An ABA standard basketball can have a diameter of up to 25cm how much space is there for air inside a standard basketball

Perimeter of the polygon is 92 cm

Solution:

The reference image for the solution is attached below.

AX = 10.5 cm, BY = 11.5 cm and CZ = 12.5 cm

AW and AX are tangents to the circle from external point A.

BX and BY are tangents to the circle from external point B.

CY and CZ are tangents to the circle from external point C.

DZ and DW are tangents to the circle from external point D.

Two tangents drawn from an external point to a circle are equal in length.

AW = AX, BX = BY, CY = CZ and DZ = DW

⇒ AW = AX

AW = 10.5 cm

⇒ BX = BY

BX = 11.5 cm

⇒ CY = CZ

CY = 12.5 cm

Given ∠B ≅ ∠D

If two angles are congruent, then the corresponding sides are congruent.

BX = BY = DZ = DW

DZ = DW = 11.5 CM

Perimeter = AW + AX + BX + BY + CY + CZ + DZ + DW

                = 10.5 + 10.5 + 11.5 + 11.5 + 12.5 + 12.5 + 11.5 + 11.5

                = 92

Perimeter of the polygon is 92 cm.

The answer is 'A' twinks!!

The equation that represents her total cost for the day is

A) 45x + 100 = 370

An equation is written in the form of variables and constants separated by the operation of multiplication and division,

An equation states that terms in different forms on both sides of the equality sign are equal.

Multiplication and division do not separate the terms of an equation.

Given, Kayla rented a boat.

There was a one-time charge of $100 plus an hourly rate of $45.

Assuming no. of hours to be x and total cost to be c(x).

∴ c(x) = 45x + 100.

Her total cost for the day was $370.

370 = 45x + 100.

learn more about equations here :

https://brainly.com/question/10413253

#SPJ5

Answer:

3 books per box, 3 left over.

Step-by-step explanation:

Brian has 18 books, and he is packing them into 5 boxes. 18 is only divisible by 1, 3, 9, and 18.

1 x 5 = 5 books =  13 left over. Still enough to pack more books.

3 x 5 = 15 books = 3 left over. This is the answer, but let's check the others as well.

9 x 5 = 45 books. He doesn't have that many! He can't put 9 books in each box because it would only take 2 boxes he's using all 5.

18 x 5 = 90 books. We can clearly see this isn't the answer. He would only be able to put all 18 books in 1 box but he wants to use all 5 boxes.

There are 3 books in each box with 3 left over.

Brian has 3 books in each of the 4 boxes, with 3 books remaining as leftovers after packing them equally.

Brian is moving his 18 books and has packed an equal number of books into each of the 5 empty boxes. To find out how many books are in each box, we divide the total number of books by the number of boxes, which is = 18 books ÷ 5 boxes = 3 books per box with 3 books left over. Therefore, there will be 3 full boxes and one box with 3 leftover books.

https://brainly.com/question/30233369

#SPJ2

Answer:

The correct answer to the question is;

a. number of clusters.

Step-by-step explanation:

Clustering is the process of looking for smaller similar groups of observation within a set of data.

K-means clustering is a vector quantization method used in data mining cluster analysis. The objective of k-means clustering is to a given number of observations into k number of clusters whereby an observation is grouped in a cluster having the closest mean value, hence being representative of tha particular cluster. This is in atempt to make observations in a particular group to be similar.

In k-means clustering, the number of clusters is specified as k.

In k-means clustering, k represents the number of clusters, indicated by answer choice (a). This technique involves partitioning the data into k compact and separate clusters, with the k initial cluster centers often chosen randomly.

