6 TH GRADE MATH , WHAT IS THE SLOPE? TY

Step-by-step explanation:

i)

A = {2, 3, 5, 7}

as "1" can only be divided by one number (instead of the usual 2 numbers for prime numbers), if it's not part of that set.

out of U = {1, 2, 3, 4, 5, 6, 7, 8}

B = {1, 3, 5}

I am not sure what you mean by "y E 7".

I don't think you mean the E7 algebraic group.

I decided you mean y <> 7 (not equal to 7).

ii)

A u B (united) = {1, 2, 3, 5, 7}

A n B (elements in common) = {3, 5}

iii)

a set with n elements has 2^n subsets and (2^n) - 1 proper subsets (all subsets minus the equal one).

our U here has 8 elements, so the number of subsets is

2⁸ = 256.

the number of relations from A to B is 2^|A×B| = 2^(|A|·|B|).

|A| = 4

|B| = 3

so the number of relations from A to B are

2^(4×3) = 2¹² = 4096

remember, for the number of possible relations we have 4×3 = 12 possible combinations of elements of A and engender of B.

each of these combinations can be in the set of relations or not, which gives us 2 options per combination.

that gives us 2¹² relations.

the coordinates of point P are ((Ax + Bx) / 2, (Ay + By) / 2).

How to find?

If we want to find the coordinates of point P on the directed line segment from A to B such that P is the length of the line segment from A to B, we can use the following formula:

P = (1 - t)A + tB

where A and B are the coordinates of the two endpoints of the line segment, t is a scalar between 0 and 1, and P is the coordinates of the point we are trying to find.

When t = 1, we get the coordinates of point B, and when t = 0, we get the coordinates of point A. When t is between 0 and 1, we get a point on the line segment from A to B.

To find the point P that is the length of the line segment from A to B, we set t = 1/2, which gives us:

P = (1 - 1/2)A + (1/2)B

= (1/2)A + (1/2)B

So the coordinates of point P are the average of the coordinates of A and B:

Px = (Ax + Bx) / 2

Py = (Ay + By) / 2

where (Ax, Ay) and (Bx, By) are the coordinates of points A and B, respectively.

Therefore, the coordinates of point P are ((Ax + Bx) / 2, (Ay + By) / 2).

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Complete Question:

What are the Cartesian coordinates of point P on the directed line segment from point A to point B such that point P is located at a distance equal to the length of the line segment from point A to point B?

Answer:

3, 4, 5

Step-by-step explanation:

Take any odd number and square it.

3²=9

Divide that number by 2.

9/2=4.5

The original number (3 in this case), as well as the numbers 0.5 above and below the new number (4.5 in this case), are your set of numbers.

3, 4, 5

Hope that helps :)

Answer:

It looks like there might be a typo in the expression given. Assuming that "J" and "K" are just placeholders, we can write the expression as:

f(n) = 45 * |4 - 5(n-1)|

To find the recursive formula for this sequence, we need to determine how each term relates to the previous term. We can start by looking at the first few terms of the sequence:

f(1) = 45 * |4 - 5(1-1)| = 45 * |4 - 5(0)| = 45 * |4| = 180

f(2) = 45 * |4 - 5(2-1)| = 45 * |4 - 5(1)| = 45 * |-1| = 45

f(3) = 45 * |4 - 5(3-1)| = 45 * |4 - 5(2)| = 45 * |-6| = 270

From this, we can see that the sign of the expression inside the absolute value changes with each term, alternating between positive and negative. Furthermore, the magnitude of this expression increases by 5 with each term. We can use these observations to write the recursive formula:

f(1) = 180

f(n) = f(n-1) + (-1)^(n-1) * 5 * 45 for n >= 2

This formula says that the first term in the sequence is 180, and each subsequent term is found by adding or subtracting 225 (5 * 45) from the previous term, depending on whether n is odd or even.

