A movie ticket is 12.00 plus an 8.5 sales tax how much will one ticket movie cost

After evaluating the conclusion is that one serving of orange juice has 35.7 mg more vitamin C per serving than one serving of cranberry  juice.

According to the provided data , a bottle of orange juice has 750 mg of vitamin C and provides 6 servings. A bottle of cranberry juice has 134 mg of vitamin C and provides 1.5 servings.

Now to evaluate how many milligrams of vitamin C are in 1 serving of each type of juice, we have to  perform division to evaluate the total amount of vitamin C by the number of servings.

For orange juice

750 mg / 6 servings

= 125 mg/serving

For cranberry juice

134 mg / 1.5 servings

= 89.3 mg/serving

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The different possible exams for the 6 essay questions from which only 4 appear is equal to 15.

n is the total number of items in the set = 6 essay questions

r is the number of items we want to choose = 4 questions

Using combinations,

which is a way of counting the number of ways to choose a certain number of items from a larger set without regard to order.

Choose 4 out of the 6 essay questions, without regard to the order in which they appear on the exam.

Use the formula for combinations,

C(n, r) = n! / (r! × (n - r)!)

Plugging in the values, we get,

⇒C(6, 4) = 6! / (4! × (6 - 4)!)

⇒C(6, 4) = 6! / (4! ×2!)

⇒C(6, 4) = (6 × 5 × 4 × 3) / (4 × 3 × 2 × 1)

⇒C(6, 4) = 15

Therefore, there are 15 different exams possible, each consisting of 4 out of the 6 essay questions provided by the professor.

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To calculate the relative frequency for each category, divide the number of marked mountain goats in each category by the total number of marked mountain goats.

Total marked mountain goats = 71 (male) + 93 (female) = 164
Total marked mountain goats = 103 (adult) + 61 (baby) = 164

Relative frequency for male goats = Male goats / Total marked mountain goats = 71/164
Relative frequency for female goats = Female goats / Total marked mountain goats = 93/164
Relative frequency for adult goats = Adult goats / Total marked mountain goats = 103/164
Relative frequency for baby goats = Baby goats / Total marked mountain goats = 61/164

Your answer:
Relative frequency for male goats = 71/164
Relative frequency for female goats = 93/164
Relative frequency for adult goats = 103/164
Relative frequency for baby goats = 61/164

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Answer:

17.2 cm²

Step-by-step explanation:

Alr let me try

The angle is 1.2 you got it right. The rest is in the pics

Answer: A(shaded)=17.15 cm²

Step-by-step explanation:

What you did so far is correct.

Given:

r=5

s=6

Solve for Ф  angle:

s=[tex]\frac{part circle}{wholecircle} 2\pi r[/tex]         This way help you find the portion/percent you want

6=(Ф/360) (2[tex]\pi[/tex]5)       >solve for Ф  Divide by (2[tex]\pi[/tex]5) and multiply by 360

Ф=68.75

Solve for pie/sector

Now that you have the angle,  you can use the same concept for area

Area of sector = [tex]\frac{part circle}{wholecircle} \pi r^{2}[/tex]

Area of sector = [tex]\frac{68.75}{360} \pi 5^{2}[/tex]

Area of sector = 15.0 cm²

Now let find y so we can plug into area of triangle

use tan Ф = opposite/adjacent

tan 68.75 = y/5

y=5 * tan 68.75

y=12.86 cm

Area of triangle = 1/2 b h          b=y=12.86     h =5

Area of triangle = 1/2* 12.86*5

Area of triangle = 32.15 cm²

Now subtract area of sector from triangle

A(shaded)=A(triangle)-A(sector)

A(shaded)=32.15- 15.0

A(shaded)=17.15 cm²

The probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.

Since we are given the probabilities of different outcomes, we can use the definition of conditional probability to find p(a|b), which represents the probability that the outcome is a divisor of 4 given that it is prime.

The formula for conditional probability is:

p(a|b) = p(a ∩ b) / p(b)

where p(a ∩ b) represents the probability of both events happening simultaneously.

