Find the LCD of the given rational equation

For the given function the value of f(x) when x = 2 is 0.5, option (d) is correct.

The given function f(x) = 2/x² represents a hyperbolic curve with a vertical asymptote at x=0 and a horizontal asymptote at y=0. As x approaches infinity or negative infinity, the value of the function approaches 0.

The given function is:

f(x) = 2 ÷ x²

We are asked to find the value of f(x) when x = 2.

Substituting x = 2, we get:

f(2) = 2 ÷ 2²

= 2 ÷ 4

= 1/2

= 0.5

Alternatively, we can use the fact that 2² = 4 and simplify the expression as follows:

f(x) = 2 ÷ x²

= 2 ÷ 4

= 1/2

Then, substituting x = 2, we get the same result:

f(2) = 1/2

f = 0.5

Hence, option (d) is correct.

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The model predicts that the number of stores will be at a minimum of approximately 8,123 stores in the year 2009, and will increase from there without bound.

A cubic function is a type of polynomial function with the highest degree term being 3. In other words, a cubic function is a function of the form[tex]f(x) = ax^3 + bx^2 + cx + d[/tex], where a, b, c, and d are constants. The graph of a cubic function is a curve that can take on various shapes depending on the values of the coefficients.

b) To find the predicted number of stores in 2020, we need to find the value of the function when x = 20 (since 2020 is 20 years after 2000). We can substitute x = 20 into the function and evaluate:

[tex]y = 10.4x^3 - 403x^2 + 5585x - 8695\\y = 10.4(20)^3 - 403(20)^2 + 5585(20) - 8695\\y = 140,385[/tex]

Therefore, the model predicts that the number of stores in 2020 will be 140,385.

c) To determine if the number of stores ever decreases over the period of years ending in 2020 according to the model, we can look at the behavior of the function. The function is a cubic function with a positive leading coefficient, which means it will have a global minimum (i.e., the lowest point on the graph) and then increase without bound as x approaches positive or negative infinity.

To find the global minimum, we can take the derivative of the function and set it equal to zero:

[tex]y' = 31.2x^2 - 806x + 5585\\0 = 31.2x^2 - 806x + 5585[/tex]

x ≈ 8.68 or x ≈ 22.12

The first solution, x ≈ 8.68, is within the range of interest (2000 to 2020), so we can use it to find the global minimum:

[tex]y = 10.4x^3 - 403x^2 + 5585x - 8695[/tex]

y ≈ 8,123

Therefore, the model predicts that the number of stores will be at a minimum of approximately 8,123 stores in the year 2009, and will increase from there without bound. So according to the model, the number of stores never decreases over the period of years ending in 2020.

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1. Find the vertex (turning point) of the parabola by using the formula x = -b/2a, where a = 4 and b = 8. This gives x = -1, which is the x-coordinate of the vertex. To find the y-coordinate, substitute x = -1 into the equation: y = 4(-1)² + 8(-1) + 3 = -1.

2. Plot the vertex at the point (-1, -1).

3. To find the x-intercepts, set y = 0 in the equation and solve for x. This gives x = (-8 ± √(8² - 4(4)(3)))/(2(4)) = (-8 ± √16)/8 = -1/2 or -3. Plot these points on the x-axis.

4. To find the y-intercept, set x = 0 in the equation and solve for y. This gives y = 3, so the y-intercept is at the point (0, 3).

5. Since the coefficient of x² is positive, the parabola opens upwards. Sketch the curve passing through the points found above.

6. Label the points of intersection and the turning point on the graph.

Answer:

Time = Distance / Speed

Time is taken from home to park = d / 10

Abel cycled back from the park to his home at an average speed of 8 km/h. The time taken for this part of the journey can be calculated using the same formula:

Time = Distance / Speed

Time is taken from park to home = d / 8

According to the given information, the total time taken for the whole journey is 1 1/5 hours, which is equivalent to 6/5 hours.

