Find the inverse and original steps of the inverse equation

The club made 20 pieces of pasta.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

The cafeteria donated four pies, and each pie was cut into four pieces. Therefore, the total number of pieces donated is:

4 pies × 4 pieces/pie = 16 pieces

The total number of pieces to sell is given as 36. So the number of pieces the club made is:

Number of pieces made = Total number of pieces to sell - Number of pieces donated

Number of pieces made = 36 - 16

Number of pieces made = 20

Therefore, the club made 20 pieces of pasta.

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Given the above conditions with regard to Rolle's Theorem, the answers are:

Therefore, the solution is:

F(0) = 0

f(4) = (-8) e^4

C = 1

To find F(0), we can simply substitute x=0 into the expression for f(x):

f(0) = (0^2 - 4(0)) e^0 = 0

To find f(4), we can substitute x=4:

f(4) = (4^2 - 4(4)) e^4 = (-8) e^4

Next, we need to find the critical point(s) of f(x) on the interval [0,4]. To do this, we find the derivative of f(x) and set it equal to zero:

f'(x) = (2x-4) e^x + (x^2-4x) e^x = (x^2-2x) e^x

Setting f'(x) equal to zero, we have:

(x^2-2x) e^x = 0

Since e^x is never zero, we can divide both sides by e^x to get:

x^2 - 2x = 0

Factoring out x, we have:

x(x-2) = 0

Therefore, the critical points on [0,4] are x=0 and x=2.

Since both critical points are in the open interval (0,4), Rolle's Theorem applies. This means there exists a number c in (0,4) such that f'(c) = 0. To find this value of c, we can simply take the average of the critical points:

c = (0 + 2)/2 = 1

Therefore, the solution is:

F(0) = 0

f(4) = (-8) e^4

C = 1

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The total garden area obtained by adding the new area of the garden is obtained as 36  sq. ft.

Your polygon's four sides must be two congruent pairings in order to have two pairs with parallel sides.

The length of the left and right sides, as well as the base and top, will be equal. A four-sided polygon with two sets of parallel, congruent sides with four internal angles of 90° each is required to be a rectangle.

Given data:

current area of garden = 24 square feet.

Coordinates of other garden:

(−17, 15), (−20, 15), (−17, 11), and (−20, 11)

Length of new garden = -20 + 17 = 3 ft.

Width of new garden = 15 - 11 = 4 ft.

Area of new garden = length * width

Area of new garden = 3ft * 4 ft

Area of new garden = 12 sq. ft.

total area of the garden = current area +  new garden area

total area of the garden = 24 sq. ft. + 12 sq. ft.

total area of the garden = 36  sq. ft.

Thus, the total area of the garden obtained by adding the new area of the garden is obtained as 36  sq. ft.

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f[g(-2)] = 2 as the function f(x) = x - 5 must then be used to evaluate the function f[g(-2)] .

A function is a mathematical construct that transforms a set of inputs (referred to as the domain) into a single output from another set (called the range). A function is, in other words, a rule that allots precisely one output to each input in the domain. The notation f(x) is frequently used to denote functions, where f stands for the function name and x for the input value. As an illustration, the function f(x) = 2x + 1 doubles any input x, adds 1, and returns the result.

given

To begin, we must calculate g(-2) using the formula g(x) = x2 + 3:

[tex]g(-2) = (-2)^2 + 3 = 4 + 3 = 7[/tex]

The function f(x) = x - 5 must then be used to evaluate the function f[g(-2)]:

f[g(-2)] = f(7) = 7 - 5 = 2

Hence, f[g(-2)] = 2.

f[g(-2)] = 2 as the function f(x) = x - 5 must then be used to evaluate the function f[g(-2)] .

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The next two numbers in the sequence are 35 and 107.

The given number chain seems to be following the pattern of adding the previous number multiplied by a specific factor.

To find the next two numbers in the sequence, we will use the same pattern.

1 ► 3 ► 11 ► ?

To get the next number in the sequence, we multiply the previous number by 3 and add 2.

So, the next number in the sequence would be:

11 x 3 + 2 = 35

Therefore, the next number in the sequence is 35.