Looking at the table of outcomes and their probabilities, we can see that there are four prime numbers: 2, 3, 5, and 7. Of these, only 2 is a divisor of 4.

Therefore, p(a ∩ b) is the probability that the outcome is 2, which is 0.1.

The probability of the outcome being prime is the sum of the probabilities of the four prime outcomes, which is:

p(b) = 0.1 + 0.2 + 0.3 + 0.2 = 0.8

Substituting these values into the formula for conditional probability, we get:

p(a|b) = p(a ∩ b) / p(b) = 0.1 / 0.8 = 0.125

Therefore, the probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.

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Answer:Therefore, the cost of a utility pole with a diameter of 42 cm and a height of 740 cm is approximately $1957.43.

Step-by-step explanation:First, we need to calculate the volume of the cylinder-shaped utility pole:

The radius of the pole is half the diameter, so it's 42 cm / 2 = 21 cm.

The height of the pole is 740 cm.

The volume of a cylinder is given by the formula V = πr²h, where π is approximately 3.14, r is the radius, and h is the height.

Substituting the values we have, we get V = 3.14 x 21² x 740 = 1,034,462.4 cm³.

Now we can calculate the mass of the pole:

The density of the metal is 6.3 g/cm³, which means that 1 cm³ of the metal has a mass of 6.3 g.

The volume of the pole is 1,034,462.4 cm³, so its mass is 6.3 x 1,034,462.4 = 6,524,772.72 g.

Next, we convert the mass to kilograms and calculate the cost:

1 kg is equal to 1000 g, so the mass of the pole in kilograms is 6,524,772.72 g / 1000 = 6524.77 kg.

The cost of the metal is $0.30 per kilogram, so the cost of the pole is 6524.77 kg x $0.30/kg = $1957.43.

Based on the linear regression formula y = 1.3485x + 840.51, you can calculate the monthly rental price (y) for a one-bedroom apartment by plugging in the square footage (x) of the apartment.

The linear regression formula for the 15 similar one-bedroom apartments is y = 1.3485x + 840.51, where x represents the square footage and y represents the monthly rental price. This means that for every square foot increase in the apartment size, the monthly rental price is predicted to increase by $1.35.

The y-intercept of the formula is $840.51, which represents the predicted monthly rental price for an apartment with 0 square footage (this is not possible in reality, but is used in the formula for mathematical purposes). To get the rental price, round your answer to the nearest whole number. For example, if an apartment has 500 square feet, you'd calculate: y = 1.3485(500) + 840.51 ≈ 1344.76, which rounds to $1,345 as the monthly rental price.

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The colony will need to triple in number after about 7.58 days.

Since the number of bacteria doubles every 5 days, we can write the relationship between the number of bacteria and time as an exponential function:

N(t) = N0 x 2^(t/5)

where N0 is the initial number of bacteria and t is the time in days.

To find out how long it will take for the colony to triple in number, we need to solve the equation:

N(t) = 3N0

Substituting the expression for N(t) from above, we get:

N0 x 2^(t/5) = 3N0

Dividing both sides by N0, we get:

2^(t/5) = 3

Taking the logarithm of both sides (with base 2) gives:

t/5 = log2(3)

Multiplying both sides by 5, we get:

t = 5 x log2(3)

Using a calculator or a computer program to evaluate the logarithm, we get:

t ≈ 7.58

Therefore, the colony will need to triple in number after about 7.58 days.

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a) The graph of the given lines is as attached

b) The area of the enclosed triangle is: 8 square units

The general form of expression of linear equations in slope intercept form is expressed as:

y = mx + c

where:

m is slope

c is y-intercept

We are given the equations as:

y = x + 5

y = 5

x = 4

The graph of these three linear equations is as shown in the attached file

2) The area of the given triangle enclosed by the three lines is gotten from the formula:

A = ¹/₂ * b * h

where:

A is area

b is base

h is height

Thus:

A = ¹/₂ * 4 * 4

A = 8 square units

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The correct statement regarding the maximum values of the quadratic functions is given as follows:

Function 1 has the larger maximum at (4, 1).

