How did knowing what addition and subtraction mean help you figure out how the value of an expression can decrease even if the value of a variable increases?

Answer:

Okay, let's solve this step-by-step:

   There is a spinner with 14 equal areas, numbered 1 through 14

   We want to find the probability that the result is a multiple of 3 or a multiple of 4

   There are 14 possible outcomes (numbers 1 through 14) when the spinner is spun.

   Of these 14 numbers:

   4 are multiples of 3: 3, 6, 9, 12

   4 are multiples of 4: 4, 8, 12, 16

   However, 12 is also a multiple of both 3 and 4, so we have counted it twice.

   We need to subtract 1 from each to account for this:

   Multiples of 3: 3

   Multiples of 4: 4

   So there are 3 possible multiples of 3 and 3 possible multiples of 4.

   In total, there are 3 + 3 = 6 possible multiples of 3 or 4.

   To calculate probability:

   Probability = (Number of favorable outcomes) / (Total possible outcomes)

   = 6 / 14

   = 3/7

   Converting to a percent: 3/7 = 42.9%

   Rounded to the nearest whole percent: 43%

Therefore, the probability that the result is a multiple of 3 or a multiple of 4 is 43%.

Step-by-step explanation:

There is a volume of  14,000 gallons of water in the tank after 4 minutes.

To answer your question, let's consider the given terms: the rate at which water is poured into the tank (2000t + 1000 gallons per minute) and the initial volume of water in the tank (5000 gallons).

Since the rate is given in terms of time t, we can find the volume of water poured in after 4 minutes by plugging t = 4 into the rate equation:

Volume poured in 4 minutes = 2000(4) + 1000
Volume poured = 8000 + 1000
Volume poured = 9000 gallons

Now, add this to the initial volume of water in the tank:

Total water in tank after 4 minutes = Initial volume + Volume poured
Total water = 5000 + 9000
Total water = 14,000 gallons

So, after 4 minutes, there are 14,000 gallons of water in the tank.

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The sequences that appear to be converging are:

2/7, 11/16, 26/37, 47/65, 74/101, 107/144, 146/197...

8/7, 64/49, 512/343, 4096/2401, 37768/16807, 262144/117649...

32.45, 31.80, 31.655, 31.654, 31.653, 31.652,...

converging refers to a sequence or series of numbers approaching a limit as more terms are added. It indicates that the values of the sequence or series are getting closer and closer to a specific value or point without ever reaching it.

a sequence is a list of numbers or objects arranged in a specific order. Each element in the sequence is called a term, and the position of the term in the sequence is called its index.

According to the given information:

The first two sequences are examples of converging sequences. In the first sequence, the numerators and denominators increase by a fixed amount for each term, but the ratio of the terms approaches a limit as the sequence progresses. Similarly, in the second sequence, each term is the previous term multiplied by a fixed factor, and the terms approach a limit as the sequence progresses.

The fifth sequence is also an example of a converging sequence, where the terms are decreasing and getting closer to a specific value as the sequence progresses.

The third sequence is not converging since the terms are increasing without bound, and the fourth sequence is oscillating between two values and not getting closer to any specific value.

The sixth sequence starts with a fixed value and increases by a fixed percentage for each term, so the terms are also increasing without bound and not converging to a specific value.

In summary, converging sequences have terms that approach a specific value as the sequence progresses, while non-converging sequences either increase or oscillate without getting closer to any specific value.

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Answer:

There are 20 R100 notes and 80 R50 notes.

Step-by-step explanation:

Let's call the number of R100 notes "x" and the number of R50 notes "4x" (since there are 4 times as many R50 notes as R100 notes).

We can set up an equation to represent the total value of the notes:

100x + 50(4x) = 6000

Simplifying:

100x + 200x = 6000

300x = 6000

x = 20

So there are 20 R100 notes and 4 times as many R50 notes:

4x = 4(20) = 80

Therefore, there are 20 R100 notes and 80 R50 notes.

Answer:

The possible radius and height for the inside of the cylinder that meets the constraints are r = 4 cm and h = 9.4 cm.

