Solve for length of segment c.

L = ∫√(cos^2(3θ) + 9sin^2(3θ)) dθ.

This expression represents the arc length of the rose curve r = cos(3θ).

To understand how to set up an expression for the arc length of the rose curve r = cos(3θ), we first need to understand the concept of arc length in polar coordinates.

In Cartesian coordinates, the distance between two points can be calculated using the Pythagorean theorem. However, in polar coordinates, the distance between two points is given by the arc length formula, which involves integrating a function.

Consider a curve defined by the polar equation r = f(θ). To find the arc length of the curve between two angles θ1 and θ2, we divide the interval [θ1, θ2] into small pieces, and approximate the length of each piece as the hypotenuse of a right triangle.

The base of the triangle is a small change in θ, and the height is a small change in r. By taking the limit as the length of the intervals goes to zero, we can integrate to find the exact length of the curve.

The arc length formula for polar coordinates is given by:

L = ∫√(r^2 + (dr/dθ)^2) dθ.

This formula calculates the length of the curve r = f(θ) between θ1 and θ2. The expression inside the square root is the Pythagorean theorem for polar coordinates, and dr/dθ is the derivative of r with respect to θ.

Now, let's use this formula to find the arc length of the rose curve r = cos(3θ).

First, we need to find the derivative of r with respect to θ, which is given by:

dr/dθ = -3sin(3θ).

Now, we can plug in r and dr/dθ into the arc length formula:

L = ∫√((cos(3θ))^2 + (-3sin(3θ))^2) dθ.

Simplifying the expression inside the square root, we get:

L = ∫√(cos^2(3θ) + 9sin^2(3θ)) dθ.

This expression represents the arc length of the rose curve r = cos(3θ). By evaluating this integral between the appropriate limits of integration, we can find the exact length of the curve.

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The measure of angle JML is 180 degrees because in a circle, an angle formed by two chords intersecting inside the circle.

In a circle, the measurement of angle formed by two chords intersecting inside the circle is half the sum of the arcs intercepted by the angle. Using this property, we can find the measure of angle JML.

Since angle JKL is a right angle, its intercepted arc is the diameter of the circle. Therefore, its measure is 180 degrees.

By the same property, we know that angle JML is half the sum of the arcs intercepted by it. The arcs intercepted by angle JML are arcs JL and KM.

Since angle JKL is a right angle, arc JL is also 180 degrees.

Since J, K, L, and M are concyclic, we know that angle JKM is supplementary to angle JLM. Therefore, arc KM is the supplement of arc JL and has measure 360 - 180 = 180 degrees.

Thus, the sum of the intercepted arcs is 180 + 180 = 360 degrees, and angle JML is half of this, so its measure is 180 degrees.

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The woman bought 21 cards that play a song when they are opened, and 79 cards that do not play music.

Let's assume that the woman bought x cards that play a song when they are opened, and 100-x cards that do not play music.

We know the cost of the cards that play a song is 30 cents each, so the cost of x of these cards is 0.3x dollars.

Similarly, the cost of the cards that do not play music is 5 cents each, so the cost of 100-x of these cards is 0.05(100-x) dollars.

The total cost of all the cards is $10.25, so we can set up the following equation

0.3x + 0.05(100-x) = 10.25

Simplifying the equation, we get

0.3x + 5 - 0.05x = 10.25

0.25x = 5.25

x = 21

Therefore, the woman bought 21 cards that play a song when they are opened, and 100-21 = 79 cards that do not play music.

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a. Null hypothesis (H0): μ1 = μ2 and Alternative hypothesis (H1): μ1 < μ2. b. The test statistic is -2.57 and p-value is  0.005 for the hypothesis test. c. At α = 0.05 it can be concluded that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

a. First, let's formulate the hypotheses:

Null hypothesis (H0): μ1 = μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is equal to the mean hours worked per week by all U.S. CPAs during tax season.

Alternative hypothesis (H1): μ1 < μ2, which means that the mean hours worked per week during tax season by CPAs in states with flat state income tax rates is less than the mean hours worked per week by all U.S. CPAs during tax season.

b. Now, let's compute the test statistic and p-value using the given sample data:

Sample mean (x) = 55 hours

Population mean (μ) = 60 hours

Population standard deviation (σ) = 27.4 hours

Sample size (n) = 150

We'll use the z-test for this hypothesis test:

z = (x - μ) / (σ / √n) = (55 - 60) / (27.4 / √150) ≈ -2.57

To find the p-value, we need to look up the z-value in the standard normal table, which gives us a p-value of approximately 0.005.

c. Lastly, let's draw our conclusion using α = 0.05:

Since the p-value (0.005) is less than α (0.05), we reject the null hypothesis (H0). This suggests that CPAs in states with flat state income tax rates work fewer hours per week during tax season compared to the average U.S. CPAs.