Total time is taken = Time from home to park + Time from park to home

6/5 = d/10 + d/8

To simplify the equation, let's find the least common multiple (LCM) of 10 and 8, which is 40:

(6/5) * 40 = (d/10) * 40 + (d/8) * 40

48 = 4d + 5d

48 = 9d

d = 48/9

d = 16/3 km

Now, to find the time taken from the park to Abel's home, we substitute the distance value:

Time from park to home = (16/3) / 8

Time from park to home = (16/3) * (1/8)

Time from park to home = 16/24

Time from park to home = 2/3 hour

Since 2/3 of an hour is equal to 40 minutes, Abel took 40 minutes to cycle from the park to his home.

Therefore, Abel took 40 minutes (or 2/3 of an hour) to cycle from the park to his home.

Answer:

x =  7, y = 3, z = -4

Step-by-step explanation:

[tex]\sf V_{3 Cylinders} =2.8274(in^{3} ).[/tex]

From the provided information about the ice cubes (height and diameter), it's safe to assume that they are cylinder shaped ice cubes. They could be circular cone ice cubes, but the safest assumption is cylinders. Now, to find the volume of a cylinder we use the following formula:

[tex]\sf V_{Cylinder} =\pi r^{2} h[/tex] or [tex]\dfrac{\pi d^{2} h}{4}[/tex].

For the fist equation, "r" stands for radius, and "h" for height.

On the alternate equation, "d" stands for diameter.

Since we're given the diameter and height, let's use the alternate equation:

[tex]\sf V_{Cylinder} =\dfrac{\pi (1(in))^{2} 1.2(in)}{4}=\dfrac{3}{10} \pi (in^{3} )=0.9425(in^{3} ).[/tex]

This can be done by just multiplying the volume of a single cube ([tex]\dfrac{3}{10} \pi (in^{2} )[/tex]) by 3.

[tex]\sf V_{3 Cylinders} =(3)\dfrac{3}{10} \pi (in^{3} )=\dfrac{9}{10} \pi (in^{3} )=\boxed{\sf 2.8274(in^{3} )}.[/tex]

Assuming the ice cubes have the shape of a circular cone, this is the process for calculating the volume of all 3 cubes:

[tex]\sf V_{Cone} =\dfrac{1}{3} \pi r^{2} h\\ \\\\ \\ V_{3Cones} =3*\dfrac{1}{3} \pi r^{2} h\\ \\ \\= \pi r^{2} h\\ \\ \\=\pi (\dfrac{1(in)}{2} )^{2} (1.2(in))\\ \\\\ =\dfrac{3}{10} \pi (in^{3} )=\boxed{\sf 0.9425(in^{3} )}[/tex]

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A)  The continuous rate of growth of this bacterium population = 0.35

B) The initial population of the culture = 920

C) The number of bacteria at time t=5 are 5294

Here, the function [tex]n(t)=920\times e^{(0.35\times t)}[/tex] represents the number of bacteria in a culture at time t

A) We know that in exponential function f(x) = [tex]ae^{kx}[/tex], k is the rate of growth

Here, in this function n(t) = [tex]920\times e^{(0.35\times t)}[/tex],  the continuous rate of growth is 0.35

B) To find the initial population of the culture

Substitute t = 0, in n(t)

n(0) = [tex]920\times e^{(0.35 \times 0 )}[/tex]

n(0) = 920 × e⁰

n(0) = 920

This is the initial population.

C) Now we find the number of bacteria at time t=5

[tex]n(t)=920\times e^{(0.35\times t)}[/tex]

Substitute t = 5 in above equation.

n(5) =[tex]920 \times e^{(0.35 \times 5)}[/tex]

n(5) = 5294.23

n(5) ≈ 5294

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The number of bacteria in a culture is given by the function

n(t) =920e^0.35t Where t is measured in hours

A) what is the continuous rate of growth of this bacterium population? Your answer is __ percent

B) what is the initial population of the culture (at t=0) your answer is __

C) how many bacteria will the culture contain at time t=5 ? Your answer is __ Round to the nearest bacteria

The 99% confidence interval estimate of the population standard deviation is [12.54, 54.62] mi/h.

