The sum of the differences must be zero for any distribution consisting of n observations.

A. True

B. False

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.

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

Answer:

Diameter

Step-by-step explanation:

Knowing that a diameter is the largest chord as it passes through the center point of a circle, if the definition of the chord is a line segment with it's two endpoints on the circle (here by circle we mean circumference) then a diameter is a line segment that passes through the center and has endpoints on the circumference.

Each friend will get 1.35 pounds of pecans.

To find this, simply divide 16.2 by 12 to find the weight of pecans everyone gets. Thus making 1.35 pounds of pecans the answer.

I hope this helps!

Answer:

The 3rd option Summation(4^i-4)

Step-by-step explanation:

Summation(4^i-4)

When i = 1

4^I-4 = 4^1-4 = 4^-3 = 1/64

When i = 2

4^i-4 = 4^2-4 = 4^-2 = 1/16

When i = 3

4^i-4 = 4^3-4 = 4^-1 = 1/4

When i = 4

4^i-4 = 4^4-4 = 4^0 = 1

When i = 5

4^i-4 = 4^5-4 = 4^1 = 4

Answer:

[tex]\sum _{i=1}^54^{i-4}[/tex]

Step-by-step explanation:

[tex]a_{1} \\[/tex] = First term

In this case, our first term is [tex]\frac{1}{64}[/tex]

The ratio of all of the adjacent terms is 4

[tex]a_{1} \\[/tex] = [tex]4^1^-^4[/tex]

[tex]a_{1}=\frac{1}{64}[/tex]

~Hope this helps!~

Answer:

 GENERAL EXPLICIT SEQUENCE IS GIVEN   [tex]a_n = (14.74)(r)^{n-1)}[/tex]

Step-by-step explanation:

Let n be the number of year the data is recorded in.

a: The number of raccoons taken initially.

r: The multiplying factor

[tex]a_n[/tex]  : The number of raccoon in the nth year.

As given:  [tex]a_6 = 45, a_8 = 71[/tex]

Now, as the given situation can be expressed as GEOMETRIC SERIES:

[tex]a_n = a r^{(n-1)}[/tex]

Applying the same to given terms, we get:

[tex]a_6 = a r^{(6-1)} = ar^5 = 45\\\implies ar^5 = 45[/tex]

[tex]a_8 = a r^{(8-1)} = ar^7 =71\\\implies ar^7 = 71[/tex]

Dividing both equations, we get:

[tex]\frac{ar^7}{ar^5} = \frac{71}{45} \\\implies r^2 = 1.58\\\implies r = 1.25[/tex]

So, the first term [tex]a = \frac{45}{(1.25)^5} = 14 .74 \approx 15[/tex]

So, the GENERAL EXPLICIT SEQUENCE IS GIVEN as:  [tex]a_n = (14.74)(r)^{n-1)}[/tex]

Answer:

is d

Step-by-step explanation:

Answer:

2.13%

Step-by-step explanation:

Inaccurate measurement = 5.5cm

Actual measurement = 5.62cm

Difference= actual - inaccurate

=> 5.62 - 5.5

Therefore, the measurement Jocelyn obtained is off by 0.12cm (difference)

Note: error = difference

% diff = error÷ actual measurement × 100

= 0.12/5.62 × 100

% diff. = 2.13 (to the nearest decimal place)

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.

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Answer:

The answer is 160

Step-by-step explanation:

The value of x is (πr - 20).

The arc length of a circle is the distance between two points on the curve of the circle.

We have,

The measure of an angle formed by two tangents outside the circle.

= Difference of the intercepted arc / 2 ______(1)

Now,

Larger arc = (2πr - x)

Small arc = x

Angle = 20

Substituting in (1).

20 = (2πr - x - x) / 2

40 = 2πr - 2x

20 = πr - x

x = πr - 20

Thus,

The value of x is (πr - 20).

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Answer:

1220

Step-by-step explanation:

Given that in a game, a player earns 100 points for each question answered correctly and earns −30 points for each question answered incorrectly.

Let x be the no of questions correctly answered .  Then 20-x would be the question wrongly answered since total number of questions = 14+6 =20

Points gained for correct answer = 100(x) = 100x

Points lost for wrong answer = -30(20-x) = -600+30x

So total points gained when x questions are answered right

= [tex]100x-600+30x\\= 130x-600[/tex]

A player answered 14 questions correctly and 6 questions incorrectly.