In k-means clustering, k represents the number of clusters into which the data is to be partitioned. This method involves assigning each data point to the nearest cluster, while keeping the clusters as small as possible. The initial positions of the k clusters are typically chosen at random, and then the mean position of all the points in each cluster is recomputed, and this becomes the new center for the cluster. This process is repeated until the cluster assignments no longer change significantly, meaning the clusters are as compact and as separate as possible. The mean refers to the mean of the data points within each cluster once the clusters have formed. The standard deviation is a measure of the variability of the original distribution of the data. Sample size, denoted as n, is the number of observations in the dataset. Therefore, the answer to the question about what k represents in k-means clustering is (a) the number of clusters.

Answer:

Part 1) The volume of the object is [tex]32\pi\ cm^{3}[/tex]  or [tex]100.48\ cm^{3}[/tex]

Part 2) see the procedure

Step-by-step explanation:

The picture of the question in the attached figure

Part 1)  What is the volume, in cubic centimeters, of the object?

we know that

The volume of the object is equal to the volume of the cylinder plus the volume of the cone

Find the volume of the cone

The volume of the cone is equal to

[tex]V=\frac{1}{3}\pi r^{2} h[/tex]

we have

[tex]r=4/2=2\ cm[/tex] -----> the radius is half the diameter

[tex]h=3\ cm[/tex]

substitute the values

[tex]V=\frac{1}{3}\pi (2^{2})(3)=4\pi\ cm^{3}[/tex]

Find the volume of the cylinder

The volume of the cylinder is equal to

[tex]V=\pi r^{2} h[/tex]

we have

[tex]r=4/2=2\ cm[/tex] -----> the radius is half the diameter

[tex]h=(10-3)=7\ cm[/tex]

substitute the values

[tex]V=\pi (2^{2})(7)=28\pi\ cm^{3}[/tex]

Part 2) Then explain how you found the volume of the total shape

The volume of the total shape is equal to the volume of the cylinder plus the volume of the cone

[tex]4\pi\ cm^{3}+28\pi\ cm^{3}=32\pi\ cm^{3}[/tex] ------> exact value

Find the approximate value of the volume

assume

[tex]\pi=3.14[/tex]

[tex]32(3.14)=100.48\ cm^{3}[/tex]

Answer:

Step-by-step explanation:

Answer:

3:7 or .4285

Step-by-step explanation:

the odds against are 4:7 so the odds with must be 7-4=3

which will be written in 3:7 form

Answer: y = 6

y = -3x - 12

y = 3x + 24

Use elimination method to get 2y = 12 so y = 6.

Answer: the last option is the correct answer.

Step-by-step explanation:

The given system of equations is expressed as

y = - 3x - 12- - - - - - - - - - - - -1

y = 3x + 24- - - - - - - -- - - - - - -2

We would apply the method of elimination. Since x has the same coefficient and the sign is opposite, we would eliminate x by adding equation 1 to equation 2. It becomes

y + y = - 3x + 3x - 12 + 24

2y = 12

Dividing the left hand side and the right hand side of the equation by 2, it becomes

2y/2 = 12/2

y = 6

Answer:

The answer to your question is below

Step-by-step explanation:

Process

1.- Convert the lengths to the same denominator (12)

Ridge trail        3 /4 mile =     3(3) / 12 =  9/12

Valley trail        7/12 mile =                   = 7/12

Crystal brook    2/3 mile =    4(2) / 12  = 8/12

2.- Order from shortest to longest

a) Valley trail

b) Crystal brook

c) Ridge trail

Answer:

Ridge trail, Valley trail, Crystal Brook trail

Step-by-step explanation:

3/4 --> 9/12

2/3 --> 8/12

7/12

The subject of this question is Mathematics. The student needs to determine the number of birds and beasts the victualer needs to get for the upcoming wedding feast.

The subject of this question is Mathematics.

The student is being asked to determine how much of each bird and beast the victualer needs to get for the upcoming wedding feast. The king specifies that there should be a total of 180 heads of bird and beast, and the number of feet should be 500. The student needs to calculate the number of each bird and beast that satisfies these conditions.

