Calculate the total amount to be repaid on a simple interest loan of \( \$ 4,000 \) for 3 years at an interest rate of \( 11 \% \). p.a. (in the format \( \$ 0.00 \) )?

The FIFO principle and enter the units with the lower unit cost first in the Cost of Goods Sold Unit Cost column and in the Inventory Unit Cost column.

To determine the cost of goods sold for each sale and the inventory balance after each sale using the first-in, first-out (FIFO) method, we need to follow these steps:

1. Identify the units sold and their respective costs:
  - Start with the units available for sale at the beginning of the year.
  - For each sale, allocate the units sold from the oldest inventory (first-in) at their corresponding cost.

2. Calculate the cost of goods sold (COGS) for each sale:
  - Multiply the number of units sold by their respective cost.
  - This will give you the cost of goods sold for each sale.

3. Update the inventory balance after each sale:
  - Subtract the units sold from the total units available for sale.
  - Multiply the remaining units by their respective cost to get the ending inventory value.

Remember to follow the FIFO principle and enter the units with the lower unit cost first in the Cost of Goods Sold Unit Cost column and in the Inventory Unit Cost column.

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The values of angle are sec(θ) = `√2`csc(θ) = `-√2`tan(θ) = `-1`cot(θ) = `-1`

The given value of θ is θ = -3π/4. We need to determine the exact value of sec(θ), csc(θ), tan(θ), and cot(θ).Solution:sec(θ)equals `1/cos(θ)`csc(θ)equals `1/sin(θ)`tan(θ)equals `sin(θ)/cos(θ)`cot(θ)equals `cos(θ)/sin(θ)`First, we need to find the value of cos(θ) and sin(θ)We know thatθ = -3π/4, hence it lies in the third quadrant. This means the point (-1, -1) lies on the terminal arm of the angle θ.cos(θ) equals `cos(-3π/4)`= `cos(π/4)` = `1/√2`sin(θ) equals `sin(-3π/4)` = `-sin(π/4)` = `-1/√2`Now that we know the values of cos(θ) and sin(θ), we can easily determine the values of sec(θ), csc(θ), tan(θ), and cot(θ).sec(θ) equals `1/cos(θ)`= `1/(1/√2)` = `√2`csc(θ) equals `1/sin(θ)` = `1/(-1/√2)` = `-√2`tan(θ) equals `sin(θ)/cos(θ)` = `(-1/√2)/(1/√2)` = `-1`cot(θ) equals `cos(θ)/sin(θ)` = `(1/√2)/(-1/√2)` = `-1`Therefore,sec(θ) = `√2`csc(θ) = `-√2`tan(θ) = `-1`cot(θ) = `-1`

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The distance between the two parallel streets is 200 feet.


To find the distance between the two parallel streets, we can subtract the frontage of one street from the frontage of the other street. In this case, the frontage on one street is 100 feet and on the other street is 300 feet. Subtracting 100 from 300 gives us 200 feet, which is the distance between the streets.


To find the distance between the two parallel streets, we can subtract the frontage of one street from the frontage of the other street. In this case, the frontage on one street is 100 feet and on the other street is 300 feet. Subtracting 100 from 300 gives us 200 feet, which is the distance between the streets.

This means that the two parallel streets are 200 feet apart. The four-sided lot owned by Amanda lies between these two streets, and it has a total area of 43,000 square feet.

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The exact values of each real number \( y \) when \( y = \arctan(-1) \) are \( y = 135° \) and \( y = 315° \).

The inverse tangent function, denoted as \(\arctan(x)\) or \( \tan^{-1}(x) \), is the angle whose tangent is equal to \( x \). In other words, if we have \( y = \arctan(x) \), it means that \( x = \tan(y) \).

In this case, we have \( y = \arctan(-1) \). The exact value of \( y \), we need to find an angle whose tangent is equal to -1.

Let's consider the unit circle, where the tangent function is defined. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.

On the unit circle, the tangent of an angle is equal to the y-coordinate divided by the x-coordinate of the point where the angle intersects the circle.

For \( y = \arctan(-1) \), we need to find an angle whose tangent is -1.

Since the tangent is negative in the second and fourth quadrants, we need to find an angle in either of those quadrants where the tangent is -1.

In the second quadrant, the tangent is negative for angles between 90° and 180°.

In the fourth quadrant, the tangent is negative for angles between 270° and 360°.