Here x =14

Hence we substitute x =14 to get total points earned

Total points earned

[tex]= 130(14)-600\\= 1820-600\\=1220[/tex]

Answer:

6

Step-by-step explanation:

f(-4) = g(-10) = f(-10+k)

f(-2) = g(-8) = f(-8+k)

-4 = -10+k

k = -4+10

k = 6

If the line f passes through (-4,0) and (-2,2) it means:

f(-4)=0 and f(-2)=2

If the line g passes through (-10,0) and (-8,2) it means:

g(-10)=0 and g(-8)=2

We can see that

                                                g(-10)=0=f(-4)

                                              g(-10)=0=f(-10+6)

Also,

                                                  g(-8)=2=f(-2)

                                                 g(-8)=2=f(-8+6)

Therefore, k=6

Answer: the swimming pool can hold 31707.72 gallons of water.

Step-by-step explanation:

The swimming pool is shaped like a cylinder. The formula for determining the volume of a cylinder is expressed as

Volume = πr²h

Where

r represents the radius of the cylindrical swimming pool.

h represents the height of the pool.

π is a constant whose value is 3.14

Therefore, volume of the swimming pool is

Volume = 3.14 × 15² × 6 = 4239 cubic feet

if one cubic foot holds 7.48 gallons of the water, then the number of gallons of water that the swimming pool can hold is

4239 × 7.48 = 31707.72 gallons of water

Answer:

the swimming pool can hold 31707.72 gallons of water.

Step-by-step explanation:

Answer:

(a) The value of C is 1.

(b) In 2010, the population would be 1.07555 billions.

(c) In 2047, the population would be 1.4 billions.

Step-by-step explanation:

(a) Here, the given function that shows the population(in billions) of the country in year x,

[tex]P(x)=Ca^{x-2000}[/tex]

So, the population in 2000,

[tex]P(2000)=Ca^{2000-2000}[/tex]

[tex]=Ca^{0}[/tex]

[tex]=C[/tex]

According to the question,

[tex]P(2000)=1[/tex]

[tex]\implies C=1[/tex]

(b) Similarly,

The population in 2025,

[tex]P(2025)=Ca^{2025-2000}[/tex]

[tex]=Ca^{25}[/tex]

[tex]=a^{25}[/tex]                    (∵ C = 1)

Again according to the question,

[tex]P(2025)=1.2[/tex]

[tex]a^{25}=1.2[/tex]

Taking ln both sides,

[tex]\ln a^{25}=\ln 1.2[/tex]

[tex]25\ln a = \ln 1.2[/tex]

[tex]\ln a = \frac{\ln 1.2}{25}\approx 0.00729[/tex]

[tex]a=e^{0.00729}=1.00731[/tex]

Thus, the function that shows the population in year x,

[tex]P(x)=(1.00731)^{x-2000}[/tex]     ...... (1)

The population in 2010,

[tex]P(2010)=(1.00731)^{2010-2000}=(1.00731)^{10}=1.07555[/tex]          

Hence, the population in 2010 would be 1.07555 billions.

(c) If population P(x) = 1.4 billion,

Then, from equation (1),

[tex]1.4=(1.00731)^{x-2000}[/tex]

[tex]\ln 1.4=(x-2000)\ln 1.00731[/tex]

[tex]0.33647 = (x-2000)0.00728[/tex]

[tex]0.33647 = 0.00728x-14.56682[/tex]

[tex]0.33647 + 14.56682 = 0.00728x[/tex]

[tex]14.90329 = 0.00728x[/tex]

[tex]\implies x=\frac{14.90329}{0.00728}\approx 2047[/tex]

Therefore, the country's population might reach 1.4 billion in 2047.

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.

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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:

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:

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

Answer:

C option is correct 219.

Step-by-step explanation:

Total fans = 365

Fans who bought popcorn = 3/5

No. of Fans who bought popcorn = 365 x 3/5

                                                       = 219

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

-(x+2)^2 --> -(x+2)(x+2) --> -(x^2+4x+4) = -x^2-4x-4

Answer:

[tex]x^{2} -4x+4[/tex]

Step-by-step explanation:

[tex]y =-(x+2)^{2}[/tex]

The negative sign multiplies the positive in the bracket

[tex]y=(x-2)^{2}[/tex]

[tex](x-2) X (x-2)[/tex]

[tex]x^{2} -2x-2x+4[/tex]

That gives us

[tex]x^{2} -4x+4[/tex]

Answer:

Answer is; Evaluating

Step-by-step explanation:

Student self-assessment involves students evaluating their own work and learning progress.