A confidence interval is the mean of your estimate plus and minus the variation in that estimate. This is the range of values you expect your estimate to fall between if you redo your test.

x = [64, 60, 56, 60, 52, 59, 58, 59, 70, 68]

s = 5.963

n = 10

CI = [(n-1)*s²/chi2_upper, (n-1)*s²/chi2 lower]

CI = [(9*5.963^m²)/19.022, (9*5.963²)/2.700]

CI = [12.54, 54.62]

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The volume of the rectangular prism is given as follows:

V = 0.9375 cubic inches.

The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, according to the equation presented as follows:

Volume = length x width x height.

The dimensions for this problem are given as follows:

Hence the volume is given as follows:

V = 1 x 1.25 x 0.75

V = 0.9375 cubic inches.

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Answer:

8 = 2m + 4

2m = 4, so m = 2

So the equation of this line is y = 2x + 4.

D is the correct answer.

The solutions of the equation tan(3) + 1 = sec(3) in the interval [0, 2) are approximately:

3 ≈ 0.523599 radians or 30 degrees (in the first quadrant)

3 + π ≈ 3.66444 radians or 210 degrees (in the third quadrant)

The equation tan(3) + 1 = sec(3) can be rewritten as:

tan(3) + 1 = 1/cos(3)

Multiplying both sides by cos(3), we get:

sin(3) cos(3) + cos(3) = 1

Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite the left-hand side as:

sin(2*3) + cos(3) = 1

Simplifying this expression, we get:

2sin(3)cos(3) + cos(3) = 1

Factorizing out cos(3), we get:

cos(3)(2sin(3) + 1) = 1

Dividing both sides by 2sin(3) + 1, we get:

cos(3) = 1/(2sin(3) + 1)

We know that sin^2(3) + cos^2(3) = 1, so we can substitute cos^2(3) = 1 - sin^2(3) into the above equation and simplify:

1/(2sin(3) + 1) = cos(3) = sqrt(1 - sin^2(3))

Squaring both sides and simplifying, we get:

(2sin(3) + 1)^2 = 1 - sin^2(3)

Expanding the left-hand side and simplifying, we get:

4sin^3(3) - 3sin^2(3) - 3sin(3) + 1 = 0

This is a cubic equation in sin(3), which can be solved using various methods, such as Cardano's formula or numerical methods. However, the solutions are quite complicated and involve complex numbers.

In the interval [0, 2), we can use a numerical method, such as Newton's method, to find an approximate solution. Starting with an initial guess of sin(3) = 0.5, we can iteratively apply the formula:

sin(3)_n+1 = sin(3)_n - f(sin(3)_n)/f'(sin(3)_n)

where f(sin(3)) = 4sin^3(3) - 3sin^2(3) - 3sin(3) + 1 and f'(sin(3)) = 12sin^2(3) - 6sin(3) - 3.

After several iterations, we find that sin(3) ≈ 0.464758. Substituting this into the equation cos(3) = 1/(2sin(3) + 1), we get cos(3) ≈ 0.885005.

Therefore, the solutions of the equation tan(3) + 1 = sec(3) in the interval [0, 2) are approximately:

3 ≈ 0.523599 radians or 30 degrees (in the first quadrant)

3 + π ≈ 3.66444 radians or 210 degrees (in the third quadrant)

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a)There is higher variability of Y with larger values of X.

b) There is non linear relation between X and Y.

a) Based on the given figure, we can make the following observations:

Therefore, the correct statement is There is higher variability of Y with larger values of X.

b) Based on the features, it is likely that a linear model may not be suitable for these data, and a non-linear model or other regression techniques may be more appropriate to capture the relationship between X and Y accurately.

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To use the change of base formula, we first identify the base of the original logarithm and the desired base of the new logarithm.

The change of base formula is a useful tool in logarithmic functions that allows us to rewrite a logarithm with one base as an equivalent logarithm with a different base. The formula is as follows:

logᵦ a = logᵦ(x) / logₐ(x)

where a is the value being logged, β is the base of the original logarithm, and x is any positive number other than 1.