Thus, we can say that \( y = \arctan(-1) \) has two possible solutions:

\( y = 135° \) or \( y = 315° \).

Therefore, the exact values of \( y \) when \( y = \arctan(-1) \) are \( y = 135° \) and \( y = 315° \).

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The thermal conductivity of polyurethane foam is k = 0.02 W/m°C, and that of cork is k = 0.045 W/m°C. A spheroidal vessel made of A1st 316 stainless steel sheet with a diameter of 3 m and a thickness of 9 mm is considered. The inside temperature of the vessel is -80°C, and the outside temperature is 31°C.The thermal conductivity of polyurethane foam is k = 0.02 W/m°C, and that of cork is k = 0.045 W/m°C. Let's now address the given problem:

Calculation of (a):The total thermal resistance of the insulated vessel wall is given by;$$R_{total}=R_{metal}+R_{PU foam}+R_{cork}$$Where, $$R_{metal}=\frac{ln(\frac{r_2}{r_1})}{2πk}$$Here, $$r_1= 3m /2 = 1.5m$$Thickness of the wall, t = 9 mm = 0.009 mOutside diameter of the vessel, $$D_o= r_2 + 2t = 3 m$$$$r_2=D_o-2t=2.982 m$$Hence,$$R_{metal}= \frac{ln(\frac{2.982}{1.5})}{2π*14*10^{-3}}= 0.076 K/W$$Similarly,$$R_{PU foam}=\frac{0.1}{0.02}=5 K/W$$and$$R_{cork}=\frac{0.15}{0.045}=3.33 K/W$$Therefore,$$R_{total}= 0.076 + 5 + 3.33= 8.406 K/W$$Thus, the total thermal resistance of the insulated vessel wall is 8.406 K/W.

Calculation of (b):The rate of heat flow to the vessel is given by the following formula:$$\dot{Q}=\frac{\Delta T}{R_{total}}$$where, $$\Delta T= T_{outside}-T_{inside}=(31-(-80))=111°C$$Thus,$$\dot{Q}= \frac{111}{8.406}=13.209 W$$Hence, the rate of heat flow to the vessel is 13.209 W.Calculation of

(c):The heat flux and temperature at the interface between polyurethane and cork layers are the same. Therefore,$$R_{interface}=\frac{0.1}{0.02+0.045}= 1.429 K/W$$The heat flux at the interface between polyurethane and cork layers can be calculated by;$$q_{interface}= \frac{T_1-T_2}{R_{interface}}$$where, $$T_1=31°C$$and$$T_2=?$$Here,$$q_{interface}= \frac{31-T_2}{1.429}$$or, $$T_2= 31 - 1.429q_{interface}$$The temperature and heat flux at the interface between the polyurethane and cork layers are not given. Thus, their values cannot be computed.

Calculation of (d):Percentage error is given by,$$\% Error = \frac{R_{total(exact)}- R_{total(approx)}}{R_{total(exact)}}×100$$Here,$$R_{total(exact)}= 8.406 K/W$$Since the heat transfer resistance of the metal wall is neglected in the approximation,$$R_{total(approx)}= R_{PU foam}+R_{cork}= 5+3.33= 8.33 K/W$$Thus,$$\% Error= \frac{8.406-8.33}{8.406}×100 =0.905 \%$$Therefore, the percentage error in calculation is 0.905%.

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The types of concurrent constructions that are needed to find the centroid of a triangle include the following: B. intersection of the lines drawn to the midpoint of each side of the triangle to its opposite vertex.

In Mathematics and Geometry, the centroid theorem states that the centroid of a triangle is located at two-third (2/3) of the distance from the vertex to the midpoint of the (opposite) sides.

Generally speaking, the centroid of a triangle simply refers to the point where the three (3) medians of the triangle meet or intersect. This ultimately implies that, a centroid is a point of intersection of the lines from each vertex of a triangle to the midpoint of the opposite sides.

In this context, the types of concurrent constructions would be modeled by answer option B.

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Complete Question:

What types of concurrent constructions are needed to find the centroid of a triangle.

A. intersection of the lines drawn from each vertex of a triangle and perpendicular to its opposite side.

B. intersection of the lines drawn to the midpoint of each side of the triangle to its opposite vertex.

C. intersection of the lines drawn to bisect each vertex of the triangle.

D. intersection of the lines drawn perpendicular to each side of the triangle through its midpoint

In a polygon with 35 sides, you can draw 32 diagonals from one vertex.