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The correct answer is: b

8 points within any 36-month period

6 points within any 24-month period

4 points within any 12-month period

In most US states, drivers are assigned points for certain traffic violations or accidents. If a driver accumulates too many points within a certain period of time, they may be labeled as a "negligent operator" and face penalties such as license suspension or revocation. The point thresholds for being labeled as a negligent operator may vary by state, but the options given in the question are generally accurate.

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The sample mean weight is 16.235 ounces, and the margin of error is 0.360 ounces.

The sample mean weight of grapes is the midpoint of the confidence interval, which is given by:

sample mean = (lower bound + upper bound) / 2

sample mean = (15.875 + 16.595) / 2

sample mean = 16.235

Therefore, the sample mean weight of grapes is 16.235 ounces.

The margin of error is half the width of the confidence interval, which is given by:

margin of error = (upper bound - lower bound) / 2

margin of error = (16.595 - 15.875) / 2

margin of error = 0.360

Therefore, the margin of error is 0.360 ounces.

The correct answer is: The sample mean weight is 16.235 ounces, and the margin of error is 0.360 ounces.

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"Twisted till" and "claw curled" are examples of alliteration, which is a poetic device that involves the repetition of consonant sounds at the beginning of words.

The author repeats "hands folded in as if he wished for something" to exaggerate the praying mantis's movement and make it seem more human.

This repetition is a literary technique used to emphasize the mantis's behavior and draw the reader's attention to it.

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Corresponding angles have the same position on the parallel lines, alternate interior angles are inside and opposite, alternate exterior angles are outside and opposite, and consecutive interior angles are on the same side.

When two parallel lines are intersected by a transversal, there are several types of angle pairs that are formed. Corresponding angles are pairs of angles that are located in the same position on the parallel lines relative to the transversal. They have the same measure and are congruent.

Alternate interior angles are pairs of angles that are located on opposite sides of the transversal and inside the parallel lines. They are congruent and have the same measure. Alternate exterior angles are pairs of angles that are located on opposite sides of the transversal and outside the parallel lines. They are congruent and have the same measure.

Consecutive interior angles are pairs of angles that are located on the same side of the transversal and inside the parallel lines. They add up to 180 degrees.

To classify each angle pair, we need to determine their positions relative to the parallel lines and the transversal. By knowing the classifications, we can identify each angle pair and their properties.

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Answer: 7x=30.5

Step-by-step explanation: If you were to answer this equation with the given information it would not be correct. 7(4.25)=29.75.  You need to go through BEDMAS to answer this.

You what you need to do is divide 30.5 by 7 to get the amount she needs to jog for a week, if you do that you get 4.37 miles each day to get to 30.5 in a week.

A polynomial whose GCF is 2x is 2x(x² + 2x + 3), a polynomial with a factor of (x + 1) is x = (-1 ± sqrt(-3))/2, a polynomial with a factor of (2y - 3x) is -3x(2y² + 1) + 5, a polynomial that is a difference of perfect squares is (4x + 3y)(4x - 3y), and a trinomial with a factor of (y + 4) and a GCF of 3y is (3y + 4x)(y + 4).

1. To create a polynomial whose GCF is 2x, we can start by choosing two terms that have 2x as a common factor. For example, 2x³ and 4x². To make it a polynomial, we can add another term, say 6x. The polynomial in standard form is:

2x³ + 4x² + 6x

To write it in factored form, we can factor out 2x from all terms:

2x(x² + 2x + 3)

2. To create a polynomial with a factor of (x + 1), we can start by choosing two terms that multiply to x², such as x and x. To make it a trinomial with (x + 1) as a factor, we can add another term, such as 1. The polynomial in standard form is:

x² + x + 1

To write it in factored form, we can use the quadratic formula to find the roots:

x = (-1 ± sqrt(1 - 4))/2

x = (-1 ± sqrt(-3))/2

Since the roots are complex, the polynomial cannot be factored further over the real numbers.

