Sarah is a psychologist at an practise. she earns a basic salary of R3000 per month as well as 20% commission on income up to R 5000. She receives an additional 10% bonus on top of the normal commission rate on earning above R 5000. If Sarah did work to the value of R 12000 ,how much did she earn in total???​

Using probability, we can find that:

E(Y)= 2.4, Var (Y) = 0.64

E(Y) = 480, Var(Y) = 25,600

The probability of an event is the proportion of favourable outcomes to all other potential outcomes. To determine how likely an event is, use the following formula:

Probability (Event) = Positive Results/Total Results = x/n

Given,

The weekly CPU time is as follows:

f(y) where, 0≤y≤4

Here, probability density function is 4-y is correct, or else we get negative expected values.

We have to find E(Y) and var(Y)

E(Y) = 2.4

var (Y) = E(Y²)-(E(Y)) ²

= 6.4 - (2.4) ²

= 0.64

The CPU time is costing the firm $200 per hour.

Now, we find E(Y) and var(Y) of the weekly cost for the CPU time.

Y = 200E

E(Y) = 200 × 2.4

= 480

var(Y) = 200V(Y)

= 200 × 0.64

= 25600

We can observe that the weekly cost is not exceeding $600 as weekly cost for CPU time = 480.

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The concept of Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different theorems and axioms.

The concept of Euclidean geometry as required to be discussed is basically introduced for flat surfaces or plane surfaces. The postulates of the Euclidean geometry are as follows!

1 : A straight line may be drawn from any one point to any other point.

2 :A terminated line can be produced indefinitely.

3 : A circle can be drawn with any centre and any radius.

4 : All right angles are equal to one another (Congruent).

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Step-by-step explanation:

Speed =  distance / time    

  you are given distance = 23 miles  and time = .5 hr

      distance / time =  23 miles / .5 hr =   46 mph

The probability of the union of events A, B, and C is 0.7 where P(A)=0.2, P(B)=0.2 and P(C)=0.3.

In set theory, the union of two or more sets is a set that contains all the distinct elements of the sets being considered. Formally, the union of sets A and B, denoted as A ∪ B, is the set of all elements that are in either set A or set B, or in both.

When A, B, and C are mutually exclusive events, it means that they cannot happen at the same time. Therefore, the probability of the union of these events is equal to the sum of their individual probabilities.

In this case, we are given that:

P(A) = 0.2

P(B) = 0.2

P(C) = 0.3

To find the probability of the union of these events, we need to add their probabilities:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C)

Substituting the given probabilities, we get:

P(A ∪ B ∪ C) = 0.2 + 0.2 + 0.3

P(A ∪ B ∪ C) = 0.7

Therefore, the probability of the union of events A, B, and C is 0.7.

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The question is A, B, and C are mutually exclusive. P(A) = 0.2, P(B) = 0.2, P(C) = 0.3. Find P(A ∪ B ∪ C).

P(A ∪ B ∪ C) = ?

Answer:

Let's assume that Mr. Nkalle invested an amount of x in Scheme A and (20900 - x) in Scheme B.

The simple interest earned on Scheme A in 2 years would be:

SI(A) = (x * 9 * 2)/100 = 0.18x

The simple interest earned on Scheme B in 2 years would be:

SI(B) = [(20900 - x) * 8 * 2]/100 = (3344 - 0.16x)

The total simple interest earned in 2 years is given as N$3508:

SI(A) + SI(B) = 0.18x + (3344 - 0.16x) = 3508

0.02x = 164

x = 8200

Therefore, Mr. Nkalle invested N$8200 in Scheme A and N$12700 (20900 - 8200) in Scheme B. So the amount invested in Scheme B was N$12700.

Yes, there is an analogous ratio between 2:1 and 20:10.

We just cancel by a common factor. So 4:2=2:1 . The simplest representation of the ratio 4 to 2 is the ratio 2 to 1. Also, since each pair of numbers has the same relationship to one another, the ratios are equivalent.

