A biologist is studying a certain species of tree in a forest. The tree density for this species in this forest is about 50 trees per acre. The forest is 93 hectares in area; one hectare is equal to 2.47 acres. What is the approximate density of this species in trees per hectare?

The inverse of the function f(x) = x³ + 1 is f⁻¹(x) = ∛(x - 1).

Given the function in the question;

f(x)  = x³ + 1

To determine the inverse of the function  f(x) = x³ + 1,

we can follow these steps:

Step 1: Replace f(x) with y

y = x³ + 1

Step 2: Solve for x in terms of y

y = x³ + 1

y - 1 = x³

x = ∛(y - 1)

Step 3: Replace x with f⁻¹(x) to get the inverse function

f⁻¹(x) = ∛(x - 1)

Therefore, the inverse of is f⁻¹(x) = ∛(x - 1).

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Answer:

(-4, 4)

Step-by-step explanation:

5x + y = -16

2x + y = -4

Eliminate the y variable.

5x + y = -16

-1 (2x + y = -4)

Solve:

5x + y = -16

-2x - y = 4

3x = -12

Divide both sides by 3.

x = -4

5(-4) + y = -16

-20 + y = -16

Add 20 to both sides.

y = 4

The ordinary number form of the mentioned scientific notation form of the number is 0.00604.

Scientific notation of representation of a number refers to converting a number to its readable form. It is applicable on both small and large numbers, where value of zeroes are represented in exponential form for easy interpretation.

The exponent of -3 is interpreted as three zeroes in the denominator. The division with zero will further shorten the number by adding zeroes to left hand side of the digit after decimal. Hence, the ordinary form of the number will be 0.00604.

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The valid assumptions for constructing a confidence interval for the difference between two population means are: Data is Quantitative, Random Sample and Both sample sizes are greater than 30 or the data is from a Normal Distribution.

To answer your question, when constructing a confidence interval for the difference between two population means, certain assumptions must be met. These assumptions include:

1. Data is Quantitative: Since we are dealing with population means, the data should be quantitative, meaning it consists of numerical values. This is a correct assumption.
2. Data is from a Convenience Sample: This is not a valid assumption. To ensure the reliability of the confidence interval, data should be collected through random sampling, which ensures that each individual in the population has an equal chance of being included in the sample.
3. Random Sample: This is a correct assumption. A random sample is necessary to ensure the sample's representativeness and accuracy in estimating the population means.
4. Both sample sizes are greater than 30 or the data is from a Normal Distribution: This is a valid assumption. If both sample sizes are greater than 30, the Central Limit Theorem can be applied, which states that the sampling distribution of the sample means will be approximately normal. If the data is already from a normal distribution, the normality assumption is met.
5. Data is Categorical: This assumption is incorrect. As previously mentioned, the data should be quantitative for this analysis.
6. There are at least 15 successes and 15 failures: This assumption is not relevant for confidence intervals for the difference between two population means. This criterion is related to proportions rather than means.

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If the null hypothesis is not rejected, we cannot conclude that there is a significant difference between the two time periods.

Assuming that the question is "Has real gross domestic product per capita significantly increased between 1988 and 1990?" We can use a paired-sample t-test to answer this question.

Here are the steps to conduct the paired-sample t-test:

Null hypothesis: The mean difference between the real gross domestic product per capita in 1988 and 1990 is zero. In other words, there is no significant difference between the two time periods.

Alternative hypothesis: The mean difference between the real gross domestic product per capita in 1988 and 1990 is not zero. In other words, there is a significant difference between the two time periods.

Determine the level of significance, alpha (α). Let's assume α = 0.05.

Calculate the difference between the two time periods (1990 - 1988) for each observation.

Calculate the mean and standard deviation of the differences.

Calculate the standard error of the mean difference (SEd) by dividing the standard deviation of the differences by the square root of the sample size.

Calculate the t-statistic by dividing the mean difference by the SEd.

Calculate the degrees of freedom (df) using the formula: df = n - 1, where n is the sample size.

Determine the critical t-value from the t-distribution table with df = n - 1 and α = 0.05.

Compare the calculated t-value with the critical t-value. If the calculated t-value is greater than the critical t-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Interpret the results. If we reject the null hypothesis, we can conclude that there is a significant difference between the two time periods, and the direction of the difference can be determined by the sign of the mean difference.