To solve this problem, we can set up a system of equations. Let x be the number of birds and y be the number of beasts. We have the following two equations:

x + y = 180 (equation 1)

2x + 4y = 500 (equation 2)

We have equation 1 to represent the total number of heads, and equation 2 to represent the total number of feet. We can solve this system of equations to find the values of x and y that satisfy both equations.

In equation 1, we can solve for x in terms of y: x = 180 - y. We can substitute this value of x in equation 2 to get:

2(180 - y) + 4y = 500

Simplifying this equation gives: 360 - 2y + 4y = 500

Combining like terms: 2y = 140

Dividing both sides by 2: y = 70

Substituting this value of y in equation 1 gives: x = 180 - 70 = 110

So, the victualer needs to get 110 birds and 70 beasts for the upcoming wedding feast.

Learn more about Mathematics here:

https://brainly.com/question/41753146

#SPJ11

After 6 years the number of subscribers are 16500.

percentage, a relative value indicating hundredth parts of any quantity.

Given that Subscriptions to a fashion magazine have gone down by a consistent 5% each year.

The magazine currently has 25,000 subscribers.

We need to find the number of subscribers after 6 years.

[tex]A=Pe^{rt}[/tex]

where r is rate of interest

t is the time

p is the initial subscribers

[tex]A=25000e^{0.05(6)}[/tex]

[tex]A=25000e^{0.3}[/tex]

A=25000(1.34)

A=33500

As it is going down so we need to subtract 33500-25000

8500

So from initial subscribers 8500 are left.

250000-8500

16500

Hence, after 6 years the number of subscribers are 16500.

To learn more on Percentage click:

https://brainly.com/question/28269290

#SPJ5

Answer:

Option B) Statistic    

Step-by-step explanation:

Parameter and Statistic

For the given research:

Population:

All young adults who watch the television show Parks and Recreation

Sample:

All young adults at the mall who watch the television show Parks and Recreation

Thus, the he mean age of these young adults at the mall who watch Parks and Recreation is a statistic as it describes a sample of the whole population.

Thus, the correct answer is :

Option B) Statistic

Final answer:

To determine the number of ways a smaller focus group can be chosen, we use combinations to calculate the possible selections for each political affiliation separately and then multiply these numbers. For part (a), there are 1470 ways to select 1 Republican and 4 Democrats. For part (b), there are 945 ways to choose 2 Republicans, 2 Democrats, and 1 Independent.

Explanation:

To find out how many ways a smaller focus group can be chosen from a larger group, we use combinations, which is a part of probability and combinatorics in mathematics.

Part (a): 1 Republican and 4 Democrats

To select 1 Republican out of 7, we calculate this as 7 choose 1, denoted as 7C1.

To select 4 Democrats out of 10, we calculate 10 choose 4, denoted as 10C4.

The total number of ways to choose the group is the product of these two combinations:

7C1 × 10C4 = 7 × (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 7 × 210 = 1470 ways

Part (b): 2 Republicans, 2 Democrats, and 1 Independent

Choosing 2 Republicans from 7, we have 7C2.

For the Democrats, 10C2. Since there are only 2 Independents, they are both selected, so no need to choose:

7C2 × 10C2 × 2C2 = (7 × 6) / (2 × 1) × (10 × 9) / (2 × 1) × 1 = 21 × 45 × 1 = 945 ways

Answer:

The sugar does the average person in North Carolina eat in a year =  175.71 pounds.

Step-by-step explanation:

The person eat sugar in 70 year life span = 12300 pounds

The same person eat sugar in one year = [tex]\frac{12300}{70}[/tex]

                                                                  = 175.71 pounds

Thus the person eat about 175.71 pounds of sugar in a year.