By dividing the terms of each ratio by their greatest common factor, we may simplify both ratios to explain why.

As the greatest common factor for the ratio 2:1 is 1, additional simplification is not necessary.

The greatest common factor for the ratio 20:10 is 10. When we multiply both terms by 10, we get:

20 ÷ 10 : 10 ÷ 10

= 2 : 1

As a result, both ratios have the same reduced form, 2:1, making them equal.

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Answer:

8.7 m
Option B

Step-by-step explanation:

The tree, the shadow and the line of sight from the tip of the shadow to the top of the tree form a right triangle

The angle formed is 30°

Using the relationship
[tex]\tan 30 = \dfrac{h}{15}[/tex]

we get
[tex]h = 15 \times \tan 30[/tex]

[tex]h = 15 \times \dfrac{1}{\sqrt{3}}[/tex]

[tex]h = 8.66 m[/tex]

Rounded up this would be 8.7 m which is option B

The first term of the GP is 4 and the common ratio is 3. We can now substitute these values into the formula for the sum of the first 10 terms to get 118096.

What is Geometric Progression?

A progression of numbers with a constant ratio between each number and the one before

Let the first term of the geometric progression be denoted by "a" and the common ratio be denoted by "r".

We know that the 4th term is 108, so we can use the formula for the nth term of a GP to write:

a*r³ = 108 .....(1)

We also know that the 8th term is 8748, so we can write:

a*r⁷ = 8748 .....(2)

To find the sum of the first 10 terms, we can use the formula for the sum of a finite geometric series:

S = a(1 - rⁿ)/(1 - r)

where S is the sum of the first n terms of the GP. We want to find the sum of the first 10 terms, so we plug in n = 10:

S = a(1 - r¹⁰)/(1 - r)

We now have two equations (1) and (2) with two unknowns (a and r). We can solve for a and r by dividing equation (2) by equation (1) to eliminate a:

(ar⁷)/(ar³) = 8748/108

r⁴ = 81

r = 3

Substituting r = 3 into equation (1) to solve for a, we have:

a*3³ = 108

a = 4

Therefore, the first term of the GP is 4 and the common ratio is 3. We can now substitute these values into the formula for the sum of the first 10 terms to get:

S = 4(1 - 3¹⁰)/(1 - 3)

S = 4(1 - 59049)/(-2)

S = 4(59048)/2

S = 118096

Therefore, the sum of the first 10 terms of the GP is 118096.

118096.

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The actual intake of carbohydrates is 138% as compare to recommended intake.

Recommended intake of carbohydrates or any other nutrient are,

Based on the information provided,

Consumed 159 grams of carbohydrates,

Recommended intake is between 115 and 166 grams.

Calculate the percentage of actual intake compared to the recommended intake, use the following formula,

Percentage = (Actual Intake / Recommended Intake) x 100%

Substituting the values in the formula we have,

⇒Percentage = (159 / 115) x 100%

⇒Percentage ≈ 138.3%

Therefore,  the actual intake of carbohydrates is about 138% of the recommended intake, indicating that consumption of more carbohydrates than recommended.

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If a₁ = 5 and an 5an-1 then The value οf a₄ is 625.

An arithmetic sequence is a sequence οf numbers in which each term after the first is fοund by adding a fixed cοnstant number, called the cοmmοn difference, tο the preceding term. Fοr example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a cοmmοn difference οf 3, since each term after the first is fοund by adding 3 tο the preceding term.

The nth term οf an arithmetic sequence can be fοund using the fοrmula:

an = a1 + (n-1)d

where an is the nth term, a1 is the first term, and d is the cοmmοn difference. The sum οf the first n terms οf an arithmetic sequence can be fοund using the fοrmula:

Sn = n/2 (a1 + an)

We are given that a₁ = 5, and that the nth term is 5 times the (n-1)th term. We can use this infοrmatiοn tο find the value οf a₄ as fοllοws:

a₂ = 5a₁ = 5(5) = 25

a₃ = 5a₂ = 5(25) = 125

a₄ = 5a₃ = 5(125) = 625

Therefore, the value of a₄ is 625.