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The ratio relationship of a car's fuel efficiency can be expressed as equivalent rates of miles per gallon and gallons per mile, or as a unit rate of either 30 miles per gallon or 1/30 gallons per mile.

A real-world situation involving a ratio relationship could be the fuel efficiency of a car. For example, suppose a car travels 60 miles using 2 gallons of gas.

To express this ratio relationship as equivalent rates, we can write:

60 miles ÷ 2 gallons = 30 miles per gallon

2 gallons ÷ 60 miles = 0.0333 gallons per mile

To express the ratio as a unit rate, we can divide the numerator and denominator by the same amount to simplify the fraction. Let's choose to divide by 2 to get:

60 miles ÷ 2 gallons = 30 miles per gallon

2 gallons ÷ 60 miles = 1/30 gallons per mile

In both cases, the ratio represents the same relationship between miles traveled and gallons of gas used, but it is expressed in different units or rates.

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The probability of a pitcher throwing exactly 5 pitches in an at-bat is 0.1 or 10%.

What is probability and how is it calculated?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain. The probability of an event A is calculated as the ratio of the number of outcomes that correspond to event A to the total number of possible outcomes.

Calculating probability of a pitcher throwing exactly 5 pitches :

To calculate the probability of a pitcher throwing exactly 5 pitches in an at-bat, we need to add up the frequencies of all the at-bats that have exactly 5 pitches. From the given table, we see that there are 8 at-bats that have exactly 5 pitches.

The total number of at-bats is the sum of the frequencies of all pitch counts.

Total number of at-bats = 12+16+32+12+8 = 80

Therefore, the probability of a pitcher throwing exactly 5 pitches in an at-bat is:

P(5) = Frequency of 5-pitch at-bats / Total number of at-bats

P(5)= 8/80 = 0.1 or 10%

Hence, the probability that a pitcher will throw exactly 5 pitches in an at-bat is 10%.

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There are 3 seniors who take Math and Spanish, but not Physics.

What is Venn Diagram?

A diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.

To solve this problem, we can use a Venn diagram to organize the information given about the students.

First, we know that there are 150 seniors in total, so we can label the outer rectangle of the Venn diagram as having a total of 150 students.

Next, we can label the three circles as Math, Spanish, and Physics, with the numbers given in the problem:

80 seniors take Math, so we label the Math circle with 80.

41 seniors take Spanish, so we label the Spanish circle with 41.

54 seniors take Physics, so we label the Physics circle with 54.

We also know that 10 seniors take Math and Spanish, 19 take Math and Physics, 12 take Physics and Spanish, and 7 take all three. We can label these overlaps in the Venn diagram as follows:

The overlap of Math and Spanish is 10.

The overlap of Math and Physics is 19.

The overlap of Physics and Spanish is 12.

The overlap of all three is 7.

To find the number of seniors who take Math and Spanish, but not Physics, we need to subtract the number of seniors who take all three (because they are counted in all three circles) and the number of seniors who take Math, Spanish, and Physics (because they are counted in the overlap of all three circles). Then we add the number of seniors who take only Math and Spanish (because they are not counted in the Physics circle).

Using the formula for the number of elements in the union of three sets, we have:

Math or Spanish or Physics = Math + Spanish + Physics - (Math and Spanish) - (Math and Physics) - (Spanish and Physics) + (Math and Spanish and Physics)

So, the number of seniors who take Math and Spanish, but not Physics is:

Math and Spanish - (Math and Spanish and Physics) + (Math or Spanish or Physics - (Math + Spanish + Physics) + (Math and Spanish and Physics))

= 10 - 7 + (80 + 41 + 54 - 2(10) - 2(19) - 2(12) + 7)

= 3

Therefore, there are 3 seniors who take Math and Spanish, but not Physics.

