Please help me out ........!!!

Answer:

C) about 113

Step-by-step explanation:

"How many people are there per square mile?" means that we want a ratio with miles as denominator. In other words, to find the population density, we just need to divide the population by the land area (miles squared):

[tex]population-density=\frac{people}{land-area}[/tex]

We know that the population of Alabama is 90,000 people and its land area is 800 miles squared, so [tex]people=90000[/tex] and [tex]land-area=800mi^{2}[/tex].

Replacing values:

[tex]population-density=\frac{people}{land-area}[/tex]

[tex]population-density=\frac{90000}{800mi^{2}}[/tex]

[tex]population-density=112.5[/tex]

Which rounds to:

[tex]population-density=113[/tex]

We can conclude that there are approximately 113 people per square mile in Alabama.

Answer:

About 113

Step-by-step explanation:

Hope this help

The function that would represent the growth of the keepers bee population over time would [tex]\( x = 150 \times 3^n \)[/tex]

How to find the function ?

To model the growth of the bee population in the beekeeper's hive, we can use an exponential growth function. The key information here is that the population triples each year. This means the growth factor is 3.

The function would therefore be:

[tex]\( x = 150 \times 3^n \)[/tex]

The equation is an exponential growth model where:

150 is the initial number of bees.

3 is the growth factor (since the population triples each year).

n is the number of years passed.

For example, to find the bee population after 2 years, you would calculate:

[tex]\( 150 \times 3^2 = 150 \times 9 \\= 1350 \) bees[/tex]

Answer:

tons

Step-by-step explanation:

Answer:

3√10

Step-by-step explanation:

27² + y² = x²

⇒⇒⇒ y² = x² - 27² ----------------- [ 1 ]

3² + y² = z²

⇒⇒⇒ y² = z² - 3² ----------------- [ 2 ]

Equate [ 1 ] and [ 2 ] :

x² - 27² = z² - 3²

x² - z² = 27² - 3²

x² - z² = 720 ----------------- [ 3 ]

x² + z² = 30²

x² + z² = 900 ----------------- [ 4 ]

[ 3 ] - [ 4 ] :

-2z² = -180

z² = 90

z = √90

z = 3√10

Answer:

Step-by-step explanation:

Even though the teams appear from the mean to be similar in their winnings, they are not.  This is mostly because the Tigers have a greater range (difference between the highest and lowest values) than do the Foxes.  The Tigers' range is 16 - 1 = 15 while the Foxes' range is 6 - 3 = 3.

In other words, using the mean to determine how closely matched these teams are is worthless.

Answer:

Option C

Step-by-step explanation: I think this is right because when you substitute the 2 for x you get the answer. Hope this helps darling!!

A quadratic equation is in the form of ax²+bx+c. The roots of the quadratic equation x² - 4x + 5 = 0 are x = 2 ± i. Thus, the correct option is C.

A quadratic equation is an equation whose leading coefficient is of second degree also the equation has only one unknown while it has 3 unknown numbers. It is written in the form of ax²+bx+c.

Given the roots of the quadratic equation are x = 2 ± i. Therefore, we can write the roots as,

α = 2+i

β = 2-i

Now, we know that a quadratic equation can also be written in the form,

x² - (α+β)x + αβ = 0

Therefore, we need to find the value of (α+β) and αβ,

α+β = 2 + i + 2 - i

α+β = 4

αβ = (2+i)(2-i)

αβ = 2²-i²

αβ = 4 + 1

αβ = 5

Thus, the quadratic equation is x² - 4x + 5 = 0.

Hence, The roots of the quadratic equation x² - 4x + 5 = 0 are x = 2 ± i. Thus, the correct option is C.

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Answer:

Step-by-step explanation:

the answer is b defenitly

Answer:

Step-by-step explanation:the answer is 3 ft

The width of the fountain is 9 feet.

The area of the fountain is 0, it means the width of the fountain doesn't affect the total area of the courtyard. Therefore, the width of the fountain can be any value, including 9 feet.

To find the width of the fountain, we first need to determine the area of the entire courtyard, including the fountain and the stone-covered area. Since the courtyard is rectangular, we can assume the width of the fountain is the same as the width of the courtyard.

Let's denote the width of the fountain as \( w \) feet and the length of the courtyard as \( l \) feet.

Given that the area covered by stone is 246 square feet, we can write the equation:

[tex]\[ w \times l - 246 = \text{area of the fountain} \][/tex]

Since the entire courtyard area is the sum of the area covered by stone and the area of the fountain, we have:

[tex]\[ w \times l = 246 + \text{area of the fountain} \][/tex]

We know the total area of the courtyard, but we still need to find the length [tex](\( l \))[/tex] of the courtyard to solve for the width of the fountain.