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Answer:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

To find the Z-score that has 48.4% of the distribution's area to its left, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look for the closest probability value to 0.484. In the table, we find that the closest probability value is 0.4838, which corresponds to a Z-score of approximately 1.96.

Alternatively, we can use a calculator with a built-in normal distribution function. Using the inverse normal distribution function, also known as the quantile function, we can find the Z-score that corresponds to a given probability. For a probability of 0.484, we get:

invNorm(0.484)= 1.96

Therefore, the Z-score that has 48.4% of the distribution's area to its left is approximately 1.96.

Answer:

y = 4 sin(π x) + 5

Step-by-step explanation:

A sine function with a midline of y=5, an amplitude of 4, and a period of 2 can be written in the following form:

y = A sin(2π/ x) +

where A is the amplitude, is the period, is the vertical shift (midline), and x is the independent variable (usually time).

Substituting the given values, we get:

y = 4 sin(2π/2 x) + 5

Simplifying this expression, we get:

y = 4 sin(π x) + 5

Therefore, the sine function with the desired characteristics is:

y = 4 sin(π x) + 5

Answer:

Find f(g(x)) f(x)=7x-8 , g(x)=3x-2. f(x)=7x−8 f ( x ) = 7 x - 8 , g(x)=3x−2 g ( x ) = 3 x - 2. Step 1. Set up the composite result function. f(g(x)) f ( g ...

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Your total grade point average (GPA) for the semester is 2.67.

GPA stands for Grade Point Average. It is a numerical calculation used to measure the academic success of a student. It is calculated by taking the average of all grades received by a student across all courses taken in a given semester or academic year. Each course is assigned a specific number of credit hours, and each grade is assigned a numerical value based on the school’s grading scale. The numerical value of each grade is then multiplied by the number of credit hours for the course, and the total of all courses is added together to determine a student’s GPA.

A higher GPA is generally indicative of higher academic performance while a lower GPA is generally indicative of lower academic performance. A student’s GPA is used by schools, employers, and other organizations to evaluate a student’s academic record.

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This would give you a total of 29.0. Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.

GPA stands for Grade Point Average. It is an academic measure of a student's performance in a course or program of study. It is calculated by dividing the total number of grade points earned by the total number of credit hours taken. A grade point average is typically expressed as a number on a 4.0 scale. A 4.0 GPA is considered to be the highest possible grade point average, while anything below a 2.0 GPA is usually considered to be failing.

This is calculated by adding up the total number of credit hours (10 credit hours) and then multiplying each grade by the respective number of credit hours. A=4.0, B=3.0, C=2.0.
So, you would multiply 4.0 by 3 for FYE 105, 3.0 by 3 for ENG 101, 2.0 by 3 for MAT 150, 2.0 by 3 for BIO 112, and 4.0 by 1 for BIO 113.
This would give you a total of 29.0.
Divide this by 10 credit hours and you get a GPA of 2.85 for the semester.

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Sin 30 =3/x

1/2=3/x

x=6

(a) The natural coefficient of variation for one patient is the ratio of the standard deviation to the mean, expressed as a percentage:

C0 = (standard deviation / mean) x 100% = (3 / 5) x 100% = 60%.

(b) If the admission times of patients are independent, the mean and variance of admitting a group of 50 emergency patients can be calculated as follows:

mean = n x mean time = 50 x 5 = 250 minutes

variance = n x variance of individual patient / sample size = 50 x (3)^2 / 50 = 9

The coefficient of variation for a group of 50 emergency patients is the ratio of the standard deviation to the mean, expressed as a percentage:

C0 = (standard deviation / mean) x 100% = (3 / 5) x 100% = 60%.