Step-by-step explanation:

[tex] \frac{y - y1}{x - x1} = \frac{y2 - y1}{x2 - x1} [/tex]

[tex] \frac{y - 0}{x - ( - 1)} = \frac{3 - 0}{1 - ( - 1)} [/tex]

[tex] \frac{y}{x + 1} = \frac{3}{1 + 1} [/tex]

[tex] \frac{y}{x + 1} = \frac{3}{2} [/tex]

[tex]y = \frac{3}{2} (x + 1)[/tex]

[tex]y = \frac{3}{2} x + \frac{3}{2} [/tex]

option B

The concluding part of the question as obtained from the textbook;

The number on the 20 beach balls that come up in the simulated sample:

42, 1701, 638, 397, 113, 1243, 912, 380, 769, 1312, 76, 547, 721, 56, 4, 1411, 1766, 677, 201, 1840

A) Based on this sample, how many

defective beach balls might the

manufacturer expect in the next

shipment?

B) What is the difference between the

number of defective beach balls in the

actual shipment and the number

predicted in the next shipment?

Answer:

A) 500 defective beach balls.

B) Difference between the

number of defective beach balls in the

actual shipment and the number

predicted in the next shipment = 350

Step-by-step explanation:

The beach balls are labelled 1 to 2000 with the 150 defective ones labelled 1 to 150.

Then a random sample of 20 beach balls is picked, and the numbers are presented as

42, 1701, 638, 397, 113, 1243, 912, 380, 769, 1312, 76, 547, 721, 56, 4, 1411, 1766, 677, 201, 1840

Note that only the defective beach balls have numbers 1 to 150.

A) The number of beach balls with numbers from 1 to 150 in the sample is 5 (numbers 42, 113, 76, 56, 4). This is the number of defective beach balls in the sample.

Probability of getting a defective ball in the next shipment = (5/20) = 0.25

If every shipment contains 2000 beach balls, then there will be (0.25 × 2000) defective beach balls in the next sample; 500 defective beach balls.

B) Number of defective beach balls in actual shipmemt = 150

Number of predicted defective beach balls in the next shipment = 500

difference = 500 - 150 = 350.

Hope this Helps!!!

Answer:

C.) 500

Step-by-step explanation:

In a shipment of 2,000 beach balls, 150 are defective. The manufacturer generates a random sample to simulate 20 beach balls to inspect in the next shipment. The integers 1 to 150 represent defective beach balls.

   42         1701         638         397          113  

1243          912         380          769       1312

   76           547         721            56              4

1411         1766        677          201       1840

12. Based on this sample, how many defective beach balls might the manufacturer expect in the next shipment?

Group of answer choices

A.) 100

B.) 200

C.) 500

D.) 1000

Answer:

i = 351, j = 0

Step-by-step explanation:

The while loop runs 5 times as indicated by the variable, count.

The variable, i, has an initial value of 10. The first line of the loop code doubles the value of i while the second increments it by 1. This is done 5 times. We have

Iteration 1: i = 10 + 10 + 1 = 21

Iteration 1: i = 21 + 21 + 1 = 43

Iteration 1: i = 43 + 43 + 1 = 87

Iteration 1: i = 87 + 87 + 1 = 175

Iteration 1: i = 175 + 175 + 1 = 351

The third line of the loop code decrements j by 1. However, the fourth line sets j = 0 by subtracting it from itself. Hence, j is always 0 at the end of the loop, no matter its initial value or the number of iterations.

Thus, at the end of the code snippet, i = 351 and j = 0.

Answer:

C) 9

Step-by-step explanation:

The response is very low

Answer: 950, option D

Step-by-step explanation:

Adequate response rate from participants prevents non response bias.

Non-response bias can be said to occur when response rates from participants is below 60%.

Meanwhile, all other options are less than or equals 20% respond rate, while 950 implies 100% response rate.

Answer:

By knowing the standard deviation, one gets the idea of how the value is scattered or dispersed about the mean.

Step-by-step explanation:

Let us first define standard deviation.

As it is known that the standard deviation is a measure of dispersion which express the spread of observation in terms of the average of deviations of observations from some central values.

Measure of dispersion gives us an idea about homogeneity or heterogeneity of the distribution.

Standard deviation is supposed almost an ideal measure of dispersion except the general nature of extracting the square root.  