To get the number after 35, we apply the same pattern:

35 x 3 + 2 = 107

Therefore, the next number in the sequence is 107.

Hence, the next two numbers in the sequence are 35 and 107.

So the complete number sequence is: 1 ► 3 ► 11 ► 35 ► 107 ►

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The critical value Zc is ±2.170.

What is a confidence interval?

An estimated range for an unknown parameter is known as a confidence interval. The 95% confidence level is the most popular, however other levels, such as 90% or 99%, are occasionally used when computing confidence intervals.

Here, we have

Given: a confidence interval at the level of confidence shown below.

c=0.97

We have to find the critical value Zc.

c = 0.97

we get a = 1 - c

a = 1 - 0.97

a = 0.03

Critical value Zc = ±2.170

Hence, the critical value Zc is ±2.170.

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The summary table of the function obtained using differential calculus, and the graph of the function are presented as follows;

[tex]\begin{tabular}{|c|c|c|} \cline{1-3} Interval & Increasing/Decreasing &Concave Up/Down\\ \cline{1-3}(-\infty, 0)&Decreasing&Concave Up \\ \cline{1-3} (0, 3/2) & Decreasing & Concave Down \\ \cline{1-3} (3/2, \infty) & Increasing& Concave Up\\ \cline{1-3}\end{tabular}[/tex]

Please find attached, the graph of the function, created using MS Excel

Differential calculus is the part of calculus that is concerned with the rate of change of quantities.

The steps to determine the intervals where f(x) = x⁴ - 2·x³ is increasing or decreasing, or constant are presented as follows;

The derivative of the function is first found using calculus, differentiation as follows;

f'(x) = 4·x³ - 6·x²

The critical points at which the slope of the graph changes from increasing are found at the points where f'(x) = 0 as follows;

f'(x) = 0 = 4·x³ - 6·x²

4·x³ - 3·x² = 0

2·x²·(2·x - 3) = 0

Therefore, at the critical points, x = 0, and x = 3/2

The first derivative test can be used to determine the interval the function is increasing or decreasing as follows;

f(x) is increasing at x < a, if f'(x) > 0 for x < a and f(x) is decreasing at x > a if f(x) < 0 at x > a, where a is a critical point

f(x) is decreasing at x < a, if f'(x) < 0 for x < a and f(x) is increasing at x > a if f(x) > 0 at x > a,

The concavity of the function at x = a if f'(x) = 0 at x = a can be determined using the second derivative as follows;

f''(x) = 12·x² - 12·x

f''(x) > 0 at x = a if f(x) is concave up

f''(x) < 0 at x = a if f(x) is concave down

When a = 0, we get;

f''(0) = 12 × 0² - 12 × 0 = 0

f''(0.5) = 12 × 0.5² - 12 × 0.5 = -3 (Concave down)

f''(-0.5) = 12 × (-0.5)² - 12 × (-0.5) = 9 (Concave up)

f''(0) = 0 at x = 0, therefore, x = 0 is an inflection point

f'(-1) = 4·(-1)³ - 6·(-1)² = -10

f'(1) = 4·(1)³ - 6·(1)² = -2

Therefore, the graph changes from decreasing and is also decreasing at x > 0

Similarly we get;

f'(2) = 4·(2)³ - 6·(2)² = 8 (Increasing)

Therefore;

f'(1) = 4·(1)³ - 6·(1)² = -2

There

f''(3/2) = 12 × (3/2)² - 12 × (3/2) = 9

The second derivative at x = 3/2 = 9 > 0, therefore;

The function, f(x) is concave up at x = 3/2

Therefore, we get the following table;

Please find attached the graph of the function created using MS Excel

and the following points;

x   [tex]{}[/tex]      f(x)

-0.9    [tex]{}[/tex]2.1141

-0.8 [tex]{}[/tex]   1.4336

-0.1 [tex]{}[/tex]    0.0021

0 [tex]{}[/tex]       0

0.4 [tex]{}[/tex]    -0.1024

0.5 [tex]{}[/tex]    -0.1875

1.5 [tex]{}[/tex]     -1.6875

2 [tex]{}[/tex]        0

2.5 [tex]{}[/tex]     7.8125

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Since the limit is just 33 μg/ft³, it exceeded the given limit. Thus, with this condition, there would be strong odor and irritation of the eyes.

