore, we can round x up or down to the nearest whole number. If we round down, we get x = 11, which means that 11 students took history and 33 students took mathematics.

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Answer:

3x(x - 4y)

Step-by-step explanation:

3x² - 12xy ← factor out 3x from each term

= 3x(x- 4y)

Answer:To simplify the expression, you need to combine the like terms, which are the terms that have the same variable and power. In this case, the like terms are -3a and 5a:

(-3a + 56) + (5a + 40)

= (-3a + 5a) + (56 + 40)

= 2a + 96

Therefore, the simplified expression is 2a + 96.

Enjoy (:

Step-by-step explanation:

Find the percent of each number :-

1.) 64% of 75 tiles :

[tex] \implies \sf \: \dfrac{64}{100} \times 75 \\ \\\implies \sf \:0.64 \times 75 \\ \\ \implies \sf \: 48 \\ [/tex]

Hence, 64% of 75 tiles is 48 tiles.

2.) 20% of 70 plants.

[tex] \implies \sf \: \dfrac{20}{100} \times 70 \\ \\\implies \sf \:0.2 \times 70 \\ \\ \implies \sf \: 14 \\ [/tex]

Hence, 20% of 70 plants is 14 plants.

3.) 32% of 25 pages .

[tex] \implies \sf \: \dfrac{32}{100} \times 25 \\ \\\implies \sf \: 0.32 \times 25 \\ \\ \implies \sf \: 8 \\ [/tex]

Hence, 32% of 25 pages is 8 pages.

4.) 85% of 40 e -mails.

[tex] \implies \sf \: \dfrac{85}{100} \times 40 \\ \\\implies \sf \:0.85 \times 40 \\ \\ \implies \sf \: 34 \\ [/tex]

Hence, 85% of 40 e -mails is 34 e-mails.

5.) 72% of 350 friends.

[tex] \implies \sf \: \dfrac{72}{100} \times 350 \\ \\\implies \sf \:0.72 \times 350 \\ \\ \implies \sf \: 252 \\ [/tex]

Hence, 72% of 350 friends is 252 friends.

6.) 5% of 220 files.

[tex] \implies \sf \: \dfrac{5}{100} \times 220 \\ \\\implies \sf \:0.05 \times 220 \\ \\ \implies \sf \: 11 \\ [/tex]

Hence, 5% of 220 files is 11 files.

Answer: $2,162.06

Step-by-step explanation:

To calculate the total monthly payment to the bank, we need to add up the monthly mortgage payment, the monthly portion of the semi-annual property tax bill, and the monthly portion of the annual homeowner's insurance bill.

First, we need to find the monthly portion of the semi-annual property tax bill. To do this, we divide the semi-annual property tax bill by 6 (since there are 6 months in half a year):

Monthly property tax payment = Semi-annual property tax bill / 6

Monthly property tax payment = $3,239 / 6

Monthly property tax payment = $539.83

Next, we need to find the monthly portion of the annual homeowner's insurance bill. To do this, we divide the annual homeowner's insurance bill by 12 (since there are 12 months in a year):

Monthly homeowner's insurance payment = Annual homeowner's insurance bill / 12

Monthly homeowner's insurance payment = $980 / 12

Monthly homeowner's insurance payment = $81.67

Now we can add up the monthly mortgage payment, the monthly property tax payment, and the monthly homeowner's insurance payment to find the total monthly payment to the bank:

Total monthly payment = Monthly mortgage payment + Monthly property tax payment + Monthly homeowner's insurance payment

Total monthly payment = $1,711.22 + $539.83 + $81.67

Total monthly payment = $2,332.72

Therefore, the Dotkoms pay the bank $2,332.72 each month.

Answer:

Step-by-step explanation:

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Using proportions, it is found that they pay the bank $2332.72 each month.

What is a proportion?

A proportion is a fraction of a total amount.

Their payments are given by:

Monthly mortgage of $1,711.22.

Semi-annual property tax bill is $3,239, that is, it is paid every 6 months, hence 3239/6 = $539.83.

Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).

The null hypothesis, which is an assertion or assumption that there is no significant difference or association between two or more variables or populations, is used in statistical hypothesis testing. It is frequently indicated by the letter H0 and is typically the hypothesis that is tested against a competing hypothesis. The objective of the hypothesis test is to either reject or fail to reject the null hypothesis based on the evidence or data seen. The null hypothesis serves as the default or baseline assumption. If the alternative hypothesis is supported by evidence, the null hypothesis is likely to be rejected.

given

The test's null and alternate hypotheses are as follows:

H0: p 0.025 (The percentage of American women with university educations who had never previously been married at the start of their 40s and are now 45 and married is less than or equal to 2.5%)

Ha: p > 0.025 (More than 2.5% of American women with college degrees who were unmarried at the start of their 40s and are now 45 and married are never before married).

Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).

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Answer:

2

Step-by-step explanation:

The basic 30-60-90 triangle ratio is:

     Side opposite the 30° angle: x

     Side opposite the 60° angle: x√3

     Side opposite the 90° angle: 2x

Here, the side opposite the 90° angle is 4.

