If f(x) and f^1(x)
are inverse functions of each other and f(x) - 2x+5, what is f^-1(8)?

-1
3/2
41/8
23

To solve this integral, we'll use a trigonometric substitution. Let x = (1/6)tan(θ), which implies dx = (1/6)sec^2(θ)dθ.

Now, we can rewrite the integral as:

∫√(1 + 36(1/6tan(θ))^2) (1/6)sec^2(θ)dθ

Simplify the expression inside the square root:

∫√(1 + 6^2tan^2(θ)) (1/6)sec^2(θ)dθ

Now, recall the trigonometric identity: 1 + tan^2(θ) = sec^2(θ). Using this identity, we have:

∫√(sec^2(θ)) (1/6)sec^2(θ)dθ

Simplify and integrate:

(1/6)∫sec^3(θ)dθ

Unfortunately, the integral of sec^3(θ) is non-elementary, so we cannot find a closed-form expression for it. However, you can look up the techniques used to evaluate this integral, such as integration by parts or reduction formulas, if you need a more detailed solution.

Remember to convert the result back to the original variable x using the substitution x = (1/6)tan(θ), and don't forget to add the constant of integration, C, at the end.

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(a) The three different trigonometric ratios that could have been used to find the distance x between Emma and the base of the tree are sin(60°) = 43.3/50, cos(60°) = x / 50, and tan(60°) = 43.3 / x. The distance is 25 ft.

(b) Using the Pythagorean theorem, the distance x between Emma and the base of the tree is 25 ft.

(a) We can use sine, cosine, and tangent trigonometric ratios to find the distance x between Emma and the base of the tree.

1. Sine:
sin(60°) = opposite/hypotenuse = (tree height) / 50
tree height = 50 * sin(60°) = 43.3 ft (since it's given that the tree is 43.3 ft tall)

2. Cosine:
cos(60°) = adjacent/hypotenuse = x / 50
x = 50 * cos(60°) = 25 ft

3. Tangent:
tan(60°) = opposite/adjacent = (tree height) / x
x = (tree height) / tan(60°) = 43.3 / tan(60°) ≈ 25 ft

(b) To find the distance x between Emma and the base of the tree using the Pythagorean theorem, we can consider the triangle formed by Emma, the base of the tree, and the top of the tree.

Let's call the distance from the base of the tree to the top of the tree (tree height) y.

Emma's distance to the cat (50 ft) is the hypotenuse, the distance x is one leg, and the tree height y (43.3 ft) is the other leg of the right triangle.

Using the Pythagorean theorem: x² + y² = hypotenuse²
x² + 43.3² = 50²
x² + 1874.89 = 2500
x² = 625.11
x ≈ 25 ft

So, the distance between Emma and the base of the tree is approximately 25 ft.

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A line segment of length 15 cm at a bearing of 50° from P to locate the position of Q on the drawing. Use the Law of Cosines the distance d between Q and L, which is approximately 71.2 km. Use the Law of Sines the angle x opposite d, which is approximately 29.5°, giving the bearing of Q from L.

Using the given scale of 1 cm represents 3 km, we can draw a line segment of length 15 cm (since 5 km/h x 12 h = 60 km) on a bearing of 50° from P to locate the position of Q. The point Q can be marked on the drawing using the x tool.

We can use the Law of Cosines to find the distance d between Q and L. Let a = 60 km (distance from P to Q), b = 36 km (distance from P to L), and C = 130° (the angle between a and b, which is equal to the sum of the angles at Q and L). Then

d² = a² + b² - 2ab cos(C)

d² = (60)² + (36)² - 2(60)(36)cos(130°)

d ≈ 71.2 km

Therefore, the distance of port Q from lighthouse L is approximately 71.2 km.

We can use the Law of Sines to find the angle x opposite the distance d between Q and L. Let a = 60 km (distance from P to Q), b = 36 km (distance from P to L), and sin(A) = sin(130°)/d. Then

sin(x)/60 = sin(130°)/d

sin(x) = (60/d)sin(130°)

x ≈ 29.5°

Therefore, the bearing of port Q from lighthouse L is approximately 29.5°.