(c) The effective mean and coefficient of variation of the admission time for a group of 50 trauma patients can be calculated using the following formula:

te = n x mean time / (1 - p1 x p2)

where p1 is the probability of machine failure and p2 is the probability of repair completion. Assuming the machine can fail at any time, p1 can be calculated as 1 / (mean time between failures / mean admission time) = 1 / (80 x 60 / 5) = 0.001042. Assuming the repair time is also exponentially distributed, p2 can be calculated as 1 / mean repair time = 1 / 4 = 0.25. Therefore, te = 50 x 5 / (1 - 0.001042 x 0.25) = 250.14 minutes. The variance of the admission time can be calculated using the formula:

σe^2 = n x variance of individual patient / (1 - p1 x p2)^2 = 50 x (3)^2 / (1 - 0.001042 x 0.25)^2 = 10.81. The coefficient of variation for a group of 50 trauma patients is the ratio of the standard deviation to the mean, expressed as a percentage:

Ce = (standard deviation / mean) x 100% = (sqrt(10.81) / 250.14) x 100% = 2.60%.

(d) The variability class of the squared coefficients of variation can be determined as follows:

C0² (Part a): (0.6)^2 = 0.36 (low variability)

C0² (Part b): (0.6)^2 = 0.36 (low variability)

Ce² (Part c): (0.026)^2 = 0.000676 (low variability)

(e) The manager of the center can improve the inflated effective admission time in Part c by implementing preventive maintenance measures to reduce the probability of machine failure, such as regular inspection and cleaning of the X-Ray machine, and by improving the repair process to reduce the mean repair time, such as hiring more skilled technicians or improving the repair procedures.

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The value of the triple integral[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex] by using spherical coordinates [tex]2\pi(e^{-1}-e^{-9})[/tex].

Given that the triple integral is-

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

E is the region bounded by the spheres which are,

[tex]x^2+y^2+z^2=1\\\\x^2+y^2+z^2=9[/tex]

In spherical coordinates we have,

x = r cosθ sin ∅

y = r sinθ sin∅

z = r cos∅

dV = r²sin∅ dr dθ d∅

E contains two spheres of radius 1 and 3 () respectively, the bounds will be like this,

1 ≤ r ≤ 3

0 ≤ θ ≤ 2π

0 ≤ ∅ ≤ π

Then

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

[tex]\int\int\int _{E} \frac{e^{-r^2}}{r}r^2Sin\phi drd\phi d\theta\\\\2\pi \int_{0}^{\pi} \int_1^3 re^{-r^2} dr d\phi\\\\2\pi \int_1^3 re^{-r^2} dr\\\\2\pi(e^{-1}-e^{-9})[/tex]

The complete question is-

Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.

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By conducting subtraction we know that Michael ran 5 miles more on day 3 than on day 2.

The four arithmetic operations are addition, multiplication, division, and subtraction.

Removal of items from a collection is represented by the operation of subtraction.

For instance, in the following image, there are 5 2 peaches, which means that 5 peaches have had 2 removed, leaving a total of 3 peaches.

The number that the other number is deducted from is known as a minuend.

Subtrahend: The amount that needs to be deducted from the minuend is known as a subtrahend.

Difference: A difference is an outcome obtained by deducting a subtractor from a minimum.

So, more miles Michael ran on day 3 than on day 2:

= 7 - 2

= 5 miles

Therefore, by conducting subtraction we know that Michael ran 5 miles more on day 3 than on day 2.

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The correct option is D. $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

In math, addition is the process of adding two or more integers together. The numbers having added are known as addends, while the outcome of the addition process, or the final response, is known as the sum. It is among the most fundamental mathematical operations we employ on a daily basis.

Quarterly profit for Mr. Brown's Goodwill Store in 2012 (in dollars)

1 $9,841.28

2 $8,957.67

3 $7,429.84

4 $11,095.67

Total profit = profit of 3rd quarter + profit of 4th quarter

Total profit = $7,429.84 + $11,095.67

Total profit =  $18,525.51.

Thus, $18,525.51. is the total profit made in the third as well as fourth quarters by Mr. Brown's store.

