Using the same situation you described in #1, now interpret what the part-to-whole ratio 2:5 means in the situation. Use complete sentences in your answer.

The die was rolled 20 times, so the total number of possible outcomes is 20.

The die landed on 5 a total of 4 times, so the probability that a die will land on 5 is 4/20 = 0.2 or 20%.

The die landed on 1 a total of 5 times, so the probability that a die will land on 1 is 5/20 = 0.25 or 25%.

If a probability is unlikely, then the probability is less than 0.5 or 50%. This is because the probability of an event ranges from 0 (impossible) to 1 (certain), so any probability less than 0.5 means that the event is less likely to happen than not.

There are 32 fishes with length between 30 - 40 cm in length

To obtain the frequency value , multiply the frequency density by the upper limit of the frequency bar.

upper limit = 40

frequency density = 0.8

Fishes between 30 - 40 cm long :

Therefore, there are 32 fishes that are between 30 - 40 cm long .

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The coordinates of polygon A'B'C'D' after the rotation of 180º are given as follows:

A'(0,0), B'(-5,-2), C'(-5,5), D'(0,3).

The five more known rotation rules are given as follows:

The original coordinates for this problem are given as follows:

A(0,0), B(5,2), C(5,-5), D(0,-3).

Exchanging the sign of each of the coordinates, the coordinates after the rotation are given as follows:

A'(0,0), B'(-5,-2), C'(-5,5), D'(0,3).

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Answer:

385 envelopes will be needed.

Step-by-step explanation:

y = the total number of envelopes needed.

x =  the number of months

y = 5x + 325

y = 5(12) + 325

y = 60 + 325

y = 385

Helping in the name of Jesus.

The width of the pen is approximately 0.47 feet, and the length would be approximately 10.47 feet.

Let's assume the width of the rectangular dog pen is represented by "w" feet. According to the given condition, the length is 10 feet longer than the width, so the length would be "w + 10" feet.

The area of a rectangle is calculated by multiplying its length by its width. In this case, the area is given as 100 square feet. We can use this information to set up an equation:

[tex]w \times (w + 10) = 100[/tex]

Expanding the equation, we get:

[tex]w^2 + 10w = 100[/tex]

Rearranging the equation and setting it to zero, we have a quadratic equation:

[tex]w^2 + 10w - 100 = 0[/tex]

To solve this equation, we can factorize it or use the quadratic formula. Factoring doesn't yield integer solutions, so we'll use the quadratic formula:

[tex]w = (-b \pm \sqrt(b^2 - 4ac)) / (2a)[/tex]

Plugging in the values, we get:

w = (-10 ± √(10^2 - 4 * 1 * -100)) / (2 * 1)

Simplifying further:

w = (-10 ± √(100 + 400)) / 2

w = (-10 ± √500) / 2

w = (-10 ± 10√5) / 2

w = -5 ± 5√5

Since the dimensions cannot be negative, we take the positive value:

w ≈ 0.47 feet

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The scale factor from shape A to shape B is determined as 0.75.

The value of the missing length x in shape B is 5.25 cm.

The size by which the shape is enlarged or reduced is called as its scale factor.

Scale factor from shape A to shape B is calculated as;

Scale factor =   3/4 = 9/12 = 0.75

The value of the missing length x in shape B is calculated by applying the principle of similar triangles as follows;

the given lengths of shape A and shape B;

A = 4 cm and 7 cm

B = 3 cm and x cm

The value of the missing length x in shape B is calculated as;

x/3 = 7/4

x = (3 x 7/4)

x = 5.25 cm

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Answer:

  745

Step-by-step explanation:

You want the original number of books in the library if adding 298 children's books increased the fraction of children's books from 2/5 to 4/7.

We often like to work problems like this in terms of "ratio units." Here, we'll let x represent the number of books in a ratio unit. This means the number of children's books is originally 2x, and the total number of books is originally 5x. Then we have ...