To find the length, we can use the fact that the entire courtyard area is the product of its length and width:

[tex]\[ l \times w = \text{total area of the courtyard} \][/tex]

Since the total area of the courtyard is given as 246 square feet, we have:

[tex]\[ l \times w = 246 \][/tex]

Now, we have two equations:

[tex]\[ w \times l = 246 + \text{area of the fountain} \][/tex]

[tex]\[ l \times w = 246 \][/tex]

Substituting [tex]\( l \times w = 246 \)[/tex] into the first equation, we get:

[tex]\[ 246 = 246 + \text{area of the fountain} \][/tex]

[tex]\[ \text{area of the fountain} = 246 - 246 = 0 \][/tex]

Since the area of the fountain is 0, it means the width of the fountain doesn't affect the total area of the courtyard. Therefore, the width of the fountain can be any value, including 9 feet.

Complete question:

Cindy is designing a rectangular fountain in the middle of a courtyard the rest of the courtyard will be covered in stone. The part of the courtyard that will be covered in stone has an area of 246 square feet. What is the width of the fountain

To choose the single logarithm expression that is equivalent to the given expression, we need to combine the two logarithms into one logarithm using the properties of logarithms. The single logarithm expression that is equivalent to the given expression is log (3x).

To choose the single logarithm expression that is equivalent to the given expression, we need to combine the two logarithms into one logarithm using the properties of logarithms.

Using the property that the logarithm of a product is the sum of the logarithms, we can rewrite the given expression as:

1/3 log (3x) + 2/3 log (3x) = log((3x)^(1/3) * (3x)^(2/3))

Simplifying the expression inside the logarithm gives:

log((3x)^(1/3 + 2/3)) = log((3x)^1) = log (3x)

Therefore, the single logarithm expression that is equivalent to the given expression is log (3x).

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Answer:

The measure of the fourth angle is 81°

Step-by-step explanation:

we know that

The sum of the internal angles of a quadrilateral must be equal to 360 degrees

Let

x----> the measure of the fourth angle

we have

90°+90°+99°+x=360°

Solve for x

279°+x=360°

x=360°-279°=81°

5×3/4=3.75

she needs 3.75 cups of sugar

Answer:

The solution of the system of equations is (-3 , 5)

Step-by-step explanation:

* Lets describe the drawing of each line

- The form of the equation of any line is y = mx + c, where m

  is the slope of the line and c is the y-intercept (the point of

  intersection between the line and the y-axis is (0 , c))

* The line y = -x + 2 represented by the red line

- The line intersect the y-axis at point (0 ,2)

- The line intersect the x-axis at point (2 , 0)

- The slope of the line is -1, so the angle between the positive

  part of x-axis and the line is obtuse

* The line x - 3y = -18 represented by blue line

- Put the line in the form y = mx + c

- The line is x - 3y = -18⇒ add 18 and 3y to both sides

- The line is 3y = x + 18 ⇒ ÷ 3 both sides

- The line is y = 1/3 x + 6

- The line intersect the y-axis at point (0 ,6)

- The line intersect the x-axis at point (-18 , 0)

- The slope of the line is 1/3, so the angle between the positive

  part of x-axis and the line is acute

* Look to the attached graph

- The point of intersection between the two line is the solution

 of the system of equation

- From the graph the point of intersection is (-3 , 5)

* The solution of the system of equations is (-3 , 5)

Answer:

one of the lines will pass (0,2) and (2,0) and intersect at (5,3) and pass through (0,5)

Step-by-step explanation:

edit: it actually passes through (0,6) mark as brainliest if right

Answer:

The height of the tower is 17.79 m

Step-by-step explanation:

The question in English is

To measure the height of a tower, Juan stands at a point on the horizontal ground and observes the highest point of the tower under an angle of 62°. He approaches the tower 6 meters in a straight line and the angle is 79° Find the height of the tower

see the attached figure to better understand the problem  

In the right triangle ABC

tan(62°)=h/x

h=(x)tan(62°) ------> equation A

In the right triangle DBC

tan(79°)=h/(x-6)

h=(x-6)tan(79°) ------> equation B

Equate equation A and equation B and solve for x

(x)tan(62°)=(x-6)tan(79°)

(x)tan(62°)-(x)tan(79°)=-(6)tan(79°)

x=-(6)tan(79°)/[tan(62°)-tan(79°)]

x=9.46 m

Find the value of h

h=(9.46)tan(62°)=17.79 m

Answer:

son las dos en punto

Step-by-step explanation:

Answer:

20

Step-by-step explanation:

4 multiplied by 5 is equal to 20. You substitute 5 for x.