The standard definition of a quadratic function is given as follows:

y = ax² + bx + c.

The x-coordinate of the vertex of a quadratic function is given as follows:

x = -b/2a.

Hence, for each function, the x-coordinate of the vertex is given as follows:

Each function has a negative leading coefficient, hence the vertex represents a maximum value, and the y-coordinate is given as follows:

1 > -14, hence the correct statement is given as follows:

Function 1 has the larger maximum at (4, 1).

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To conduct the science experiment, the temperature needs to be decreased from 36°C at a rate of 4°C every hour.

So, after 1 hour, the temperature will be 36°C - 4°C = 32°C. After 2 hours, the temperature will be 32°C - 4°C = 28°C. Continuing this pattern, after 10 hours, the temperature will be 36°C - (4°C x 10) = 36°C - 40°C = -4°C.

However, this temperature is below freezing and unlikely to be accurate for a science experiment.

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The expression given in a simplified form is n³⁰

Index forms are described as those mathematical forms that are used to write numbers that are too large or small in more convenient forms.

They are also expressed as a number or variable that is raised to an exponents.

Index forms are also referred to as standard forms or scientific notations.

Following the rules of index forms, we have that;

From the information given, we have that;

(n⁵)⁶

Multiply the exponents

n³⁰

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$23,000 is more than the $19,000 needed for tuition and fees for the first two years, the minimum amount the student should deposit in the savings account every month is $300 (option C).

The minimum monthly deposit the student should make in order to save enough money for her first two years of college tuition and fees in four years, we need to calculate how much she would need to have in her savings account at the end of four years.

Since the student estimated she needs $9,500 per year for tuition and fees, she will need $19,000 for her first two years. Since she already has $5,000 in her savings account, she needs to save an additional $14,000 in four years.

Looking at the table, we can see that the account value in five years with a monthly deposit of $200 is $17,000. To find out how much the account value would be in four years, we need to calculate the future value of $17,000 with a 4-year time frame and an annual interest rate of 0%, which gives:

Future value = $17,000 x (1 + 0%)^(4 x 12/12) = $17,000

Since the account value with a $200 monthly deposit is only $17,000 after 5 years, which is not enough to cover the $19,000 needed for tuition and fees for the first two years, the student needs to make a higher monthly deposit.

Looking at the table again, we can see that the account value in five years with a monthly deposit of $300 is $23,000. To find out how much the account value would be in four years, we need to calculate the future value of $23,000 with a 4-year time frame and an annual interest rate of 0%, which gives:

Future value = $23,000 x (1 + 0%)^(4 x 12/12) = $23,000

$23,000 is more than the $19,000 needed for tuition and fees for the first two years, the minimum amount the student should deposit in the savings account every month is $300 (option C).

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The shape of the cross section of a right rectangular pyramid sliced vertically (down) by a plane not passing through the vertex of the pyramid m is a trapezoid. (A)

This is because when a pyramid is sliced vertically, the resulting cross section is always a two-dimensional representation of the pyramid's base.

Since the base of a right rectangular pyramid is a rectangle, slicing it vertically will result in a trapezoid-shaped cross section. The top and bottom sides of the trapezoid will be parallel, and the other two sides will be slanted.

In a right rectangular pyramid, the vertex m is located directly above the center of the rectangle base. When a plane is passed through this vertex, it will result in a triangular cross section. However, when a plane is passed through a different point, as described in the question, it will result in a trapezoidal cross section.(A)

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The inequality that is true is Negative 2 and three-fourths less-than negative 1 and one-half.

The first inequality from the number line can be shown to be :

( 1 / 4 ) < - 1 1 / 2

This is not possible as a negative cannot be larger than a positive.

The second inequality is:

- 2. 75 < - 1. 5

This is true as larger negative numbers are lower than smaller negative numbers.

The third inequality is:

- 2. 25 > - 1. 25

This is not possible for the reason explained.

In conclusion, option B is correct.

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The amount in millimeters of lace they have left after finishing the quilts is 400 millimeters.