  (2x +298)/(5x +298) = 4//7

Cross multiplying gives ...

  7(2x +298) = 4(5x +298)

  3(298) = 6x . . . . . . . . . . . . . subtract (14x+4(298))

  149 = x

  745 = 5x . . . . . . the original number of books in the library

There were 745 books in the library at first.

__

Additional comment

If we let x represent the original number of library books, then the arithmetic involves more fractions. You would have an equation like ...

  2/5x +298 = 4/7(x +298)

<95141404393>

Answer:

Step-by-step explanation:

To determine the mean and median of the distribution of masses, we first need to calculate the midpoint of each class interval. The midpoint is found by taking the average of the lower and upper limits of each interval. Then, we multiply the midpoint by the corresponding frequency and sum up the results. Finally, we divide the sum by the total frequency to find the mean.

Here is the calculation:

Class Interval   |  Midpoint (x)  |  Frequency (f)  |  f * x

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

30-34.9               |    32.5               |         5                 |   162.5

35-39.9               |    37.5               |         9                 |   337.5

40-44.9               |    42.5               |         7                 |   297.5

45-49.9               |    47.5               |         6                 |   285

50-54.9               |    52.5               |         4                 |   210

55-59.9               |    57.5               |         4                 |   230

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

Total                      |                                  |       35               |   1,522.5

Mean = (Sum of (f * x)) / (Sum of f) = 1,522.5 / 35 ≈ 43.5 kg

To find the median, we need to arrange the masses in ascending order. The cumulative frequency (cf) is calculated by summing up the frequencies as we move down the list. The median is the value that falls in the middle when the cumulative frequency reaches half of the total frequency.

Arranged Masses: 30, 30, 30, 30, 30, 35, 35, 35, 35, 35, 35, 35, 35, 35, 40, 40, 40, 40, 40, 40, 40, 45, 45, 45, 45, 45, 45, 50, 50, 50, 50, 55, 55, 55, 55

Cumulative Frequency: 5, 14, 21, 27, 31, 35

Since the total frequency is 35, the median will be the value at the (35/2 = 17.5)th position. Since it falls between the 17th and 18th values, we take the average of those two values.

Median = (40 + 40) / 2 = 40 kg

Therefore, the mean of the distribution is approximately 43.5 kg, and the median is 40 kg.

There are 6 numbers from -10 to +10 that can be obtained by adding and using each number or its opposite exactly once.

To find the number of possible sums that can be obtained by adding the positive and negative integers from 1 to 4, we can consider the possible combinations.

The four positive integers are 1, 2, 3, and 4, and their sum is +10.

The four negative integers are -1, -2, -3, and -4, and their sum is -10.

To determine the possible sums, we can combine the positive and negative integers in various ways. Each number or its opposite can be used exactly once.

Here are the possible combinations:

1 + (-2) + (-3) + (-4) = -8

1 + (-2) + (-4) + (-3) = -8

1 + (-3) + (-2) + (-4) = -8

1 + (-3) + (-4) + (-2) = -8

1 + (-4) + (-2) + (-3) = -8

1 + (-4) + (-3) + (-2) = -8

From the given example, we can observe that a sum of -8 can be obtained. Similarly, we can calculate the sums for other combinations.

In total, there are 6 different sums that can be made in this way.

Therefore, if you add and use each number or its opposite exactly once, you can get 6 numbers from -10 to +10.

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a) The sum of the areas of all the squares is 2 unit².

b) The sum of the perimeters of all squares is [tex]\frac{4\sqrt{2} }{\sqrt{2} -1}[/tex] unit.

In Mathematics and Geometry, the area of a square can be calculated by using this mathematical equation (formula);

A = x²

Where:

Area of 1st square, A₁ = 1² unit².

Area of 2nd square, A₂ = (1/√2)² unit².

Area of 3rd square, A₃ = (1/2)² unit².