A cube's volume is defined as the total number of cubic units that it takes up in its entirety. To calculate the cube's volume, we need to know how long each cube side is. In order to compute volume, use the following equation: Side x Side x Side equals volume.

33 μg/ft³ is the maximum allowed formaldehyde concentration. Our foundation should be this. It is now assumed that wood panelling generates 3,365 μg of formaldehyde per square foot. The wood panelling measures 5 feet by 6 feet; therefore the area is equal to 30 ft². As a result, the formaldehyde emission is calculated as follows: Emitted formaldehyde = (3,365 μg/ft²) (30 ft²) = 100,950 μg

The size of the room is then measured:

V = 10 ft × 10 ft × 6 ft = 600 ft³

Let's compare the ratio to our basis:

Ratio = 100,950 μg/600 ft³ = 168.25 μg/ft³

Since the limit is just 33 μg/ft³, it exceeded the given limit. Thus, with this condition, there would be strong odor and irritation of the eyes.

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Some of the radicals below that are like terms are; √2, 5√2, − 3√2, √32.

The like terms can be combined as 7√2.

The reason why the leftover terms are NOT like terms with the others is that the radicals lack the same root number also  radicand (expression under the root).

Radicals with the same root number AND radicand are similar radicals (expression under the root)

A number's radical and root are the same thing. A cube root, a square root, or an nth root could be the root. Therefore any number or phrase that uses a root is a radical. The word "radical" is derived from the Latin word radix, which means root. The radical can be used to explain a number's square, cube, fourth, and other forms of roots. The number that comes before the radical is known as the index number or degree.

3a. Some of the radicals that are like terms are; √2, 5√2, − 3√2, √32 and the reason why √32 is part of it is that √32 = 4√2

3b. The like terms can be combined as ;

(√2+ 5√2 − 3√2 + √32)

= (√2+ 5√2 − 3√2 + 4√2)

=7√2

3c. The reason why the leftover terms are NOT like terms with the others is that the radicals  do not have the same root number as well as  radicand (expression under the root).

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We can conclude after answering the provided question that Because each element in the domain is paired with exactly one element in the range, this mapping diagram represents a function.

The domain of a function is the range of possible values that it can accept. These numbers represent the x-values of a function such as f. (x). The domain of a function is the range of possible values that can be used with it. This set includes the value returned by the method after inserting the x value. A function with y as the dependent variable and x as the independent variable has the formula y = f. (x). When a single value of y can be produced successfully from a value of x, that value of x is said to be in the function's domain.

Each element in the domain (set of x values) is paired with exactly one element in the range (set of y values), as represented by arrows connecting the two sets in the mapping diagram. For example, the domain element 2 is paired with the range element 4, and the domain element -1 is paired with the range element 1.

Because each element in the domain is paired with exactly one element in the range, this mapping diagram represents a function. The function is not one-to-one because multiple domain elements are paired with the same range element (for example, both -2 and 3 are paired with 1).

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The equation of the oscillating circle of the curve y = ¹/₄ x² +2 at point (2,1) is (x - 5/2)² + (y - 1)² = 1/4.

To find the equation of the oscillating circle of the curve y = ¹/₄ x² +2 at point (2,1), we need to follow these steps:

Find the first and second derivatives of the given curve:

y = ¹/₄ x² + 2

y' = ¹/₂ x

y'' = ¹/₂

Use the first and second derivatives to find the equation of the tangent line to the curve at point (2,1):

y - 1 = 1/2(x - 2)

y = 1/2 x + 0.5

Find the perpendicular bisector of the tangent line at point (2,1):

The slope of the tangent line is 1/2, so the slope of the perpendicular bisector is -2.

y - 1 = -2(x - 2)

y = -2x + 5

Find the intersection of the perpendicular bisector with the x-axis:

When y = 0, -2x + 5 = 0, so x = 5/2.