Step-by-step explanation:

The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height. We want to find a radius and height that will give us a volume between 475 and 480 cubic centimeters, with a diameter between 8 and 10 centimeters. First, we’ll use the lower limits of the constraints: a volume of 475 cubic centimeters and a diameter of 8 centimeters. The diameter is 8 centimeters, so the radius is 4 centimeters. We can plug in these values to the formula for volume and solve for h: 475 = 3.14 x 4^2 x h 475 = 50.24h h = 9.44 So a possible radius and height for the inside of the cylinder that meets the constraints are: r = 4 cm and h = 9.4 cm.

Percentiles provide information about the placement of data points in a quantitative dataset, histograms display the distribution of data, categorical data is grouped into categories, and ordinal data has an inherent order but no meaningful numerical differences.

Based on the terms you provided, we are looking for information about percentiles, histograms, categorical data, and ordinal data.
A percentile is a measure of placement in a sorted quantitative dataset.

It indicates the relative standing of a data point within the dataset.

For example, a score in the 80th percentile means that it is higher than 80% of the scores in the dataset.
A histogram is a graphical representation of the distribution of a dataset, typically used for continuous data.

The vertical axis of a histogram displays frequency or density, while the horizontal axis represents the data values divided into bins or intervals.
Categorical data is a type of data that can be divided into groups or categories, such as gender, ethnicity, or favorite color.

A numerical summary for categorical data usually involves counts or percentages for each category.
Ordinal data is a type of data where the order matters, but the difference between values is not meaningful, such as rankings or ratings.

A numerical summary for ordinal data can include measures of central tendency, such as the median or mode.

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Answer:

P(heads) = 1/2

There are 4 prime numbers less than 10: 2, 3, 5, 7.

P(prime number) = 4/10 = 2/5

1/2 × 2/5 = 1/5

Thus, the obtained area of sector for the given circle is found as 43.96 in².

Two radii that meet at the centre to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle assessment and radius measurement are both crucial for solving circle-related difficulties.

The curved portion that runs along the circle's perimeter and joins the ends of a two radii that make up a sector is known as the sector arc.

given data:

Formula for the area of sector:

area of sector = Ф /360 * (πr²)

area of sector = 140/360 * (3.14 *6²)

area of sector = 7/18 * 3.14 *36

area of sector = 43.96 in²

Thus, the obtained area of sector for the given circle is found as 43.96 in².

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Answer:

Starting with the 3-4-5 right triangle, multiply all lengths by 50, obtaining 150, 200, and 250. So the length of the longer leg is 200 meters.

The greatest distance, in feet, a person could be from the parking lot light and be detected is 4 ft. So, correct option is B.

The equation (x + 10)^2 + (y−8)^2=16 describes a circle with center (-10,8) and radius 4. The distance from the center of a circle to any point on its circumference is equal to the radius. Therefore, the greatest distance a person could be from the parking lot light and still be detected is equal to the radius of the circle, which is 4 feet.

This means that any point on the circumference of the circle with a distance of 4 feet or less from the center (-10,8) would be within the range of the motion detector.

Option (a) 2 feet is too small and would fall inside the circle, meaning the person would not be detected. Option (c) 9 feet and option (d) 256 feet are too large and fall outside the circle, meaning the person would also not be detected.

Therefore, the correct answer is option (b) 4 feet.

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Answer: 80

Step-by-step explanation:

Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.

Simple interest denotes interest that is simply charged on the principal amount, which is the original amount borrowed or deposited. The interest charge is going to be applied once, regardless of how frequently it is applied. Many loans base their calculations on simple interest, but you should double-check before signing anything.

Given data:

Formula for the estimating simple interest:

SI = PRT/100

SI = 1250*3.5*2 / 100

SI = 87.5

Amount = principal + simple interest

A = 1250 + 87.5

A = 1337.5

Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.

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Answer: 22

Step-by-step explanation:

So since Margo missed 24.6% of 90 free throws, you have to find 24.6% of 90. The easiest way to do this is to do 90 multiplied by 0.246.

To multiply by percentages in general, move the decimal point to the left by two places. For example, if it said 82.6, the decimal point would go to the left two places and be 0.826.