Therefore, the sum of the areas of all the squares is given by;

Sum of all areas = A₁  + A₂ + A₃ + ...... + Aₙ

Sum of all areas = 1² + (1/√2)² + (1/2)² + .....

Common ratio, r = 1/2 and the first term is 1.

Sum of all areas to infinity = 1/(1 - 1/2)

Sum of all areas to infinity = 2 unit².

Part b.

In Mathematics and Geometry, the perimeter of a square can be calculated by using the following formula;

P = 4x

Perimeter of 1st square, P₁ = 4(1) unit.

Perimeter of 2nd square, P₂ = 4(1/√2) unit.

Perimeter of 3rd square, P₃ = 4(1/2) unit.

Therefore, the sum of the perimeter of all the squares is given by;

Sum of all perimeters = 4(1) + 4(1/√2) + 4(1/2) + .....

Sum of all perimeters = 4(1 + 1/√2 + (1/√2)² + ....)

Common ratio, r = 1/√2 and the first term is 1.

Sum of all perimeters to infinity = [tex]4(\frac{1}{1-\frac{1}{\sqrt{2} } } )[/tex]

Sum of all perimeters to infinity = [tex]\frac{4\sqrt{2} }{\sqrt{2} -1}[/tex] unit.

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Here is your answer.

A tank contains 5832 litres of water.

Total amount of water in tank = 5832 litres.

Each day one-third of the water in the tank is removed and not replaced.

Since one third of the water is removed for six days.

1st day = 1/3rd of 5832 litres is removed

[tex] = \dfrac{5832}{3} = 1944[/tex]

Amount of water left = 5832 - 1944 = 3888 litres

2nd day = 1/3rd of 3888 litres is removed

[tex] = \dfrac{3888}{3} = 1296 [/tex]

Amount of water left = 3888 - 1296 = 2592 litres

3rd day = 1/3rd of 2592 litres is removed

[tex] = \dfrac{2592}{3} = 864 [/tex]

Amount of water left = 2592 - 864 = 1728 litres

4th day = 1/3rd of 1728 litres is removed

[tex] = \dfrac{1728}{3} = 576 [/tex]

Amount of water left = 1728 - 576 = 1152 litres

5th day = 1/3rrd of 1152 litres is removed

[tex] = \dfrac{1152}{3} =384 [/tex]

Amount of water left = 1152 - 384 = 768 litres

6th day = 1/3rrd of 768 litres is removed

[tex] = \dfrac{768}{3} = 256 [/tex]

Amount of water left = 1768 - 256 = 512 litres

So, 512 litres water remains in the tank at the end of 6 days

An equation for the income of Americans in terms of year is y = 0.4x - 792.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (7.8 - 7.4)/(1999 - 1998)

Slope (m) = 0.4/1

At data point (1998, 7.4) and a slope of 0.4, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 7.4 = 0.4(x - 1998)

y = 0.4x - 799.2 + 7.2

y = 0.4x - 792.

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Complete Question:

Personal income of Americans, in billions of dollars 1998 7.4 billion 2000 8.4 billion 1999 7.8 billion 2001 8.7 billion. Find the equation y= mx + b for the income of Americans in terms of year x, using the points (1998, 7.4) , and (1999, 7.8)

Answer:

Compound interest is the interest imposed on a loan or deposit amount.

The most commonly used concept in our daily existence is compound interest.

The compound interest for an amount depends on both Principal and interest gained over periods.

Formula to calculate compound interest.

[tex]{ \boxed{ \rm{C.I = P \bigg(1 + \dfrac{r}{n} \bigg) ^{nt} - P}}}[/tex]

Here,

P is Principal

r is rate of interest

t is total time span.

n is number of times interest got compounded annually.

Let's assume the original mass of the saline water (salt water) is x grams. According to the given information, the mass ratio of salt to water is initially 1:10.

This means the mass of salt in the saline water is (1/11) * x grams, and the mass of water is (10/11) * x grams.

After adding 22 grams of salt, the mass ratio of salt to water becomes 2:9.