3. To create a polynomial with a factor of (2y - 3x), we can start by multiplying two terms that have 2y and 3x as coefficients, respectively. For example, 2y² and -3x. To make it a polynomial, we can add another term, say 5. The polynomial in standard form is:

-6xy² - 3x + 5

To write it in factored form, we can factor out -3x from the first two terms:

-3x(2y² + 1) + 5

4. To create a polynomial that is a difference of perfect squares, we can start by choosing two terms that are perfect squares and have a subtraction sign between them. For example, 16x² and 9y². The polynomial in standard form is:

16x² - 9y²

To write it in factored form, we can use the difference of squares formula:

(4x + 3y)(4x - 3y)

5. To create a trinomial with a factor of (y + 4) and a GCF of 3y, we can start by multiplying two terms that have 3y as a common factor. For example, 3y and 4x. To make it a trinomial with (y + 4) as a factor, we can add another term, say 12. The polynomial in standard form is:

12y + 3y² + 12 + 4xy

To write it in factored form, we can factor out the GCF 3y from the first two terms and factor out (y + 4) from the last two terms:

3y(y + 4) + 4x(y + 4)

(3y + 4x)(y + 4)

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Answer:The probability that a student takes algebra 2 = 8%

Step-by-step explanation: hope it helps :)

Answer:

They can make 5 complete teams of 8 children even after one child goes home.

Step-by-step explanation:

If there are 43 children and they want to make teams of 8, we can find out how many complete teams they can make by dividing the total number of children by the number of children per team:

This means that they can make 5 complete teams of 8 children, with 3 children left over.

However, since one child goes home, there are only 42 children left. We can repeat the division:

This means that they can make 5 complete teams of 8 children, with 2 children left over. Therefore, they can make 5 complete teams of 8 children even after one child goes home.

The value of the variable is 9

It is important to note that  inequalities are described as non- equal comparison of numbers or expressions.

The signs of inequalities represents;

From the information given, we have that;

7x - 1  is less than or equal to 62

This is represented as;

7x - 1≤ 62

collect the like terms, we have;

7x ≤ 62 + 1

Add the values

7x ≤ 63

Divide both sides by the coefficient, we get;

x ≤ 63/7

x ≤ 9

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Answer:

B.

Resources, as in details of the problem and then you do check the answers and present the solution.

Answer:

[tex] \sqrt{3 {x}^{3} } = x \sqrt{3x} [/tex]

We note that x>0 here.

Answer:

The answer is x√3x

Step-by-step explanation:

√3x³=x√3x

Answer:

4/6 or 2/3

Step-by-step explanation:

probability is successful out of total. The total is 1,2,3,4,5,6, or 6 ways, and the successful is 2,4,5,6, or 4 ways

The probability that Sam is chosen first, Mike second, and Cindy third in a random order is 37!/40 (Option B).

The question is: Sam, Mike, and Cindy are in a lottery drawing for housing with 40 other students to choose their dorm rooms. If the students are chosen in random order, what is the probability that Sam is chosen first, Mike second, and Cindy third?

To find the probability, we need to consider the total possible ways the students can be chosen and the specific arrangement we want (Sam first, Mike second, and Cindy third). There are a total of 43 students, so there are 43! (43 factorial) ways to arrange them.

For the specific arrangement we want:
- There is 1 way to choose Sam first (out of 43 students).
- After choosing Sam, there is 1 way to choose Mike second (out of the remaining 42 students).
- After choosing Mike, there is 1 way to choose Cindy third (out of the remaining 41 students).

So, there is a total of 1 × 1 × 1 = 1 way to have the specific arrangement we want.

Now, we can calculate the probability by dividing the number of ways to get the specific arrangement by the total number of arrangements:

Probability = (1 way for the specific arrangement) / (43! total arrangements) = 1/(43!)

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To find out how much lace Ms. Regan needs to purchase, we first need to calculate the circumference of the circular quilt. We know that the area of the quilt is 28.26 square feet, and we can use the formula A = πr^2 to find the radius of the quilt.

28.26 = 3.14 x r^2

r^2 = 9

r = 3

Now that we know the radius is 3 feet, we can use the formula C = 2πr to find the circumference of the quilt.

C = 2 x 3.14 x 3

C = 18.84 feet

Therefore, Ms. Regan needs to purchase 18.84 feet of lace to go around the outside of her circular quilt.