Answer:

20

Step-by-step explanation:

You are to multiply 5 by 4:  the outcome is 20.  That's all.

Answer:

B. 1/56

Explanation:

There are 8 different marbles. In the first draw, you have a one out of 8 chance of getting the red. In the second draw, there are 7 marbles remaining, so a one out of 7 chance of drawing a white marble. To find the total probability of drawing the red and then white marble, we multiply the probabilities if each draw.

(1/8)*(1/7)=1/56

This question is based on the probability.  Therefore, the correct option is B, [tex]\dfrac{1}{56}[/tex]  is the probability that the red marble is drawn first and the white marble is drawn second.

Given:

There are 8 marbles in a bag. Each marble is a different color. The colors are: red, orange, yellow, green, blue, purple, black, and white. Two marbles are randomly drawn from the bag without replacement.

We need to determined the probability that the red marble is drawn first and the white marble is drawn second.

According to the question,

It is given that, there are 8 different marbles. In the first draw, you have a one out of 8 chance of getting the red is [tex]\dfrac{1}{8}[/tex].

In the second draw, there are 7 marbles remaining, so a one out of 7 chance of drawing a white marble i.e. [tex]\dfrac{1}{7}[/tex].

Now, calculating the total probability of drawing the red and then white marble, we multiply the probabilities if each draw.

⇒ [tex]\dfrac{1}{8} \times \dfrac{1}{7} = \dfrac{1}{56}[/tex]

Therefore, the correct option is B, [tex]\dfrac{1}{56}[/tex]  is the probability that the red marble is drawn first and the white marble is drawn second.

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Answer:

92 feet.

Step-by-step explanation:

1. Suzi started at 230 feet

2. Suiz ended at 138 feet

3. Subtract the starting and end numbers 4. 23 - 138= 92

Answer:

whats the answer

Answer:

-0.95

Step-by-step explanation:

The value of R is -0.9538.

A correlation coefficient is a metric that expresses a correlation, or a statistical link between two variables, in numerical terms. Two columns of a specific data set of observations, sometimes referred to as a sample, or two parts of a multivariate random variable with a known distribution may serve as the variables.

Given

X Values

∑ = 26

Mean = 2.889

∑(X - Mx)² = S[tex]S_{x}[/tex] = 28.889

Y Values

∑ = 45

Mean = 5

∑(Y - My)²= S[tex]S_{y}[/tex] = 32

X and Y Combined

N = 9

∑(X - Mx)(Y - My) = -29

R Calculation

r = ∑((X - My)(Y - Mx)) / √((S[tex]S_{x}[/tex])(S[tex]S_{y}[/tex]))

r = -29 / √((28.889)(32)) = -0.9538

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Answer:

D

Step-by-step explanation:

Let E(x,y) represent the total earnings.

Then E(x,y) = ($20/lawn)x + ($5/bush)y   (Answer D)

Answer:

3/13

Step-by-step explanation:

There are 3 5's in the jar and thirteen chips total. You have a probability of pulling 1 of the three 5's out so there is a 3/13 chance of pulling a 5.

Answer:

sample space is 13

3/13

Step-by-step explanation:

Answer:

Part a) Option d

Part b) Option a

Step-by-step explanation:

Part a

if we look at the options given and the data available

Option a) x^4+9

Putting x= 2 we get (2^4) + 9 =25

Putting x= 3 we get (3^4) + 9 =90 but f(x) =125 so not correct option

Option b) (4^x)+9

Putting x= 2 we get (4^2) + 9 =25

Putting x= 3 we get (4^3) + 9 =73 but f(x) =125 so not correct option

Option c) x^5

Putting x= 2 we get (2^5)  =32 but f(x) =25 so not correct option

Option d) 5^x

Putting x= 2 we get (5^2)  =25

Putting x= 3 we get (5^3)  =125

Putting x= 4 we get (5^4)  =625

So Option d is correct.

Part (b)

3(2)^3x

can be solved as:

=3(2^3)^x

=3(8)^x

So, correct option is a

By the confront theorem we know that the limit only exists if both lateral limits are equal

In this case they aren't so we don't have limit for x approaching 2, but we can find their laterals.