This means the mass of salt is now (2/11) * (x + 22) grams, and the mass of water is (9/11) * (x + 22) grams.

Since the mass of salt remains the same in both scenarios, we can set up an equation:

(1/11) * x = (2/11) * (x + 22)

Simplifying the equation:

x = 2 * (x + 22)

x = 2x + 44

x - 2x = 44

-x = 44

x = -44 (disregard this negative value)

Since the value of x is negative, it is not a valid solution in this context.

It seems there may be an error or inconsistency in the given information or calculations. Please double-check the values and ratios provided to ensure accuracy.

Answer:

Step-by-step explanation:

You want to show that the point (4, -4) is on the graph of x² +y² = 32, and you want the tangent and normal lines at that point.

The point is on the curve because when x = 4 and y = -4 the resulting statement is 16 +16 = 32, which is a true statement.

The tangent to a circle is most easily found by first considering the normal. The normal line will be the line through the point on the circle and the center of the circle. Here, the center is (0, 0). The slope of the normal line is ...

  m = y/x = -4/4 = -1

The tangent line's slope is the opposite reciprocal of this:

  m = -1/(-1) = 1

The point-slope form of the equation for the tangent line is ...

  y -k = m(x -h) . . . . . equation for line with slope m through point (h, k)

  y -(-4) = 1(x -4) . . . . . equation for line with slope 1 through point (4, -4)

  x -y = 8 . . . . . . . . . . add 4-y to put in standard form

The equation of the tangent line is x - y = 8.

In the section above, we found the slope of the normal line is -1. We also noted that the normal line goes through the point (0, 0). Then the point-slope form of the normal line is ...

  y -0 = -1(x -0)

  y = -x

The equation of the normal line is y = -x.

__

Additional comment

The normal line can also be written as x + y = 0.

The tangent line can also be written as y = x -8.

<95141404393>

Answer:

Step-by-step explanation:

1st 10 days - 8 GB

2nd 10 days - 8 GB

3rd 10 days - 8 GB

________________

30 days    -  24 GB

By using the minimum and maximum data entries, and the number of classes, Lower Class Limit for the second class: 21.17

Upper Class Limit for the second class: 21.17 + 12.17 ≈ 33.34

The class width is approximately 12.17.

To find the class width, we need to calculate the range of the data, which is the difference between the maximum and minimum values. In this case, the minimum value is 9 and the maximum value is 82.

Range = Maximum value - Minimum value

Range = 82 - 9

Range = 73

Next, we divide the range by the number of classes to determine the class width. In this case, there are 6 classes.

Class Width = Range / Number of classes

Class Width = 73 / 6

Class Width ≈ 12.17

The class width is approximately 12.17.

To find the lower and upper class limits, we start with the minimum value and add the class width to obtain the upper limit for each class. The lower limit for each class is the upper limit of the previous class.

Lower Class Limit for the first class: 9

Upper Class Limit for the first class: 9 + 12.17 ≈ 21.17

Lower Class Limit for the second class: 21.17

Upper Class Limit for the second class: 21.17 + 12.17 ≈ 33.34

We continue this process to determine the lower and upper class limits for each class.

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Answer:

The correct answer is B.

When x increases, y increases, and when x decreases, y decreases.

a) The value of f(5) of the given graph is: Positive

b) The point at which f(x) = 0 is at x = 1.

c) The interval for which the function f(x) ≤ 0 is from the interval: -2 ≤ x ≤ 1

The domain of a function is defined as the set of input values that can be plugged into a function. This set is the x values in a function such as f(x). The range of a function is defined as the set of output values that the function assumes. This set is the values that the function brings out after we plug an x value in.

a) The value of f(5) of the given graph is:

f(5) = 1

Thus, f(5) is positive

b) The point at which f(x) = 0 is the x-intercept which in this case it is at x = 1.

c) The interval for which the function f(x) ≤ 0 is from the interval:

-2 ≤ x ≤ 1

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Based on the information, there are 25 cars with good brakes and good tires.

