69 POINTS NEED HELP ASAP QUESTION IS DOWN BELOW

The solutions for x are x= +[tex]\sqrt{12.5[/tex] and -[tex]\sqrt{12.5[/tex] which has been obtained by solving the equations given in the question.

Define Equation?

An equation can be defined in numerous ways. An equation is a claim that demonstrates the equivalence of two mathematical expressions, according to algebra.

In the initial equation, we get: by substituting y=x:

[tex]x^2[/tex] +[tex]y^2[/tex]= 25

[tex]x^2[/tex] +[tex]x^2[/tex] = 25 (since y = x)

2[tex]x^2[/tex] = 25

[tex]x^2[/tex] = 12.5

When the two sides are square, we get:

x = ±[tex]\sqrt{12.5[/tex]

Therefore, the solutions for x are x = +[tex]\sqrt{12.5[/tex] and x = -[tex]\sqrt{12.5[/tex]

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The constant of proportionality for vanilla to chocolate ice cream is 2/3.

The constant of proportionality represents the fixed relationship between two quantities in a proportional relationship.

In this case, we are given that there are 3 containers of chocolate ice cream for every 2 containers of vanilla ice cream.

To find the constant of proportionality in terms of vanilla to chocolate, we need to express this relationship as a fraction with the same units.

We can start by setting up the ratio of vanilla to chocolate ice cream, which would be 2:3.

This can then be expressed as

2/3

Therefore, the constant of proportionality for vanilla to chocolate ice cream is 2/3.

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Zara should buy 400  pairs of jeans ordered from Hong Kong and 100 pairs ordered from Colombia for a total cost of $6,000.

See below for other solutions

Let x be the number of pairs of jeans ordered from Hong Kong and y be the number of pairs ordered from Colombia.

Objective function:

Maximize profit, C(x, y) = 11x + 16y

Subject to the following constraints:

See attachment for the graph, where we have the following corner points

(x, y) = (100, 400), (300, 200), (400, 100)

By substitution, we have

C(100, 400) = 11(100) + 16(400) = 7500

C(300, 200) = 11(300) + 16(200) = 6500

C(400, 100) = 11(400) + 16(100) = 6000

The minimum cost above is $6000

So, Zara should buy 400  pairs of jeans ordered from Hong Kong and 100 pairs ordered from Colombia.

The amount paid for them is $6,000

If the Hong Kong supplier were able to reduce its percentage of defective pairs of jeans from 7% to 5%, we would need to update the constraint accordingly:

0.05x + 0.02y ≤ 0.05(x+y)

The new optimal solution would be to order 425 pairs of jeans from the Hong Kong supplier and 75 pairs of jeans from the Colombian supplier, for a total cost of $5875

i.e. C(425, 75) = 11(425) + 16(75) = 5875

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To find the speed of the car, we need to use the formula:

speed = (RPM * tire diameter * pi * gear ratio) / (336 * 12)

where:

RPM is the engine speed in revolutions per minute

tire diameter is the diameter of the tires in inches

pi is the mathematical constant pi (approximately 3.14)

gear ratio is the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear

336 is a constant representing the number of inches per minute that a car will travel at 1 mile per hour

12 is a constant representing the number of inches in a foot

Plugging in the given values, we get:

speed = (2690 * 27 * 3.14 * 3.2) / (336 * 12)

simplifying the equation, we get:

speed = 63.8 mph (rounded to one decimal place)

Therefore, the car will travel at a speed of approximately 63.8 miles per hour

Answer:

x = 20.4

Step-by-step explanation:

cos(32) = adjacent/ hypotenuse

24 cos (32) = x

x = 20.4

The Correct answer is  D) Most students read more often than they draw.as the median of the "read" plot is lower than the median of the "draw" plot.

Based on the box plots, we can see that:

The median for the "read" data is around 8 days per month, and the interquartile range (IQR) is between 4 and 12 days.

The median for the "draw" data is around 12 days per month, and the IQR is between 8 and 16 days.

So, we can conclude that:

A) Most students read less than 12 days a month. This is true, since the median for the "read" data is below 12 days per month, and the box plot shows that most of the data is concentrated in the lower half of the range.

B) Most students draw at least 12 days a month. This is not true, since the median for the "draw" data is around 12 days per month, and the box plot shows that there is a considerable amount of data below that median.

C) Most students draw more often than they read. This is not true, since the median for the "read" data is lower than the median for the "draw" data.