We then apply the formula by taking the logarithm of the value being logged with the desired base, and dividing it by the logarithm of the same value with the original base.

For example, let's say we want to rewrite the logarithm log₂ 8 in base 10. Using the change of base formula, we have:

log₂ 8 = log₁₀ 8 / log₁₀ 2

The value of log₁₀ 8 is simply 0.9031, and the value of log₁₀ 2 is 0.3010, so we have:

log₂ 8 = 0.9031 / 0.3010 ≈ 3

Therefore, log₂ 8 is equivalent to log₁₀ 8 ≈ 3. By using the change of base formula, we can simplify logarithmic expressions and evaluate them using common logarithms or natural logarithms.

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The given total area including the frame is y = (8+2x)(10+2x).

As per the question the thickness of the frame is x.

The area of a rectangle is given by A=L*B

For a picture with frame it will be A=(L+2t)*(B+2t)

1. Comparing the above equation with the given equation it can be concluded that the Length is 10 and the width is 8.

2. The given equation is 100=(8+2x)(10+2x)

Upon solving it simplifies to [tex]x^2+9x-5=0[/tex]

x=0.525.

Thus the thickness x=0.525.

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Answer:

30 square units

Step-by-step explanation:

The area of a rectangle is its length multiplied by its width.

We can see that the length of Rectangle A is 6 units, and its width is 5 units.

Multiplying these together to get the rectangle's area:

6 units × 5 units = 30 square units

An equation that models the position of the object at time t is:

s(t) = -2cos(2πt/5).

The general form for the equation that will model a wave is:

±a (sin/cos) (2π(x - p)/T)

where:

a is the amplitude

p is the phase shift

T is the period.

The ± will become +ve provided that the graph starts in the positive direction, and the will become -ve provided it starts in the negative direction.

The (sin/cos) will become sine provided the graph starts at 0 before it is being shifted. Then, it becomes cosine provided that the graph starts at the amplitude.

In this case, our graph begins at negative, and the at the amplitude that has no phase shift, the ±ve will become -ve, (sin/cos) will now become cos, and p will become zero. Plugging in the values that were given in the problem, we see that a = 2 and T = 5.

Thus, this equation is: s(t) = -2cos(2πt/5).

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when T = 0, L = 150 (the original number of pages  to read) and when T = 6 hours (time to read the entire book), L = 0 (no pages left to read).

A chart can be defined as a pictorial representation or  diagram that presents data or values ​​in an organized manner. Points on a graph often represent a relationship between two or more things.

We can use a linear equation of the form to describe a function that relates the number of pages read to the time elapsed since the start of reading:

L = mT + b

where L is the number of pages remaining, T is the elapsed time  in hours, m is the page read rate  in pages per hour, and b is the original number of pages remaining when T = 0.  

In this case, we know that Ray is reading at a constant rate of 25 pages per hour, so m = -25 (since we are counting  the remaining pages). We also know that when T = 0 (start of  reading),  150 pages remain unread, so b = 150. Combining everything we get:

L = -25T + 150

This formula describes a function that combines the number of pages being read with the time elapsed since the start of  reading. Note that when T = 0, L = 150 (the original number of pages  to read) and when T = 6 hours (time to read the entire book), L = 0 (no pages left to read).

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Answer:

B

Step-by-step explanation:

take square on both sides

7x-4 = 4x

x = 4/3

The mean amount of electricity generated in Texas by wind-powered turbines from 2007 to 2010 is 4,353 megawatts, to the nearest megawatt.

The data for this question can be found in the following table:

Year Electricity Generated (Megawatts)

2007 4,353

2008 7,113

2009 9,403

2010 10,089

To find the mean, we simply add up the values in the table and divide by the number of values, which is 4.

Mean = (4,353 + 7,113 + 9,403 + 10,089) / 4 = 4,353 megawatts

Therefore, the mean amount of electricity generated in Texas by wind-powered turbines from 2007 to 2010 is 4,353 megawatts, to the nearest megawatt.