Note: In order to conduct the paired-sample t-test, we need to assume that the differences are normally distributed and the variances of the two populations are equal.

After conducting the test, we can conclude whether there is a significant difference in real gross domestic product per capita between 1988 and 1990. If the null hypothesis is rejected, we can conclude that there is a significant difference between the two time periods. If the null hypothesis is not rejected, we cannot conclude that there is a significant difference between the two time periods.

Complete question: Two of the variables in this file ( rgdpch and rgdp88) measure real Gross Domestic Product (GDP) per capita in 1990 and 1988. Please conduct the appropriate t-test to answer the following question:

Is there statistically significant evidence that per capita GDP in 1988 is different from that of 1990? Please show your results and reasoning to justify your answer.

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Answer:

243π

Step-by-step explanation:

Area of the entire circle = πr² = π18² = 324π

Area of sector = (90/360)π(18)² = 81π

Shaded area = 324π - 81π = 243π

try (-18,-5)

my step by step explanation is you would distrubute the -5 and add it to the -13 and you would do the same thing for the orher side

hopefully this is correct

and may i have brainliest

To determine the volume of the box, we need to find the length, width, and height of the box. We can do this by subtracting twice the length of the square cutouts from the original length, twice the width of the square cutouts from the original width, and the height will be the length of the square cutouts.

Let's assume that the side length of the square cutouts is x. Then the length of the box is (24 - 2x), the width of the box is (20 - 2x), and the height of the box is x. Thus, the volume of the box can be expressed as:

V(x) = (24 - 2x) * (20 - 2x) * x

To simplify this expression, we can expand it using the distributive property and then combine like terms:

V(x) = 4x^3 - 88x^2 + 480x

The approximate domain for V(x) would be [0, 10] since the side length of the square cutouts cannot be greater than half the length or width of the original piece of metal, which is 12 and 10 respectively.

To calculate V(x), we simply substitute x into the formula for V(x):

V(x) = 4x^3 - 88x^2 + 480x

Let's say we want to calculate V(2):

V(2) = 4(2)^3 - 88(2)^2 + 480(2)

= 16 - 352 + 960

= 624

Therefore, the volume of the box when the side length of the square cutouts is 2 inches is 624 cubic inches.

The coordinates of point Q that partitions line segment AB into the ratio 1:3 is (-0.75, 4.5).

In this scenario, line ratio would be used to determine the coordinates of the point Q on the directed line segment AB that partitions the segment into a ratio of 1 to 3.

In Mathematics, line ratio can be used to determine the coordinates of Q and this is modeled by the following mathematical expression:

Q(x, y) = [(mx₂ + nx₁)/(m + n)],  [(my₂ + ny₁)/(m + n)]

By substituting the given points into the formula for line ratio, we have;

Q(x, y) = [(1(6) + 3(-3))/(1 + 3)],  [(1(0) + 3(6))/(1 + 3)]

Q(x, y) = [(6 - 9)/(4)],  [(0 + 18)/4]

Q(x, y) = [-3/4],  [18/(4)]

Q(x, y) = [-0.75, 4.5]

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Based on the information given, it appears that the consumer rejected the null hypothesis that the mean price for a gallon of unleaded gasoline in the United States in a certain year is $3.110. This means that the consumer believed the mean price to be different from $3.110. However, in reality, the mean price for a gallon of unleaded gasoline in the United States in that year was $3.210.

Yes, an error was made in this situation. The consumer's hypothesis test was aimed at determining whether the mean price for a gallon of unleaded gasoline in the United States in a certain year is different from $3.110. Since the consumer rejected the null hypothesis, they concluded that the mean price is indeed different from $3.110.

Therefore, the consumer made a type I error. A type I error is made when a null hypothesis is rejected when it is actually true. In this case, the consumer rejected the null hypothesis that the mean price for a gallon of unleaded gasoline in the United States in a certain year is $3.110, when in fact, the true mean price was $3.210.

In reality, the mean price for a gallon of unleaded gasoline in the United States in that year is $3.210. Since the true mean is different from the hypothesized mean of $3.110, the consumer's decision to reject the null hypothesis was correct.

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1. What are the units of measurement we have learned so far for length?
2. Can you give an example of something that could be measured in inches?
3. Can you give an example of something that could be measured in feet?
4. Can you give an example of something that could be measured in yards?
5. How do we decide which unit of measurement to use for different objects?
6. Can you think of a real-life situation where measuring length accurately is important?