In summary, to find out how much lace Ms. Regan needs to purchase, we need to calculate the circumference of the circular quilt. We do this by first finding the radius using the formula A = πr^2. Once we know the radius, we can use the formula C = 2πr to find the circumference. In this case, the circumference is 18.84 feet, so Ms. Regan needs to purchase that amount of lace.

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The coordinates of point P along the directed line segment AB with a ratio of 1:4 are (-5, -2).

Since the ratio of AP to PB is 1:4, we can use the midpoint formula to find the coordinates of point A. The midpoint formula is

((x₁ + x₂)/2, (y₁ + y₂)/2)

Plugging in the coordinates of points P and B, we get:

((4(-7) - 2)/5, (4(-5) + 0)/5) = (-30/5, -20/5) = (-6, -4)

we can use the point-slope formula to find the equation of the line segment AB:

(y - (-4)) = (1/5)(x - (-6))

Simplifying this equation, we get:

y = (1/5)x + 2

Finally, we can use the given ratio of 1:4 to find the coordinates of point P. Since the ratio of AP to PB is 1:4, we can use the ratio formula to find the coordinates of point P:

(x, y) = (4(-5) + (-2))/5, (4(-2) - (-4))/5) = (-30/5, 12/5) = (-6, 2.4)

Rounding off to one decimal place, we get the coordinates of point P as (-5, -2).

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P(Dead l WH) = 0.23/0.48 = 0.4792

Here the western red cedar is relatively young.

(a) P(Dead l RC) = 0.02/0.20 = 0.10

P(Dead l DF) = 0.16/0.32 = 0.50

P(Dead l WH) = 0.23/0.48 = 0.4792

Here the western red cedar is relatively young.

(b) P(sapling l RC) = 0.08/0.20 = 0.4

P(sapling l DF) = 0.00/0.32 = 0

P(sapling l WH) = 0.04/0.48 = 0.08333

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The median of the random variable X with pdf f(x) = 3x−4 for x ≥1 (and 0 is approximately 1.482.

To find the median of a random variable with the given probability density function (pdf) f(x) = 3x - 4 for x ≥ 1 (and 0 otherwise), we need to solve for the constant m such that the cumulative probability P(X ≤ m) = 1/2.

First, we find the cumulative distribution function (CDF) by integrating the pdf:

F(x) = ∫(3x - 4) dx, where the limits of integration are from 1 to x.

F(x) = [(3/2)x² - 4x] evaluated from 1 to x.

Now, set the CDF equal to 1/2 to find the median:

1/2 = [(3/2)m² - 4m] - [(3/2)(1)² - 4(1)]

1/2 = (3/2)m² - 4m - (1/2)

1 = 3m² - 8m

0 = 3m² - 8m - 1

To find the value of m, we solve the quadratic equation above. Unfortunately, it cannot be factored easily, so we use the quadratic formula:

m = (-b ± √(b² - 4ac)) / 2a

In this case, a = 3, b = -8, and c = -1. Plugging in these values:

m ≈ (8 ± √(64 + 12)) / 6 ≈ 1.482

Since the median must be greater than or equal to 1, we take the positive root of the equation: m ≈ 1.482. Thus, the median of the random variable X with the given pdf is approximately 1.482.

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The percentile rank for a score of 80 is 70%.

The percentile rank for a score of 68 is 57%.

Carmela's percent score is 77.25%.

Carmela's percentile rank is 75%.

Carmela is eligible for a job with the government.

a) For a score of 80, there are 21 out of 30 scores that are equal to or less than 80

Therefore, the percentile rank for a score of 80 is (21/30) x 100% = 70%.

b) For a score of 68, there are 17 out of 30 scores that are equal to or less than 68.

Therefore, the percentile rank for a score of 68 is (17/30) x 100% = 57%.

2a) Carmela's percent score is (618/800) x 100% = 77.25%.

b) Carmela's percentile rank:

520 individuals scored less than Carmela's score of 618.

Therefore, her percentile rank is (520/700) x 100%

Carmela's percentile rank = 75%.

c) To be in the top 20% of individuals writing the exam, Carmela's score needs to be greater than or equal to the score of the 80th percentile.

The score of the 80th percentile is 0.8 * 700 = 560.

Therefore, the top 20% of individuals scored 560 or higher.

Carmela's score of 618 places her in the top 20% of individuals and makes her eligible for a job with the government.

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The equation of the tangent line to the curve is y = 8.013x - 1.185.

To find the equation of the tangent line to the curve at the point (0.6), we first need to find the slope of the tangent line, which is the derivative of the curve at that point.