Determine how much lace is needed for each quilt. Each quilt has a perimeter that we need to cover with lace. The perimeter is the sum of all sides of a rectangle, which is (2 x width) + (2 x length).

Each quilt is 2.2 meters wide and 2.7 meters long. So, the perimeter of one quilt is (2 x 2.2) + (2 x 2.7) = 4.4 + 5.4 = 9.8 meters.

Since there are 2 quilts, the total lace needed for both quilts is 9.8 meters x 2 = 19.6 meters.

They bought a 20-meter roll of lace. To find out how much lace is left, subtract the total lace used from the initial length of the roll: 20 meters - 19.6 meters = 0.4 meters.

To convert this to millimeters, multiply by 1,000: 0.4 meters x 1,000 = 400 millimeters.

So, Franklin and his aunt have 400 millimeters of lace left after finishing the quilts.

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The difference between the abscissa of P and Q is 1.

The abscissa of a point is its x-coordinate, or horizontal distance from the origin (usually measured along the x-axis).

In the given problem, the abscissa of point P is -2, which means it is located 2 units to the left of the origin on the x-axis. The abscissa of point Q is -3, which means it is located 3 units to the left of the origin on the x-axis.

To find the difference between the abscissas of P and Q, we simply subtract the abscissa of Q from the abscissa of P:

(abscissa of P) - (abscissa of Q) = (-2) - (-3) = -2 + 3 = 1

Therefore, the difference between the abscissas of P and Q is 1 unit.

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At the end of the weekend, Elisa had the most biscuits (168). So, the answer is (E) Elisa baked the most biscuits on Saturday.

To determine who baked the most biscuits on Saturday, we need to calculate how many biscuits each child had at the end of the weekend.

Anna had 24 biscuits, Berta had 25, Charlie had 26, David had 27, and Elisa had 28.

Let's start with the child who had twice as many biscuits as on Saturday. We can divide their total number of biscuits by 2 to get the number they had on Saturday.

If we try this calculation for each child, we find that only Elisa's total number of biscuits (28) is evenly divisible by 2. Therefore, Elisa must be the child who had twice as many biscuits as on Saturday, meaning she had 14 biscuits on Saturday.

We can use a similar process to determine how many biscuits each child had on Saturday:

- The child who had three times as many biscuits as on Saturday must have had a total of 42 biscuits, which means they had 14 biscuits on Saturday.
- The child who had four times as many biscuits as on Saturday must have had a total of 56 biscuits, which means they had 14 biscuits on Saturday.
- The child who had five times as many biscuits as on Saturday must have had a total of 70 biscuits, which means they had 14 biscuits on Saturday.
- The child who had six times as many biscuits as on Saturday must have had a total of 84 biscuits, which means they had 14 biscuits on Saturday.

Now we can add up the number of biscuits each child had on Saturday:

- Anna had 24 biscuits.
- Berta had 25 biscuits.
- Charlie had 26 biscuits.
- David had 27 biscuits.
- Elisa had 14 biscuits.

Therefore, David baked the most biscuits on Saturday with 27.
To determine who baked the most biscuits on Saturday, we need to consider the information given about the multiplication factors (twice, 3 times, 4 times, 5 times, and 6 times) and the initial number of biscuits baked by each child.

1. Anna baked 24 biscuits.
2. Berta baked 25 biscuits.
3. Charlie baked 26 biscuits.
4. David baked 27 biscuits.
5. Elisa baked 28 biscuits.

Now, let's apply the multiplication factors and see which child had the most biscuits at the end of the weekend:

1. Anna: 24 x 2 = 48
2. Berta: 25 x 3 = 75
3. Charlie: 26 x 4 = 104
4. David: 27 x 5 = 135
5. Elisa: 28 x 6 = 168

At the end of the weekend, Elisa had the most biscuits (168). So, the answer is (E) Elisa baked the most biscuits on Saturday.