A second-grade class learning about appropriate units to measure length could benefit from the following set of questions in a group discussion:

1. Which unit of length is the smallest: inches, feet, or yards? Why?
2. If you wanted to measure the length of a pencil, which unit would you choose to use and why?
3. How many inches are in a foot? How many feet are in a yard?
4. What are some objects that would be best measured in inches? Feet? Yards?
5. Can you think of a time when you might need to convert between inches, feet, and yards? How would you do that?
6. Can you think of a real-life situation where measuring length accurately is important?

These questions encourage students to think critically about the different units of length and how they are applied in various situations. Additionally, the questions promote discussion on conversions between the units, helping them to better understand the relationship between inches, feet, and yards.

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The rate of the river current is 5 mph.
Let x represent the rate of the river current. When Kellen travels upstream, she goes against the current, so her effective speed will be (15 - x) mph. When she travels downstream, she goes with the current, so her effective speed will be (15 + x) mph.

We're given that the time it takes to travel 2 miles upstream is the same as the time it takes to travel 4 miles downstream. We can express time as distance divided by speed.

So, we have the equation:

(2 mi) / (15 - x) = (4 mi) / (15 + x)

Now, we need to solve for x:

Cross-multiply:

2(15 + x) = 4(15 - x)

Distribute:

30 + 2x = 60 - 4x

Add 4x to both sides:

6x = 30

Divide by 6:

x = 5

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Answer:

The answer is approximately 50ft

Step-by-step explanation:

arrc length=0/360/2pir

L=150/360×2×22/7×19

L=1254001/2520

L≈50ft

The "Player 1 is more likely to hit the ball than Player 2 because P(Player 1) > P(Player 2)" is true.

We need to convert them to a common denominator. The least common multiple of 11, 9, and 7 is 693.

Player 1: 7/11 = 504/693

Player 2: 6/9 = 462/693

Player 3: 5/7 = 495/693

Comparing the probabilities, we can see that:

Player 1 has a probability of 504/693 of hitting the ball.

Player 2 has a probability of 462/693 of hitting the ball.

Player 3 has a probability of 495/693 of hitting the ball.

Since the denominator is the same for all three players, we can directly compare the numerators.

Comparing the numerators, we can see that:

504 > 462 > 495

Therefore, Player 1 is more likely to hit the ball than Player 2 and Player 3.

Therefore, "Player 1 is more likely to hit the ball than Player 2 because P(Player 1) > P(Player 2)" is true.

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The coordinate system with x-coordinates between -180 and 180 is likely to be the geographic coordinate system, specifically longitude values. This system represents positions on the Earth's surface using two angles: longitude (x-coordinates) and latitude (y-coordinates).

Longitude values range from -180 degrees (180 degrees West) to 180 degrees (180 degrees East) along the Earth's equator. Latitude values, on the other hand, range from -90 degrees (90 degrees South) to 90 degrees (90 degrees North). The origin (0, 0) of this coordinate system is the intersection of the Prime Meridian (Greenwich Meridian) and the Equator.

The units for these coordinates are typically expressed in decimal degrees (DD). These values can be converted into other units such as degrees-minutes-seconds (DMS) or radians, depending on the application or preference. When working with geographic coordinate systems, it is essential to consider the Earth's curvature and potential distortions when calculating distances or areas.

In summary, if a data set has x-coordinates between -180 and 180, the coordinate system is most likely the geographic coordinate system, representing longitude values. The coordinates are typically expressed in decimal degrees, and their positions relate to the Earth's surface.

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The next step in solving the equation, if the first step was to subtract 5, is to A. add x.

To solve the equation -x + 6 = 5 - 3x, given that its first step is "subtract 5", one should move on to the next step which requires adding x to both sides of the equation.

Upon completing this newly introduced action, the resulting equation can be viewed as follows:

-x + 6 - 5 = 5 - 3x - 5

-x + 1 = -3x

-x + x + 1 = -3x + x

1 = -2x

By isolating x term on the right side of the equation through addition, resolving for x will become much more manageable and feasible in the succeeding steps.

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In order to determine if the golf instructional expert's claim is accurate, the LPGA would need to analyze the data collected from the random sample of 45 students.

Based on the given information, the new golf instructional expert claims that her students can hit their drivers an average of 300 yards. However, the LPGA is skeptical about this claim and believes that the students actually hit the ball less than that. To test this hypothesis, the LPGA took a random sample of 45 of her students.