To find the number of diagonals, we use the formula n(n-3)/2, where n is the number of sides of the polygon. Plugging in n=35, we get (35)(35-3)/2 = 32 diagonals. To find the number of diagonals from one vertex in a polygon with 35 sides, we can use the formula n(n-3)/2, where n represents the number of sides.

Plugging in n=35, we get (35)(35-3)/2 = 32 diagonals. This formula calculates the number of possible connections between one vertex and the other vertices in the polygon, excluding the sides and the adjacent vertices.

Each diagonal connects the vertex with one of the other 34 vertices, excluding itself and its two adjacent vertices. Therefore, in a polygon with 35 sides, you can draw 32 diagonals from one vertex.

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Answer:

g(x) = (5x + 1)/2.

Step-by-step explanation:

To find the inverse of the function f(x) = (2x-1)/5, we can follow these steps:

Step 1: Replace f(x) with y: y = (2x-1)/5

Step 2: Swap x and y: x = (2y-1)/5

Step 3: Solve the equation for y.

Multiply both sides of the equation by 5 to eliminate the fraction:

5x = 2y - 1

Add 1 to both sides of the equation:

5x + 1 = 2y

Divide both sides of the equation by 2:

(5x + 1)/2 = y

Therefore, the inverse of the function f(x) = (2x-1)/5 is given by g(x) = (5x + 1)/2.

Answer:

-45x + 18x²

Step-by-step explanation:

-9x (5 - 2x)

|

Multiply each term in the brackets by -9x

|

-9x × 5 - 9x × (-2x)

|

Calculate the product

|

(-9x × 5) = -45

(-) & (-) = (+)

9x × 2x = 18x²

|

Solution

|

-45x + 18x²

The slope-intercept equation of the function f is: f(x) = (-3/7)x - 3

To obtain the slope-intercept equation of the function f, which is perpendicular to the line 7x - 3y - 9 = 0 and has the same y-intercept, we need to determine the slope and y-intercept of the provided line.

The provided line equation is in the form

Ax + By + C = 0, where A = 7, B = -3, and C = -9.

To obtain the slope of the provided line, we can rearrange the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

So we solve the equation for y:

7x - 3y - 9 = 0

-3y = -7x + 9

y = (7/3)x - 3

From the equation, we can see that the slope of the provided line is 7/3.

Since the function f is perpendicular to this line, the slope of f will be the negative reciprocal of 7/3. So the slope of f will be -3/7.

We know that the y-intercept of f is the same as the provided line's y-intercept. Hence, from the provided line equation, we can see that the y-intercept is -3.

Therefore, the slope-intercept equation is: f(x) = (-3/7)x - 3

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a) The exact real interest rate is approximately -5.75%.

b) The exact real interest rate is approximately 4.24%.

c) The exact real interest rate is approximately -2.38%.
d) The exact real interest rate is approximately -99.85%.
e) When the nominal interest rate (i) is greater than the inflation rate (π), the real interest rate (R) is positive.

Step by step:

a) To find the exact real interest rate (R), we can use the formula R = (1+π)/(1+i) - 1, where π is the inflation rate and i is the nominal interest rate.
Given that i = 5.5% and π = 4.5%, we can substitute these values into the formula:

R = (1+0.045)/(1+0.055) - 1
R = 1.045/1.055 - 1
R ≈ 0.9425 - 1
R ≈ -0.0575

Therefore, the exact real interest rate is approximately -5.75%.


b) For i = 18% and π = 23%:
R = (1+0.23)/(1+0.18) - 1
R = 1.23/1.18 - 1
R ≈ 1.0424 - 1
R ≈ 0.0424

Therefore, the exact real interest rate is approximately 4.24%.


c) For i = 5% and π = 2.5%:
R = (1+0.025)/(1+0.05) - 1
R = 1.025/1.05 - 1
R ≈ 0.9762 - 1
R ≈ -0.0238

Therefore,  

d) For i = 1.05% and π = 1.2%:
R = (1+0.012)/(1+0.0105) - 1
R = 1.012/1.0105 - 1
R ≈ 0.0015 - 1
R ≈ -0.9985

Therefore, the exact real interest rate is approximately -99.85%.

e) From the results obtained, we can draw the following conclusions about interest rates and inflation:

- When the nominal interest rate (i) is greater than the inflation rate (π), the real interest rate (R) is positive.