Find the center of the oscillating circle:

The center of the oscillating circle is (5/2,1).

Find the radius of the oscillating circle:

The radius of the oscillating circle is the distance between the center and the point of tangency, which is:

r = √[(2 - 5/2)² + (1 - 1)²]

r = [1/4 + 0] = 1/2.

Write the equation of the oscillating circle in standard form:

The equation of the oscillating circle is:

(x - 5/2)² + (y - 1)² = (1/2)².

Simplifying, we get:

(x - 5/2)² + (y - 1)² = 1/4.

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The complete question is below:

Find the equation of oscillating circle of curve y=1/4 x^2 +2 at (2,1)

Answer:

the answer is : 2

By answering the presented question, we may conclude that There are polygon four vertex designated HT , HB, TE and EB in the provided picture.

A polygon is a two-dimensional shape that is formed by joining three or more points in a closed loop. Triangles, tetrahedra, pentagons, hexagons, and other forms are examples of polygons. Polygons are a fundamental geometric concept used to define and analyze a wide range of real-world forms and structures, such as buildings, layouts, and computer graphics. Polygons are examined not just for their perimeter, area, and angles, but also for their symmetry.

The vertex of the polygon in the image are the line segments that link the polygon's opposing vertices and are perpendicular to its base. There are four vertex designated HT , HB, TE and EB   in the provided picture.

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Answer:23200cm^3

Step-by-step explanation:

we are going to watch this figure as 2 different prisms. 1st-parallelepiped

and 2nd-trapezoidal prism;

The volume of the parallelepiped is equal to a*b*c which is the product of its sides(width, height, and length) and in this case, it is equal to V1=a*b*c=10*12*80(because the trapezoidal prisms side is parallelogram and facing sides of parallelogram are equal so c=80cm)=9600cm^3;

V2=Area of base*h

area of base is equal to area of transpazoid which is

A=((a+b)/2)*h=(12+22)/2*10=34/2*10=17*10=170cm^2

and as i said earlier h=80 because its a parallelogram so,

V2=170+80=13600cm^3

V=V1+V2=9600+13600=23200cm^3

Answer:

shorts/ and a t-shirt

Step-by-step explanation:

25° c is equal to 77° f

The composite function [g . f](x) when evaluated has its value to be 4x² + 20x + 22

Given that, we have the following functions

f(x) = 2x + 5

g(x) = x² - 3

To find [g . f](x) we make use of g(f(x)), where we first need to find f(x) and then plug it into g(x) in place of x.

Given, f(x) = 2x + 5 and g(x) = x^2 - 3.

So,

g(f(x)) = g(2x + 5) [Substitute f(x) in place of x in g(x)]

g(f(x)) = (2x + 5)² - 3 [Substitute 2x + 5 in place of x in g(x)]

g(f(x)) = (4x² + 20x + 25) - 3

g(f(x)) = 4x² + 20x + 22

This means that

[g . f](x) = 4x² + 20x + 22

Therefore, [g . f](x) = 4x² + 20x + 22

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The function oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes.

How to study graph?

To graph the function y=3cot(x) between 0 and 2, we can start by identifying the key characteristics of the cotangent function. The cotangent function is defined as the ratio of the adjacent side to the opposite side of a right triangle, with respect to a given angle. As such, the cotangent function is periodic with a period of π and has vertical asymptotes at x=nπ, where n is an integer.

The coefficient of 3 in the given function y=3cot(x) vertically stretches the function by a factor of 3, and does not affect the period or the vertical asymptotes.

Between 0 and 2, there are two vertical asymptotes at x=0 and x=π. To graph the function, we can plot a few points along the curve, using the symmetry properties of the cotangent function. Specifically, the cotangent function is odd, meaning that cot(-x)=-cot(x), and thus the graph is symmetric about the origin.

At x=π/4, we have cot(π/4)=1, and thus y=3cot(π/4)=3. Similarly, at x=3π/4, we have cot(3π/4)=-1, and thus y=3cot(3π/4)=-3. Using the symmetry property, we can also plot points at x=-π/4 and x=-3π/4, with y=3cot(-π/4)=-3 and y=3cot(-3π/4)=3, respectively.