Back to the original question...

so 90 multiplied by 0.246 is 22.14. If you can use a calculator, this is easy, if not just do the usual multiplication.

Then, this rounds to 22, and the problem is complete!

For the food preparation time data with or without slicer, the null hypothesis for this sample data should not be rejected. So, option (b) is right answer.

We have a food preparation time data of a company. The time of preparation is significantly reduced with the handy dandy slicer. Sample size, n = 12

We have a table present in above figure which has the preparation times data consists people, with slicer and without slicer. First we calculate difference between with slicer and without slicer data, that d: -2, -6, 2, -8 , 0, -1, 1, 3, 4, -6, -5, 3.

We do all work on the difference, d.

Mean value of difference d, [tex]\bar d = \frac{\sum_{i = 1}^{n} {d_i }}{n}[/tex] = -1.25

Standard deviations, [tex] \sigma = \sqrt{\frac{ \sum {d_i^2 - (\sum d_i)²}}{n - 1}}

[/tex] = 4.115

degree of freedom= 12 - 1 = 11

Let level of significance = 0.05

The null and alternative hypothesis are defined as [tex]H_0 : \mu_d = 0 [/tex]

[tex]H_a: \mu_d < 0[/tex]

Using t-test, [tex]t = \frac{ \bar d} {\frac{ \sigma}{ \sqrt{n}}}[/tex]

[tex]= \frac{ -1.25} {\frac{ 4.115}{ √12}}[/tex]

= - 1.05

Now, using t-distribution table, value of p for t = -1.05 for 11 degree of freedom is -1.796. So, p-value = -1.796 < 0.005, Hence, we should not reject the null hypothesis.

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The radius of the lid would be 3. 5 cm.

To estimate the radius of the lid, we must first determine the circumference of the jar, which is equal to the jar's perimeter. The specified circumference is 21.98 cm.

The circumference is:

C = 2πr

We can solve for circumference as:

21. 98 = 2 πr

r = 21. 98 / ( 2 π )

r = 21. 98 / (2 x 3. 1416 )

r = 21. 98 / 6. 2832

r = 3. 5 cm

In conclusion, the radius of the lid is approximately 3.5 cm.

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The volume of the cone is approximately 37.7 cubic units

What is volume?

A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder, or a sphere. Different shapes have different volumes.

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

V = (1/3) * π * r² * h

where π is the constant pi, r is the radius of the base of the cone, and h is the height of the cone.

Plugging in the given values, we get:

V = (1/3) * 3.14 * 3² * 4 ≈ 37.7

Therefore, the volume of the cone is approximately 37.7 cubic units (rounded to the nearest tenth).

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f'(x) = 2x - 2 is always positive on (-∞, 1) and always negative on (1, ∞), the function is increasing and decreasing, respectively, and therefore one-to-one. Therefore, f(x) is invertible in its natural domain.

What is the exponential function?

An exponential function is a mathematical function of the form f(x) = aˣ

where "a" is a constant called the base, and "x" is a variable. Exponential functions can be defined for any base "a", but the most common base is the mathematical constant "e" (approximately 2.71828), known as the natural exponential function.

To determine if the function f(x) = x² - 2x + 3 is increasing, decreasing, even, odd, and/or invertible in its natural domain, we need to analyze its derivative and second derivative:

f'(x) = 2x - 2

f''(x) = 2

Increasing/decreasing: Since f''(x) is always positive (i.e., 2 is positive), the function is always concave up and has a minimum at x = 1. Thus, f(x) is increasing on (-∞, 1) and decreasing on (1, ∞).

Even/odd: To determine if f(x) is even or odd, we need to check if it satisfies the properties of even and odd functions.

Even function: A function f(x) is even if f(-x) = f(x) for all x in the domain. Let's check if f(x) satisfies this property:

f(-x) = (-x)² - 2(-x) + 3 = x² + 2x + 3

f(x) = x² - 2x + 3

f(-x) ≠ f(x), so the function is not even.