Taking the derivative of y = x⁴ + 6eˣ, we get:

y' = 4x³ + 6eˣ

Now, we can find the slope of the tangent line at x = 0.6 by plugging in this value into the derivative:

y'(0.6) = 4(0.6)³ + 6e⁰.⁶ ≈ 8.013

So the slope of the tangent line at the point (0.6) is approximately 8.013.

Next, we need to find the y-coordinate of the point on the curve at x = 0.6. Plugging this value into the original equation, we get:

y = (0.6)⁴ + 6e⁰.⁶ ≈ 6.976

So the point on the curve that corresponds to x = 0.6 is approximately (0.6, 6.976).

Finally, we can use the point-slope form of the equation of a line to find the equation of the tangent line:

y - 6.976 = 8.013(x - 0.6)

Simplifying, we get:

y = 8.013x - 1.185

So the equation of the tangent line to the curve y = x⁴ + 6eˣ at the point (0.6) is y = 8.013x - 1.185.

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We can conclude that Jack's claim is correct. QRST is indeed a parallelogram, since opposite angles in QRST are congruent, and opposite sides in a parallelogram are also congruent

To determine whether Jack's claim that QRST is a parallelogram is correct, we need to use the properties of parallelograms. One of the properties of a parallelogram is that opposite angles are congruent. Therefore, we need to check if the opposite angles in QRST are congruent.

If m∠R = 72 and m∠T = 108, then the sum of these angles is 180 degrees (72 + 108 = 180). This indicates that angles R and T are supplementary.

If m∠S = 72, then we need to find the measure of angle Q. Since QRST is a quadrilateral, the sum of its interior angles is 360 degrees.

m∠Q + m∠R + m∠S + m∠T = 360

Substituting the given values, we get:

m∠Q + 72 + 72 + 108 = 360

Simplifying the equation, we get:

m∠Q = 108

Therefore, angles Q and S are congruent (both measuring 72 degrees) and angles R and T are supplementary (measuring 72 and 108 degrees, respectively). Since opposite angles in QRST are congruent, and opposite sides in a parallelogram are also congruent, we can conclude that Jack's claim is correct. QRST is indeed a parallelogram.

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The inequality to solve is 56 - x > 0. The solution is x < 56. Therefore, the domain of the function g(x) is x < 56. So, the answer is option B.

The function is defined as g(x) = log(56 - x).

The domain of a logarithmic function is all the values that make the argument of the logarithm positive. In other words, the argument of the logarithm (56 - x) must be greater than 0.

So, we solve the inequality 56 - x > 0 for x

56 - x > 0

Subtract 56 from both sides

-x > -56

Divide both sides by -1, and remember to reverse the inequality

x < 56

Therefore, the domain of the function g(x) is all real numbers x such that x < 56. So, the correct answer is B).

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--The given question is incomplete, the complete question is given

" The domain of g(x) = log 56 - x) can be found by solving the inequality

A. 56-x<0 ,B. 56-x>0,C. 56-x>=0, D. 56-x<=0 "--

The derivative of g(x) is (2x/3∛x) + (8x+4)/9.

To find g'(x), we first need to apply the power rule of differentiation to the first term in the expression for g(x), which is ∛x². Recall that the power rule states that if f(x) = xⁿ, then f'(x) = n*xⁿ⁻¹. In this case, n = 1/3, so we have:

d/dx [∛x²] = (1/3) * d/dx [x²] = (1/3) * 2x = 2x/3∛x

Next, we need to apply the chain rule of differentiation to the second term in the expression for g(x), which is (2x+1)²/9. Recall that the chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). In this case, we have:

h(x) = 2x+1

g(u) = u²/9

u = h(x) = 2x+1

So, applying the chain rule, we have:

d/dx [(2x+1)²/9] = 2/9 * (2x+1) * d/dx [2x+1] = 4/9 * (2x+1)

Putting these two results together, we have:

g'(x) = d/dx [∛x² + (2x+1)²/9] = 2x/3∛x + 4/9 * (2x+1)

Simplifying this expression, we get:

g'(x) = 2x/3∛x + 8x/9 + 4/9

g'(x) = (2x/3∛x) + (8x+4)/9

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Complete Question:

Consider g(x) = ∛x² + (2x + 1)² / 9

Calculate g'(x)

The solutions for f(x) = g(x) are x = 5 and x = 3.

To solve f(x) = g(x) using substitution method, we need to substitute g(x) in place of x in the equation f(x) = x² - 7x + 13.