D) Most students read more often than they draw. This is true, since the median for the "read" data is lower than the median for the "draw" data, and the box plot shows that most of the "read" data is concentrated in the lower half of the range, while most of the "draw" data is concentrated in the upper half of the range.

The box plots below show data from a survey of students under 14 years old. They were asked on how many days in a month they read and draw. Based on the box plots, which is a true statement about students? pg780337 A Most students read less than 12 days a month. B Most students draw at least 12 days a month. C Most students draw more often than they read. D Most students read more often than they draw.

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Answer:

84 feet is equal to 1008 inches. So, the 84-foot building would be 1008 inches divided by 21 feet, which is 48 inches tall in the model.

Step-by-step explanation:

2.2 percentage answer you silly

Answer:

0.33

Step-by-step explanation:

There are a total of 5 + 15 + 10 + 15 = 45 people in the restaurant.

The probability of selecting a manager or a cook is the sum of the probabilities of selecting a manager and selecting a cook, since these events are mutually exclusive (a person cannot be both a manager and a cook at the same time).

The probability of selecting a manager is 5/45, since there are 5 managers out of 45 people in total.

The probability of selecting a cook is 10/45, since there are 10 cooks out of 45 people in total.

Therefore, the probability of selecting either a manager or a cook is:

P(manager or cook) = P(manager) + P(cook)

P(manager or cook) = 5/45 + 10/45

P(manager or cook) = 15/45

P(manager or cook) = 1/3

So, the probability that the person selected at random is either a manager or a cook is 1/3 or approximately 0.333

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the product of (x + 3/x)² is x² + 6 + 9/x².

Why it is and what is Algebra?

To find the product of (x + 3/x)², we can use the formula for squaring a binomial:

(a + b)² = a² + 2ab + b²

In this case, we have a = x and b = 3/x, so we can substitute these values into the formula and simplify:

(x + 3/x)² = x² + 2(x)(3/x) + (3/x)²

= x² + 6 + 9/x²

Therefore, the product of (x + 3/x)² is x² + 6 + 9/x².

Algebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. It involves the use of letters and symbols to represent numbers and quantities in equations and expressions. The primary goal of algebra is to solve mathematical problems and equations using various operations such as addition, subtraction, multiplication, and division.

Algebraic concepts can be applied to a wide range of mathematical and real-world problems, including geometry, physics, engineering, economics, and statistics.

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Complete question:

Find the product of (x + 3/x)²2.

The interquartile range of the number of novels written by each of Amir's 10 favorite authors is 6.

What is the interquartile range ?

In statistics, the interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. Quartiles divide the data set into four equal parts, with the interquartile range being the difference between the upper (third) and lower (first) quartiles. The IQR represents the middle 50% of the data, and is used as a measure of spread or dispersion in a data set, particularly when the data set contains outliers.

To find the interquartile range (IQR), we need to first find the first quartile (Q1) and the third quartile (Q3).

To find Q1:

Arrange the data in order from smallest to largest: 1 3 3 4 5 7 8 10 12 23

Find the median of the lower half of the data (the first five numbers): (3+3)/2 = 3

This is the first quartile (Q1)

To find Q3:

Arrange the data in order from smallest to largest: 1 3 3 4 5 7 8 10 12 23

Find the median of the upper half of the data (the last five numbers): (8+10)/2 = 9

This is the third quartile (Q3)

Now we can calculate the IQR:

IQR = Q3 - Q1 = 9 - 3 = 6

Therefore, the interquartile range of the number of novels written by each of Amir's 10 favorite authors is 6.

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With 98% confidence, we can say that the population variance is between 7.36 and 71.09.

What is probability?

Probability is a branch of mathematics that deals with the study of chance or randomness in events. It is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

To construct the confidence interval for the population variance, we will use the chi-squared distribution.

First, we need to calculate the chi-squared values for the lower and upper bounds of the confidence interval using the following formulas:

χ²_L = (n - 1) * s² / χ²(α/2, n-1)

χ²_U = (n - 1) * s² / χ²(1-α/2, n-1)

where n is the sample size, s² is the sample variance, α is the level of significance, and χ²(α/2, n-1) and χ²(1-α/2, n-1) are the chi-squared values with α/2 and 1-α/2 degrees of freedom, respectively.