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c = number of chickens

d = number of ducks

c = 4d because there are 4 times as many chickens as ducks

c = d+72 because there are 72 more chickens

4d = d+72 after using substitution

4d-d = 72

3d = 72

d = 72/3

d = 24

c = 4d = 4*24 = 96  ...or...  c = d+72 = 24+72 = 96

Rounded to the nearest cubic centimeter, the volume of the regular hexagonal prism is approximately 82 [tex]cm^3.[/tex]

To calculate the volume of the regular hexagonal prism, we need to find the area of the base and multiply it by the height.

The base of the prism is a regular hexagon with side length 3 centimeters. The formula for the area of a regular hexagon is:

[tex]Area = (3√3/2) * (side length)^2.[/tex]

Substituting the given side length of 3 centimeters:

[tex]Area = (3√3/2) * 3^2[/tex]

= (3√3/2) * 9

= (27√3/2).

Now, let's calculate the volume by multiplying the base area by the height:

Volume = Area * height

= (27√3/2) * 7

≈ 81.729[tex]cm^3[/tex].

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Step-by-step explanation:

since both spins are independent events (one does not have any impact on the other), the sequence does not matter. the probability of first blue and then gray is the same as first gray and then blue.

it is the probability of getting 1 gray and 1 blue result.

a probability is always the ratio

desired cases / totally possible cases.

since 6/10 of the area of the spine are gray, and 4/10 of the area are blue, the probability for any single spin to result in gray is 6/10 = 3/5 = 0.6.

and the probability to result in blue is 4/10 = 2/5 = 0.4

the probability to get 1 gray and 1 blue is then the product of both probabilities :

0.6 × 0.4 = 0.24

it is like rolling a die twice and asking for e.g. two 6s or any other combination of 2 specific numbers. that probability is

1/6 × 1/6 = 1/36

FYI :

the probability of getting gray twice is then

0.6 × 0.6 = 0.36

the probability of getting blue twice is

0.4 × 0.4 = 0.16

as mentioned to get gray first and blue second is

0.6 × 0.4 = 0.24

to get blue first and gray second

0.4 × 0.6 = 0.24

and these are all the possible results you can get in 2 spins.

therefore, the probability for any of them is

0.36 + 0.16 + 0.24 + 0.24 = 1

The minimum amount of glass needed to construct the Louvre pyramid is approximately 3133 square meters.

The surface area of a square base pyramid can be calculated by adding the area of the base to the sum of the areas of the four triangular faces.

The area of the square base is simply the width of the base squared:

Area_base = (35 m)^2 = 1225 square meters

The area of each triangular face can be calculated using the formula:

Area_triangle = (1/2) * base * height

For the Louvre pyramid, the base of each triangular face is equal to the width of the base of the pyramid, which is 35 meters. The height of each triangular face can be found using the Pythagorean theorem:

[tex]height^2 = (1/2 * width)^2 + height^2[/tex]

[tex]height = sqrt((1/2 * 35 m)^2 + (20.6 m)^2)[/tex]

height = 29.2 m

Therefore, the area of each triangular face is:

Area_triangle = (1/2) * (35 m) * (29.2 m) = 508.5 square meters

The total surface area of the pyramid is:

Surface_area = Area_base + 4 * Area_triangle

Surface_area = 1225 + 4 * 508.5

Surface_area = 3133 square meters

Since the pyramid is made of glass, we need to calculate the minimum amount of glass needed to construct it. Assuming the glass is thin and flat, we can simply use the surface area of the pyramid to estimate the amount of glass needed.

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The following table shows the weight of a cat in pounds and the corresponding number of cans of cat food to be given each day.

To complete the table, we need to use the information that the directions on the small cans of cat food say to feed a cat 1 can of food each day for every 4 pounds of body weight.

For example, for a cat weighing 4 pounds, we need to give 1 can of food each day.

For a cat weighing 5 pounds, we need to give more than 1 can but less than 2 cans of food each day.

To find the exact number of cans, we can use the formula:

cans per day = weight in pounds / 4

Substituting the given values, we get:

cans per day = 5 / 4

cans per day = 1.25

Therefore, for a cat weighing 5 pounds, we need to give 1.25 cans of food each day. We can round this to the nearest tenth to get 1.3 cans per day.