This means that 2x = 4 and, thus, x = 2.

Since b is the side opposite the 30° angle, b = x, so it is 2.

The fraction Bret needs to turn the logo is 1 / 2.

Bret needs to put a logo on a business card.  He prints the logo the wrong way up by mistake. Bret needs to turn the logo the correct way up.

Therefore, the fraction he needs to turn the logo can be represented as follows:

For a complete turning, it will be 360 degrees. according to the diagram he has already turned the logo 180 degrees.

Therefore, for Bret to turn the logo correctly, he needs to turn it 180 degrees.

Hence, the fraction he needs to turn the logo is as follows:

180 / 360 = 18  /36 = 1 / 2

Hence, Bret needs to turn the logo half way(1 / 2)up.

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The first of these two events is the number displayed on the cube. The result of the coin toss is the second. There should be two columns in this tree, one for each occurrence.

a list or tree digrams to represnts the sample space in the given below figure . Observe the first column first. Let the number cube's result be. The following six values are possible:. Six branches, one for each outcome, result from that proceed to the following column. Let the result of the coin toss be. There would always be two possible results for, regardless of the number obtained for.

The first of these two events is the number displayed on the cube. The result of the coin toss is the second. There should be two columns in this tree, one for each occurrence.

Observe the first column first. Let N represent the number cube's result. Six values are possible: 1, 2, 3, 4, 5, and 6. Six branches, one for each outcome, result from that. proceed to the following column. Let the result of the coin toss be. There would always be two possible results for, regardless of the number obtained for.

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the actual question is :

The number cube is rolled and a coin is tossed . How can you sketch a tree diagram to represent the sample space ?

The difference in length between the two pieces of scrap wood is 7/8 inches.

To get the difference we just need to take the difference between the two lenghs.

Remember that we only have pieces of scraph wood if we have an "x" over the correspondent value in the line diagram.

By looking at it we can see that the longest pice measures 5 inches, while the shortest one (there are two of these) measure (4 + 1/8) inches.

The difference is:

5 - (4 + 1/8) = 7/8

The longest piece is 7/8 inches longer.

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Step-by-step explanation:

(-1, 1) is a solution.

because

-(-1) + 6×1 = 7

1 + 6 = 7

7 = 7

correct.

every ordered pair of x and y values that make the equation true is a solution.

(5, 2) would be another solution. and so on.

The steps for finding out the third derivative of y (given as y''') are explained and given below.

To find the third derivative of y (y'''), you would need to differentiate the function y three times with respect to the independent variable. Here are the steps:

Start by differentiating y once to get the first derivative, y'.

Differentiate y' again to get the second derivative, y''.

Finally, differentiate y'' to get the third derivative, y'''.

You can use the chain rule, product rule, quotient rule, and other differentiation rules as needed to find each derivative.

Here's an example of finding the third derivative of y for the function y = x^4 + 2x^3 - 5x:

y' = 4x^3 + 6x^2 - 5

y'' = 12x^2 + 12x

y''' = 24x + 12

So the third derivative of y is y''' = 24x + 12.

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Answer:

The simplest possible equation for the line on the graph would be x = - 2

Answer:

The given function is an exponential growth function, not an exponential decay function because as the exponent x increases, the value of y also increases instead of decreasing.

To describe its end behavior using limits, we need to find the limit of the function as x approaches infinity and as x approaches negative infinity.

As x approaches infinity, the exponent -x approaches negative infinity, and the base (1/6) is raised to increasingly larger negative powers, causing the function to approach zero. So, the limit as x approaches infinity is 0.

As x approaches negative infinity, the exponent -x approaches infinity, and the base (1/6) is raised to increasingly larger positive powers, causing the function to approach infinity. So, the limit as x approaches negative infinity is infinity.

Therefore, the end behavior of the function is that it approaches zero as x approaches infinity and approaches infinity as x approaches negative infinity.

The 3-dimensional vector space represented in the form subspace dimensions A and B is given by option B. V = A + B.

V be 3-dimensional vector space.

Subspace of dimensions of A and B are 2 and 1 respectively.

And A ∩ B = 0.

It follows that every vector in A is linearly independent of every vector in B.

This implies,

Any vector v in V can be expressed uniquely as a sum of a vector in A and a vector in B.

Let v be an arbitrary vector in V.

A has dimension 2, it has a basis of two linearly independent vectors.

Let {a1, a2} be such a basis.

B has dimension 1, it has a basis consisting of a single nonzero vector b.

Then, any vector v in V can be expressed uniquely as

v = c1a1 + c2a2 + cb,

where c1, c2, and c are scalars.

Thus,

V = A + B.

Therefore, the correct answer to represents 3 dimensional vector space V  as option(B).  V = A + B.

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In response to the stated question, we may state that As a result, the probability of correctly answering none of the 20 questions is roughly 0.0000262, or 0.00262%.

Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

The likelihood of getting any one question accurate is 1/4 if you merely guess on each question. The binomial distribution formula may be used to calculate the chance of correctly answering none of the 20 questions:

P(X=0) = (n choose X) * pX * (1-p) (n-X)

[tex]If n=20, X=0, and p= 1/4\\P(X=0) = (20 pick 0) (20 choose 0) * (1/4)^0 * (3/4)^20\\P(X=0) = 1 * 1 * 0.0000262\\P(X=0) = 0.0000262[/tex]

As a result, the probability of correctly answering none of the 20 questions is roughly 0.0000262, or 0.00262%.

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Answer:

Step-by-step explanation:

Given,

Let length be x and breadth (x - 5).

Perimeter of rectangle is calculated by :

[tex] \: \: \boxed{ \pmb{ \sf{Perimeter_{(rectangle)} = 2(l + b)}}} \\ [/tex]

On substituting the values we get :

[tex]\dashrightarrow \: \: 130 = 2(x + x - 5) \\ [/tex]

[tex]\dashrightarrow \: \: 130 = 2(2x - 5) \\ [/tex]

[tex]\dashrightarrow \: \dfrac{130}{2} = (2x - 5) \\ [/tex]

[tex]\dashrightarrow \: \: 65 = 2x - 5 \\ [/tex]

[tex]\dashrightarrow \: \: 65 + 5 = 2x \\ [/tex]

[tex]\dashrightarrow \: \: 70 = 2x \\ [/tex]

[tex]\dashrightarrow \: \: \frac{70}{2} = x \\ [/tex]

[tex]\dashrightarrow \: \: 35 = x \\ [/tex]

Hence,

The regular expression is (0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)], which matches any string that is at least 6 symbols long and contains at least one 0 and at least one 1.

One possible regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1} is:

(0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)]

This regular expression matches any string that satisfies the following conditions:

The string contains at least one 0 and at least one 1.

The string is at least 6 symbols long.

The string can have any number of 0s and 1s before and after the first 0 or 1, but it must contain at least one of each before and after the first 0 or 1.

For example, this regular expression matches strings like "0101010", "1000001", "1110010", but does not match strings like "101", "11111", "0000000".

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Complete question:

Alphabet = {0,1}.

Define a regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1}

The scale of the map in the question is 1 inch = 50 miles.

A scale in a map is a relation that tells us how many units each unit in the map represents. In this case, we know that the distance between two points A and B on the map is 5 inches, while the actual distance between these two places is 250 miles.

Then we start with the relation:

5 inches = 250 miles.

But to get the scale of the map we need to see how many miles one inch represents in the map, then we can divide both sides of the equation by 5 to geT:

5 in = 250 mi

1 in = 250mi/5

1 in = 50 mi

The scale of the map is 1 inch to 50 miles.

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The functions that have all real numbers for their range are the ones in the option B.

Remember that for a function y = f(x), we define the range as the set of possible outputs of the function, that is, possible values of y.

Here we can see a lot of functions, first, the option with the absolute value function can be discarded because we know that:

|x| ≥ 0

So it never takes negative values.

We also can discardthe option with the sine and cosine, because the range of these two functions is [-1, 1].

The only remaining option is B, and the range of these 3 functions is (-∞, ∞), so that is the correct option in this case.

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The derivative of S(t)= 1+e −t is  S'(t) = S(t)(1 - S(t)). So, the correct answer is (iii).

To find S'(t), we can use the chain rule:

S'(t) = (d/dt) [1 + e^(-t/2)]^-2 * d/dt [1 + e^(-t/2)]

Using the chain rule again for the second derivative:

d/dt [1 + e^(-t/2)] = (-1/2)e^(-t/2)

d/dt [1 + e^(-t/2)]^-2 = -2(1 + e^(-t/2))^-3 * (-1/2)e^(-t/2) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3

Substituting into the expression for S'(t), we have:

S'(t) = [(1/2) e^(-t/2) / (1 + e^(-t/2))^3] * [1 - (1/2)e^(-t/2)]

S'(t) = (1/2) e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]

S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]

Taking the derivative of S(t), we have:

S'(t) = e^(-t/2) / (1 + e^(-t/2))^2

Comparing this to the given choices, we can see that:

S'(t) = S(t) is not true, since S(t) = 1 + e^(-t/2) and S'(t) is a different function.

S'(t) = (S(t))^2 is not true, since (S(t))^2 = (1 + e^(-t/2))^2 is a different function from S'(t).

S'(t) = S(t)(1 - S(t)) is true, since we can substitute S(t) and S'(t) from above and simplify:

S'(t) = e^(-t/2) / (1 + e^(-t/2))^3 * [2 - e^(-t/2)]

S(t)(1 - S(t)) = [1 + e^(-t/2)] * [1 - (1 + e^(-t/2))] = e^(-t/2) / (1 + e^(-t/2))

Therefore, S'(t) = S(t)(1 - S(t)) is true.

S'(t) = -S(-t) is not true, since S(-t) = 1 + e^(t/2) and -S(-t) is a different function from S'(t).

So the correct choice is (iii): S'(t) = S(t)(1 - S(t)).

To know more about derivative:

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