Connecting these points with smooth curves, we can see that the graph of y=3cot(x) oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes at x=0 and x=π.

Overall, the graph of y=3cot(x) between 0 and 2 can be summarized as follows:

There are two vertical asymptotes at x=0 and x=π.

The function oscillates between two horizontal asymptotes, y=3 and y=-3, as x approaches the vertical asymptotes.

The graph is symmetric about the origin.

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The Area of the rhombus with base 21 cm and height 17.3 cm is 363.3 cm².

What is area of Rhombus?

To calculate the base of the rhombus, we will follow the steps below;

write the formula for calculating the area of a rhombus

area of a rhombus = b × h

where h is the height of the rhombus and b is the base of the rhombus

Given:

Base = 21 cm

Height = 17.3 cm

We know area of Rhombus =  base x height

                                             = 21 x 17.3

                                             = 363.3 cm²

So, the area of Rhombus is 363.3 cm².

Answer:

  3 6 4 2 1 5

Step-by-step explanation:

You want to know the sequence of solving the problem of finding the time until the rope hits the ground when thrown at 10 m/s down a 50 m cliff.

The equation describing the height of the rope is of the form ...

  h(t) = -4.9t² +v₀t +h₀

where v₀ is the initial upward velocity (meters per second), and h₀ is the initial height (meters).

Here, the initial velocity is described as 10 m/s downward, so we expect v₀ to have a value of -10. However, the offered equation(s) show v₀ = +10, so we will be solving the problem assuming the rope is initially thrown upward.

For v₀ = 10 and h₀ = 50, the equation for the height of the rope is ...

  h(t) = -4.9t² +10t +50 . . . . . . . step 1

We want to find the time (t) when the height is 0, so we substitute h(t) = 0 into the above equation:

  0 = -4.9t² +10t +50 . . . . . . . . step 2

This can be solved nicely using the quadratic formula. We recognize that a=-4.9, b=10, c=50, so the filled-in formula is ...

  [tex]t=\dfrac{-10\pm\sqrt{10^2-4(-4.9)(50)}}{2(-4.9)}\qquad\textsf{step 3}[/tex]

Simplifying this and taking the square root, we have ...

  [tex]t\approx\dfrac{-10\pm32.86}{-9.8}\qquad\textsf{step 4}[/tex]

Evaluating this expression gives ...

  t ≈ -2.33  or  t ≈ 4.37 . . . . . . . . . step 5

Writing the answer as a sentence is the last step:

  The rope will hit the ground in about 4.37 seconds. . . . . . step 6

The sequence of step numbers is ...

  3 6 4 2 1 5

__

Additional comment

The equation assume ballistic motion, with no air resistance or other impediment to the rope falling only due to gravity. We assume that Krystal plans to use the rope for rappelling, so it is probably coming uncoiled as it falls. That process usually changes the motion from one that is purely defined by the force of gravity.

We also note that if the initial throwing direction is downward, then the solution to the equation h(t) = -4.9t² -10t +50 is actually t ≈ 2.33 s.

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If it's a decimal round to the nearest tenth then the value οf x is 4/3.

In mathematics, a prοpοrtiοn is an equatiοn that relates twο ratiοs. It is expressed in the fοrm οf a/b = c/d, where a, b, c, and d are numbers οr quantities.

In a prοpοrtiοn, the crοss-prοducts are equal, meaning that the prοduct οf the numeratοr οf the first ratiο and the denοminatοr οf the secοnd ratiο is equal tο the prοduct οf the denοminatοr οf the first ratiο and the numeratοr οf the secοnd ratiο.

the given prοblem can be sοlved by using ratiοs οf the cοrrespοnding sides.

we knοw that, the ratiο οf cοrrespοnding sides is equal.

sο, we have :

[tex]$\frac{8}{x+2} = \frac{12}{5}[/tex]

8 × 5 =  12 × (x+2)

40   = 12x + 24

12x  = 40 - 24

= 16

∴ x = 16/12 = 4/3

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The function with the greater initial value in this problem is given as follows:

Function f(x), the graphed function.