Similarly, we can use the formula to complete the rest of the table:

KIT-E-KAT weight in pounds cans per day

4   1

5   1.3

6   1.5

7   1.8

8   2

9   2.3

10  2.5

11   2.8

12  3

13  3.3

14  3.5

15  3.8

Therefore, the completed table shows the weight of a cat in pounds and the corresponding number of cans of cat food to be given each day.

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If each bag of hot chips requires around 500 ml of soda to drink with, then 46,000 ml of soda would be needed to buy 20 bags of 4 bags of hot chips and 3 bottles of soda.

To determine how much soda is needed if you buy 20 bags of 4 bags of hot chips and 3 bottles of soda, we need to know the volume of each bottle of soda.

Assuming each bottle of soda contains 2 liters of soda, the total volume of soda needed can be calculated as follows:

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To find the probability that the number of successes x is exactly a certain value in a binomial distribution with n trials and a probability of success of p, we can use a binomial probability table. The table will provide us with the probability of getting x successes out of n trials, given a specific value of p.

For example, let's say we want to find the probability of getting exactly 3 successes in a binomial distribution with 10 trials and a probability of success of 0.5. We can use a binomial probability table to find the probability of getting exactly 3 successes, which is 0.117.

It is important to note that the probability of getting a specific number of successes in a binomial distribution is dependent on both the number of trials and the probability of success. Therefore, if we change either of these values, the probability of getting a certain number of successes will also change.

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The unknown number x = (6.1 ± sqrt(6.

Let's call the two numbers x and y.

We are given:

AM (Arithmetic Mean) between x and y exceeds their GM (Geometric Mean) by 2:

(x + y)/2 - sqrt(xy) = 2

GM between x and y exceeds their HM (Harmonic Mean) by 1.8:

sqrt(xy) - 2xy/(x+y) = 1.8

We can solve for one variable in terms of the other from the second equation and substitute into the first equation to solve for the remaining variable. Let's solve for y in terms of x from the second equation:

sqrt(xy) - 2xy/(x+y) = 1.8

sqrt(xy)(x+y) - 2xy = 1.8(x+y)

sqrt(xy)x + sqrt(xy)y - 2xy = 1.8x + 1.8y

sqrt(xy)(x+y-1.8) = 0.2x + 0.8y

x+y-1.8 = (0.2/0.8)sqrt(xy)(x+y-1.8)

x+y-1.8 = 0.25sqrt(xy)(x+y-1.8)

4x + 4y - 7.2 = sqrt(xy)(x+y)

Now we can substitute this expression for sqrt(xy)(x+y) into the first equation and solve for x:

(x + y)/2 - sqrt(xy) = 2

(x + y)/2 - (4x + 4y - 7.2)/4 = 2

2(x + y) - (4x + 4y - 7.2) = 8

12.2 = 2x + 2y

6.1 = x + y

Now we can substitute x + y = 6.1 into the expression we derived for sqrt(xy)(x+y) to solve for sqrt(xy):

4x + 4y - 7.2 = sqrt(xy)(x+y)

4x + 4y - 7.2 = sqrt(xy)(6.1)

sqrt(xy) = (4x + 4y - 7.2)/6.1

Finally, we can substitute both x + y = 6.1 and sqrt(xy) = (4x + 4y - 7.2)/6.1 into the equation sqrt(xy) - 2xy/(x+y) = 1.8 and solve for y:

sqrt(xy) - 2xy/(x+y) = 1.8

(4x + 4y - 7.2)/6.1 - 2xy/6.1 = 1.8

4x + 4y - 7.2 - 12.2xy = 11.38

4x + 4y - 11.38 = 12.2xy

4x + 4(6.1 - x) - 11.38 = 12.2xy (substituting x + y = 6.1)

xy = 4.08

Now we know that xy = 4.08, and we can use this to solve for x and y:

y = 6.1 - x

xy = 4.08

x(6.1 - x) = 4.08

6.1x - x^2 = 4.08

x^2 - 6.1x + 4.08 = 0

We can solve for x using the quadratic formula:

x = (6.1 ± sqrt(6.