Thus for the given question, if we want to compare the two different groups of students whose mean score is 85. Here the standard deviation for both the groups interprets an idea about how the individual score for each group scattered or varied about the mean score i.e. 85.

Answer:

Step-by-step explanation:

Let the speed of cyclist be C and bus be B.

Using the concept of relative speed:-

For incoming bus,speed = B + C

For overtaking bus,speed = B - C

Now,distance travelled b/w 2 incoming bus = (B + C)*4

Now,distance travelled b/w 2 overtaking bus = (B - C)*12

so,

(B + C)*4 = (B - C)*12

or, B=2C

Hence,distance b/w two buses in same direction,d=4*(3/2)B=6 B

Therefore, time gap=d/B=6 minutes

Time interval between consecutive buses is 6 minutes, determined using relative speeds and given overtaking and meeting intervals.

To solve this problem, let's denote the following:

- [tex]\( v_c \)[/tex] is the speed of the cyclist.

- [tex]\( v_b \)[/tex] is the speed of the buses.

- [tex]\( t \)[/tex] is the time interval between consecutive buses.

Step-by-Step Solution:

1. Relative Speed in the Same Direction:

  When a bus overtakes the cyclist, it means the bus is catching up to the cyclist from behind. The relative speed of the bus with respect to the cyclist is:

  [tex]\[ v_b - v_c \][/tex]

  According to the problem, a bus overtakes the cyclist every 12 minutes:

  [tex]\[ v_b - v_c = \frac{d}{12} \][/tex]

  where [tex]\( d \)[/tex] is the distance between consecutive buses.

2. Relative Speed in the Opposite Direction:

  When the cyclist meets an oncoming bus, they are moving towards each other. The relative speed is:

  [tex]\[ v_b + v_c \][/tex]

  According to the problem, the cyclist meets an oncoming bus every 4 minutes:

  [tex]\[ v_b + v_c = \frac{d}{4} \][/tex]

3. Setting Up Equations:

  From the given information, we have two equations:

  [tex]\[ v_b - v_c = \frac{d}{12} \][/tex]

  [tex]\[ v_b + v_c = \frac{d}{4} \][/tex]

4. Solving for [tex]\( d \)[/tex] and [tex]\( t \)[/tex]  :

  Let's add these two equations to eliminate [tex]\( v_c \)[/tex]:

  [tex]\[ (v_b - v_c) + (v_b + v_c) = \frac{d}{12} + \frac{d}{4} \][/tex]

  [tex]\[ 2v_b = \frac{d}{12} + \frac{d}{4} \][/tex]

  [tex]\[ 2v_b = \frac{d}{12} + \frac{3d}{12} \][/tex]

  [tex]\[ 2v_b = \frac{4d}{12} \][/tex]

  [tex]\[ 2v_b = \frac{d}{3} \][/tex]

  Multiply both sides by 3:

  [tex]\[ 6v_b = d \][/tex]

  Now, substitute [tex]\( d = 6v_b \)[/tex] back into either equation, let's use [tex]\( v_b + v_c = \frac{d}{4} \)[/tex]:

  [tex]\[ v_b + v_c = \frac{6v_b}{4} \][/tex]

  [tex]\[ v_b + v_c = 1.5v_b \][/tex]

  [tex]\[ v_c = 0.5v_b \][/tex]

5. Finding the Time Interval [tex]\( t \)[/tex]:

  We already established that the distance between consecutive buses is [tex]\( d = 6v_b \)[/tex].

  The time interval [tex]\( t \)[/tex] between consecutive buses can be found using the speed of the buses:

  [tex]\[ t = \frac{d}{v_b} \][/tex]

 [tex]\[ t = \frac{6v_b}{v_b} \][/tex]

  [tex]\[ t = 6 \text{ minutes} \][/tex]

So, the time interval between consecutive buses is 6 minutes.