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The equation in standard form of the eclipse is expressed as:

D. x²/9 + y²/8 = 1.

The standard form equation for an ellipse with center at the origin, a vertex at (a, 0), and a focus at (c, 0) is:

x²/a² + y²/b² = 1, where c² = a² - b².

In this case, the center is at the origin, a vertex is at (3, 0), and a focus is at (1, 0).

Since the center is at the origin, we know that a vertex is located a distance of a units to the right and left of the center, so a = 3.

Since the focus is located a distance of c units to the right and left of the center, we know that c = 1.

Using c² = a² - b², we can solve for b²:

1² = 3² - b²

b² = 8

So the equation in standard form is:

x²/9 + y²/8 = 1.

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Answer:

[tex]100( {1.05}^{5} ) = 127.63[/tex]

[tex]127.63 - 100 = 27.63[/tex]

The closest answer here is B. $25

Answer:

B is the answer of the given above question

Answer:

| only

Step-by-step explanation:

Thw correct option based on the information given about the parabola will be B. X.

The observed parabola has its vertex situated at (0, 0), declining until (0, 0) then again rising. Its equation falls in the form of f(x) = a(x - 0)² + 0, which simplifies to ax².

In order to detect the value of 'a' we can utilize any point on the parabola itself. Let's take (1, 1):

1 = a(1)²

1 = a

Therefore, the formula for the presented parabola is exsiting in the form of f(x) = x².

Now consider g(x) = (x + 1)² such that it's merely a horizontal movement of function f(x) = x². The vertex of g(x) stands at (-1, 0) and linearly declines until (-1, 1) prior to onching higher again. All this leaves us with option X as the only conceivable graph for g(x).

Therefore, the answer is B) X.

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The total charge if 138 guests attend the conference is $5450

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

Using the above as a guide, we have the following:

Total = ballroom + food * number of guests

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

Total = 620 + 35 * 138

Evaluate

Total = 5450

Hence, the total charges is $5450

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The most common point total scored was 10 with a frequency of 9 or 25% of the total frequency.

To create the score frequency distribution, we have to list the unique values in the data set and count how many times each value appears.

1                                   1                       1/24 =  0.042 = 4.2%

3                                  2                      2/24 = 0.083  = 8.3%

4                                  1                       1/24 = 0.042  = 4.2%

5                                  1                       1/24 = 0.042  = 4.2%

6                                  1                       1/24 = 0.042 = 4.2%

7                                  2                      2/24 = 0.083 = 8.3%

8                                  1                       1/24 = 0.042   = 4.2%

9                                  2                      2/24 = 0.083  = 8.3%

10                                 6                      6/24 = 0.25   = 25%

11                                  1                       1/24 = 0.042 = 4.2%

12                                 2                      2/24 = 0.083 = 8.3%

13                                 2                      2/24 = 0.083 = 8.3%

14                                 2                      2/24 = 0.083  = 8.3 %

                                 ----------                                  

                                    24                                    

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The constant of variation that exists between the weight and the volume of the objects would be = 7/5. That is option C.

A measurement is said to vary directly with another when the increase in proportion of the object leads to a direct decrease to the other one.

That is for the given object;

Weight is directly proportional to Volume

Therefore, weight = k Volume

K = constant of variation

make k the subject of formula;

k= weight/volume

but weight = 28

volume = 20

k = 28/20

= 7/5

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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 1000\\ P=\textit{original amount deposited}\\ r=rate\to 1.3\%\to \frac{1.3}{100}\dotfill &0.013\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus fifty two} \end{array}\dotfill &52\\ t=years\dotfill &3 \end{cases}[/tex]

[tex]1000 = P\left(1+\frac{0.013}{52}\right)^{52\cdot 3} \implies 1000=P(1.00025)^{156} \\\\\\ \cfrac{1000}{(1.00025)^{156}}=P\implies 961.76\approx P[/tex]