Through self-assessment evaluation, students can:

* See where their knowledge is weak

* See where to focus their attention in learning.

* Set realistic goals

* Revise their work

* Track their own progress.

In Rachel's case, after writing her midterm exams, she already set her goal grade. So when she got her midterm grades, she needed to evaluate her work to find out her performance.

Therefore, according to the question, Rachel is EVALUATING her performance.

Answer:

The probability density function is f(x)=0.125.

Step-by-step explanation:

We have he continuous random variable x, which has a uniform distribution over the interval from 20 to 28. We calculate the probability density function has what value in the interval between 20 and 28.  

We use the formula:

[tex]\boxed{f(x)=\frac{1}{b-a}}[/tex]

We have a=20 and b=28, we get

[tex]f(x)=\frac{1}{28-20}\\\\f(x)=\frac{1}{8}\\\\f(x)=0.125\\[/tex]

The probability density function is f(x)=0.125.

The value of the probability density function (pdf) in the interval between 20 and 28 is 0.125.

The probability density function (pdf) of a continuous uniform distribution is represented by a horizontal line. Since the random variable x has a uniform distribution over the interval from 20 to 28, the pdf will also have a constant value over this interval.

To find the value of the pdf in the interval between 20 and 28, we need to calculate the area under the pdf curve in this interval. Since the pdf is a horizontal line, the area is simply equal to the height multiplied by the width of the interval.

Since the width of the interval is 8 (28 - 20) and the pdf is a uniform distribution, the height is 1/8. Therefore, the value of the pdf in the interval between 20 and 28 is 1/8 or 0.125.

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Answer:

x = y = {36, 78, 12, 24}

Step-by-step explanation:

The loop executes 4 times, as indicated by the length of array x.

The first line in the content of the loop assigns every element in array y to array x. Because both arrays now have the same content, the second line of code is quite redundant and is assigning the new values of x to y. Since these new values of x are the old values of y, there is no change in the contents of y. They are just being replaced by themselves.

In other words, the second line is not needed for anything. In fact, if the loop has much more contents, the second makes it work twice as much, reducing efficiency.

Answer: the maximum number of socks she will be able to make from the leftover yarn is 10

Step-by-step explanation:

Let x represent the number of sweaters.

Let y represent the number of socks.

The baby sweaters take 10 feet of yarn and the baby socks take 5 feet of yarn. This means that the total number of feet of yarn needed to make x sweaters and y socks is expressed as

10x + 5y

She has 100 feet of yarn and wants to make 5 sweaters. This means that

10 × 5 + 5y = 100

50 + 5y = 100

5y = 100 - 50 = 50

y = 50/5

y = 10

Answer:

A.) it costs 140$ for 7 tickets B.) 460$ = 23 tickets

Step-by-step explanation:

If it is 20 dolors for 1 ticket if you buy 7 you do 7 x 20 = 140.

But if you have 460$ then you do the opisit you do 460/20=23

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

Final answer:

The perimeter of a square with side length (6x-1) is expressed as P = 24x - 4. Substituting x with 3, the perimeter is calculated to be 68 inches.

Explanation:

The expression for the perimeter of a square is given by the formula P=4s, where 's' is the side length of the square. For a square with side length (6x-1) inches, the perimeter would be:

P = 4(6x-1)

This expression can be simplified to:

P = 24x - 4

When x equals 3, we substitute 3 in place of x:

P = 24(3) - 4

P = 72 - 4

P = 68 inches

Therefore, the perimeter of the square when x is 3 is 68 inches.

Answer:

The answers to the question is

(a) Jamie is gaining altitude at 1.676 m/s

(b) Jamie rising most rapidly at t = 15 s

At a rate of 2.094 m/s.