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Answer: D. Square

Step-by-step explanation:

The shape created by the cross section of the cut through a square pyramid is a square.

To see why, imagine the pyramid sitting on a table with its square base flat against the surface. The cut goes through the vertex, which is the point at the top of the pyramid. Since the cut is perpendicular to the base, it divides the pyramid into two smaller pyramids with congruent, but not identical, bases. Each of these smaller pyramids has a triangular base that is an isosceles right triangle. The two triangles share a common hypotenuse, which is the line of the cut.

The cross section of the cut is the shape formed where the two triangles meet along the hypotenuse. Since both triangles are congruent and the hypotenuse is the same for both, the cross section is a square. The sides of the square are equal to the base of the original pyramid, which is one of the legs of the isosceles right triangles formed by the cut. Therefore, the answer is D, a square.

Answer:

8(3/4y - 2) + 6(-1/2x + 4) + 1 can be simplified as:

6(-1/2x) = -3x

8(3/4y) = 6y

8(-2) = -16

6(4) = 24

1 remains as 1.

So the expression becomes:

6y - 3x - 16 + 24 + 1

which simplifies to:

6y - 3x + 9

Therefore, the expressions that are equivalent to 8(3/4y - 2) + 6(-1/2x + 4) + 1 are:

6y - 3x + 9

The volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.

To find the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis, we use the formula:

V = ∫[a,b] πr²dy

where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.

First, we need to find the equation of the curve that is generated when we rotate y = 4 - 2x² around the y-axis. To do this, we use the formula for the equation of a curve generated by revolving y = f(x) around the y-axis, which is:

x² + y² = r²

where r is the distance from the y-axis to the curve at any point (x, y).

Substituting y = 4 - 2x² into this formula, we get:

x² + (4 - 2x²) = r²

Simplifying, we get:

r² = 4 - x²

Therefore, the radius of the circular cross-sections perpendicular to the y-axis is given by:

r = √(4 - x²)

Now, we can integrate πr²dy from y = 0 to y = 2:

V = ∫[0,2] π(√(4 - x²))²dy

V = ∫[0,2] π(4 - x²)dy

V = π∫[0,2] (4y - y²)dy

V = π(2y² - (1/3)y³)∣[0,2]

V = π(8 - (8/3))

V = (16/3)π

Therefore, the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.

b. To find the volume of the solid generated by revolving the region enclosed by the curves x = √y+9, x = 0, and y = 1 through 360° about the y-axis, we use the same formula as in part (a):

V = ∫[a,b] πr²dy

where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.

First, we need to solve the equation x = √y+9 for y in terms of x:

x = √y+9

x² = y + 9

y = x² - 9

Next, we need to find the equation of the curve that is generated when we rotate y = x² - 9 around the y-axis. Using the same formula as in part (a), we get:

r = x

Now, we can integrate πr²dy from y = 1 to y = 10 (since x = 0 when y = 1 and x = 3 when y = 10):

V = ∫[1,10] π(x²)²dy

V = π∫[1,10] x⁴dy

V = π(1/5)x⁵∣[0,3]

V = π(243/5)

Therefore, the volume of the solid generated by revolving the region.

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Answer:

(a) Find (f + g)(x)

To find (f + g)(x), we add the two functions f(x) and g(x):

(f + g)(x) = f(x) + g(x) = (4x + 9) + (9x - 5) = 13x + 4

The domain of (f + g)(x) is all real numbers, since there are no restrictions on x that would make (f + g)(x) undefined.

(b) Find (f - g)(x)

To find (f - g)(x), we subtract the function g(x) from f(x):

(f - g)(x) = f(x) - g(x) = (4x + 9) - (9x - 5) = -5x + 14

The domain of (f - g)(x) is all real numbers, since there are no restrictions on x that would make (f - g)(x) undefined.