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Answer:

We need to find dz/dt given:

z = cos(x + 8y), x = 7t^5, y = 5/t

Using the chain rule, we can find dz/dt by taking the derivative of z with respect to x and y, and then multiplying by the derivatives of x and y with respect to t:

dz/dt = dz/dx * dx/dt + dz/dy * dy/dt

First, let's find dz/dx and dz/dy:

dz/dx = -sin(x + 8y)

dz/dy = -8sin(x + 8y)

Now, let's find dx/dt and dy/dt:

dx/dt = 35t^4

dy/dt = -5/t^2

Substituting these values, we get:

dz/dt = (-sin(x + 8y)) * (35t^4) + (-8sin(x + 8y)) * (-5/t^2)

Simplifying this expression, we get:

dz/dt = -35t^4sin(x + 8y) + 40sin(x + 8y)/t^2

Substituting x and y, we get:

dz/dt = -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2

Therefore, dz/dt is given by -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2.

Answer:

see step by step

Step-by-step explanation:

He can spin a lot of times the wheel, Notice IF THE SPINNER IS FAIR the probability must be equally, since the spinner have 4 options it must have 0.25 (25%) probability in each of the colors. Of course, since the probability is random the theoretically probability need to be closer to 0.25 for each one, to evade the randomness Klevon can spin the spinner a lot of times.


(for example, if he spins 100 times, he can get 28, 31, 25 and 20 and the spinner can still be fair, he can spin another 100 times to see the results, but if he have 5,5,80, 5 in the results, the probability is really low of this happening, so its probability unfair, but still, posible)

By using the partial quotients method 1032/32 = 32 R8.

In mathematics, an expression is a combination of numbers, symbols, and operators that represents a mathematical quantity or relationship. It can be a single number, a variable, or a combination of both, and can also include mathematical operations such as addition, subtraction, multiplication, division, exponents, and roots. Expressions can be evaluated or simplified using mathematical rules and formulas.

Step 1: Estimate how many times 32 goes into 1032. It's helpful to find a multiple of 32 that is close to 1032. In this case, 32 x 30 = 960, which is less than 1032, and 32 x 31 = 992, which is greater than 1032. So we can estimate that 32 goes into 1032 around 30 to 31 times.

Step 2: Write 30 on top of a division symbol, and multiply 30 by 32. Write the result, 960, under 1032, and subtract.

    30

 -------

32|1032

   960

  ------

    72

Step 3: Write 72 next to the 30 on top of the division symbol. Then, add 72 to the partial quotient (30) to get 102. Write 102 under the partial difference (72) and bring down the next digit, which is 2.

    30  72

 -------

32|1032

   960

  ------

    72  

    64  

  ------

     8  

step 4: Estimate again how many times 32 goes into the new partial difference, 82. Since 32 x 2 = 64 is less than 82 and 32 x 3 = 96 is greater than 82, we estimate 32 goes into 82 two to three times.

Step 5: Write 2 on top of the division symbol, and multiply 32 by 2. Write the result, 64, under 82, and subtract.

    30  72  2

 ------------

32|1032

   960

  ------

    72  

    64  

  ------

     8  

tep 6: Write 2 next to the 30 and 72 on top of the division symbol, and add 2 to the partial quotient to get 32. Write 32 under the partial difference and bring down the next digit, which is 0.

    30  72  2

 ------------

32|1032

   960

  ------

    72  

    64  

  ------

     8  

     0  

Step 7: Since there are no more digits to bring down, we have the final answer. The quotient is 32 with a remainder of 8. Therefore, 1032 divided by 32 is equal to 32 with a remainder of 8.

Therefore, by using the partial quotients method 1032/32 = 32 R8 .

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The correct option is D. $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

In math, addition is the process of adding two or more integers together. The numbers having added are known as addends, while the outcome of the addition process, or the final response, is known as the sum. It is among the most fundamental mathematical operations we employ on a daily basis.