We want

[tex]2x<6\cdot 2=12[/tex]

So, if we divide both sides by 2, we have

[tex]x<6[/tex]

So, we can accept all values for [tex]x[/tex], as long as they are less than 6 (excluded)

Answer:

a) 2x < 6(2)

x < 6

x can take any real value less than 6

b) divide both sides by 2;

Or multiply both sides by ½

Answer:

Again, A.

Step-by-step explanation:

Just graph the equation, but you can tell just by looking at it that (0,11) is the y-intercept meaning that it's one of the answer choices.

Step-by-step explanation:

a. The first four terms are:

-4 (⅓)¹⁻¹ = -4

-4 (⅓)²⁻¹ = -4/3

-4 (⅓)³⁻¹ = -4/9

-4 (⅓)⁴⁻¹ = -4/27

b. This is a geometric series.  Since the absolute value of the common ratio is less than 1 (|⅓| < 1), the series converges.

c. The series converges to the sum of:

S = a₁ / (1 − r)

S = -4 / (1 − ⅓)

S = -6

Answer:

Cross out Ab

Step-by-step explanation:

Answer: AD; 6

Step-by-step explanation:

using the shortest route algorithm. The next shortest route to AD would be AD; 6.

To approximate the proportion of adults consuming more than 9.92 gallons of wine annually, we can use the empirical rule.

To approximate the proportion of adults who consume more than 9.92 gallons, we can use the empirical rule. The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations. Since we want to find the proportion of adults consuming more than 9.92 gallons, which is more than 1 standard deviation above the mean, we can estimate that it would be approximately 32% based on the empirical rule.

https://brainly.com/question/35669892

#SPJ3

Answer:

undefined or 1

Step-by-step explanation:

(-3g^2)^0

Where g=0. (given)

now,

[tex](-3g^2)^0 \\ ( - 3(0) {}^{2} ) {}^{0} \\

[/tex]

undefined or 1

[tex]\text{Solve by plugging in 0 to g}\\\\(-3\left(0\right)^2)^0\\\\\text{Solve:0}\\\\(-3\left(0\right)^2)^0\\\\(-3(0))^0\\\\(0)^0=1\\[/tex]

Answer:

Neither the ranges nor the interquartile ranges for the data sets are the same.

Step-by-step explanation:

In a visual display, the boxplot presents five sample statistics: the minimum, the lower quartile, the median, the upper quartile and the maximum, and the box length gives an indication of the sample variability and the line across the box shows where the sample is centred, with an end at each quartile. The length of the box is thus the interquartile range of the sample and, whether the sample is symmetric or skewed, either to the right or left, the "shape" of the sample, and by implication, the shape of the population from which it was drawn, considering appropriate analyses of the data.

The question is incomplete! The complete question along with answer and explanation is provided below.  

Question:  

Which statement is true about the box plots? (attached in the image)

A Both the ranges and the interquartile ranges for the data sets are the same.

B. Neither the ranges nor the interquartile ranges for the data sets are the same.

C. The interquartile ranges for the box plots are the same, but their ranges are different.

D. The ranges for the box plots are the same, but their interquartile ranges are different.

Answer:

D. The range of both box-pots is same (9) but the interquartile range of box-plots is different (6 and 5)

Step-by-step explanation:

To answer this question, first we have to understand what a box plot is!

A box plot is a type of graph which shows 5 statistical characteristics of a data set.

1. Maximum and 2. Minimum values of data

3. Upper Interquartile and 4. Lower interquartile of data

5. Median of the data

Now lets analyze the attached box-plot so that we can conclude what is true about them and what is not!

We have two box-plots for two teacher's classes Marc and Sue and they show the number of incorrect questions in exam.