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The polynomials are multiplied to give the expression x³ + 3x² - 22x - 24

We need to know that algebraic expressions are described as expressions that are composed of terms, variables, their coefficients, factors and constants.

Also, these expressions are made up of mathematical operations. They are listed as;

From the information given, we have the expression;

(x+6)(x2−3x−4)

expand the bracket, we get;

x³ - 3x² - 4x + 6x² - 18x - 24

add the like terms, we get;

x³ + 3x² - 22x - 24

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Probability of picking an even or prime card at random is approximately 88.89%.

To find the probability of picking an even or prime card at random, we first need to identify the possible cards that meet these conditions.

In a standard deck of cards, there are 52 cards, but we only consider the numerical values, which are 2-10 for each of the four suits (hearts, clubs, spades, and diamonds).

Even numbers: 2, 4, 6, 8, 10

Prime numbers: 2, 3, 5, 7

Combining the even and prime numbers, we have: 2, 3, 4, 5, 6, 7, 8, 10. Note that 2 appears only once in the combined list.

Now we find the probability by dividing the number of favorable outcomes (even or prime numbers) by the total number of possible outcomes (cards numbered 2-10).

P(even or prime) = (Number of even or prime cards) / (Total number of cards from 2-10)

P(even or prime) = 8 / 9

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

P(even or prime) = (8 / 9) × 100 ≈ 88.89%

So, the probability of picking an even or prime card at random is approximately 88.89%.

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The pattern of the data indicates a linear relationship or strong positive correlation between the average house prices and average monthly payments.

The monthly payment for a house costing $450,000 is $3,600.

A formulate connecting the average monthly payment with the average house price in slope-intercept form is y = 0.008x.

In Mathematics, a proportional relationship can be represented by this equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) as follows:

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 8/1000

Constant of proportionality (k) = 1/125 or 0.008.

Therefore, a formula that connects the two variables is given by;

y = kx

y = 0.008x

When average house price (x) = $450,000, the average monthly payment (y) is given by:

y = 0.008(450,000)

y = $3,600.

In conclusion, we can logically deduce that the pattern of the data shows a linear relationship or strong positive correlation between the average house prices (x) and average monthly payments (y) because as the average house prices (x) increases, the average monthly payments (y) also increases.

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The algebraic expressions representing the books Jan and Jackie have left are J - 4 and J - 2 - 4, respectively, where J is the initial number of books checked out by both.

Let's use "J" to represent the number of books Jan and Jackie checked out from the library. After 3 weeks, Jan turned in 4 books, so she has J - 4 books left.

Jackie returned 2 books after the first week, so she has J - 2 books left. When she returned 4 more books later, she had J - 2 - 4 = J - 6 books left. Therefore, the algebraic expressions for the number of books Jan and Jackie have left are

Jan: J - 4

Jackie: J - 6

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Sine and cosine are both equivalent to x/y, depending on the angle of the point on the circle.

To find two trigonometric functions equivalent to the ratio x/y, we can use the unit circle. Let's assume that x and y are the coordinates of a point on the circle, where x represents the horizontal displacement, and y represents the vertical displacement.

From this, we can find that the sine of the angle that x and y make with the x-axis is y, and the cosine of that angle is x. Therefore, the two trigonometric functions equivalent to the ratio x/y are sine and cosine.

We can write this as sin(angle) = y/y and cos(angle) = x/y. It's important to note that the value of the angle depends on the position of the point on the unit circle.

So, sine and cosine are both equivalent to x/y, depending on the angle of the point on the circle.

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Answer:We can start by resolving the forces acting on the ornamental star in the vertical direction and in the horizontal direction. Since the star is stationary, these forces must balance each other out.

Let's call the tension in the higher chain T1 and the tension in the lower chain T2. We can use trigonometry to find the vertical and horizontal components of each tension force.