Quarterly profit for Mr. Brown's Goodwill Store in 2012 (in dollars)

1 $9,841.28

2 $8,957.67

3 $7,429.84

4 $11,095.67

Total profit = profit of 3rd quarter + profit of 4th quarter

Total profit = $7,429.84 + $11,095.67

Total profit =  $18,525.51.

Thus, $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

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4775 g968r648 747474874 483892874 23773259635y84b2375789325 7437594365825 4378574937587 49388959365n 98437858746587 32o4iy548569

Answer:

?

Step-by-step explanation:

Step-by-step explanation:

We have:

x = a sin 2t

Differentiating with respect to t, we get:

dx/dt = 2a cos 2t

Next, we have:

y = a(cos 2t + log tan t)

Differentiating with respect to t, we get:

dy/dt = -2a sin 2t + (1/tan t)(1/ln 10)

Using the identity:

sin^2 t + cos^2 t = 1

We have:

sin 2t = 2sin t cos t

And:

cos 2t = cos^2 t - sin^2 t

cos 2t = 2cos^2 t - 1

Using these identities, we can rewrite dx/dt and dy/dt in terms of x and y:

dx/dt = 2a sqrt(1 - x^2/a^2)

dy/dt = -2a sqrt(1 - x^2/a^2) + (1/ln 10)(y - a cos 2t)

Therefore, we have:

dx/dy = dx/dt ÷ dy/dt

Substituting the expressions for dx/dt and dy/dt, we get:

dx/dy = (2a sqrt(1 - x^2/a^2)) / (-2a sqrt(1 - x^2/a^2) + (1/ln 10)(y - a cos 2t))

Simplifying, we get:

dx/dy = (-2 sqrt(1 - x^2/a^2)) / (2 sqrt(1 - x^2/a^2) - (1/ln 10)(y - a cos 2t))

Answer:

Step-by-step explanation:

B because after having the star and the circle comes the square.

The equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l).

The equation ak + bl = 0 is a linear equation in two variables and is solved using the method of elimination. The equation can be written in the form ax + by = c, where a, b, c are constants. To solve this equation, both sides of the equation should be divided by the coefficient of one of the variables (a or b). This will result in a equation of the form x + qy = r, where q and r are constants. Then, the equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l). The two variables can then be calculated using the point of intersection by substituting the x and y values into the two equations. In this way, the two variables k and l can be found such that ak + bl = 0.

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What are the integers k and l such that ak + bl = 0?

Answer:

resultant force

Step-by-step explanation:

the body will move to the direction where greater force is applied

Answer:

I HOPE ITS HELPFUL

Step-by-step explanation:

Cost price of 300 articles = ₹1500Hence cost of each item = 1500/300 = ₹5With 20% profit, selling price of each item = 5 * 120/100 = ₹6Hence, total selling price of 260 items = 260 * 6 = ₹1560Selling price of balance items = 6/2 = ₹3Total selling price for balance items = 40 * 3 = ₹120Total selling price = 1560 + 120 = ₹1680Profit gained = 1680 - 1500 = ₹180Profit percentage = 180/1500 * 100 = 12

Answer: The variance of a sample of size n can be calculated using the formula:

variance = (sum of squared deviations from the mean) / (n - 1)

where the deviation from the mean is the difference between each individual value and the sample mean, and the sum of squared deviations is the sum of the squares of these differences.

However, we don't have the individual weights or the sample mean, only the standard deviation. We can use a related formula that expresses the standard deviation in terms of the variance:

standard deviation = sqrt(variance)

Solving for variance, we get:

variance = (standard deviation)^2

Plugging in the given standard deviation of 12.3194 kg, we get:

variance = (12.3194 kg)^2 = 151.8992 kg^2

Therefore, the variance of the weights of the 10 subjects is 151.8992 kg^2.

Step-by-step explanation:

Answer:

To shift the graph of f(x) = |x| to have a domain of [-3, 6], we need to move the left endpoint from -6 to -3 and the right endpoint from 3 to 6.