- When the nominal interest rate (i) is equal to the inflation rate (π), the real interest rate (R) is approximately zero.

- When the nominal interest rate (i) is less than the inflation rate (π), the real interest rate (R) is negative.

- Higher inflation rates generally lead to lower real interest rates, as the purchasing power of money decreases.

- Lower inflation rates generally lead to higher real interest rates, as the purchasing power of money increases.

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Answer:

Within 10 feet of a fire hydrant

Step-by-step explanation:

Answer:

10 feet.

Step-by-step explanation:

we can express the function as [x - √(x²+ y²)]/[x² + y²]^(1/2).

We have converted the given trigonometric function into an algebraic expression. [x - √(x²+ y²)]/[x² + y²]^(1/2)

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The weekday of May 25, 1995, we can use a calendar or consult a date calculator. For May 25, 1995, it was a Thursday.

The date and weekday that is 10,000 days earlier than today, we can calculate the difference and subtract it from the current date.

Let's calculate:

Get the current date: June 20, 2023 (assuming today's date).

Subtract 10,000 days from the current date: June 20, 2023 - 10,000 days = May 25, 1995.

So, the date that is 10,000 days earlier than today is May 25, 1995.

To find the weekday of May 25, 1995, we can use a calendar or consult a date calculator. For May 25, 1995, it was a Thursday.

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The mean absolute deviation is  calculated for both the 2-month and 3-month moving averages. The mean absolute deviation based on the 2-month average of March through April is $0.055, while the mean absolute deviation based on the 3-month average of April is $0.048. Based on the mean absolute deviation, the 3-month moving average performs better.

To calculate the forecast for March and April using a 2-month moving average, we take the average of the prices from January and February for the forecast of March, and the average of the prices from February and March for the forecast of April.

To calculate the forecast for April using a 3-month moving average, we take the average of the prices from February, March, and April.

The mean absolute deviation is calculated by taking the absolute difference between the forecasted prices and the actual prices for each month, and then calculating the average of these differences. This gives us a measure of the average deviation from the actual prices.

Based on the mean absolute deviation values, we can compare the performance of the 2-month and 3-month moving averages. A lower mean absolute deviation indicates better performance in terms of accuracy in predicting the chip prices.

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Answer:

60%

Step-by-step explanation:

A z-score is a standardized measure that represents the distance between a particular value (X) and the mean (μ) of a population in terms of standard deviations (σ). It allows us to determine how far a given data point deviates from the average value of a distribution.

To calculate the z-score for a specific value, we need to know the population mean (μ) and the standard deviation (σ). In this case, we are given a population with a mean of 80 and a standard deviation of 9.

To find the z-score for X = 84, we use the formula: z = (X - μ) / σ. Plugging in the values, we get:

z = (84 - 80) / 9 = 4 / 9 = 0.44

Therefore, the z-score for X = 84 in a population with a mean of 80 and a standard deviation of 9 is 0.44. This indicates that the value of 84 is 0.44 standard deviations above the mean.

Z-scores provide a standardized way of comparing data points across different distributions. They help us understand the relative position of a particular value within a population. By calculating z-scores, we can analyze and interpret data in a standardized and meaningful manner.

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Algebraically, this means that the dependent variable y is a linear function of the independent variable x, with no coefficient multiplying x.

When β₂=0, the regression model simplifies to yᵢ = β₁xᵢ + eᵢ. This means that the dependent variable y is solely determined by the intercept β₁ and the error term e, with no effect from the independent variable x.

This means that the best estimate for β₁, when β₂ = 0, is the mean of the dependent variable y.This is because when β₂ = 0, the value of x does not contribute to the variation in y. The line is parallel to the x-axis and has a constant intercept β₁

.

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The function y = sin(x) has its minimum values at the values of x when y = -1, if −2π ≤ x ≤ 2π. The minimum value of a sinusoidal function is the lowest point that occurs at the trough (or bottom) of the wave.

The general formula of a sinusoidal function is f(x) = A sin (B(x - C)) + D where, A is the amplitude, B is the number of cycles, C is the horizontal shift (phase), and D is the vertical shift. In the case of y = sin(x), the amplitude (A) is 1, the number of cycles (B) is 1, the horizontal shift (C) is 0, and the vertical shift (D) is 0.