By taking a random sample, the LPGA can estimate the average driving distance of all of the expert's students. If the sample's average is significantly less than 300 yards, then the LPGA's hypothesis would be supported. On the other hand, if the sample's average is close to 300 yards, then the new golf instructional expert's claim would be supported.

It is important to note that the average driving distance of a random sample may not necessarily represent the average driving distance of all of the expert's students. However, taking a random sample is a reliable way to estimate the population's average with a certain level of confidence.

To address your question, let's break it down into steps:

1. A golf instructional expert claims that her students can hit their drivers an average of 300 yards.
2. The LPGA believes this claim is inaccurate and thinks the students' actual average distance is less than 300 yards.
3. To test the claim, the LPGA takes a random sample of 45 students from the golf instructional expert's students.

In order to determine if the golf instructional expert's claim is accurate, the LPGA would need to analyze the data collected from the random sample of 45 students. They would calculate the average distance these students can hit their drivers and compare it to the claimed 300-yard average. If the actual average distance is significantly less than 300 yards, the LPGA would have evidence to dispute the golf instructional expert's claim.

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Complete Question:

A new golf instructional expert is on the scene. She claims that her students can hit their drivers an average of 300 yards. The LPGA does not think that this is accurate and that the students actually hit the ball less than that. The LPGA took a random sample of 45 of her students. Assume an error level of 5%. What is the null hypothesis?

[tex]\blue{\boxed{\bold{\mathbb{SOLUTION}}}}[/tex]

Area of the biggest circle:

[tex]\pi r^{2}=\pi \times 6^{2}=\pi \times 36 \ cm^2[/tex].

Area of the rectangle:

[tex]4 \times (2+2+2+2) = 4 \times 8 = 32 \ cm^2[/tex].

Area of the 6 semicircles:

[tex]6 \times \dfrac{1}{2} \times \pi \times r^2 = 3 \times \pi \times 2^2 = \pi \times 12 \ cm^2.[/tex]

Total shaded area:

Biggest circle area - rectangle area - 6 semicircles area

[tex]= \pi \times 36 - 32 - \pi \times 12\\= \pi \times (36-12) - 32\\= \pi \times 24 - 32\\= 3.14 \times 24 - 32\\= 75.36 - 32\\= \boxed{43.36 \ cm^2}[/tex]

Therefore, the shaded part area is [tex]43.36 \ cm^2[/tex].

[tex]\aqua{\boxed{\mathfrak{Thank \: You}}}\\\aqua{\boxed{\bold{answered \: by: \: akbarsdtazm}}}[/tex]

There is approximately a 3.84% chance that a randomly selected adult has an IQ greater than 144.4.

To find the probability that a randomly selected adult has an IQ greater than 144.4, assuming their scores are normally distributed with a mean of 102.7 and a standard deviation of 23.5, we will use the z-score formula.

First, calculate the z-score:
z = (X - μ) / σ
where X is the IQ score (144.4), μ is the mean (102.7), and σ is the standard deviation (23.5).

z = (144.4 - 102.7) / 23.5
z ≈ 1.77

Now, use a z-table to find the probability corresponding to this z-score. The z-table value for a z-score of 1.77 is approximately 0.9616. Since we want the probability of an IQ greater than 144.4, we will find the area to the right of this z-score.

Probability = 1 - 0.9616 = 0.0384

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Answer: 568mm ²

Step-by-step explanation:

area of trapezoid = [h(b₁+b₂)]/2

b₁ and b₂ are the lengths of the top and bottom sides, h is the height; 16

a) The probability that a person waits fewer than 12.5 minutes is 0.8333 or 83.33%.

b) On average, a person must wait 7.5 minutes, with a standard deviation of approximately 4.3301 minutes.

c) Ninety percent of the time, the minutes a person must wait falls below 13.5 minutes.

To find the probability that a person waits fewer than 12.5 minutes, you can use the formula for a uniform distribution:

P(X < x) = (x - a) / (b - a)

Where X is the random variable (waiting time), a is the minimum waiting time (0 minutes), b is the maximum waiting time (15 minutes), and x is the given time (12.5 minutes).

P(X < 12.5) = (12.5 - 0) / (15 - 0)

= 12.5 / 15

= 5/6 ≈ 0.833

So, the probability that a person waits fewer than 12.5 minutes is approximately 0.833 or 83.3%.