For T1:

The vertical component is T1cos(40°) and the horizontal component is T1sin(40°).

For T2:

The vertical component is T2cos(30°) and the horizontal component is T2sin(30°).

Now, resolving the forces in the vertical direction, we have:

T1cos(40°) + T2cos(30°) - mg = 0

where m is the mass of the star and g is the acceleration due to gravity.

Substituting the values:

T1cos(40°) + T2cos(30°) - (40 kg)(9.81 m/s^2) = 0

T1cos(40°) + T2cos(30°) = 392.4 N

Next, resolving the forces in the horizontal direction, we have:

T1sin(40°) = T2sin(30°)

Now we have two equations with two unknowns:

T1cos(40°) + T2cos(30°) = 392.4 N

T1sin(40°) = T2sin(30°)

Solving these equations simultaneously, we get:

T1 = 266.4 N

T2 = 205.1 N

Therefore, the tension in the higher chain (T1) is 266.4 N and the tension in the lower chain (T2) is 205.1 N.

Step-by-step explanation:

Answer:

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Step-by-step explanation:

If Alex can stack exactly 16 cookies with a diameter of 5 cm inside a cylindrical container with the same diameter, then the height of the cylinder will be equal to the height of 16 cookies stacked on top of each other, which is 16 multiplied by the height of one cookie.

The diameter of each cookie is 5 cm, so the radius is 2.5 cm. The volume of each cookie is πr²h, where r is the radius and h is the height, so the volume of one cookie is:

V1 = π(2.5 cm)²h

The volume of 16 cookies will be:

V16 = 16π(2.5 cm)²h

Since the volume of the cylindrical container is 100% cm³, we have:

V16 = Vcyl

where Vcyl is the volume of the cylindrical container. Therefore:

16π(2.5 cm)²h = Vcyl

The height of 16 cookies stacked on top of each other is 16 times the height of one cookie, so:

h = 16(1 cm) = 16 cm

Substituting this value into the equation above and solving for the radius, we get:

r = √(Vcyl / (16πh)) = √(100 cm³ / (16π(16 cm))) ≈ 1.03 cm

The surface area of the cylindrical container is given by the formula:

A = 2πr² + 2πrh

Substituting the values we found for r and h, we get:

A = 2π(1.03 cm)² + 2π(1.03 cm)(16 cm) ≈ 142 cm²

Therefore, the surface area of the container is approximately 142 square centimeters. Rounded to the nearest square centimeter, the answer is 142 square centimeters.

The surface area of the given cylindrical container is 290 cm².

The volume of a cylinder is given by the formula:

V = πr²h

where r is the radius of the cylinder and h is its height.

We know that the volume of the cylindrical container is 100π cm³ and that it has the same diameter as the cookies, which is 5 cm.

Since the diameter of the container is 5 cm, its radius is 2.5 cm.

We can rearrange the formula for volume to solve for

h = V/πr²

h = 100π/π(2.5)²

h = 16

So, the height of the container is 16 cm.

To find the surface area of the container, we can use the formula:

A = 2πrh + 2πr²

where r is the radius of the container and h is its height.

Substituting the values we have, we get:

A = 2π(2.5)(16)+2π(2.5)²

A = 92.5π

A ≈ 290.45

Rounding to the nearest square centimeter,

A = 290

Thus, the surface area of the container is 290 cm².

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To find the volume of a cone, we use the formula:

[tex]V = (1/3)\pi r^2h[/tex]

where V is the volume, r is the radius of the circular base, h is the height of the cone, and [tex]\pi[/tex] is approximately 3.14159.

In this problem, the height of the cone is given as 10 ft and the radius of the circular base is given as 6 ft.