Therefore, the lowest point of the wave, which is the minimum value of the function y = sin(x), occurs when the value of sin(x) is -1. This occurs at the values of x when x = -π/2 + 2nπ or x = 3π/2 + 2nπ, where n is an integer. If we consider the interval −2π ≤ x ≤ 2π, then the values of x when y = sin(x) has its minimum values are x = -π/2, 3π/2.

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The first step in finding the rational zeros of a polynomial is to use the Rational Root Theorem. According to this theorem, any rational zero of a polynomial with integer coefficients will have the form p/q, where p is a factor of the constant term (in this case, -18) and q is a factor of the leading coefficient (in this case, 1).

Let's find the factors of -18: -1, 1, -2, 2, -3, 3, -6, 6, -9, 9, -18, 18.
Now let's find the factors of 1: -1, 1.

Now we can check all possible combinations of these factors to find the rational zeros. By dividing the polynomial f(x) by each of these possible zeros, we can see if any of them result in a remainder of zero. If the remainder is zero, then that value is a zero of the polynomial.

The rational zeros of f(x) = x^4-6x^3-7x^2+54x-18 are:
-1, 1, -2, 2, -3, 3, -6, 6, -9, 9, -18, 18.

Now, to find all the roots of the equation, we can use the depressed equation. The depressed equation is obtained by dividing the original equation by (x - r), where r is a root of the equation.

Let's take one of the rational zeros, -1, as an example. We divide f(x) by (x + 1) to obtain the depressed equation.

The depressed equation is: g(x) = x^3 - 7x^2 + 14x - 18.

We can continue this process for each rational zero we found earlier, and for any non-rational zeros we might have missed.

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Proportion should a 10% cream be mixed with a 1% cream of the same active ingredient to make a 3% cream  the proportion in which the 10% cream should be mixed with the 1% cream to make a 3% cream is X:Y = 1:3.5, which can be approximated as X:Y = 2:7.

To determine the proportion in which a 10% cream should be mixed with a 1% cream to make a 3% cream, we can again use the concept of weighted averages.

Let's assume we mix X parts of the 10% cream with Y parts of the 1% cream.

The equation for the weighted average can be written as:

(Percentage A * Weight A) + (Percentage B * Weight B) = Desired Percentage * Total Weight

In this case, the equation would be:

(10% * X) + (1% * Y) = 3% * (X + Y)

Simplifying the equation, we get:

0.1X + 0.01Y = 0.03X + 0.03Y

Rearranging the terms, we have:

0.03X - 0.1X = 0.03Y - 0.01Y

-0.07X = 0.02Y

Dividing both sides by 0.02Y, we get:

(-0.07X) / (0.02Y) = 1

Simplifying further:

-3.5X = Y

Therefore, the proportion in which the 10% cream should be mixed with the 1% cream to make a 3% cream is X:Y = 1:3.5, which can be approximated as X:Y = 2:7.

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Answer:

The fact that a corporation has limited liability means that the owners or shareholders of the corporation are not personally responsible for the debts or obligations of the corporation beyond the amount of their investment. In other words, the liability of the owners or shareholders is limited to the amount of money they have invested in the corporation. This is one of the key advantages of forming a corporation, as it provides a level of protection for the owners or shareholders in the event that the corporation incurs significant debts or is sued for damages.

The average rate of change of a function over an interval is the slope of the secant line connecting two points on the function. In this case, we want to find the average rate of change of the function \( Q(t) = -3x^2 + 4x + 7 \) over the interval \([x, x+h]\).

To find the average rate of change, we need to calculate the difference in \( Q \) values and the difference in \( x \) values, and then divide the difference in \( Q \) by the difference in \( x \).

Let's start by finding \( Q(x+h) \) and \( Q(x) \):

\( Q(x+h) = -3(x+h)^2 + 4(x+h) + 7 \)
\( Q(x) = -3x^2 + 4x + 7 \)

Now, let's calculate the difference in \( Q \) values:

\( \Delta Q = Q(x+h) - Q(x) \)
\( \Delta Q = (-3(x+h)^2 + 4(x+h) + 7) - (-3x^2 + 4x + 7) \)
\( \Delta Q = -3(x^2 + 2xh + h^2) + 4x + 4h + 7 + 3x^2 - 4x - 7 \)
\( \Delta Q = -3x^2 - 6xh - 3h^2 + 4x + 4h + 7 + 3x^2 - 4x - 7 \)
\( \Delta Q = -6xh - 3h^2 + 4h \)