For a uniform distribution, the mean (average waiting time) can be calculated using the formula μ = (a + b) / 2, and the standard deviation using the formula σ = [tex]\sqrt{\frac{{(b - a)^2}}{{12}}}[/tex].

Mean (μ) = (0 + 15) / 2 = 7.5 minutes

Standard deviation (σ) = [tex]\sqrt{\left(\frac{{15 - 0}}{2}\right)^2}[/tex]

Standard deviation (σ) = [tex]\sqrt{\frac{{15^2}}{{12}}}[/tex] ≈ 4.33 minutes

On average, a person must wait 7.5 minutes for a bus, with a standard deviation of approximately 4.33 minutes.

To find the value below which 90% of the time falls, we multiply the desired percentile (90%) by the total range of possible values (15 - 0 = 15) and add it to the lower bound, giving us:

= 0.9 x 15 + 0

= 13.5 minutes

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The correct match of each dilation to its function is: j(x) = (1/2x)^2: vertical stretch, k(x) = 2x²: vertical compression, h(x) = (2x)²: horizontal compression, and g(x) = x²: no dilation, it is the parent function.

Each of the given functions is a dilation of the parent function f(x) = x^2, which means that they are obtained by stretching or compressing the graph of the parent function either vertically or horizontally.

- j(x) = (1/2x)^2: This function is a vertical stretch of the parent function, because the constant factor of 1/2 in front of the x causes the function to be stretched vertically by a factor of 2.

- k(x) = 2x²: This function is a vertical compression of the parent function, because the constant factor of 2 in front of the x^2 causes the function to be compressed vertically by a factor of 1/2.

- h(x) = (2x)²: This function is a horizontal compression of the parent function, because the constant factor of 2 inside the x^2 causes the function to be compressed horizontally by a factor of 1/2.

- g(x) = x²: This is the parent function, and it is not dilated in any way. Its graph is a parabola that opens upwards and has a vertex at the origin.

Thus, this is the correct match for the given scenario.

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The solution is : 140 cm of the original piece is left after Joe cuts three pieces.

Given that,

Given the piece of string

3.5m

As we know that

3.5m = 350 cm

so the length of a piece of string = 350 cm

As joe cuts three pieces from it, and each 70 cm long.

so the total length of three pieces which were

cut by joe = 70cm + 70cm + 70cm = 210 cm

So, subtracting the total length of the three pieces that were cut from the total length of the string:

350 cm - 210 cm = 140 cm

Therefore, 140 cm of the original piece is left after Joe cuts three pieces from it.

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complete question:

A piece of string is 3-5 m long. Joe cuts three pieces from it,

each 70 cm long. How much of the original piece is left?​

-5, view below for what Maureen did wrong.

Given:

   -1.2+(-3.8)

A positive times a negative is a negative:

   -1.2-3.8

Subtract:

   -5

   Instead of distributing the positive sign into the -3.8 and subtracting, Maureen added 1.2 to -3.8, leading to -2.6 instead of -5.

In this situation, the best statistical method to use would be a hypothesis test, specifically a one-sample t-test. This test will help determine if the average weight loss after following the gym's 45-minute workout routine for a month is significantly greater than 5 pounds.

In this situation, the best statistical method to use would be a hypothesis test. The gym's belief that their workout can lead to weight loss greater than 5 pounds can be tested by setting up a null hypothesis (the workout does not lead to weight loss greater than 5 pounds) and an alternative hypothesis (the workout does lead to weight loss greater than 5 pounds). The gym can then collect data on weight loss from participants who complete the workout for a month and use statistical analysis to determine if there is enough evidence to reject the null hypothesis and support the alternative hypothesis.

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To determine which vitamin leads to weight gain, a hypothesis test can be conducted using SPSS with a level of significance of 5%. The group with the significantly higher mean weight gain would indicate which vitamin leads to weight gain.

To determine which vitamin leads to weight gain at a level of significance of 5%, a hypothesis test needs to be conducted. The null hypothesis would be that there is no significant difference in weight gain between the three groups, and the alternative hypothesis would be that there is a significant difference.

To conduct the test in SPSS, the first step would be to input the data for each group and calculate the mean weight gain for each group. Then, a one-way ANOVA test can be conducted to determine if there is a significant difference between the means. The level of significance is set at 5%.