First, we need to find the slant height of the cone. We can use the Pythagorean theorem:

[tex]l = \sqrt{(r^2 + h^2)[/tex]

[tex]l = \sqrt{(6^2 + 10^2)[/tex]

[tex]l = \sqrt{\\(36 + 100)[/tex]

[tex]l = \sqrt{136[/tex]

[tex]l = 11.66 ft[/tex]

Now we can substitute the values into the formula for the volume:

[tex]V = (1/3)\pi r^2h[/tex]

[tex]V = (1/3)\pi (6^2)(10)[/tex]

[tex]V = 120\pi /3[/tex]

[tex]V = 40\pi[/tex]

[tex]V= 125.6 cubic feet[/tex]

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Answer:

568.2 in

Step-by-step explanation:

To find this we first have to divide 1894/10 then we get 189.4 which we multiply by 3 to find how much more sand we need.

To find the population at any given term n, continue to apply the recursive rule.
Pₙ = Pₙ₋₁ + 50
Using this recursive rule and the initial population, you can find the population at any given term n.

We are given a population growth model with a recursive rule and an initial population. Let's break down the information and find the population at any given term n.

Recursive rule: Pₙ = Pₙ₋₁ + 50
Initial population: P₀ = 30

Now let's find the population at any term n, using the recursive rule:

Step 1: Determine the base case, which is the initial population.
P₀ = 30

Step 2: Apply the recursive rule to find the next few terms.
P₁ = P₀ + 50 = 30 + 50 = 80
P₂ = P₁ + 50 = 80 + 50 = 130
P₃ = P₂ + 50 = 130 + 50 = 180

Step 3: To find the population at any given term n, continue to apply the recursive rule.
Pₙ = Pₙ₋₁ + 50

Using this recursive rule and the initial population, you can find the population at any given term n.

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Juanita will keep $187.50 as profit from her ice cream shop's sales this month.

Juanita will keep 25% of the monthly sales as profit, which is equivalent to $750 x 0.25 = $187.50. This means that out of the total monthly sales of $750, Juanita will keep $187.50 as her profit, while the remaining $562.50 will go towards the expenses of running the ice cream shop.

Profit is the amount of money that a business earns after deducting all its expenses. It is the excess revenue that remains after all the costs of doing business have been taken into account. Profit is essential for businesses as it helps them to sustain their operations, invest in growth opportunities, and generate returns for their owners or shareholders.

In Juanita's case, her profit is 25% of the monthly sales, which is a significant amount that can help her to keep her ice cream shop running smoothly. By keeping a portion of the sales as profit, Juanita can use the money to pay for her expenses, such as rent, utilities, and supplies, while still having enough left over to reinvest in her business or save for her personal use.

Overall, understanding the concept of profit is crucial for entrepreneurs and business owners to ensure that they can generate enough revenue to cover their expenses and make a profit that will help them to grow and succeed in the long run.

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There would be 26,790 books. of the 30,000 books in the collection should Mrs. Ross expect to have no damage

Out of the random sample, 902 books had no damage. To estimate the number of books with no damage in the entire collection of 30,000 books, we can use the proportion of books in the sample that had no damage.

The proportion of books in the sample with no damage is:

902 / (902 + 80 + 43) = 0.893

Therefore, we can expect approximately:

30,000 x 0.893 = 26,790

books in the collection to have no damage. Rounded to the nearest whole number, the answer is:

26,790 books.

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According to the model, each additional year of education adds $1200 to a person's yearly income in their first job after completing schooling.

We're given the equation y = 1200x + 7000, where x represents the number of years of education, and y represents the yearly income.

To find how much 1 year of education adds to a person's yearly income, we need to look at the coefficient of the variable x, which is 1200.

So according to the model, 1 year of education adds $1,200 to a person's yearly income.

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Answer:

If the pattern is to continue, the next value would be a decrease of 7 from the previous value because 20-13=7. Therefore, the value for this week would be:

13 - 7 = 6

So, if you were to continue the pattern, you would have bought 6 cupcakes this week.

We used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation

First, we used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation and solving for ∅, which resulted in ∅ ≈ 22.5 degrees.

Next, we applied the arc length formula, L = ∅/360 × 2 × π × r, and plugged in the values we had, including the calculated ∅ and the given radius (12 feet).