Substituting the given values, we get:

χ²_L = (9 - 1) * 17.3 / χ²(0.01, 8) ≈ 3.355

χ²_U = (9 - 1) * 17.3 / χ²(0.99, 8) ≈ 29.587

Next, we can use these chi-squared values to construct the confidence interval for the population variance:

(V_L, V_U) = [(n - 1) * s² / χ²_U, (n - 1) * s² / χ²_L]

Substituting the given values, we get:

(V_L, V_U) = [(9 - 1) * 17.3 / 29.587, (9 - 1) * 17.3 / 3.355]

Simplifying, we get:

(V_L, V_U) ≈ [7.36, 71.09]

Therefore, with 98% confidence, we can say that the population variance is between 7.36 and 71.09.

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Answer: The selling price of cat litter in the container shown when the markup is 20% is $3.00 per pound.

Step-by-step explanation:

One cubic foot of cat litter weighs about 48 pounds, so the weight of one pound of cat litter is 1/48 cubic feet.

The store pays $2.50 per pound for cat litter, so the cost of one pound of cat litter is $2.50.

To find the selling price with a 20% markup, we add 20% of the cost to the cost.

20% of $2.50 is $0.50.

So, the selling price per pound is $2.50 + $0.50 = $3.00.

Hope this helps, and have a great day!

Answer: If Jamie is selling bags of carrots for $5 each and already has $30 what would he use to calculate how much money he made.

After cutting the squares, the length of each shorter side (and the side length of each triangular quilt piece) is approximately 14√2 inches i.e. B.

What are squares?

A square is a two-dimensional shape with four straight sides that are of equal length and four right angles (90-degree angles) at the corners. It is a special case of a rectangle, where all four sides are the same length. A square can also be thought of as a special type of rhombus, where the angles are all right angles.

Now,

Since the hypotenuse of each triangle is 28 inches long, we know that this is also the longest side of the right triangle formed by the two shorter sides of the triangle. Let's call one of the shorter sides x.

Using the Pythagorean Theorem, we can find the length of the other shorter side:

c²= a² + b²

28² = x² + x²

784 = 2x²

x² = 392

x = √(392)

x=14√2 in

Therefore, the length of each shorter side (and the side length of each triangular quilt piece) is approximately 14√2 inches (rounded to two decimal places).

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Therefore , the solution of the given problem of percentage comes out to be 15 customers per hour make up the unit cost. 15 clients per hour, to be exact.

A number or statistic that is stated as an amount of 100 is referred to as "a%" in statistics. The forms "pct," "pct," or "pc" are additionally uncommon. The common way to indicate it is with a representation "%," though. The ratio to the total sum is flat, and there are no indicators. Percentages are actually integers because they almost always add up to 100. To suggest that a number indicates a percentage, either the word "fraction" or a representation for percentage (%) must come before the number.

Here,

We must divide the overall number of customers by the number of hours in order to determine the unit rate (in customers per hour).

There are 120 total clients.

There are 8 hours in all.

Unit rate is equal to the sum of all clients times the number of hours.

=> Unit rate: 120 / 8

=>  Rate per unit: 15

Consequently, 15 customers per hour make up the unit cost. 15 clients per hour, to be exact.

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Answer:

Measures of Central Tendancy

Mean: 429

Median: 344

Mode: 134,185,193,217,471,580,799,853

Range: 719

Step-by-step explanation:

The mean of a data set is commonly known as the average. You find the mean by taking the sum of all the data values and dividing that sum by the total number of data values. The formula for the mean of a population is

[tex]\mu = \frac{{\sum}x}{N}[/tex]

The formula for the mean of a sample is

[tex]\bar{x} = \frac{{\sum}x}{n}[/tex]

Both of these formulas use the same mathematical process: find the sum of the data values and divide by the total. For the data values entered above, the solution is:

[tex]\frac{3432}{8} = 429[/tex]

The median of a data set is found by putting the data set in ascending numerical order and identifying the middle number. If there are an odd number of data values in the data set, the median is a single number. If there are an even number of data values in the data set, the median is the average of the two middle numbers. Sorting the data set for the values entered above we have:

[tex]134, 185, 193, 217, 471, 580, 799, 853[/tex]

Since there is an even number of data values in this data set, there are two middle numbers. With 8 data values, the middle numbers are the data values at positions 4 and 5. These are 217 and 471. The median is the average of these numbers. We have

[tex]{\frac{ 217 + 471 }{2}}[/tex]

Therefore, the median is

[tex]344[/tex]

The mode is the number that appears most frequently. A data set may have multiple modes. If it has two modes, the data set is called bimodal. If all the data values have the same frequency, all the data values are modes. Here, the mode(s) is/are

[tex]134,185,193,217,471,580,799,853[/tex]

Answer:

translation

Step-by-step explanation:

becuz it can't be rotation as it isn't rotated in any way.

it also cant be reflection as it isnt facing the other way.

so its translation

the organization needs to sell at least 22 tickets to break even.