If the p-value is less than 0.05, the null hypothesis can be rejected, indicating that there is a significant difference between the means. Further post-hoc tests can then be conducted to determine which specific groups differ significantly.

In conclusion, to determine which vitamin leads to weight gain, a hypothesis test can be conducted using SPSS with a level of significance of 5%. The group with the significantly higher mean weight gain would indicate which vitamin leads to weight gain.

To determine which vitamin leads to weight gain in the specific population, you can perform an analysis using SPSS at a 5% level of significance. First, input the weight gain data for the three vitamin groups into SPSS. Then, conduct an ANOVA (Analysis of Variance) test to compare the means of the three groups. If the p-value obtained from the test is less than the level of significance (0.05), it indicates a significant difference between the groups. Further post-hoc tests (such as Tukey's HSD) can then be conducted to identify which vitamin group leads to a significant weight gain compared to the others.

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Answer:

Step-by-step explanation:

First get y byitself to oneside

[tex]2x-y\leq 5[/tex]

Subtract -2x to bothside

[tex]-y\leq 5-2x[/tex]

Divide -1 to make y positive. Dividing or Multiplying changes direction.

[tex]y\geq 2x-5[/tex]

y >= any value above the positive of the line.

Answer:

504

Step-by-step explanation:

use pemdas

multiply 8 and 7 first then u have 56 so multiply it by 9

The expression 9x (8x7) can be rewritten as (9x8) x7 or 72x7 by using the associative law of multiplication. This law states that the way numbers are grouped doesn't affect the result of multiplication.

The question asks to rewrite 9x (8x7) using the associative law of multiplication. The associative law of multiplication, in basic terms, states that the way in which numbers are grouped in multiplication does not change the result.

To apply this law to the given expression, we just need to change the grouping of the numbers. Instead of grouping (8x7) together, we could group (9x8) together. So, the content loaded rewrite of 9x (8x7) using the associative law of multiplication would be: (9x8) x7 or we can write it as 72x7.

This does not change the result of the multiplication, thus demonstrating the associative property. This is a helpful principle to understand, as it is commonly applied in a variety of mathematical operations and can simplify complex equations.

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The answer is (3) 0.

For the first integral, we have:

I = ∭W x^2 + y^2 dV

Using cylindrical coordinates, we have:

x = r cos(theta)

y = r sin(theta)

z = z

where 0 <= r <= 5, 0 <= theta <= 2pi, and -5 <= z <= 1.

Also, dV = r dz dr d(theta)

Substituting these into the integral, we have:

I = ∭W r^2 dV

= ∫(0 to 2pi) ∫(0 to 5) ∫(-5 to 1) r^2 (r dz dr d(theta)) dz dr d(theta)

= ∫(0 to 2pi) ∫(0 to 5) [(r^4/4)(1 - (-5))] dr d(theta)

= ∫(0 to 2pi) ∫(0 to 5) (13r^4/4) dr d(theta)

= ∫(0 to 2pi) [(13/20)r^5] from 0 to 5 d(theta)

= ∫(0 to 2pi) (13/20)(5^5) d(theta)

= (13/20)(3125)(2pi)

= 112pi

Therefore, the answer is (1) 112π.

For the second integral, we have:

I = ∭W y dV

Using cylindrical coordinates, we have:

x = r cos(theta)

y = r sin(theta)

z = z

where 2 <= r <= 4, 0 <= theta <= 2pi, and 0 <= z <= x+6.

Also, dV = r dz dr d(theta)

Substituting these into the integral, we have:

I = ∭W y dV

= ∫(0 to 2pi) ∫(2 to 4) ∫(0 to r cos(theta)+6) r sin(theta) (r dz dr d(theta)) dz dr d(theta)

= ∫(0 to 2pi) ∫(2 to 4) [(r^3 sin(theta)/3)(r cos(theta)+6)] dr d(theta)

= ∫(0 to 2pi) ∫(2 to 4) [(r^4/3) cos(theta) + 2r^3/3] sin(theta) dr d(theta)

= ∫(0 to 2pi) [(4^4/3 - 2^4/3)(cos(theta)/4) + 2(4^3/3 - 2^3/3)/3] sin(theta) d(theta)

= ∫(0 to 2pi) [(16/3)(cos(theta)/4) + (32/3)] sin(theta) d(theta)

= 0

Therefore, the answer is (3) 0.

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