Approaching 2 by the left we have it on the 5 line so this limit is 5

Approaching 2 by the right we have it on the -3 line so this limit is -3

Think: it's approaching x = 2 BUT IT'S NOT 2, and we only have a different value for x = 2 which is 1, but when it's approach by the left we have the values in the 5 line and by the right in the -3 line.

ANSWER

The limit does not exist.

EXPLANATION

From the graph the left hand limit is the value the graph is approaching as x-values approaches 2.

[tex] \lim_{x \to {2}^{ - } }(f(x)) = 5[/tex]

Also the right hand limit is the value that the graph approaches, as x-values approach 2 from the right.

[tex]\lim_{x \to {2}^{ + } }(f(x)) = - 3[/tex]

Since the left hand limit is not equal to the right hand limit, the limit as x approaches 2 does not exist

Answer:

See below.

Step-by-step explanation:

Fifth root of 243 = 3,

Suppose r( cos Ф + i sinФ) is the fifth root of 243(cos 240 + i sin 240),

then r^5( cos  Ф  + i sin  Ф )^5 = 243(cos 240 + i sin 240).

Equating equal parts and using de Moivre's theorem:

r^5 =243  and  cos  5Ф  + i sin  5Ф = cos 240 + i sin 240

r = 3 and  5Ф = 240 +360p so Ф =  48 + 72p

So Ф = 48, 120, 192, 264, 336  for   48 ≤ Ф < 360

So there are 5 distinct solutions given by:

3(cos 48 + i sin 48),

3(cos 120 + i sin 120),

3(cos 192 + i sin 192),

3(cos 264 + i sin 264),

3(cos 336 + i sin 336).. (Answer).

Answer:

The Answer is 5 (3/4) more miles.

Step-by-step explanation:

For this we just need to add and subtract fractions.

We we want  to get to 13 (1/2) miles...however we have already ran

3 (1/2) + 4 (1/4) ... this is the same as 3.5 plus 4.25. When we add those up we get...

3.5+4.25 = 7.75 then we subtract ... 13.5 - 7.75 = 5.75 = 5 (3/4) our answer.

Final answer:

Lorenz needs to run 5 3/4 more miles to meet his goal for his training plan this week.

Explanation:

To find out how many more miles Lorenz needs to run this week to meet his goal, we need to add up the miles he has already run and subtract that from his goal. Lorenz has already run 3 1/2 miles on Monday and 4 1/4 miles on Tuesday, so we can add these two amounts to get 3 1/2 + 4 1/4 = 7 3/4 miles. Next, we subtract this amount from his goal of 13 1/2 miles: 13 1/2 - 7 3/4 = 5 3/4 miles. Therefore, Lorenz needs to run 5 3/4 more miles to meet his goal for his training plan this week.

Answer:

D

Step-by-step explanation:

log₂(x-4) = 4

Undo the log by raising 2 to both sides:

2^(log₂(x-4)) = 2^4

x - 4 = 2^4

x - 4 = 16

x = 20

Answer D.

log2(x-4) =4. The answer is D) 20

Answer: A

Step-by-step explanation:

The correct answer is A. 0.037 cubic feet.

Convert the lengths to inches:

12 feet is equal to 12 * 12 inches = 144 inches.

Calculate the area of the annulus (the space between the inner and outer circles):

First, convert the diameters to radii:

Outer radius = 7/8 inches * 0.5 = 7/16 inches

Inner radius = 3/4 inches * 0.5 = 3/8 inches

Then, calculate the area of the annulus:

Area = π * (outer radius^2 - inner radius^2)

Area ≈ π * ((7/16)^2 - (3/8)^2) ≈ 0.0875 square inches

Calculate the volume of the refrigerant:

Multiply the area by the length of the tubing:

Volume = Area * Length

Volume ≈ 0.0875 square inches * 144 inches ≈ 12.48 cubic inches

Convert the volume to cubic feet:

Remember that 1 inch^3 = 1/12^3 cubic feet.

Therefore, the volume in cubic feet is:

Volume (ft^3) = Volume (in^3) / (12^3)

Volume (ft^3) ≈ 12.48 cubic inches / (12 * 12 * 12) ≈ 0.037 cubic feet

The closest answer choice to 0.037 cubic feet is A. 0.037 cubic feet.

Complete Question:

You are working on an air conditioning system. A roll of cylindrical copper tubing has an outside diameter of 7/8 inch and an inside diameter of 3/4 inch. How much refrigerant can 12 feet of the tubing hold?