Odd function: A function f(x) is odd if f(-x) = -f(x) for all x in the domain. Let's check if f(x) satisfies this property:

f(-x) = (-x)² - 2(-x) + 3 = x² + 2x + 3

-f(x) = -(x² - 2x + 3) = -x² + 2x - 3

f(-x) ≠ -f(x), so the function is not odd.

Invertible: To determine if f(x) is invertible, we need to check if it has an inverse function.

A function has an inverse function if it is one-to-one, which means that it passes the horizontal line test.

To check if f(x) is one-to-one, we can analyze its derivative.

Hence,  f'(x) = 2x - 2 is always positive on (-∞, 1) and always negative on (1, ∞), the function is increasing and decreasing, respectively, and therefore one-to-one. Therefore, f(x) is invertible in its natural domain.

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Complete question:

For the following, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain: f(x) = x² - 2x + 3.

If ethan threw away 16 Paris of solid-coloured socks and 16 Paris of patterned socks, Ethan had 140 pairs of patterned socks at first.

Let's start by assuming that Ethan had x pairs of patterned socks. According to the problem statement, Ethan had 3/4 as many solid-coloured socks as patterned socks. Therefore, the number of solid-coloured socks he had can be represented as 3/4*x.

Ethan threw away 16 pairs of solid-coloured socks and 16 pairs of patterned socks. So, after the clean-up, he had (3/4*x)-16 pairs of solid-coloured socks and (x-16) pairs of patterned socks.

The problem states that 1/5 of his socks were now solid-coloured socks. So, we can set up an equation:

(3/4x-16) = (1/5)(3/4*x + x - 32)

Simplifying and solving for x, we get:

x = 140

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Answer:

Therefore, the answer is: (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2

Step-by-step explanation:

To prove that (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2 for all values of x, we can simply expand the left-hand side of the equation and simplify it to match the right-hand side.

Expanding the left-hand side using the distributive property, we get:

(2x-1)(x+2)(3x-1) = (2x^2+3x-2)(3x-1)

= 6x^3 + 7x^2 - 9x + 2

This matches the right-hand side of the equation, so we have proven that (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2 for all values of x.

The slope of the cathedral ceiling is 0.83.

To calculate the slope of the cathedral ceiling, we need to find the rise over the run. The rise is the difference in ceiling heights between the two points. The run is the horizontal distance between the two points.

From the problem statement, we know that the ceiling is 8 feet high at the west wall and 10.5 feet high three feet from the west wall. So the rise is:

rise = 10.5 - 8 = 2.5 feet

The run is the horizontal distance between the two points, which is:

run = 3 feet

Now we can calculate the slope:

slope = rise/run = 2.5/3 = 0.83

Therefore, the slope of the cathedral ceiling is 0.83.

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Using the definition of the derivative, we have:
f'(a) = lim (h -> 0) [(f(a+h) - f(a)) / h]

We're asked to use the definition of the derivative, which includes the terms:

limit, h, f(a+h), f(a), and the fraction f(a+h) - f(a) / h.
The definition of the derivative of a function f(x) at a specific point x=a is given by:
f'(a) = lim (h -> 0) [(f(a+h) - f(a)) / h]
f(a+h) represents the value of the function f(x) at the point (a+h).
f(a) represents the value of the function f(x) at the point a.
f(a+h) - f(a) calculates the difference in the function values at these two points.
h is the small change in the x-coordinate (x-axis) between these two points, and we let h approach 0 to ensure we're finding the instantaneous rate of change.
(f(a+h) - f(a)) / h represents the average rate of change between the points (a, f(a)) and (a+h, f(a+h)).
By taking the limit as h approaches 0, we find the instantaneous rate of change at the point x=a, which is the derivative of the function f(x) at x=a.