Step-by-step explanation:

(a) The time to make one complete revolution = period T = 15 seconds

Here will be required to develop the periodic motion equation thus

One complete revolution = 2π,

therefore the  we have T = 2π/k = 15

Therefore k = 2π/15

The diameter = radius of the wheel = (diameter of wheel)/2 = 5

also we note that the center of the wheel is 6 m above ground

We write our equation in the form

y = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex]

When Jamie is 9 meters above the ground and rising we have

9 = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex] or 3/5 = [tex]sin(\frac{2*\pi*t}{15} )[/tex] = 0.6

which gives sin⁻¹(0.6) = 0.643 =[tex]\frac{2*\pi*t}{15}[/tex]

from where t = 1.536 s

Therefore Jamie is gaining altitude at

[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] 1.676 m/s.

(b) Jamie is rising most rapidly when   the velocity curve is at the highest point, that is where the slope is zero

Therefore we differentiate the equation for the velocity again to get

[tex]\frac{d^2y}{dx^2} = -5*(\frac{\pi *2}{15} )^2*sin(\frac{2\pi t}{15})[/tex] =0, π, 2π

Therefore [tex]-sin(\frac{2\pi t}{15} )[/tex] = 0 whereby t = 0 or

[tex]\frac{2\pi t}{15}[/tex] = π and t =  7.5 s, at 2·π t = 15 s

Plugging the value of t into the velocity equation we have

[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] - 2/3π m/s which is decreasing

so we try at t = 15 s and we have [tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi *15}{15}) = \frac{2}{3} \pi[/tex]m/s

Hence Jamie is rising most rapidly at t = 15 s

The maximum rate of Jamie's rise is 2/3π m/s or 2.094 m/s.

(a) When Jamie is 9 meters above the ground and rising, she is gaining altitude at approximately 1.68 meters per second. (b) Jamie is rising most rapidly when the cosine function is at its maximum, which happens at the lowest point of the Ferris wheel, and the rate is approximately 2.094 meters per second.

Part (a): Rate at Which Jamie is Gaining Altitude

1. Identify the position function of Jamie on the Ferris wheel:

  The height h of Jamie above the ground as a function of time t can be modeled by the equation of a sinusoidal function:

 [tex]\[ h(t) = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \][/tex]

  Here, 6 meters is the height of the center of the Ferris wheel above the ground, and 5 meters is the radius of the wheel.

2. Differentiate the height function to find the rate of change of height:

  To find the rate at which Jamie is gaining altitude, we need to differentiate h(t) with respect to t:

[tex]\[ h'(t) = \frac{d}{dt} \left( 6 + 5\sin\left(\frac{2\pi}{15}t\right) \right) = 5 \cdot \frac{2\pi}{15} \cos\left(\frac{2\pi}{15}t\right) \][/tex]

  Simplifying,

 [tex]\[ h'(t) = \frac{2\pi}{3} \cos\left(\frac{2\pi}{15}t\right) \][/tex]

3. Determine t when Jamie is at 9 meters above the ground and rising:

[tex]\[ 9 = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \] Solving for \( \sin \left(\frac{2\pi}{15}t\right) \): \[ 3 = 5\sin\left(\frac{2\pi}{15}t\right) \] \[ \sin\left(\frac{2\pi}{15}t\right) = \frac{3}{5} \][/tex]

  Jamie is rising when [tex]\( \cos \left(\frac{2\pi}{15}t\right) > 0 \)[/tex].

4. Find the rate at which Jamie is gaining altitude at this instant:

  Substitute [tex]\(\sin \left(\frac{2\pi}{15}t\right) = \frac{3}{5}\)[/tex] into the derivative [tex]\( h'(t) \)[/tex]:

  [tex]\[ \cos \left(\frac{2\pi}{15}t\right) = \sqrt{1 - \sin^2 \left(\frac{2\pi}{15}t\right)} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \][/tex]

  Thus, the rate of change of height:

[tex]\[ h'(t) = \frac{2\pi}{3} \cdot \frac{4}{5} = \frac{8\pi}{15} \approx 1.68 \text{ meters per second} \][/tex]

Part (b): When Jamie is Rising Most Rapidly

1. Identify when Jamie is rising most rapidly:

  Jamie rises most rapidly when [tex]\( \cos \left(\frac{2\pi}{15}t\right) = 1 \)[/tex], which corresponds to the maximum value of the cosine function.

2. Rate of change of height at maximum rise:

[tex]\[ h'(t) = \frac{2\pi}{3} \cdot 1 = \frac{2\pi}{3} \approx 2.094 \text{ meters per second} \][/tex]

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.

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 :

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