Finally, we calculated the arc length (L) to be approximately 4.71 feet, which is the length of steel that must be replaced.

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Answer:

Step-by-step explanation:

18 students don't walk to school

6 boys, 12 girls

12 students walk to school (30-18)

3/4 of students who walk to school are boys = 9 boys, 3 girls

The cost of a bag of chips is $0.75 and the cost of a cheeseburger is $3.

The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

let the cost of a cheesburger be x.

let the cost of a bag of chips be y

therefore, it is given that x + y = 3.75                ...........(i)

it is also given that 2x + 3y = 8.25                     ............(ii)

multiplying the equation in( i) by 2 we get 2x + 2y = 7.50     ......(iii)

subtracting the equation in iii) from the equation in ii) we get y = $0.75

Therefore ,the cost of a bag of chips is $0.75

Substituting the value of y found in (ii) we get x = 3.

therefore ,the cost of a cheeseburger is $3.

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The complete question is ,

Nathan ordered 1 cheeseburger and 1 bag of chips for 3. 75 jack ordered 2 cheeseburgers and 3 bags of chips for 8. 25.find the value of one cheeseburger and one bag?

The survey results show that the estimates for the number of students in the school who prefer rap music range from 110 to 154. The center of the estimates is around 121, with a variation of about 66.

To estimate the number of students who prefer rap music, multiply the proportion who prefer rap in each sample by the total number of students in the school (440)

Sample 1: 0.25 x 440 = 110

Sample 2: 0.20 x 440 = 88

Sample 3: 0.30 x 440 = 132

Sample 4: 0.35 x 440 = 154

The center of the estimates is the average of the four estimates: (110 + 88 + 132 + 154) / 4 = 121. Variability can be described using the range (154 - 88 = 66) or the standard deviation of the estimates, which requires additional calculations.

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The clockwise rotation about the origin from the pre-image to the image​ include the following: a rotation of 90° clockwise.

In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Additionally, the rotation of a point 90° about the center (origin) in a clockwise direction would produce a point that has these coordinates (y, -x).

Next, we would apply a rotation of 90° clockwise to the coordinate of triangle ABC as follows;

(x, y)                               →            (y, -x)

Coordinate A = (1, 1) → Coordinate A' = (1, -(1)) = (1, -1)

Coordinate B = (1, 4) → Coordinate B' = (4, -(1)) = (4, -1)

Coordinate C = (5, 1) → Coordinate C' = (1, -(5)) = (1, -5)

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The student does 18 J of work on the textbook. Their power output is 0.0228 W. Work is calculated by multiplying force and displacement, while power is the rate at which work is done and is calculated by dividing work by time.

To calculate the work done by the student on the textbook, we use the formula

work = force x distance x cos(theta)

where force is the weight of the textbook, distance is the height it is lifted, and theta is the angle between the force and distance vectors, which is 0 degrees since the force is acting vertically upward and the distance is upward as well. Thus, we have

work = 12 N x 1.5 m x cos(0) = 18 J

To calculate the power output of the student, we use the formula:

power = work / time

where work is the work done by the student on the textbook, and time is the total time it took to lift and carry the book. Thus, we have:

time = 1.5 s + 7 s = 8.5 s

power = 18 J / 8.5 s = 2.12 W

Therefore, the student did 18 J of work on the textbook, and had a power output of 2.12 W.

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--The given question is incomplete, the complete question is given

" a student lifts a 12 N textbook 1.5 m in 1.5 s and carries the books 5 m across the room in 7 s. how much work does the student do on the textbook? and what is the power output of the student.​"--

Answer:

28 degrees and 62 degrees

Step-by-step explanation:

Set up your equation like this:

x+(x-34)=90

2x-34=90

2x=56

x=28

28+34=62

So the smaller angle is 28 degrees and the larger angle is 62 degrees.