What is the arithmetic operation?

The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.

Let C be the total production cost and n be the number of tickets sold.

From the problem, we know that the organization had a net loss of $800 after selling 10 tickets, so we have:

10(70) - C = -800

Simplifying, we get:

C = 1500

This means that the total production cost was $1500.

The revenue from selling n tickets at $70 per ticket is given by:

R = 70n

Substituting the values of R and C, we get:

70n = 1500

Solving for n, we get:

n = 1500/70 = 21.43

Since we can't sell a fraction of a ticket, we need to round up to the nearest whole number.

Therefore, the organization needs to sell at least 22 tickets to break even.

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Answer:

Start off with the 20ft sides

20x2 =40ft

40ft + 8ft = 48ft

Now we need to find the perimeter of the half circle.

Since we know the diameter of the half circle is 8ft, we can use the following formula: Diameter x Pi = Circumference

Plug in:

8ft x 3.14159 = 25.132ft

25.132 /2 gives us the perimeter of the half circle

25.132 / 2 = 12.566

Rounded = 12.57 ft

48 ft + 12.57 ft =  60.57 feet perimeter.

The perimeter of the given solution is 76.56 feet.

We need to find the perimeter of the figure by splitting the figure into a rectangle and a semi-circle. The radius for the semi-circle is found by dividing the diameter by 2, Radius = 8 ÷ 2 = 4.

Therefore, the formula for finding the Perimeter of the given rectangle is  

P = 2( l+b )

P = 2( 20 + 8)

P = 56 feet.

now, the formula for the circumference of a semi-circle is

C = [tex]\pi[/tex]r + 2r

C = 3.14( 4) + 2 (4)

C = 20.56 feet.

Therefore, the perimeter of the given figure is

The perimeter of  the rectangle + circumference of the Semi-circle

= 56 + 20.56

= 76.56 feet

Therefore, the perimeter of the given solution is 76.56 feet.

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We may be confident that this is the correct scale factor because both equation equations yield the same value of k. As a result, the dilatation has a scale factor of 5/8 and is centered at the origin.

An equation is a statement in mathematics that states the equality of two expressions. An equation has two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the value "9". The purpose of equation solving is to determine which variable(s) must be changed in order for the equation to be true. Simple or complex equations, regular or nonlinear equations, and equations with one or more elements are all possible. In the equation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are employed in a variety of mathematical disciplines, including algebra, calculus, and geometry.

Let (x,y) be a point on the plane and k be the dilation scale factor centred at the origin. The image of (x,y) under dilation is thus given by (kx, ky).

The dilation is given as (4,6) to (5/2,15/4). That is to say:

[tex]k(4) = 5/2 \sk(6) = 15/4\\k = 5/8 \sk = 5/8[/tex]

We may be confident that this is the correct scale factor because both equations yield the same value of k. As a result, the dilatation has a scale factor of 5/8 and is centered at the origin.

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Answer:

Concatenation means joining number characters from string end to string end.

Step-by-step explanation:

Example:

concatenation of "water" and "bottle" is "waterbottle"

In math 1, 234, 5678 is 12345678

In your word problem the concatenation of "x" and "f" is X=F

Hopefully, this was helpful!! :)

It switches between negative and positive every 40 seconds.  it switches between positive and negative every 80 seconds. So correct statements are B and E.

Mathematics' branch of algebra deals with symbols and the formulas used to manipulate them. It is an effective tool for dealing with issues involving mathematical expressions and equations. In algebra, variables—which are typically represented by letters—are used to represent unknowable or variable quantities.

Equations represent mathematical relationships between variables in algebra. An equation is made up of two expressions, one on either side of an equal sign, separated by an equation. Algebraic expressions can involve constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

As we can see from the first roller coaster's graph, Michael's height changes from positive to negative after 40 seconds, whereas it was positive for the first 40. It remains negative between 40 and 80 seconds. It continues to be positive from 80 to 120, and so forth.