So, we have:

f(x) = g(x)
x² - 7x + 13 = x - 2   (Substituting g(x) = x - 2)

Now, we can solve for x by simplifying and solving the resulting quadratic equation:

x² - 8x + 15 = 0

Factoring the quadratic equation, we get:

(x - 5)(x - 3) = 0

So, x = 5 or x = 3.

Therefore, the solutions for f(x) = g(x) are x = 5 and x = 3.

To check, we can substitute each value back into the equations:

f(5) = 5² - 7(5) + 13 = 25 - 35 + 13 = 3
g(5) = 5 - 2 = 3

f(3) = 3² - 7(3) + 13 = 9 - 21 + 13 = 1
g(3) = 3 - 2 = 1

So, both solutions satisfy the original equation f(x) = g(x).

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Using Pythagorean theorem, none of the triangles given are right triangles.

To determine which of the triangles are right triangles, you can use the Pythagorean theorem (a² + b² = c²), where a and b are the shorter side lengths and c is the longest side (hypotenuse).

Triangle 1:
Side lengths: 12, 19, 505
Checking: 12² + 19² = 144 + 361 = 505 ≠ 505²
Triangle 1 is not a right triangle.

Triangle 2:
Side lengths: 16, 19, 1467
Checking: 16² + 19² = 256 + 361 = 617 ≠ 1467²
Triangle 2 is not a right triangle.

Triangle 3:
Side lengths: 14, 20, 596
Checking: 14² + 20² = 196 + 400 = 596 ≠ 596²
Triangle 3 is not a right triangle.

Triangle 4:
Side lengths: 11, 23, 1421
Checking: 11² + 23² = 121 + 529 = 650 ≠ 1421²
Triangle 4 is not a right triangle.

None of the triangles given are right triangles.

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Answer:

31

Step-by-step explanation:

1. Replace the variables with numbers

8(5)-(9)

2.Conduct order of operations

8(5)-9           -->     40-9    ---> 31

Answer:

31

Explanation:

Trust

Answer:

ChatGPT

To find the balances for each type of investment after one year, we can use the formula for compound interest:

Treasury bond: A = P(1 + r/n)^(nt)

A = 0.4(30000)(1 + 0.0535/1)^(1*1) = $12,912.00

CD: A = P(1 + r/n)^(nt)

A = 0.1(30000)(1 + 0.0475/1)^(1*1) = $10,316.25

Stock plan: After the first year, 30% is still in the savings account. The other 70% is in the stock plan, which increased by 9%, so the new value is:

0.7(30000)(1 + 0.09) = $23,940.00

Savings account: A = P(1 + r/n)^(nt)

A = 0.2(30000)(1 + 0.039/1)^(1*1) = $6,351.00

To find the total gain from all of the investments combined, we need to add up the gains from each investment:

Treasury bond: $12,912.00 - $12,000.00 = $912.00 gain

CD: $10,316.25 - $9,000.00 = $1,316.25 gain

Stock plan: After the second year, the stock plan decreased in value by 5%, so the new value is:

0.7($23,940.00)(1 - 0.05) = $19,149.00

After the third year, the stock plan increased by 7%, so the final value is:

0.7($19,149.00)(1 + 0.07) = $20,129.57

The gain from the stock plan is:

$20,129.57 - $21,000.00 = -$870.43 loss (since the stock plan decreased in value overall)

Savings account: $6,351.00 - $6,000.00 = $351.00 gain

Total gain = $912.00 + $1,316.25 - $870.43 + $351.00 = $708.82

If you had invested 40% in stock and 30% in Treasury bonds, the calculations would be:

Treasury bond: A = P(1 + r/n)^(nt)

A = 0.3(30000)(1 + 0.0535/1)^(1*3) = $12,853.81

Stock plan: After the first year, 40% is still in the savings account. The other 60% is in the stock plan, which increased by 9%, so the new value is:

0.6(30000)(1 + 0.09) = $16,200.00

After the second year, the stock plan decreased in value by 5%, so the new value is:

0.6($16,200.00)(1 - 0.05) = $15,390.00

After the third year, the stock plan increased by 7%, so the final value is:

0.6($15,390.00)(1 + 0.07) = $16,019.16

Total gain = ($12,853.81 - $12,000.00) + (-$981.84) + ($1,019.16) = $890.13

Therefore, investing 40% in stock and 30% in Treasury bonds