Next, let's calculate the difference in \( x \) values:

\( \Delta x = (x+h) - x \)
\( \Delta x = h \)

Finally, let's divide \( \Delta Q \) by \( \Delta x \) to find the average rate of change:

\( \frac{\Delta Q}{\Delta x} = \frac{-6xh - 3h^2 + 4h}{h} \)

We can simplify this expression by factoring out \( h \) from the numerator:

\( \frac{\Delta Q}{\Delta x} = \frac{h(-6x - 3h + 4)}{h} \)

Canceling out \( h \) from the numerator and denominator, we get:

\( \frac{\Delta Q}{\Delta x} = -6x - 3h + 4 \)

So, the average rate of change of the function \( Q(t) = -3x^2 + 4x + 7 \) over the interval \([x, x+h]\) is \( -6x - 3h + 4 \).

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Answer:

The volume of Cone B is twice the volume of Cone A.

Step-by-step explanation:

The volume of a cone is given by

V = 1/3 pi r^2 h  where r is the radius and h is the height.

Cone A

d = diameter

r = radius

h = height

V = 1/3 pi r^2 h

Cone B

The diameter is double the diameter of A.

2d  so the radius is 2r

The height is half the height of A.

1/2 h

Substitute into the equation for volume.

V = 1/3 pi ( 2r)^2 (1/2 h)

V = 1/3 pi (4r^2) (1/2h)

V = 1/3 pi 2r^2 h

The volume of Cone B is twice the volume of Cone A.

Answer:

Cone B has a greater volume.

Step-by-step explanation:

Cone B has the greatest volume.

The volume of a cone is calculated using the following formula:

[tex]\boxed{\bold{\tt{Volume\: of\:cone = \frac{1}{3}\pi*r^2h}}}[/tex]

where:

For Cone A.

[tex]\boxed{\bold{\tt{Volume\: of\:cone\: (A)= \frac{1}{3}\pi*r^2h}}}[/tex]

For Cone B

In this case, the radius of Cone B is double the radius of Cone A, and the height of Cone B is half the height of Cone A. This means:

radius(r)=2r

height(h)= [tex]\tt{\frac{1}{2}*\frac{h}{2}}[/tex]

[tex]\boxed{\bold{\tt{Volume\: of\:cone\: (B)= \frac{1}{3}\pi*(2r)^2*\frac{h}{2}}}}[/tex]

[tex]\boxed{\bold{\tt{Volume \: of\:cone (B)= 2*\frac{1}{3}\pi*r^2h}}}[/tex]

[tex]\boxed{\bold{\tt{Volume(B) =2*Volume\: of\:cone(A)}}}[/tex]

Since the volume of cone B is twice the volume of Cone A.

Therefore, Cone B has a greater volume.

a) Alexandra's fortnightly repayments: Approximately $340.16

b) The outstanding balance on the loan after 6 fortnights: Approximately $1914.99

To calculate Alexandra's fortnightly repayments and the outstanding balance on the loan after 6 fortnights, we can use the formula for calculating the loan repayment and the formula for calculating the compound interest.

Calculate the fortnightly interest rate:

The annual interest rate is 8.12%, so the fortnightly interest rate would be (8.12% / 26) = 0.3123%.

Calculate the fortnightly repayment amount:

To calculate the loan repayment, we can use the formula for the equal installment loan repayment amount:

Repayment = (Loan amount * (interest rate * (1 + interest rate) ^ number of payments)) / ((1 + interest rate) ^ number of payments - 1)

Here, the loan amount is $2000, the interest rate is 0.3123% (as calculated in Step 1), and the number of payments is 6 fortnights.

Repayment = ($2000 * (0.003123 * (1 + 0.003123) ^ 6)) / ((1 + 0.003123) ^ 6 - 1)

Using a calculator, the fortnightly repayment amount is approximately $340.16.

Calculate the outstanding balance after 6 fortnights:

To calculate the outstanding balance after 6 fortnights, we can use the compound interest formula:

Outstanding balance = Loan amount * (1 + interest rate) ^ number of payments - Total repayments

Here, the loan amount is $2000, the interest rate is 0.3123% (as calculated in Step 1), the number of payments is 6 fortnights, and the total repayments is the repayment amount calculated in Step 2 multiplied by 6 fortnights.