Total cars = 50

Cars with faulty brakes = 10

Cars with faulty tires = 15

Cars with faulty brakes and good tires = 2

Let's calculate the number of cars with good brakes and good tires:

Cars with faulty brakes or faulty tires = Cars with faulty brakes + Cars with faulty tires - Cars with faulty brakes and good tires

= 10 + 15 - 2

= 25

Cars with good brakes and good tires = Total cars - Cars with faulty brakes or faulty tires

= 50 - 25

= 25

Therefore, there are 25 cars with good brakes and good tires.

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The limit of the function as x approaches -2 is 20.To find the limit of the function (if it exists), we can substitute the given value into the function and simplify. Given function: f(x) = 4x² + 5x - 6x + 2

To find the limit as x approaches -2, we substitute -2 into the function:

f(-2) = 4(-2)² + 5(-2) - 6(-2) + 2
      = 4(4) - 10 + 12 + 2
      = 16 - 10 + 12 + 2
      = 20

Therefore, the limit of the function as x approaches -2 is 20.

To write a simpler function that agrees with the given function at all but one point, we can use a graphing utility. By plotting the given function and observing its behavior, we can create a simpler function that matches the original function except at one point.

However, without further information about the specific behavior of the given function, it is not possible to provide a more detailed explanation or a graph of the simpler function.

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a. The object is momentarily at rest at t = 6.

b. It moves up in the interval (0, 6) and moves down in the interval (6, 12).

c. The object changes direction at t = 6.

d. It speeds up in the interval (0, 6) and slows down in the interval (6, 12).

e. The object does not have a maximum speed within the given interval, and it is moving slowest at t = 0 and t = 12.

f. The object is farthest from the axis origin at t = 6.

To solve the given problem, we start by finding the velocity and acceleration functions based on the position function provided:

Position function: [tex]s = f(t) = 192t - 16t^2[/tex]

a. Velocity function: v(t) = ds/dt = f'(t)

Taking the derivative of the position function with respect to time, we have:

[tex]v(t) = d(192t - 16t^2)/dt = 192 - 32t[/tex]

b. Acceleration function: [tex]a(t) = d^2s/dt^2 = f''(t)[/tex]

Taking the second derivative of the position function with respect to time, we have:

a(t) = d(192 - 32t)/dt = -32

Now, let's answer the given questions:

a. The object is momentarily at rest when its velocity is equal to zero.

Setting v(t) = 0 and solving for t:

192 - 32t = 0

32t = 192

t = 6

b. The object moves up when the velocity is positive (v(t) > 0) and moves down when the velocity is negative (v(t) < 0).

Interval when the object moves up: (0, 6)

Interval when the object moves down: (6, 12)

c. The object changes direction when the velocity changes sign. In this case, it changes direction at t = 6.

d. The object speeds up when the acceleration and velocity have the same sign, and it slows down when the acceleration and velocity have opposite signs.

The object speeds up in the interval (0, 6) and slows down in the interval (6, 12).

e. The object is moving fastest when its velocity is maximum. The velocity function is v(t) = 192 - 32t, and it reaches its maximum when its derivative is equal to zero.

Setting v'(t) = 0 and solving for t:

v'(t) = -32 = 0

There is no solution for t since -32 is a constant.

The object is moving slowest when its velocity is minimum, which occurs at the endpoints of the given interval.

So, the object is moving slowest at t = 0 and t = 12.

f. To find when the object is farthest from the axis origin, we look for the maximum value of the position function.

The position function is [tex]s(t) = 192t - 16t^2,[/tex] and it reaches its maximum value when its derivative is equal to zero.

Setting s'(t) = 0 and solving for t:

s'(t) = 192 - 32t = 0

32t = 192

t = 6

So, the object is farthest from the axis origin at t = 6.

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The correct comparison between the two functions is that option A. They have the same x intercept and same end behavior.

The end behavior of a function is the statement which describes the behavior of the graph of the function towards the end of the X axis.

x-intercept of a function is the x coordinate of the point that the graph touches with the X axis.

y coordinate will be 0 there.

f touches the X axis at (1, 0).

x intercept of f = 1

g passes through the point (1, 0).

So x intercept of g = 1

Both have the same x intercept.

y intercept is the y coordinate where the graph touches the Y axis.

y intercept of f = -3

y intercept of g = -4

y intercept is not the same.

The graph of f is approaching to a constant value as it moves along the X axis.

Similarly, for g, the difference between the values of the function at consecutive points is decreasing and it tends to 0.

So the end behavior is same. Hence the correct option is A.

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Answer:

Step-by-step explanation:

You want the equation representing the relation between Children's tickets and Senior tickets, and the solution to the problem if ...

The augmented matrix for the problem, and its solution, are shown in the attachment.

<95141404393>

1.) Probability of getting 1: 0.12 or 12%.

2.) Probability of getting 2: 0.16 or 16%.

3.) Probability of getting 3: 0.25 or 25%.

4.) Probability of getting 4: 0.18 or 18%.

5.) Probability of getting 5: 0.15 or 15%.

6.) Probability of getting 6: 0.14 or 14%.

To calculate the probability of getting a specific number on a fair die, we divide the number of times that number appears by the total number of rolls. In this case, the die was rolled 100 times, and the results are given as follows:

Number: 1 2 3 4 5 6

Times: 12 16 25 18 15 14

To find the probability of getting each number, we divide the corresponding number of times by the total number of rolls (100):

Probability of getting 1 = 12 / 100 = 0.12.

Probability of getting 2 = 16 / 100 = 0.16.

Probability of getting 3 = 25 / 100 = 0.25.

Probability of getting 4 = 18 / 100 = 0.18.

Probability of getting 5 = 15 / 100 = 0.15.

Probability of getting 6 = 14 / 100 = 0.14.

Therefore, the probabilities of getting each number on a fair die, based on the given results of 100 rolls, are approximately:

1: 0.12 or 12%,

2: 0.16 or 16%,

3: 0.25 or 25%,

4: 0.18 or 18%,

5: 0.15 or 15%,

6: 0.14 or 14%.

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The equation of the level curve is [tex]x^2 + y^2 = 9[/tex].

The given function is [tex]f(x,y) = 64 - x ^{2}  - y ^{2}[/tex].

The equation for the level curve that passes through the point [tex](\sqrt2 ; \sqrt7)[/tex] can be obtained by equating [tex]f(x,y)[/tex] with [tex]f(\sqrt2, \sqrt7)[/tex].

Thus we have, [tex]f(x,y) = 64 - x ^{2}  - y ^{2} = 64 - 2 - 7 = 55[/tex]

This is the value of [tex]f(x,y)[/tex] at the point [tex](\sqrt2 ; \sqrt7)[/tex].

Now let us substitute [tex]f(x,y) = 55[/tex] and solve for [tex]y[/tex]. We get,[tex]64 - x ^{2}  - y ^{2} = 55 \\ \Rightarrow y^2 = 9 - x^2[/tex]

Now, substitute the value of [tex]y^2[/tex] in the point [tex](\sqrt2 ; \sqrt7)[/tex] to get the value of [tex]x[/tex]. Thus,[tex]7 = 9 - x ^{2} + 2 \\ \Rightarrow x ^{2} = 4[/tex]

Therefore, the point [tex](\sqrt2 ; \sqrt7)[/tex] lies on the curve [tex]y^2 = 9 - x^2[/tex] or [tex]x^2 + y^2 = 9[/tex].

Hence, the equation of the level curve is [tex]x^2 + y^2 = 9[/tex].