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The two possible sets of dimensions for the rectangular plot are:

Length = 27 meters, Width = 4 meters

Length = 4 meters, Width = 15.5 meters

Let's denote the length of the rectangular plot by L and the width by W. We know that the total length of the fencing needed is 35 m, which we can use to create an equation:

L + 2W = 35

The area of the land is 132 square meters, which we can use to create another equation:

LW = 132

We can solve the first equation for L:

L = 35 - 2W

Substituting this into the second equation, we get:

(35 - 2W)W = 132

Expanding and rearranging, we get a quadratic equation:

2W^2 - 35W + 132 = 0

We can solve for W using the quadratic formula:

W = [35 ± √(35^2 - 4(2)(132))] / (2(2))

W = [35 ± √(841)] / 4

W = [35 ± 29] / 4

Solving for W, we get two possible values:

W = 4 or W = 15.5

If W is 4 meters, then L is:

L = 35 - 2W = 27

If W is 15.5 meters, then L is:

L = 35 - 2W = 4

Therefore, the two possible sets of dimensions for the rectangular plot are:

Length = 27 meters, Width = 4 meters

Length = 4 meters, Width = 15.5 meters

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The value of x in the circle is 19.5 .

What is secant line of circle?

A secant is a straight line that intersects a circle at least twice in different places. We can draw an endless number of secants on a circle since a circle has an infinite number of points around its perimeter.

Here the given circle using formula for secant line ,

(A+B).B = (C+D).D

Here A = x , B = 8 , C= 12 and D=10. Then,

=> (x+8).8= (12+10).10

=> 8x+64 = 22*10

=> 8x+64 = 220

=> 8x = 220-64 = 156

=> x = 156/8 = 19.5

Hence the value of x in circle is 19.5.

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The probability of exactly three successes in five trials of a binomial experiment with a 90% probability of success is 0.0729 or about 7.29%.

To find the probability of exactly three successes in five trials of a binomial experiment with a 90% probability of success, we can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Plugging in the values, we get:

P(X = 3) = (5 choose 3) * (0.9)^3 * (1 - 0.9)^(5 - 3)

= 10 * (0.9)^3 * (0.1)^2

= 0.0729

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Answer: (-15, 20)

Step-by-step explanation:

we can see that the equation is the standard form of a circle:

(X-h)

^2 + (y-k)r^2

Where (h,k) is the center of the circle and r is the radius

H= -15

K= 20

R^2= 100

The correlation between the number of wins and each of the four baseball statistics would be an example of bivariate correlation analysis, which measures the strength and direction of the relationship between two variables.

The correlation between number of wins and each of the four baseball statistics would be an example of a bivariate correlation, which measures the strength and direction of the linear relationship between two variables. In this case, the two variables would be the number of wins and each of the four baseball statistics. The correlation coefficient would provide a numerical value that represents the degree of association between the two variables

A positive correlation coefficient would indicate that as one variable increases, so does the other, while a negative correlation coefficient would indicate that as one variable increases, the other decreases.

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If all these statistics were analyzed at the end of the season, the correlation between the number of wins and each of

the four baseball statistics would be an example of bivariate analysis.

Bivariate analysis involves analyzing the relationship between two variables to determine if there is any correlation or

association between them.

In this case, the two variables are the number of wins and each of the four baseball statistics.

By conducting a bivariate analysis, you can identify if there is any significant relationship between these variables and

potentially draw conclusions about their impact on team performance.

If all these statistics were analyzed at the end of the season, the correlation between the number of wins and each of

the four baseball statistics would be an example of bivariate analysis.

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The algebraic expression representing her employee's contribution to Social Security Tax is given as follows:

p > 516.67.

The algebraic expression representing her employee's contribution to Social Security Tax is obtained applying the proportions in the context of the problem.

An year is composed by 12 months, hence the monthly salary is represented as follows:

p > 100000/12

p > 8,333.33.

The Social Security tax is of 6.20%, hence the expression is given as follows:

p > 8,333.33 x 0.062

p > 516.67.

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The horizontal asymptote of the function f(x) = 2ˣ is y = 0.

In Mathematics and Geometry, a horizontal asymptote can be defined as a horizontal line (y = b) where the graph of a function approaches the line as the input values (domain or independent value) approach negative infinity (-∞) to positive infinity (∞).

In this context, the line passing through the ordered pair (0, 1) on the graph of this exponential growth function represents a horizontal asymptote and it has an equation of y = 0, as shown in the image attached above.

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