Hope this helps!! :D

Answer:

Step-by-step explanation:

Car are the preferred transportation for 168 residents.Together, Walk and Streetcar are the preferred transportation for 55 residents.Bus is the preferred transportation for 45 residents.Bicycle is the preferred transportation for 50 residents.Question 2(Multiple Choice Worth 2 points)(Appropriate Measures MC)The box plot represents the number of tickets sold for a school dance.A horizontal line labeled Number of Tickets sold that starts at 8, with tick marks every one unit up to 30. The graph is titled Tickets Sold for A Dance. The box extends from 17 to 21 on the numb

Triangle A′B′C′ is a dilation of ​triangle ABC​ about point P with a scale factor of 1/2. The dilation is a reduction.

A dilation is a transformation that changes the size of a figure, but not its shape. It can be either an enlargement or a reduction depending on the value of the scale factor.

If the scale factor is greater than 1, then the image will be larger than the original figure. This is called an enlargement.

If the scale factor is between 0 and 1, then the image will be smaller than the original figure. This is called a reduction.

In this case, the scale factor is 1/2, which is less than 1. Therefore, the image of triangle ABC is smaller than the original triangle, and the dilation is a reduction.

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Therefore, the probability that Jack wins the race and Martina finishes last is 1/6 or approximately 0.167.

There are 3 people racing, so there are 3! = 6 possible ways the race can end (assuming no ties).

These are:

Jack, Martina, Napier

Jack, Napier, Martina

Martina, Jack, Napier

Martina, Napier, Jack

Napier, Jack, Martina

Napier, Martina, Jack

Of these 6 outcomes, there is only 1 where Jack wins the race and Martina finishes last: outcome 2.

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Answer:

[tex]y = 2x + 9[/tex]

Step-by-step explanation:

It is given that the slope of the line is 2, and it passes through (-3 , 3). The equation of straight lines is y = mx + b, in which:

y = (x , y) = 3

m = slope (gradient) = 2

x = (x , y) = -3

b = y-intercept

~

Plug in the corresponding numbers to the corresponding variables:

y = mx + b

3 = (2)(-3) + b

First, multiply -3 with 2:

[tex]3 = (2)(-3) + b\\3 = (2 * -3) + b\\3 = -6 + b[/tex]

Next, isolate the variable, b. Note the equal sign, what you do to one side, you do to the other. Add 6 to both sides of the equation:

[tex]3 = b - 6\\3 (+6) = b - 6 (+6)\\b = 3 + 6\\b = 9[/tex]

Plug in 2 for slope, and 9 for y-intercept, in the given equation:

[tex]y = mx + b\\m = 2\\b = 9\\[/tex]

[tex]y = 2x + 9[/tex] is your answer.

~

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Driver's data set that has a mode that is greater than the mean or median is Fabian and a median with the lowest value of the three measures is Kadisha.

Data of Fabian: 7, 12, 11, 23, 13, 23, 30

Mean = sum of all observation / total no. of observation

Mean = (7+ 12+ 11+ 23+ 13+ 23+ 30) / 7

Mean = 17

Mode = most repeating observation

Mode = 23

For median we have to write observation in ascending order

7,11,12,13,23,23,30

Median = (N+1)/2

Where N = No. of observation

Median = (7+1)/2

Median = 4th observation

Median = 13

Here mode that is greater than the mean or median.

similarly for,

Kadisha: 8, 17, 23, 16, 17, 18, 125

Mean = 32

Median = 17

Mode = 17

Here median with the lowest value of the three measures.

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Answer:

Step-by-step explanation:

a)  Profit = Revenue - Cost = (90x - 0.5x²) - (50x + 300)

= -0.5x² + 90x - 50x - 300

= -0.5x² + 40x - 300

b)  -0.5x² + 40x - 300 = 300

-0.5x² + 40x - 600 = 0

use quadratic equation to find the roots of x (a = -0.5, b = 40, c = -600):

x = 20, 60

c)  -0.5x² + 40x - 300 = 15000

-0.5x² + 40x - 15300 = 0

use quadratic equation to find the roots of x (a = -0.5, b = 40, c = -15300):

x = 40±10√290i

Not possible to make a profit of $15,000