A translation to the right by 3 units will move the left endpoint of the graph of f(x) to -3, but it will also shift the right endpoint to 6 + 3 = 9, which is outside the desired domain.

A translation to the left by 3 units will move the right endpoint of the graph of f(x) to 3 - 3 = 0, which is outside the desired domain.

A translation upward or downward will not change the domain of the graph, so options B and D can be eliminated.

Therefore, the correct answer is C g(x) = x - 3. This translation will move the left endpoint to -3 and the right endpoint to 6, which is exactly the desired domain.

The equation of the tangent line to the curve at the point (2,4) is y = (-1/2)x + 5.

To use implicit differentiation to find an equation of the tangent line to the curve at the given point (2,4), we need an implicit equation of the curve. Let's assume the curve is given by the equation:

x² + y² = 16

We can use implicit differentiation to find the slope of the tangent line at any point on this curve. Taking the derivative of both sides with respect to x, we get:

2x + 2y (dy/dx) = 0

Simplifying for (dy/dx), we get:

dy/dx = -x/y

Now we can substitute the given point (2,4) into this equation to find the slope of the tangent line at that point:

dy/dx = -2/4 = -1/2

So the slope of the tangent line at (2,4) is -1/2. We can use this slope and the point-slope form of the equation of a line to find an equation of the tangent line. The point-slope form is:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point. Substituting the values, we get:

y - 4 = (-1/2)(x - 2)

Simplifying, we get:

y = (-1/2)x + 5

Therefore, the equation of the tangent line to the curve at the point (2,4) is y = (-1/2)x + 5.

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Complete question:

Use implicit differentiation to find an equation of the tangent line to the curve at the given point (2,4)

10√10 is the dimensions of a rectangle with area 1,000 m² whose perimeter is as small as possible.

a. The smaller value is 10√10 m.

b. The larger value is 10√10 m.

We have to determine the dimensions of a rectangle with area 1,000 m² whose perimeter is as small as possible.

P = 2w + 2L

1000 = Lw

P = 2w + 2(1000/w)

P = 2w + 2000/w

P-prime = 2 -2000/w²

0 = 2 - 2000/w²

Add 2000/w² on both side, we get

2000/w² = 2

Multiply by w² on both side, we get

2000 = 2w²

Divide by 2 on both side

w² = 2000/2

w² = 1000

Taking square root on both side, we get

w = 10√10

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The value of length of side g using the  similar triangles is found as 20 ft.

In the given figures:

DC || EA

So,

∠D = ∠A

∠C = ∠E

By Angle -Angle similarity both triangles are similar.

Thus,

Taking the ratios of their side, it will be also equal.

EA / DC = EB / BC

5 / 10 = g / 10

g = 10*10 / 5

g = 100 / 5

g = 20

Thus, the value of length of side g using the  similar triangles is found as 20 ft.

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Answer: It's a 80% increase

Step-by-step explanation:

The ordered pair that respect the given conditions are: (6,-12) and (-12,6).

A system of equations is the given term of math for two or more equations with the same variables. The solution of these equations represents the point of the intersection.

You can solve a system of equations by substitution or adding methods. In the addition method, you eliminate a variable, on the other hand, in the substitution method you replace a variable for the other.

You should convert the text of the question into equations. See below.

From equation 1, you have x=-72/y. Thus, applying the substitution method, you can solve this question by following the steps below:

1) Replace  x=-72/y into equation 2. Then, you have:

[tex]\frac{-72}{y} +y=-6\\ \\[/tex]

-72+y²=-6y

y²+6y-72=0

2) Solving the quadratic equation y²+6y-72=0 for finding y:

Δ=b²-4ac

Δ=6²-4*1*(-72)

Δ=36+288

Δ=324

[tex]y=\frac{-b\pm \sqrt{\Delta} }{2a} =\frac{-6\pm \sqrt{324} }{2*1}=\frac{-6\pm18 }{2}[/tex]. Therefore,

y1=(-6-18)/2=-12

or

y2=(-6+18)/2=12/2=6

3) Finding x

If  x=-72/y and y=-12 or y=6. You have

For y=-12, x=-72/-12, thus x=6

For y=6, x=-72/6, thus x=-12

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The quadrilaterals which have both line and rotational symmetry of order more than 1 are square, and rhombus

Symmetry is a fundamental concept in mathematics and geometry. It refers to the property of a shape that remains unchanged when it is transformed in a certain way.