For Sue's class: (on the bottom)

As you can see the maximum and minimum values are

Maximum = 12 and Minimum = 3

So the Range becomes = 12 - 3 = 9

The upper quartile Q3 is 10 and lower quartile Q1 is 5

So the Interquartile Range becomes = 10 - 5 = 5

This Interquartile Range represents the 25 to 75 percentile of the data

There is little vertical line skewed to the right represents the Median = 9

So to summarize Sue's class

Range = 9

Interquartile Range = 5

Median = 9

For Marc's class: (on the top)

As you can see the maximum and minimum values are

Maximum = 10 and Minimum = 1

So the Range becomes = 10 - 1 = 9

The upper quartile Q3 is 9 and lower quartile Q1 is 3

So the Interquartile Range becomes = 9 - 3 = 6

This Interquartile Range represents the 25 to 75 percentile of the data

There is little vertical line skewed to the left represents the Median = 5

So to summarize Marc's class

Range = 9

Interquartile Range = 6

Median = 5

Conclusion:

The range of both box-pots is same (9) but the interquartile range of box-plots is different (6 and 5).

Therefore, we can confidently conclude that option D is the correct answer.

Answer:

Step-by-step explanation:

Because angle FGH and angle HGE are congruent, that angle bisector, namely GH, separates this largr triangle into one that shares sides in proportion to each other.  

That means that [tex]\frac{x}{13.5} =\frac{68}{54}[/tex]

Cross multiply to get

54x = 918 and

x = 17

Choice A.

Answer:

x = −2.4

Step-by-step explanation:

Solve for x by simplifying both sides of the equation, then isolating the variable.

Exact Form: x = −12/5

Decimal Form: x = −2.4

Answer:

8 unit^3

Step-by-step explanation:

Given:

- The equation of the plane is:

                                     3x + 6y + 2z = 12

Find:

Use a double integral to find the volume of the solid in the first octant which is enclosed by the plane and the coordinate planes

Solution:

- Express the equation of surface ( plane ) as a subject of any coordinate axis we will use z:

                                     2z = 12 - 3x - 6y

                                     z = 6 - 1.5x - 3y

- The double integral would be set- up as:

                                    [tex]\int\limits^d_c \int\limits^a_b ({6 - 1.5x - 3y}) \, dy.dx[/tex]

- Where, a , b ,c and d are limits of integration.

- To determine the limits we will project the surface to x-y plane or z = 0 plane, the equation we have is:

                                     0 = 6 - 1.5x - 3y

                                     y = 2 - 0.5x

- For limits a and b the integration is with respect to y, so we express the limits of y in terms of x. Where lower limit b = 0, and upper limit a = 2 - 0.5x

- Similarly, the limits c and d is with respect to x are constants we have:

                                       c = 0

                                       0 = 2 - 0.5*d

                                       d = 4

- Then solve the double integral:

                     [tex]\int\limits^4_0 ({6y - 1.5xy - 1.5y^2}) \,_0 ^2^-^0^.^5^x dx \\\\\int\limits^4_0 ({6(2-0.5x) - 3x + 0.75x^2 - 1.5(2-0.5x)^2}) dx \\\\({-6(2-0.5x)^2 - 1.5x^2 +0.25x^3 + (2-0.5x)^3}) | ^4_0\\\\= ( -6(0) - 1.5(16) + 0.25*(64) + (0) + 6(4) + 0 + 0 - (8) ) \\\\= 8 unit^3[/tex]            

Final answer:

To find the volume of the solid in the first octant enclosed by the given surface and the coordinate planes, a double integral is used where z is expressed in terms of x and y, and appropriate limits for x and y are determined.

Explanation:

The volume of the solid in the first octant enclosed by the surface 3x + 6y + 2z = 12 and the coordinate planes can be found using a double integral. First, we need to express z in terms of x and y: z = 6 - (3/2)x - 3y.

The limits for x and y are determined by setting z = 0; thus, x ranges from 0 to 4, and y ranges from 0 to (6-(3/2)x)/3. Hence, the double integral equation is set up as follows:

[tex]\(\int_{0}^{4}\int_{0}^{2-(3/2)x} (6 - \frac{3}{2}x - 3y) dy dx\)[/tex]

Evaluating this double integral will give the volume of the solid.