(c) Find (f * g)(x)

To find (f * g)(x), we multiply the two functions f(x) and g(x):

(f * g)(x) = f(x) * g(x) = (4x + 9)(9x - 5) = 36x^2 + 11x - 45

The domain of (f * g)(x) is all real numbers, since there are no restrictions on x that would make (f * g)(x) undefined.

(d) Find (f / g)(x)

To find (f / g)(x), we divide the function f(x) by g(x):

(f / g)(x) = f(x) / g(x) = (4x + 9) / (9x - 5)

The domain of (f / g)(x) is all real numbers except x = 5/9, since this value would make the denominator of (f / g)(x) equal to zero, resulting in division by zero, which is undefined.

(e) Find f(g(x))

To find f(g(x)), we substitute g(x) into the expression for f(x):

f(g(x)) = 4g(x) + 9

Substituting the expression for g(x), we get:

f(g(x)) = 4(9x - 5) + 9 = 36x - 11

The domain of f(g(x)) is all real numbers, since there are no restrictions on x that would make f(g(x)) undefined.

(f) Find g(f(x))

To find g(f(x)), we substitute f(x) into the expression for g(x):

g(f(x)) = 9f(x) - 5

Substituting the expression for f(x), we get:

g(f(x)) = 9(4x + 9) - 5 = 36x + 76

The domain of g(f(x)) is all real numbers, since there are no restrictions on x that would make g(f(x)) undefined.

(g) Find f(f(x))

To find f(f(x)), we substitute f(x) into the expression for f(x):

f(f(x)) = 4f(x) + 9

Substituting the expression for f(x), we get:

f(f(x)) = 4(4x + 9) + 9 = 16x + 45

The domain of f(f(x)) is all real numbers, since there are no restrictions on x that would make f(f(x)) undefined.

(h) Find g(g(x))

To find g(g(x)), we substitute g(x) into the expression for g(x):

g(g(x)) = 9

Step-by-step explanation:

Answer:

Step-by-step explanation:

(a) Find f(g(x)).

To find f(g(x)), we first need to find g(x) and then substitute it into f(x).

g(x) = 9x - 5

f(g(x)) = f(9x - 5) = 4(9x - 5) + 9 = 36x - 11

Therefore, f(g(x)) = 36x - 11.

(b) Find g(f(x)).

To find g(f(x)), we first need to find f(x) and then substitute it into g(x).

f(x) = 4x + 9

g(f(x)) = g(4x + 9) = 9(4x + 9) - 5 = 36x + 76

Therefore, g(f(x)) = 36x + 76.

(c) Find f(f(x)).

To find f(f(x)), we need to substitute f(x) into f(x).

f(f(x)) = 4(4x + 9) + 9 = 16x + 45

Therefore, f(f(x)) = 16x + 45.

(d) Find g(g(x)).

To find g(g(x)), we need to substitute g(x) into g(x).

g(g(x)) = 9(9x - 5) - 5 = 81x - 50

Therefore, g(g(x)) = 81x - 50.

Domain of f(x) and g(x): Since both f(x) and g(x) are linear functions, their domains are all real numbers.

(e) Find the inverse of f(x).

To find the inverse of f(x), we need to switch the roles of x and f(x) and solve for f(x).

y = 4x + 9

x = 4y + 9

x - 9 = 4y

y = (x - 9) / 4

Therefore, the inverse of f(x) is f^(-1)(x) = (x - 9) / 4.

(f) Find the inverse of g(x).

To find the inverse of g(x), we need to switch the roles of x and g(x) and solve for g(x).

y = 9x - 5

x = 9y - 5

x + 5 = 9y

y = (x + 5) / 9

Therefore, the inverse of g(x) is g^(-1)(x) = (x + 5) / 9.

(g) Find the domain of f^(-1)(x).

The domain of f^(-1)(x) is the range of f(x). Since f(x) is a linear function, its range is all real numbers. Therefore, the domain of f^(-1)(x) is also all real numbers.

(h) Find the domain of g^(-1)(x).

The domain of g^(-1)(x) is the range of g(x). Since g(x) is a linear function, its range is all real numbers. Therefore, the domain of g^(-1)(x) is also all real numbers.