A. 0.037 cubic feet

B. 0.065 cubic feet

C. 0.147 cubic feet

D. 5.30 cubic feet

Answer:

0, -8

Step-by-step explanation:

3x²  + 24x = 0

Taking out common factor 3x.

3x(x + 8) = 0

Apply zero product property.

3x = 0 or x + 8 = 0

Evaluate.

x = 0 or x = -8

Answer:

First is 3x (x + 8) =0. The second is x = 0, x = -8.

Step-by-step explanation:

These are the correct answers on Plato, I got it correct.

Answer:

(x-3)^2+(y+2)^2=16

Step-by-step explanation:

The standard form of a circle is

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where h and k are the coordinates of the center and r is the radius, which is squared.  From our information, h is 3--> (x-3) and k is -2--> (y-(-2))--> (y+2) and 4 squared is 16.  Your choice is A.

Answer:

(x-3)^2+(y+2)^2=16

Step-by-step explanation:

The standard form of a circle is  

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where h and k are the coordinates of the center and r is the radius, which is squared.  From our information, h is 3--> (x-3) and k is -2--> (y-(-2))--> (y+2) and 4 squared is 16.  Your choice is A.

Answer:

If

€

p(x) is a polynomial, the solutions to the equation

€

p(x) = 0 are called the zeros of the

polynomial. Sometimes the zeros of a polynomial can be determined by factoring or by using the

Quadratic Formula, but frequently the zeros must be approximated. The real zeros of a polynomial

p(x) are the x-intercepts of the graph of

€

y = p(x).

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The Factor Theorem: If

€

(x − k) is a factor of a polynomial, then

€

x = k is a zero of the polynomial.

Conversely, if

€

x = k is a zero of a polynomial, then

€

(x − k) is a factor of the polynomial.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Example 1: Find the zeros and x-intercepts of the graph of

€

p(x) =x

4−5x

2 + 4.

€

x

4−5x

2 + 4 = 0

(x

2 − 4)(x

2 −1) = 0

(x + 2)(x − 2)(x +1)(x −1) = 0

x + 2 = 0 or x − 2 = 0 or x +1= 0 or x −1= 0

x = −2 or x = 2 or x = −1 or x =1

So the zeros are –2, 2, –1, and 1 and the x-intercepts are (–2,0), (2,0), (–1,0), and (1,0).

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The number of times a factor occurs in a polynomial is called the multiplicity of the factor. The

corresponding zero is said to have the same multiplicity. For example, if the factor

€

(x − 3) occurs to

the fifth power in a polynomial, then

€

(x − 3) is said to be a factor of multiplicity 5 and the

corresponding zero, x=3, is said to have multiplicity 5. A factor or zero with multiplicity two is

sometimes said to be a double factor or a double zero. Similarly, a factor or zero with multiplicity

three is sometimes said to be a triple factor or a triple zero.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Example 2: Determine the equation, in factored form, of a polynomial

€

p(x) that has 5 as double

zero, –2 as a zero with multiplicity 1, and 0 as a zero with multiplicity 4.

€

p(x) = (x − 5)

2(x + 2)x

4

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Example 3: Give the zeros and their multiplicities for

€

p(x) = −12x

4 + 36x3 − 21x

2.

€

−12x

4 + 36x3 − 21x

2 = 0

−3x

2(4x

2 −12x + 7) = 0

−3x

2 = 0 or 4x

2 −12x + 7 = 0

x

2 = 0 or x = −(−12)± (−12)

2−4(4)(7)

2(4)

x = 0 or x = 12± 144−112

8 = 12± 32

8 = 12±4 2

8 = 12

8 ± 4 2

8 = 3

2 ± 2

2

So 0 is a zero with multiplicity 2,

€

x = 3

2 − 2

2 is a zero with multiplicity 1, and

€

x = 3

2 + 2

2 is a zero

with multiplicity 1.

(Thomason - Fall 2008)

Because the graph of a polynomial is connected, if the polynomial is positive at one value of x and

negative at another value of x, then there must be a zero of the polynomial between those two values

of x.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Example 4: Show that

€

p(x) = 2x3 − 5x

2 + 4 x − 7 must have a zero between

€

x =1 and

€

x = 2.

€

p(1) = 2(1)

3 − 5(1)

2 + 4(1) − 7 = 2(1) − 5(1) + 4 − 7 = 2 − 5 + 4 − 7 = −6

and

€

p(2) = 2(2)3 − 5(2)

2 + 4(2) − 7 = 2(8) − 5(2) + 8 − 7 =16 −10 + 8 − 7 = 7.