The initial value of a function, often denoted as f(0), is the value of the function when the independent variable (usually denoted as x) is equal to 0. In other words, it is the value of the function at the starting point of its domain.

Hence the initial value of each function in this problem is given as follows:

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Answer:

Step-by-step explanation:

Let x be the side length of the square window. Then the area of the square window is x^2, and the area of the rectangular window is 16 * 4 = 64.

We can write the equation:

x^2 = 64

To solve for x, we can take the square root of both sides:

x = √64

x = 8

Therefore, the side length of the square window is 8 inches.

Answer:

Step-by-step explanation:

Solve from the inside outwards:

         [tex]g(-4)=\frac{-4}{2} =-2[/tex]

         [tex]f(g(-4))=f(-2)=10\times(-2)-4=-24[/tex]

Solution:  [tex]f(g(-4))=-24[/tex]

Therefore, the solutions to the quadratic equation x² + 7x - 8 = 0 are -8 and 1. The answers are D and E.

An equation is a mathematical statement that shows that two expressions are equal. It typically consists of variables, constants, and mathematical operations. Equations can be solved by manipulating the expressions to find the value of the variables that satisfy the equation. Equations can be used to model real-world situations, and they are an important tool in many fields, including mathematics, physics, engineering, and economics.

Here,

To check which values are solutions to the quadratic equation, we can substitute each value into the equation and see if it equals zero.

Substituting -1 into the equation:

(-1)² + 7(-1) - 8 = 1 - 7 - 8 = -14, which is not equal to zero.

Substituting 2 into the equation:

2² + 7(2) - 8 = 4 + 14 - 8 = 10, which is not equal to zero.

Substituting -4 into the equation:

(-4)² + 7(-4) - 8 = 16 - 28 - 8 = -20, which is not equal to zero.

Substituting -8 into the equation:

(-8)² + 7(-8) - 8 = 64 - 56 - 8 = 0, which is equal to zero. Therefore, -8 is a solution to the equation.

Substituting 1 into the equation:

1² + 7(1) - 8 = 1 + 7 - 8 = 0, which is equal to zero. Therefore, 1 is a solution to the equation.

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Answer:

C. 6 inches

Step-by-step explanation:

Base = 25 inches

Area = 75 square inches

Using the formula

A = HbB/2

Solving for Hb

Hb = 2 A/b = 2 . 75/25 = 6 inches

Answer:

The answer is (d) 200 m².

Step-by-step explanation:

Let's denote the width of the rectangle as w. Then the length of the rectangle is twice the width, which means the length is 2w.

The perimeter of a rectangle is given by the formula:

P = 2l + 2w

Substituting the values, we get:

60 = 2(2w) + 2w

Simplifying the equation, we get:

60 = 6w

Dividing both sides by 6, we get:

w = 10

So the width of the rectangle is 10 m and the length is twice the width, which is 20 m.

The area of a rectangle is given by the formula:

A = lw

Substituting the values, we get:

A = 10 x 20

Simplifying the equation, we get:

A = 200

Therefore, the area of the rectangle is 200 m².

According to the given rate the total profit made by the company in the first year is R240,000.

The problem statement does not provide enough information to determine the rate of return or interest rate. In the absence of this information, it is not possible to calculate the rate of return or interest rate earned on the capital contributions or on the profit made by the company.

According to the agreement, the profit made by the company should be divided among the partners in the ratio of their capital contributions, which in this case is 225,000:150,000:75,000 or 3:2:1.

Let's assume that the total profit made by the company in the first year is P. Dee's share of the profit is R40,000, which is 1/6 of the total profit, since her capital contribution is 1/6 of the total capital.

So, we can set up the equation:

1/6 * P = R40,000

Multiplying both sides by 6, we get:

P = R240,000

Therefore, according to the given rate the total profit made by the company in the first year is R240,000.

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differentiate between tropical and subtropical climate zone on the basis of climate and economic activities in 4 points class 10