The final closed formula answers for each part,

(a) an = n^2 + 1

(b) an = n(n + 1)(n + 2)/6

(c) an = 2n + 6

(d) an = n! + (n-1)! + ... + 2! + 1!

(a) The given sequence can be seen as the sequence of partial sums of the sequence of odd numbers: 1, 3, 5, 7, 9, . . . . That is, the nth term of the given sequence is the sum of the first n odd numbers, which is n^2. Therefore, the closed formula for the given sequence is an = n^2 + 1.

(b) The given sequence can be seen as the sequence of partial sums of the sequence of triangular numbers: 1, 3, 6, 10, 15, . . . . That is, the nth term of the given sequence is the sum of the first n triangular numbers, which is n(n + 1)(n + 2)/6. Therefore, the closed formula for the given sequence is an = n(n + 1)(n + 2)/6.

(c) The given sequence can be seen as the sequence of differences between consecutive squares: 1, 5, 9, 16, 21, . . . . That is, the nth term of the given sequence is the difference between the (n+1)th square and the nth square, which is (n + 1)^2 - n^2 = 2n + 1. Therefore, the closed formula for the given sequence is an = 2n + 6.

(d) The given sequence can be seen as the sequence of partial sums of the sequence defined recursively by a1 = 1 and an+1 = an(n + 1) for n ≥ 1. That is, the nth term of the given sequence is the sum of the first n terms of the recursive sequence. It can be shown that the nth term of the recursive sequence is n! (n factorial), and therefore the nth term of the given sequence is the sum of the first n factorials. That is, an = 1 + 1! + 2! + ... + (n-1)! + n!. Therefore, the closed formula for the given sequence is an = n! + (n-1)! + ... + 2! + 1!.

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The pairs of events of random integers are pairs of events are,

even and odd , negative and positive integers, zero and non-zero integers.

Two events are mutually exclusive if they cannot occur at the same time.

Selecting a random integer from -200 to 200,

Any two events that involve selecting a specific integer are mutually exclusive.

For example,

The events selecting the integer -100

And selecting the integer 50 are mutually exclusive

As they cannot both occur at the same time.

Any pair of events that involve selecting a specific integer are mutually exclusive.

Here are a few examples,

Selecting an even integer and selecting an odd integer.

Selecting a negative integer and selecting a positive integer

Selecting the integer 0 and selecting an integer that is not 0.

But,

Events such as selecting an even integer and selecting an integer between -100 and 100 are not mutually exclusive.

As there are even integers between -100 and 100.

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The expressiοn 5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³). Thus, οptiοn (c) 5x3 is the cοrrect respοnse.

The cοefficients (the numbers in frοnt οf the variables) οf the expressiοn 5x(x²) can be multiplied, and the expοnents οf the variables can be added, tο determine the prοduct.

The first cοefficient we have is 5 times 1, giving us 5. Sο, using the secοnd x², we have x tο the pοwer οf 2 multiplied by x tο the pοwer οf 1 (frοm the first x). Expοnents are added when variables with the same base are multiplied. Sο, x¹ multiplied by x² results in x³.

Cοmbining all οf the parts, the phrase becοmes:

5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³).

Thus, οptiοn (c) 5x³ is the cοrrect respοnse.

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The quadratic function's zeros are therefore  [tex]x = 1[/tex] and [tex]x = 0.2[/tex] . A degree two polynomial in one or so more variables that is a quadratic function.

Three points are used to determine a quadratic function, which has the form [tex]f(x) = ax2 Plus bx + c.[/tex]

[tex]Sqrt(b2 - 4ac) = [-b sqrt(b)][/tex] Where the quadratic function's coefficients are a, b, and c.