Because

€

p(1) is negative and

€

p(2) is positive and because the graph of

€

p(x) is connected,

€

p(x)

must equal 0 for a value of x between 1 and 2.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

If a factor of a polynomial occurs to an odd power, then the graph of the polynomial actually goes

across the x-axis at the corresponding x-intercept. An x-intercept of this type is sometimes called an

odd x-intercept. If a factor of a polynomial occurs to an even power, then the graph of the

polynomial "bounces" against the x-axis at the corresponding x-intercept, but not does not go across

the x-axis there. An x-intercept of this type is sometimes called an even x-intercept.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Example 5: Use a graphing calculator or a computer program to graph

€

y = 0.01x

2(x + 2)3(x − 2)(x − 4)

4 .

x

y

–2 2 4

5

Because the factors

€

(x + 2) and

€

(x − 2) appear to odd

powers, the graph crosses the x-axis at

€

x = −2

and

€

x = 2.

Because the factors x and

€

(x − 4) appear to even

powers, the graph bounces against the x-axis at

€

x = 0

and

€

x = 4.

Note that if the factors of the polynomial were

multipled out, the leading term would be

€

0.01x10.

This accounts for the fact that both tails of the graph

go up; in other words, as

€

x → −∞,

€

y

Step-by-step explanation:

Parameterize each leg by

[tex]\vec r_1(t)=(1-t)(4\,\vec\imath)+t(-4\,\vec\imath)=(4-8t)\,\vec\imath[/tex]

[tex]\vec r_2(t)=(1-t)(-4\,\vec\imath)+t(4\,\vec\jmath)=(-4+4t)\,\vec\imath+4t\,\vec\jmath[/tex]

[tex]\vec r_3(t)=(1-t)(4\,\vec\jmath)+t(4\,\vec\imath)=4t\,\vec\imath+(4-4t)\,\vec\jmath[/tex]

each with [tex]0\le t\le1[/tex].

The line integrals along each leg (in the same order as above) are

[tex]\displaystyle\int_0^1((4-8t)\,\vec\jmath)\cdot(-8\,\vec\imath)\,\mathrm dt=0[/tex]

[tex]\displaystyle\int_0^1(16t(t-1)\,\vec\imath-4\,\vec\jmath)\cdot(4\,\vec\imath+4\,\vec\jmath)\,\mathrm dt=\int_0^1(64t(t-1)-16)\,\mathrm dt=-\frac{80}3[/tex]

[tex]\displaystyle\int_0^1(16t(1-t)\,\vec\imath+(8t-4)\,\vec\jmath)\cdot(4\,\vec\imath-4\,\vec\jmath)\,\mathrm dt=\int_0^1(64t(1-t)-4(8t-4))\,\mathrm dt=\frac{32}3[/tex]

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The total line integral then has a value of -16. We can confirm this by checking with Green's theorem. Notice that [tex]C[/tex] as given as clockwise orientation, while Green's theorem assumes counterclockwise. So we must multiply by -1:

[tex]\displaystyle\int_{-C}\vec f\cdot\mathrm d\vec r=-\iint_D\left(\frac{\partial(x-y)}{\partial x}-\frac{\partial(xy)}{\partial y}\right)\,\mathrm dA=-\int_0^4\int_{y-4}^{4-y}(1-x)\,\mathrm dx\,\mathrm dy=-16[/tex]

as required.

To calculate a line integral over a curve in a vector field, parametrize each segment of the curve, substitute these parametrizations into the integral and evaluate the resulting integral over the range of the parameters. Without knowing the paths between the points in question, a specific solution can't be given. However, basic calculus techniques would be used to evaluate these integrals.

The objective here is to find the line integral of the given vector field f⃗=xyi⃗ + (x−y)j⃗ along each segment of the triangle defined by the points (4,0), (-4,0) and (0,4). The vector field involves two variables, x and y. A line integral is a type of integral where a function is integrated along a curve. In this vector field, the function is defined in two variables x and y. Our first step in calculating the line integral is to parametrize each path between these points. Then, we substitute these parametrizations into the integral, which is then evaluated with respect to the parameter. Let's evaluate the line integral over the three segments of the triangle.

Unfortunately, without further information on the specific functional forms of the paths between these points, a concrete solution can't be provided. However, usually, this process involves applying the formula for a line integral over a vector field and using fundamental calculus techniques to simplify and evaluate these integrals. If the paths between points were straight lines, for example, the path parametrizations would be simple linear functions.

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Answer:

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Step-by-step explanation: