Calculate the break even number of units if the fixed expenses are $7,000 and the contribution margin is $14/unit.

Answer:

2/3 * 660

Step-by-step explanation:

If we convert 65% to a decimal, we get 0.65.

First, it will helpful to find exactly what is 65% of 668:  

0.65 * 668 = 434.2 and thus 65% of 668 is exactly 434.2.

Similarly, if you convert 2/3 to a decimal, you get 0.66 repeating.

2/3 * 660 = 440 which is only 10.2 units larger than 434.2.  Thus, 2/3 * 660 is the most accurate estimate of 65% of 668.

By algebra properties, the simplified form of the rational equation 3 / (x + 1) - 1 / (x - 1) - 2 · x / (x² - 1) is equal to - 4 / (x² - 1). (Correct choice: C)

In this question we find a rational equation, whose simplified form must be found by algebra properties. First, write the entire expression:

3 / (x + 1) - 1 / (x - 1) - 2 · x / (x² - 1)

Second, factor all denominators:

3 / (x + 1) - 1 / (x - 1) - 2 · x / [(x + 1) · (x - 1)]

Third, use sum of fractions with different denominators:

[3 · (x - 1) - (x + 1) - 2 · x] / [(x + 1) · (x - 1)]

Fourth, simplify the resulting expression:

(3 · x - 3 - x - 1 - 2 · x) / [(x + 1) · (x - 1)]

- 4 / [(x + 1) · (x - 1)]

- 4 / (x² - 1)

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The Fourier series representation of X(t) is:

X(t) = -1 for 0 ≤ t ≤ T/2

= 1 for T/2 < t < T

To solve the Fourier series equation for the given periodic function X(t), we can follow these steps:

Step 1: Determine the fundamental frequency (ω0) and period (T) of the function X(t).

The period of X(t) is given as T.

The fundamental frequency (ω0) can be calculated as ω0 = 2π / T.

Step 2: Express X(t) as a Fourier series using complex exponential form.

The Fourier series representation of X(t) can be written as:

X(t) = A0 + Σ[Ak × cos(kω0t) + Bk × sin(kω0t)]

Since X(t) is an even function, all the sine terms will be zero. So, the series simplifies to:

X(t) = A0 + Σ[Ak × cos(kω0t)]

Step 3: Calculate the Fourier coefficients (Ak) for k = 0, 1, 2, ...

The Fourier coefficients can be calculated using the following formulas:

A0 = (1/T) × ∫[X(t)]dt

Ak = (2/T) × ∫[X(t) × cos(kω0t)]dt, for k > 0

Step 4: Calculate the Fourier coefficients A0 and Ak for the given function X(t).

Since X(t) is a piecewise function, we need to evaluate the integrals separately for each interval.

For 0 ≤ t ≤ T/2:

A0 = (1/T) × ∫[-2]dt = (1/T) × [-2t] (integrating from 0 to T/2)

= -2/T × (T/2 - 0) = -1

Ak = (2/T) × ∫[-2 × cos(kω0t)]dt = (2/T) × [-2/kω0 × sin(kω0t)] (integrating from 0 to T/2)

= -4/(kπ) × [sin(kπ) - sin(0)]

= -4/(kπ) × [0 - 0]

= 0

For T/2 < t < T:

A0 = (1/T) × ∫[2]dt = (1/T) × [2t] (integrating from T/2 to T)

= 2/T × (T - T/2) = 1

Ak = (2/T) × ∫[2 × cos(kω0t)]dt = (2/T) × [2/kω0 × sin(kω0t)] (integrating from T/2 to T)

= 4/(kπ) × [sin(kπ) - sin(kπ/2)]

= 4/(kπ) × [0 - (-1)^k]

= (-1)^(k+1) × (4/(kπ))

Step 5: Write the final Fourier series representation of X(t) using the calculated coefficients.

Since Ak = 0 for all k, except when k = 0, the series simplifies to:

X(t) = A0

= -1 for 0 ≤ t ≤ T/2

= 1 for T/2 < t < T

Therefore, the Fourier series representation of X(t) is:

X(t) = -1 for 0 ≤ t ≤ T/2

= 1 for T/2 < t < T

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The probability that Ms. Browning will select a blue marker both times is 1/9.

To calculate the probability that Ms. Browning will select a blue marker both times, we need to consider the total number of possible outcomes and the number of favorable outcomes.

Ms. Browning has 3 markers in the box: 1 red, 1 blue, and 1 yellow. Since she selects a marker, uses it, and puts it back, the total number of possible outcomes for the first selection is 3.

For each outcome of the first selection, Ms. Browning again has 3 markers to choose from for the second selection.

Therefore, the total number of possible outcomes for the second selection is also 3.

To calculate the probability of selecting a blue marker both times, we need to determine the number of favorable outcomes. There is only 1 blue marker in the box, and Ms. Browning needs to select it twice.

Therefore, the number of favorable outcomes is 1.

The probability of an event is given by the ratio of favorable outcomes to total outcomes.

In this case, the probability of selecting a blue marker both times is:

Probability = (Number of favorable outcomes) / (Number of total outcomes)

= 1 / (3 × 3)

= 1 / 9

Therefore, the probability that Ms. Browning will select a blue marker both times is 1/9.

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There are 10% of the 60 students which is 6 students who are 18 years old or younger.

To get number of students who are 18 years old or younger, we have to determine percentage of students who fall into this category.

The percentage of students who are 18 years old or younger will be derived by subtracting the sum of the percentages of students who are 19 years old, 30 years old and over 20 years old from 100%.

The percentage of students who are 18 years old or younger is:

= 100% - (20% + 30% + 40%)

= 100% - 90%

= 10%.

Translated question:

A university course has 60 students. If 20% are 19 years old, 30% 30 and 40% over 20 years old, how many students can there be 18 years old or younger?

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Answer:

x>-30

Step-by-step explanation:

1x+31>1

x+31>1

Here, since all the given distances are in yards, you can use 1 yard as 1 unit on the number line. The number line will need to be long enough to include -4.1, -3.5, and 3.5.

Zero on this number line represents the level of the boat. All positive numbers represent points above the boat and all negative numbers represent points below the boat.

The absolute value of a number is its distance from zero on the number line. In this case:

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The correct statement regarding the quadratic function f(x) = (x - 4)(x + 1) is given by:

It is increasing for x < -1 and x > 4.

We have,

Considering a concave up quadratic function, with positive leading coefficient, we have that:

It is increasing to the left of the lower root and to the right of the upper root.

It is decreasing between the roots.

In this problem, the function is given by:

f(x) = (x - 4)(x + 1).

The roots are:

x - 4 = 0 -> x = 4.

x + 1 = 0 -> x = -1.

Hence the correct statement is:

It is increasing for x < -1 and x > 4.

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question:

The graph of the function f(x) = (x – 4)(x + 1) is shown below. Which statement about the function is true?

The number of angles that are alternate interior angles with angle 9 are 2 angles.

In Mathematics and Geometry, the Alternate Interior Angles Theorem states that when two (2) parallel lines are cut through by a transversal, the pairs of alternate interior angles that are formed are congruent:

By applying the alternate interior angles theorem to parallel lines r and s, we have the following congruent angles:

∠7 ≅ ∠9 (The transversal is line s while the parallel lines are t and u).

∠9 ≅ ∠15 (The transversal is line u while the parallel lines are s and r).

In conclusion, we can logically deduce that there are only 2 angles that are alternate interior angles with angle 9.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

To solve the inequality 8x + 2 < 10x - 4, we can start by subtracting 8x from both sides to get 2 < 2x - 4. Then, we can add 4 to both sides to get 6 < 2x. Finally, we can divide both sides by 2 to get 3 < x. This means that the solution set for the inequality is all values of x that are greater than 3. In interval notation, the solution set is (3, ∞).

Step-by-step explanation:

The answers are a) 1730.485 cm³, b) 1157.33 cm³ and c) 201%

To solve this problem, we need to calculate the volumes of the cylinder and the three balls.

Let's break it down step by step.

a. Volume of the cylinder:

The cylinder's volume can be calculated using the formula V = πr²h, where r is the radius and h is the height.

Since the balls just touch the top, bottom, and sides of the cylinder, we can consider the height of the cylinder to be equal to the diameter of the balls, which is 13 cm.

The radius of each ball is half of the diameter, so the radius of each ball is 13 cm / 2 = 6.5 cm.

Substituting the values into the formula, we get:

V_cylinder = π × (6.5 cm)² × 13 cm

≈ 3.14 × 42.25 cm² × 13 cm

≈ 1730.485 cm³

Rounded to the nearest cubed centimeter, the volume of the cylinder is approximately 1730 cm³.

b. Volume of three balls:

The volume of a sphere can be calculated using the formula

V = (4/3) × πr³.

Since the diameter of each ball is 13 cm, the radius of each ball is 13 cm / 2 = 6.5 cm.

Substituting the values into the formula, we get:

V_ball = (4/3) × π × (6.5 cm)³

≈ 4/3 × 3.14 × 274.625 cm³

≈ 1157.33 cm³

The volume of each ball is approximately 1157 cm³.

Since there are three balls, the total volume of the three balls is:

V_total_balls = 3 × V_ball

≈ 3 × 1157 cm³

≈ 3471 cm³

Rounded to the nearest cubed centimeter, the total volume of the three balls is approximately 3471 cm³.

c. Percentage of the volume occupied by the three balls:

The percentage can be calculated using the formula:

Percentage = (V_total_balls / V_cylinder) × 100

Substituting the values, we get:

Percentage = (3471 / 1730) × 100

≈ 200.69%

Rounded to the nearest whole percent, the three balls occupy approximately 201% of the volume of the container.

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The correct question =

Three balls are packaged in a cylindrical container as shown below. The balls just touch the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm.

a. What is the volume of the cylinder rounded to the nearest cubed centimeter?

b. What is the total volume of the three balls rounded to the nearest cubed centimeter ?

c. What percent of the volume of the container is occupied by the three balls?

There are 40 cubes in the bag.

Using algebra to represent the number of black and white cubes in the original bag.

We know that the ratio of black to white cubes = 3:7,

So we can represent the number of black cubes as 3x

And the number of white cubes as 7x.

After adding 8 black cubes,

The ratio becomes =  5:7.

This means that the new number of black cubes is 5x,

And the new number of white cubes is 7x.

To solve for x,

Use the information that we have about the number of black cubes.

We know that we added 8 black cubes to the original bag,

So we can set up the equation:

⇒ 3x + 8 = 5x

Solving for x, we get:

⇒ 2x = 8

⇒   x = 4

Now that we know that x = 4,

We can find the total number of cubes in the bag,

original number of black cubes + 8 = 3x + 8

= 3(4) + 8

= 20

Total number of cubes = original number of black cubes + original number of white cubes

= 3x + 7x

= 10x

= 10(4)

= 40

Therefore,

The required number of cubes = 40.

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Answer: 0.0493 Chance of Probability

The probability that a car chosen at random is not a red car is 51%

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

Cars = 60%

Red cars = 15%

The probability that a car chosen at random is not a red car is calculated as

P = (1 - Red cars) * Cars

substitute the known values in the above equation, so, we have the following representation

P = (1 - 15%) * 60%

Evaluate

P = 51%

Hence, the probability that a car chosen at random is not a red car is 51%

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Answer:11

Step-by-step explanation:

The area of the triangle is 33 square centimeters.

Remember that for a triangle of base B and height H, the area is given by the formula:

A = B*H/2

So base times the height over two.

Here we can see that the base is B = 6cm, and the height of the triangle is 11cm, so we can replace these values in the formula above so we get:

A = 6cm*11cm/2 = 3cm*11cm = 33cm²

That is the area of the green triangle.

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The zeros of g(x) = (x + 2)(x − 1)(x − 2) are x = -2, 1 and 2

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

The 5 functions of g(x)

To determine the key features of the function g(x), we make use of the first

g(x) = (x + 2)(x − 1)(x − 2)

So, we have

Zeros

This is when the function equals 0

(x + 2)(x − 1)(x − 2) = 0

Evaluate

x = -2, 1 and 2

The y-intercept

This is when the value of x in the function equals 0

g(0) = (0 + 2)(0 − 1)(0 − 2)

Evaluate

g(0) = 4

End behaviour

This is the behavior of the graph of the function as it approaches the ends of the x-axis

So, we have

[tex]\mathrm{as}\:x\to \:+\infty \:,\:g\left(x\right)\to \:+\infty \:,\:\:\mathrm{and\:as}\:x\to \:-\infty \:,\:g\left(x\right)\to \:-\infty \:[/tex]

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The values of θ  for different conditions were calculated with the help of trigonometric functions.

The solution of trigonometric functions is the values of angles where the function value becomes zero.

Given function is:

A) f (θ) = 2 sin θ + √3

f (θ) = 2 sin θ + √3 = 0

2 sin θ = -  √3

sin θ = - √3 / 2

θ = nπ + π/3

where n=1,3,5,7......

So, at θ = nπ + π/3,  pogo stick's spring will be equal to its non-compressed length.

B)If the angle was doubled, the function will look like this,

f (θ) = 2 sin 2θ + √3

2 sin 2θ + √3 = 0

2 sin 2θ = -  √3

sin 2θ = - √3 / 2

θ = nπ/2 + π/6

where  n=1,3,5,7....

if n=1,  θ = 2π/3

n=2,  θ = 7π/6

n=3,  θ = 5π/6

So, These are solution in [0,2π}.

C) y = 2 cos θ + 1

And, y = sin2 θ - 1

Hence, Equate both equation,

2 cos θ + 1 = sin2θ - 1

2 cos θ = 2 sinθ cos θ - 2

cos θ = sinθ cosθ - 1

cosθ (1 - sin θ) = - 1

cosθ = - 1

θ = nπ

where n=1,2,3.....

So, θ = nπ , the lengths of springs from the original pogo stick and the toddler's pogo stick are equal.

Hence, the values of θ  for different conditions were calculated with the help of trigonometric functions.

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Answer:

x=10.5

Step-by-step explanation:

Pythagorean Theorem:

x²+17²=20²

x²+289=400

x²=400-289

x²=111

x=√111

x=10.5356

The angle (to the nearest tenth of a degree) of the terminal side through the point P is approximately 39.2 degrees.

To find the angle θ (in degrees) associated with the point P = (√7/3, √2/3) on the unit circle, we can use the trigonometric identities for sine and cosine.

Let x = √7/3 and y = √2/3 be the coordinates of the point P.

We can see that x = cos(θ) and y = sin(θ) based on the definitions of cosine and sine in relation to the unit circle.

From x = √7/3, we have cos(θ) = √7/3.

From y = √2/3, we have sin(θ) = √2/3.

To find the angle θ, we can take the inverse cosine (cos⁻¹) or inverse sine (sin⁻¹) of the given values.

cos⁻¹(√7/3)= 39.2 degrees (rounded to the nearest tenth of a degree)

sin⁻¹(√2/3) = 62.2 degrees (rounded to the nearest tenth of a degree)

Since we want the angle 0 ≤ θ < 360, we need to determine the correct quadrant for the point P.

The point P = (√7/3, √2/3) lies in the first quadrant of the unit circle (where both x and y are positive), so the angle associated with it is 39.2 degrees.

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The form that most quickly reveals the y intercept is

The y intercept is (0, 21/2)

The y-intercept is the point where a function or a curve intersects the y-axis. In other words, it is the value of the dependent variable (y) when the independent variable (x) is equal to zero.

In the form,  f(x) = 1/2 x² - 5x + 21/2 it is easier to see that eliminating x by plugging in 0 leaves 21/2 which is the y intercept

hence we can easily say that the y intercept is (0, 21/2)

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There are approximately 3.08 × 10²⁵ nanometers in each parsec.

To find out how many nanometers are there in each parsec, we need to first calculate the total number of meters in one parsec, and then convert that value to nanometers.

Given information:

3.26 light years per parsec

1 light year = 9.46 × 10¹⁵ meters

1 nanometer = 10⁻⁹ meters

To convert from light years to parsecs, we'll use the conversion factor: 3.26 light years per parsec.

1 parsec = 1 × (3.26 light years/1 parsec) = 3.26 light years

Now, we'll calculate the total number of meters in one parsec:

1 parsec = 3.26 light years = 3.26 × (9.46 × 10¹⁵ meters/light year)

Parsing the numbers:

3.26 × (9.46 × 10¹⁵) = 3.08 × 10¹⁶ meters

Finally, we'll convert the result to nanometers:

1 meter = 1 × (10⁹ nanometers/1 meter) = 10⁹ nanometers

Therefore, the total number of nanometers in one parsec is:

3.08 × 10¹⁶ meters × (10⁹ nanometers/1 meter) = 3.08 × 10²⁵ nanometers

So, there are approximately 3.08 × 10²⁵ nanometers in each parsec.

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The parent function for the given function is y=[tex]x^\frac{1}{3}[/tex].

The given function is [tex]f(x)=\sqrt[3]{27x-81}[/tex].

The parent function of a family of functions is the simplest one that satisfies the definition of that particular type of function. For example, the simplest linear function would be y = x which is the parent function of the linear family of functions.

The parent function y=[tex]x^\frac{1}{3}[/tex], horizontally shrink by a factor of 27 shift 3 units to the right.

Therefore, the parent function for the given function is y=[tex]x^\frac{1}{3}[/tex].

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Sequence 1,2,4,8,16 represents the ordered pairs that are plotted in the coordinate plane.

To determine the table that represents the ordered pairs plotted in the coordinate plane, we need to analyze the patterns in the sequences given.

Looking at the first sequence in each option:

The sequence is a geometric sequence with a common ratio of 2 (1, 2, 4, 8, 16).

The sequence does not follow a clear pattern (1, 3, 6, 9, 18).

The sequence is a geometric sequence with a common ratio of 2 (1, 2, 4, 8, 16).

The sequence does not follow a clear pattern (1, 3, 6, 9, 18).

Now, let's examine the second sequence in each remaining option:

The second sequence is an arithmetic sequence with a common difference of 4 (4, 8, 12, 16, 20).

Since only option C provides a pattern for the second sequence and matches the geometric pattern for the first sequence, we can conclude that option C represents the ordered pairs plotted in the coordinate plane.

Hence, sequence 1 1,2,4,8,16 sequence 2 4,8,12,16,20  represents the ordered pairs that are plotted in the coordinate plane.

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Being respected is generally considered better than being forced.

We have,

Whether it is better to be forced or respected depends on the context and the specific situation.

In general, being respected is considered more favorable and desirable than being forced.

Respect is earned through positive qualities such as integrity, competence, empathy, and treating others with dignity.

When people respect you, they value your opinions, trust your judgment, and are more likely to cooperate and work together effectively.

On the other hand, being forced implies the use of power, authority, or coercion to make others comply with your demands or actions.

It often leads to negative feelings, and resistance, and can damage relationships.

While there may be situations where force is necessary to maintain order or safety, it is generally preferred to find solutions that are based on mutual understanding, consent, and respect.

Ultimately, fostering an environment of mutual respect and collaboration tends to lead to healthier and more positive outcomes in personal relationships, work settings, and society as a whole.

Thus,

Being respected is generally considered better than being forced.

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The expression (tan x + 1) / (1 - tan x) can be written as tan (x - π/4)

Option D is the correct answer.

We have,

To explain why the expression (tan x + 1) / (1 - tan x) can be written as - tan (x - π/4), we can simplify the expression step by step:

- Start with the given expression: (tan x + 1) / (1 - tan x).

-We can rewrite tan x as sin x / cos x.

- Substitute sin x / cos x in the numerator and denominator:

[(sin x / cos x) + 1] / [1 - (sin x / cos x)].

- Combine the fractions in the numerator:

[(sin x + cos x) / cos x] / [1 - (sin x / cos x)].

- Simplify the denominator by multiplying by the conjugate:

[(sin x + cos x) / cos x] / [(cos x - sin x) / cos x].

- Invert and multiply to divide by a fraction:

[(sin x + cos x) / cos x] * [cos x / (cos x - sin x)].

- Cancel out the common factor of cos x in the numerator and denominator:

(sin x + cos x) / (cos x - sin x).

- Use the identity tan (x - π/4) = (sin x + cos x) / (cos x - sin x).

Therefore,

The expression (tan x + 1) / (1 - tan x) can be written as - tan (x - π/4).

By simplifying the expression, we can see that it is equivalent to:

tan (x - π/4).

Thus,

The expression (tan x + 1) / (1 - tan x) can be written as tan (x - π/4)

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The probability that a student is on the sports team, given that the student plays an instrument, is approximately 35.3% when rounded to one decimal place.

To find the probability that a student is on the sports team given that the student plays an instrument, we can use conditional probability.

Let's denote:

A = Event that a student plays an instrument

B = Event that a student is on the sports team

We are given the following information:

P(A) = 68% = 0.68 (probability of a student playing an instrument)

P(A ∩ B) = 24% = 0.24 (probability of a student playing an instrument and being on a sports team)

The conditional probability P(B|A) represents the probability of event B occurring given that event A has already occurred. It is calculated as:

P(B|A) = P(A ∩ B) / P(A)

Substituting the given values:

P(B|A) = 0.24 / 0.68 ≈ 0.3529

To express the probability as a percentage, we multiply by 100:

P(B|A) ≈ 0.3529 * 100 ≈ 35.3%

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Answer: Made it hard but here you go

Step-by-step explanation:

To find an equation for the curve of best fit, we need to perform a regression analysis on the data. This requires specialized software or tools that I do not have access to. However, based on the data provided, it appears that the value of Ashleigh’s investment is increasing linearly over time. If this trend continues, we can estimate the value of her investment in 2018 by finding the slope of the line and using it to extrapolate the value for 2018.

The slope of the line can be estimated by taking two points from the data and calculating the change in value divided by the change in time. For example, using the data for 2004 and 2009, we get a slope of (457 - 375) / (2009 - 2004) = 82 / 5 = 16.4. This means that the value of Ashleigh’s investment is increasing by approximately $16.4 per year.

Using this slope and the data for 2009, we can estimate the value of her investment in 2018 as follows: 457 + (16.4 * (2018 - 2009)) = 457 + (16.4 * 9) = 457 + 147.6 = $604.6. Therefore, I estimate that the value of Ashleigh’s investment in 2018 is approximately $604.6.

Please note that this is only an estimate and may not be accurate.