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The line intersects the plane at the point (-4, 15, -6).

The line (x, y, z) = (2, -3, 0) + k(-1, 3, -1) can be expressed parametrically as:

x = 2 - k

y = -3 + 3k

z = k

The plane [x, y, 2] = [4, -15, -8] + s[1, -3, 1] + t[2, 3, 1) can be expressed in scalar form as:

x + y - 2z = 14 + s - 2t

To find the intersection of the line and the plane, we can substitute the parametric equations of the line into the scalar equation of the plane:

(2 - k) + (-3 + 3k) - 2k = 14 + s - 2t

Simplifying this equation, we get:

-4k + 3 = 14 + s - 2t

We can also express the line and plane equations in vector form:

Line: r = (2, -3, 0) + k(-1, 3, -1) = (2-k, -3+3k, k)

Plane: r = (4, -15, -8) + s(1, -3, 1) + t(2, 3, 1) = (4+s+2t, -15-3s+3t, -8+s+t)

To find the intersection of the line and the plane, we need to find the values of k, s, and t that satisfy both equations simultaneously. We can do this by equating the vector forms of the line and plane and solving for k, s, and t:

2 - k = 4 + s + 2t

-3 + 3k = -15 - 3s + 3t

k = -8 - s - t

Substituting k into the first equation, we get:

2 + 8 + s + t = 4 + s + 2t

Simplifying this equation, we get:

t = 4

Substituting t = 4 and k = -8 - s - t into the second equation, we get:

-3 + 3(-8 - s - 4) = -15 - 3s + 3(4)

Simplifying this equation, we get:

s = -2

Substituting t = 4 and s = -2 into the first equation, we get:

k = 8 - s - t = 8 + 2 - 4 = 6

Therefore, the line intersects the plane at the point (x, y, z) = (2, -3, 0) + 6(-1, 3, -1) = (-4, 15, -6).

So the line intersects the plane at the point (-4, 15, -6).

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The quotient is -58x2 + 131x - 234 with a remainder of -5898.To solve this problem, we need to use long division. The dividend is 223 + 3x2 + 5x - 4 and the divisor is 22 + 2 + 1, which simplifies to 25.

We start by dividing 2 into 22, which gives us 11. We then write 11 above the 2 and multiply it by 25, which gives us 275. We subtract 275 from 223, which gives us -52. We bring down the 3, which gives us -523. We then repeat the process by dividing 2 into 52, which gives us 26. We write 26 above the 5 and multiply it by 25, which gives us 650. We subtract 650 from -523, which gives us -1173. We bring down the 1, which gives us -11731. We divide 2 into 117, which gives us 58.

We write 58 above the x and multiply it by 25, which gives us 1450. We subtract 1450 from -1173, which gives us -2623. We bring down the -4, which gives us -26234. We divide 2 into 262, which gives us 131. We write 131 above the 5 and multiply it by 25, which gives us 3275. We subtract 3275 from -2623, which gives us -5898. Therefore, the quotient is -58x2 + 131x - 234 with a remainder of -5898.

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The possible coordinates of point A are (-9, -7) and (-9, 11).

Coordinates refers to a set of numbers that are used to identify the position of a point in a space, usually defined by an x-axis, y-axis, and in sometimes a z-axis.

Let the y-coordinate of point A be y. Then the coordinates of point A are (-9, y).

Using the distance formula, we have:

√[(3 - (-9))² + (2 - y)²] = 15

Simplify the equation:

√[(12)² + (2 - y)² = 15

Square both sides of the equation, we get:

(12)² + (2 - y)² = 15²

144 + (2 - y)² = 225

(2 - y)² = 225 - 144

(2 - y)² = 81

We now take the square root of both sides:

2 - y = ±9

Solve for y in each case, we get:

y = 2 - 9 = -7 or y = 2 + 9 = 11

Therefore, the possible coordinates of point A are (-9, -7) and (-9, 11).

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