As a result, every 40 seconds it alternates between negative and positive.

B is accurate.

We can see from the second roller coaster's table that it stays positive from 0 to 80. It continues to be negative from 80 to 160, and so forth.

As a result, every 80 seconds it alternates between positive and negative.

E is accurate.

The complete question is:

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Median is most appropriate since it is not affected by outliers or extreme values, and gives a more representative estimate of the typical number of siblings for seventh graders. Mode can also be considered since it represents the most frequent value, but it may not be as representative as the median in this case.

What is mean?

In statistics, the mean is a measure of central tendency that is calculated by adding up all the values in a dataset and dividing by the total number of values.

It is commonly referred to as the "average." The mean is often used as a summary statistic to describe the typical value in a dataset. It is sensitive to outliers, meaning that extreme values can significantly influence the value of the mean. The mean is also widely used in hypothesis testing and statistical inference.

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Answer:

[tex]\textsf{1.}\quad \sf \overline{AZ} = 28 \; meters[/tex]

[tex]\textsf{2.}\quad \sf \overline{AM} = 28 \; meters[/tex]

[tex]\textsf{3.}\quad b = \sf 4[/tex]

[tex]\textsf{4.}\quad \sf Perimeter = 112\; meters[/tex]

[tex]\textsf{5.}\quad \sf \overline{MX} = 22\; meters[/tex]

[tex]\textsf{6.}\quad \sf \overline{AX} = 10\sqrt{3}\; meters[/tex]

[tex]\textsf{7.}\quad \sf \overline{EX} = 10\sqrt{3}\; meters[/tex]

[tex]\textsf{8.}\quad \sf \overline{AE} = 20\sqrt{3}\; meters[/tex]

Step-by-step explanation:

All sides of a rhombus are the same length. Therefore, for rhombus MAZE:

[tex]\sf \overline{AZ} = \overline{AM} = \overline{ZE} = \overline{EM}[/tex]

Given:

As the sides of a rhombus are the same length, we can equate the expressions for sides AZ and AM, and solve for b:

[tex]\begin{aligned}\overline{\sf AZ}&=\overline{\sf AM}\\8b-4&=5b+8\\8b-4-5b&=5b+8-5b\\3b-4&=8\\3b-4+4&=8+4\\3b&=12\\3b \div 3&=12 \div 3\\b&=4\end{aligned}[/tex]

Therefore, the value of b is 4.

To find the length of AZ and AM, substitute the found value of b into one of the expressions:

[tex]\begin{aligned}\overline{\sf AZ}&=8b-4\\&=8(4)-4\\&=32-4\\&=28\end{aligned}[/tex]

Therefore, as AZ = AM, then AZ = 28 and AM = 28.

[tex]\hrulefill[/tex]

As the sides of a rhombus are equal in length, each side length is 28 meters (as found previously).

The perimeter of rhombus MAZE is the sum of its side lengths. Therefore:

[tex]\begin{aligned}\sf Perimeter\;MAZE&=\sf \overline{AZ} +\overline{AM} +\overline{ZE}+ \overline{EM}\\&=28+28+28+28\\&=112\; \sf meters\end{aligned}[/tex]

Therefore, the perimeter of rhombus MAZE is 112 meters.

[tex]\hrulefill[/tex]

The point of intersection of the diagonals of rhombus MAZE is point X.

As the diagonals of a rhombus are perpendicular bisectors of each other, then:

[tex]\sf \overline{AX}=\overline{EX}\quad and \quad \overline{AX}+\overline{EX}=\overline{AE}[/tex]

[tex]\sf\overline{MX}=\overline{ZX}\quad and \quad\overline{MX}+\overline{ZX}=\overline{MZ}[/tex]

Given MZ = 44 meters, and MX is half of MZ, then:

[tex]\sf \overline{MX}=\overline{ZX}=22\;meters[/tex]

As the diagonals bisect each other at 90°, m∠MXA= 90°. Therefore, ΔMXA is a right triangle with hypotenuse AM = 28 and leg MX = 22.