Now, let's talk about quadrilaterals that have both line and rotational symmetry of order more than 1. One example of such a quadrilateral is a square.

Another example of a quadrilateral with both line and rotational symmetry of order more than 1 is a rhombus. A rhombus is a type of quadrilateral where all four sides are equal in length, and opposite angles are equal.

In summary, a square and a rhombus are examples of quadrilaterals that have both line and rotational symmetry of order more than 1.

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Complete Question:

Name the quadrilaterals which have both line and rotational symmetry of order more than 1.

The slope of the secant line joining (2, f(2)) and (7, f(7)) is 8.6.

Rise over run is the definition of a line's slope. A curve's secant line is a line that connects any two of its points. The slope of the secant line would change to the slope of the tangent line at the point when one of these points approaches the other. As a secant line is also a line, we may calculate its slope using the slope of a line formula.

The two points on the secant line are given as (2, f(2)) and (7, f(7)).

Substituting the values in the function we have:

f(x) = x² + 8x

f(2) = 2² + 8(2) = 20

f(x) = x² + 8x

f(7) = 7² + 8(7) = 63

Using the difference quotient the slope of the line is:

(f(7) - f(2)) / (7 - 2) = (63 - 20) / 5 = 8.6

Hence, the slope of the secant line joining (2, f(2)) and (7, f(7)) is 8.6.

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Answer:

Let's start by setting up an equation to represent the problem. Let x be the cost of the meal:

x + 0.1x = 126.50

Simplifying the left side of the equation:

1.1x = 126.50

Dividing both sides by 1.1:

x = 115

Therefore, the cost of the meal was £115.

Therefore, the expression [tex]\frac{7}{Tanb} +7Tanb[/tex] is equivalent to function. [tex]7*secb*cscb[/tex].

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It includes the study of trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant, as well as their properties and applications.

Trigonometry is useful in a wide range of fields, including engineering, physics, navigation, astronomy, and surveying. It is often used to solve problems involving triangles, such as determining the height of a tall object, finding the distance between two points, or calculating the trajectory of a moving object.

The origins of trigonometry can be traced back to ancient civilizations such as the Babylonians, Greeks, and Indians, who developed various methods for calculating angles and distances. Today, trigonometry is an important part of mathematics education and continues to be used extensively in many fields of study.

Given by the question.

To simplify the expression 7/tan b + 7 tan b, we need to first recall the following trigonometric identity:

tan(x) * cot(x) = 1

Using this identity, we can rewrite the expression as:

[tex]\frac{7}{Tanb} +7Tanb[/tex]

= [tex]\frac{7}{Tanb} +7Tan^{2} b*cotb\\[/tex]

=[tex]\frac{7}{Tanb} +7*(sin^{2}/cos^{2}b)*(cosb/sinb)[/tex]

= [tex]\frac{7}{Tanb} +7*(cosb/sinb)[/tex]

= [tex]7*(1/tanb+sinb/cosb[/tex]

[tex]=7*(cosb/sinb+sinb/cosb)\\=7*((cos^{2}b+sin^{2}b)/(sinb/cosb))\\ =7*(1/(sinb/cosb))\\=7*secb*cscb[/tex]

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Answer:

I don't knowI am sorry I will let someone else answer

Step-by-step explanation:

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:

https://brainly.com/question/30365299

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