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To find the exact value of (7π/6) using the unit circle, locate the angle (π/6) on the unit circle in the second quadrant, determine the coordinates, and adjust the y-coordinate to be negative. The final answer is (√3/2, -1/2). This answer is obtained by understanding the reference angle and using the coordinates of the point where the angle intersects the unit circle.

To find the exact value of (7π/6) using the unit circle, follow these steps:

1. Start by understanding the reference angle. The reference angle for (7π/6) is (π/6).

2. Locate the angle (π/6) on the unit circle. This angle lies in the second quadrant.

3. Determine the coordinates of the point where the angle intersects the unit circle. For (π/6), the x-coordinate is √3/2 and the y-coordinate is 1/2.

4. Since (7π/6) lies in the second quadrant, the x-coordinate will remain the same, but the y-coordinate will be negative. Therefore, the coordinates for (7π/6) are (√3/2, -1/2).

5. The exact value of (7π/6) is (√3/2, -1/2).

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Answer:

Step-by-step explanation:

You want the number of tickets sold at prices of $6.50 and $10 if 386 tickets were sold for $3576.50.

The number of $6.50 ticket is (386 -x) if x tickets were sold at $10. The total revenue will be ...

  10x +6.50(386 -x) = 3576.50

  3.50x +2509 = 3576.50 . . . . . simplify

  3.50x = 1067.50 . . . . . . . . . . . subtract 2509

  x = 305 . . . . . . . . . . . . . . . . . . divide by 3.50

  386 -x = 81 . . . . . . . . . find the number of cheap seats

305 tickets were sold for $10; 81 tickets were sold for $6.50.

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a. To calculate the present value of investment A at a discount rate of 21 percent, we need to find the present value of each cash flow and then sum them up. The present value of each cash flow can be calculated using the formula PV = CF / (1 + r)^n, where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of years.

Using the given cash flows and the discount rate of 21 percent, we have:

PV = 11,000 / (1 + 0.21)^1 + 11,000 / (1 + 0.21)^2 + 11,000 / (1 + 0.21)^3 + 11,000 / (1 + 0.21)^4 + 11,000 / (1 + 0.21)^5 + 11,000 / (1 + 0.21)^6

Calculating this expression will give us the present value of investment A.

b. Similarly, to calculate the present value of investment B, we use the same formula and substitute the cash flows for investment B:

PV = 11,000 / (1 + 0.21)^1 + 11,000 / (1 + 0.21)^2 + 11,000 / (1 + 0.21)^3 + 11,000 / (1 + 0.21)^4

c. For investment C, we use the formula with the cash flows for investment C:

PV = 16,000 / (1 + 0.21)^1 + 48,000 / (1 + 0.21)^2

Calculating these expressions will give us the present values of investments B and C, respectively.

The present values of investments A, B, and C are $37,461.97, $29,930.99, and $48,003.16 respectively.

To calculate the present value of the cash flows from every investment, we use the present value of annuity formula: PV = C * [(1 - (1 + r)^-n ) / r], where PV is the present value, C is the constant cash flow per period, r is the discount rate, and n is the number of periods.

(a) Investment A: C = $11,000, r = 21% or 0.21, n = 6 years. PV = $11,000 * [(1 - (1 + 0.21)^-6) / 0.21] = $37,461.97.

(b) Investment B: C = $11,000, r = 21% or 0.21, n = 4 years. PV = $11,000 * [(1 - (1 + 0.21)^-4) / 0.21] = $29,930.99.

(c) Investment C has 2 cash flows: $16,000 at end of year 1 and $48,000 at end of year 3. We calculate their present value separately and sum those up. PV = $16,000 / (1 + 0.21)^1 + $48,000 / (1 + 0.21)^3 = $13,223.14 + $34,780.02 = $48,003.16.

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The table that represents a linear function is (a)

from the question, we have the following parameters that can be used in our computation:

The table of values

By definition;

A linear function is a function that has a constant rate of change

From the options, we have

(a) as x increase by 1, y constantly increase by 4

This means that table (a) is a linear function

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