Outstanding balance = $2000 * (1 + 0.003123) ^ 6 - ($340.16 * 6)

Using a calculator, the outstanding balance after 6 fortnights is approximately $1914.99.

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The function df(x, y) = (y^2 - xy)dx - x^2 dy is an inexact differential.

To determine if a function is exact or inexact, we need to check if its partial derivatives with respect to x and y satisfy the condition ∂M/∂y = ∂N/∂x, where df(x, y) = M(x, y)dx + N(x, y)dy.

In this case, we have M(x, y) = y^2 - xy and N(x, y) = -x^2. Calculating the partial derivatives, we find:

∂M/∂y = 2y - x

∂N/∂x = -2x

Since ∂M/∂y is not equal to ∂N/∂x (2y - x ≠ -2x), the function df(x, y) = (y^2 - xy)dx - x^2 dy is inexact.

Next, we can test the integrating factor 1/(xy^2) to see if it produces an exact differential. The integrating factor is denoted by μ(x, y) and is given by μ(x, y) = e^(∫(∂M/∂y - ∂N/∂x)/N dx). If the resulting expression becomes an exact differential, the integrating factor is successful.

In this case, the integrating factor μ(x, y) = e^(∫(2y - x)/(-x^2) dx) = e^(-2y/x). However, integrating factor μ(x, y) does not produce an exact differential, and hence, it is not a suitable integrating factor for this function.

Lastly, the expression H(g) represents the enthalpy change of a substance g. The notation Hˉ∘(2000 K) - Hˉ∘(0 K) indicates the difference in enthalpy between the substance at 2000 Kelvin and 0 Kelvin. To calculate this difference, additional information or a specific equation relating enthalpy change to temperature is needed. Without further details, it is not possible to provide a numerical calculation for Hˉ∘(2000 K) - Hˉ∘(0 K) for H(g).

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To solve the system of equations, we can set the two expressions for `y` equal to each other:

2x² - x + 1 = x² + 4x + 7

Now, let's solve for `x`:

2x² - x + 1 - x² - 4x - 7 = 0

x² - 5x - 6 = 0

We can factor this quadratic equation:

(x - 6)(x + 1) = 0

So, the solutions for `x` are:

x = 6

x = -1

Now, let's find the corresponding `y` values for each `x` value:

For x = 6:

y = 2(6)² - 6 + 1

y = 2(36) - 6 + 1

y = 72 - 6 + 1

y = 67

For x = -1:

y = 2(-1)² - (-1) + 1

y = 2(1) + 1 + 1

y = 2 + 1 + 1

y = 4

So, the solutions of the system of equations are:

(x, y) = (6, 67)

(x, y) = (-1, 4)

The correct formula for the variance of Y is б²y / n. Therefore, the correct option is A. б²y / n.

The variance of Y, б2/y is given by the following formula:

Where, б² is the population variance, y is the sample mean, and n is the sample size.

Variance is a statistical measure of how dispersed a set of data points is. In statistics, variance measures the variability or spread in a dataset. In other words, it determines how far the values of a dataset are spread out from their mean.

In statistics, the formula for the variance of a population is given by:

In the above formula:

Now, let's have a look at the given options:

So, option A. б2 y/n is the correct answer.

The complete question:

The variance of Y, б2/y is given by the following formula:

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a) The value of the given summation is 55.

b) The summation does not have a finite value.

c)The value of the given summation is 2,746,560.

a. [tex]\( \sum_{i=1}^{5} i^{2} \)[/tex]

To evaluate the given expression, we need to add the squares of the numbers from 1 to 5, so\[1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}\]

Simplifying, we get\[1 + 4 + 9 + 16 + 25 = 55\]

Therefore, the value of the given summation is 55.

b.[tex]\( \sum_{i=1}^{\infty} 3 e^{i} \)[/tex]

The given summation is a divergent geometric series since the ratio between any two consecutive terms is not constant.

Therefore, the summation does not have a finite value.

c. [tex]\( \sum_{k=2}^{10} 10(3)^{k} \)[/tex]

We know that \(a(1 - r^{n}) / (1 - r)\) is the formula for the sum of the first n terms of a geometric series, where a is the first term and r is the common ratio.

Therefore,[tex]\[\sum_{k=2}^{10} 10(3)^{k} = 10\left[\frac{3^{2}(1 - 3^{9})}{1 - 3}\right] = 2,746,560\][/tex]

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