As we know the lengths hypotenuse AM and leg MX, we can use Pythagoras Theorem to calculate the length of the other leg, AX:

[tex]\begin{aligned}\sf \overline{AX}^2+\overline{MX}^2&=\sf \overline{AM}^2\\\sf \overline{AX}^2+22^2&=\sf 28^2\\\sf \overline{AX}^2&=\sf 28^2-22^2\\\sf \overline{AX}&=\sqrt{\sf 28^2-22^2}\\\sf \overline{AX}&=\sf 10\sqrt{3}\; meters\end{aligned}[/tex]

As the diagonals bisect each other, AX = EX. Therefore:

[tex]\sf \overline{EX}=\sf 10\sqrt{3}\; meters[/tex]

The length of diagonal AE is the sum of segments AX and EX. Therefore:

[tex]\begin{aligned}\sf \overline{AE}&=\sf \overline{AX}+\overline{EX}\\&=\sf 10\sqrt{3}+10\sqrt{3}\\&=\sf 20\sqrt{3}\; meters\end{aligned}[/tex]

[tex]\hrulefill[/tex]

Note: The attached diagram is drawn to scale.

According to the question the percent abundance of the medium nails in the sample is approximately 18.95%.

Whenever the set of data is presented from least to largest, the median is indeed the number in the middle. For instance, since 8 is in the middle, this would represent the median value here.

To find the percent abundance of the medium nails in the sample, we first need to calculate the total number of nails in the sample:

Total number of nails = 67 + 18 + 10 = 95

Next, we can calculate the percent abundance of the medium nails using the formula:

Percent abundance = (number of medium nails / total number of nails) x 100%

Using the values from of the table as inputs, we obtain:

Percent abundance of medium nails = (18 / 95) x 100% ≈ 18.95%

As a result, the sample's average percentage of medium nails is roughly 18.95%.

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Answer:

The height of the plant is less than 10 centimeters if less than 3 weeks have passed.

Step-by-step explanation:

If less than 3 weeks have passed, then the value of t is less than 3. We can express this as t < 3.

Using the formula h = 3t + 1, we can substitute t < 3 to get:

h = 3t + 1 < 3(3) + 1

h < 10

According to the question approximately 83.89% of professors make more than $75,000 calculated by forming the equation.

When the roots and solutions of two equations match, they are said to be equivalent. The same number, symbols, or expression must be added to or subtracted from both the equation's two sides to produce an equivalent equation. By multiplying or dividing both sides of an equation by a nonzero number, we can also obtain an analogous equation.

This issue can be resolved using the conventional normal distribution. First, we need to standardize the value of $75,000 using the formula:

z = (x - μ) / σ

where x is indeed the value we wish to standardise, is indeed the mean, and is the deviation.

z = (75,000 - 85,900) / 11,000

z = -0.99

Next, we look up the area to the right of z = -0.99 in the standard normal distribution table or by using a calculator. The area to the left of z = -0.99 is 0.1611, so the area to the right of z = -0.99 is:

1 - 0.1611 = 0.8389

Therefore, approximately 83.89% of professors make more than $75,000.

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Answer:

a) We can use similar triangles to find PB in terms of a and b. Let x be the length of AD. Then, using the fact that AP = 4PC, we have:

PC = CP = x - b

AP = 4(x - b)

Also, using the fact that DC = (1/4)AB, we have:

AD = x

AB = 4DC = x/4

BC = AB - AD = x/4 - x = -3x/4

Now, consider the similar triangles PBC and ABD:

PB/AB = BC/AD

PB/(x/4) = (-3x/4)/x

PB = -3/4(x/4) = -3x/16

Finally, substituting x = a + b, we have:

PB = -3(a + b)/16

b) Using the same similar triangles as in part (a), we have:

DP/DC = PB/BC

DP/(1/4)AB = PB/(-3x/4)

DP = -3/4(PB)(DC/BC)AB

DP = -3/4(PB)(1/4)/(AB - AD)AB

Substituting the expressions for PB, AB, and AD from part (a), we get:

DP = -3(a + b)/16 * 1/4 / (-3(a + b)/4) * (a + b)/4

DP = -3/16 * 1/4 * 4/(3(a + b)) * (a + b)

DP = -3/16

So, DP = -3/16(a + b)

c) To check if DPB forms a straight line, we need to verify if the slopes of DP and PB are equal. Using the expressions we found in parts (a) and (b), we have:

Slope of PB = Δy/Δx = (-3/16(a+b) - 0)/(0 - (-3(a+b)/16)) = 3/16

Slope of DP = Δy/Δx = (-3(a+b)/16 - (-3/16(a+b)))/(1/4 - 0) = -3(a+b)/4

Since the slopes are not equal, DPB does not form a straight line.