Here,  [tex]a = 20[/tex] , [tex]b = -19[/tex] , & [tex]c = 3[/tex]  . We obtain the quadratic formula by substituting these values: [tex]x = [-(-19) sqrt((-19)2 - 4(20)(3)] / 2(20) (20)[/tex]

When we condense this phrase, we get:

[tex]x = [19 +/- sqrt(361 - 240)] / 40 x = [19 +/- sqrt(121)] / 40\sx = [19 ± 11] / 40[/tex]

Therefore, The zeros of a quadratic equation   [tex]f(x) = 20x2 - 19x + 3[/tex] are as follows: [tex]x = (19 Plus 11) / 40 = 1 and x = (19 − 11) / 40 = 0.2.[/tex]

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The cost of the fat-free cottage cheese required in the recipe is R17.98.

How to determine the cost ?

To determine the cost, we first need to calculate the amount of fat-free cottage cheese required in the recipe. The recipe calls for 250g of fat-free cottage cheese, which can be obtained by using 2 units of 125g each. Knowing the cost of ingredients is important for Mrs Gilfillan to price the cake appropriately to cover her costs and make a profit.

What is the cost of the fat-free cottage cheese required to make the Mixed Berry and Almond Polenta Cake recipe that serves 15 espresso cups at Mrs Gilfillan's coffee shop?

The cost of 1 unit of 125g fat-free cottage cheese is R8.99.

Therefore, the cost of 2 units of 125g is calculated as:

R (2 x 8.99) =R 17.98

Hence, the cost of the fat-free cottage cheese required in the recipe is R 17.98.

This cost is in addition to the cost of the other ingredients, such as butter, sugar, ground almonds, mixed frozen berries, and polenta, as well as the cost of labor, overhead, and other expenses involved in making and selling the cake.

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The difference of the medians as a multiple of the interquartile range is 0.5,So the correct answer is option (A) 0.5.

The median is a measure of central tendency that represents the middle value in a data set when the values are arranged in numerical order.

For example, consider the data set {3, 5, 2, 6, 1, 4}. When the values are ordered from smallest to largest, we get {1, 2, 3, 4, 5, 6}. The median in this case is the middle value, which is 3.

We can first find the medians and interquartile ranges of the two dot plots.

For Mr. Wilson's class:

Median = 12

Q1 = 10

Q3 = 14

IQR = Q3 - Q1 = 14 - 10 = 4

For Mr. Watson's class:

Median = 10

Q1 = 8

Q3 = 12

IQR = Q3 - Q1 = 12 - 8 = 4

The difference of the medians is |12 - 10| = 2. Therefore, the difference of the medians as a multiple of the interquartile range is:

$$\frac{2}{4} = \boxed{0.5}$$

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So Abbie was asked to randomly select at least 368 of her high school unitary method students to estimate the percentage of students planning to go to college. With a 90% confidence level and a 5% error rate. 

The unit method is an approach to problem solving that first determines the value of a single unit and then multiplies that value to determine the required value. Simply put, the unit method is used to extract a single unit value from a given multiple. For example, 40 pens cost 400 rupees or pen price. This process can be standardized. single country. Something that has an identity element. (Mathematics, Algebra) (Linear Algebra, Mathematical Analysis, Matrix or Operator Mathematics) Adjoints and reciprocals are equivalent. 

To determine the required sample size, the following formula should be used:

[tex]n = (Z^2 * p * (1-p)) / E^2[/tex]

where:

N:sample size required

Z:The Z-score corresponds to the desired confidence level and is 1.645 at the 90% confidence level.

Pa:Estimated Percentage of Students Planning to Go to College

1-p:Percentage of students not planning to go to college

E:Desired error margin of 0.05

Since we don't know the actual percentage of students who want to go on to college, we must use estimates based on past studies and surveys. Let's assume the estimated proportion is 0.6 (her 60% of students).

After plugging in the values ​​it looks like this:

[tex]n = (1.645^2 * 0.6 * 0.4) / 0.05^2\\n = 368.03[/tex]

So Abbie was asked to randomly select at least 368 of her high school students to estimate the percentage of students planning to go to college. With a 90% confidence level and a 5% error rate. 

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