an elargemnet or expansion occurs when the absolute value of the scale factor is less than one true or false

The velocity of an ant moving along the x-axis whose position is given by x(t) = 2cos(π/2 t^2) is v(t) = -πtsin(π/2 t^2) and acceleration is a(t) = -π²t²cos(π/2 t^2) - πtsin(π/2 t^2). The ant moves in right direction in the time interval [-1, 0) and changes direction at t = 0.

(a) To find the velocity of the ant at any given time t, we need to take the derivative of the position function x(t) with respect to t. The derivative of x(t) = 2cos(π/2 t^2) is:

v(t) = -πtsin(π/2 t^2)

So the velocity of the ant at time t is given by -2πsin(π/2 t^2)

(b) To find the acceleration at any given time t, we need to take the derivative of the velocity function v(t) with respect to t. The derivative of v(t) = -πtsin(π/2 t^2) is:

a(t) = -π²t²cos(π/2 t^2) - πtsin(π/2 t^2)

So the acceleration of the ant at time t is given by  -π²t²cos(π/2 t^2) - πtsin(π/2 t^2).

(c) To determine the values of t for which the ant is moving to the right, we need to look at the sign of the velocity function v(t) = -πtsin(π/2 t^2). Since the sine function oscillates between -1 and 1, this function will be positive when sin(π/2 t^2) is negative, and negative when sin(π/2 t^2) is positive. Moving rights means positive value of x.

Therefore, the ant is moving to the right when sin(π/2 t^2) < 0. That is

(π/2)t^2 < sin⁻¹(0)

sin⁻¹(0) = nπ, where n = ..., -2, -1, 0, 1, 2, ...

t is in interval [-1, 1] so π/2t^2 is in interval [0, π/2]

sin⁻¹(0) = 0

(π/2)t^2 < 0

Therefore t < 0

That is possible values of t is [-1, 0)

(d) To determine the values of t for which the ant changes direction, we need to look at the sign of the acceleration function a(t) = -π²t²cos(π/2 t^2) - πtsin(π/2 t^2).

When ant change direction a(t) = 0.

-π²t²cos(π/2 t^2) - πtsin(π/2 t^2) = 0

πtcos(π/2 t^2) + sin(π/2 t^2) = 0

sin(π/2 t^2) = -πtcos(π/2 t^2)

tan(π/2 t^2) = -πt

On solving possible values of t in the interval [-1, 1] is 0.

So, the ant changes direction when t = 0.

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In essence, a perfect square is produced when two equal integers are multiplied by one another.

Squaring an integer produces perfect squares. For instance, the number 81 is a perfect square since it can be calculated by squaring 9: 99=81. Due to the fact that 12 is squared to produce 144, 144 is a perfect square. Because 13 is squared to produce 169, 169 is a perfect square. Language of Mathematics. Informally: The result is known as a square number, a perfect square, or simply "a square" when an integer (a "whole" number), whether positive, negative, or zero, is multiplied by itself. So all square numbers are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so forth.

given

In essence, a perfect square is produced when two equal integers are multiplied by one another.

Due to the fact that you are multiplying two equal integers (5 and 5) by one another, 25, it is a perfect square. The result of a number being multiplied by itself is referred to as the square of the number.

In the case when m and n are both natural numbers, m is a perfect square if m = n2. A perfect square is produced when two perfect squares are added together.

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The factoring of the equation is 6x(x + 8). The process of factoring involves determining what to multiply to produce an expression.

The process of factoring involves determining what to multiply to produce an expression.

It resembles "dividing" an expression into several smaller expressions.

There are three steps involved in factoring completely:

A GCF should ideally be factored out of the expression.

As much as you can, factor a trinomial.

Put as many factors as you can into a Difference Between Two Squares.

The Grouping method, the difference in two squares, and the sum or difference in cubes are the four primary methods of factoring. The Greatest Common Factor (GCF) is another.

It should be noted that the given expression must be factored based on the provided information.

The expression given is:

6x² + 48x

The common factor is 6x

6x(x + 8)

Therefore, the factoring of the equation is 6x(x + 8).

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Observational data collection might be best to study how voters might decide an upcoming ballot issue.

The majority of big data is observational data, such as bank transaction records, user viewing patterns on video streaming services, or social media posts. Observational data, however, can also be sparse data (based on just a few people).

In observational research, participants and phenomena are observed in their most natural environments. Instead of using structured environments like research labs or focus groups, this allows researchers to observe how their subjects make decisions and respond to situations in their everyday lives.

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(a) The concentration of our solution in the tank initially is 0.060 kg/L. (b) the amount of salt in the tank after 2. 5 hours is 44.45 kg. (c) The concentration of salt in the solution in the tank as time approaches infinity is 0.030 kg/L.

Divide the solute's gram weight by the total weight of the solution, then multiply the result by 100 to get the mass/mass percent of the solution. The amount of solute (measured in grams) per milliliter of solution is known as the mass/volume percent.

(A) Tank contains 60 kg of salt and 1000 L of water. So the concentration of solution in the tank initially = 60/1000 kg/l

The concentration of solution in the tank initially = 0.060 kg/L

(B) Let y(t) denote the amount of salt in the tank at any time t.

Rate In

A solution of concentration 0.03 kg of salt per liter enters a tank at the rate 9 L/min.

[tex]R_{i}[/tex]  =(concentration of salt in inflow)(input rate of solution)

= (0.03kg/L)*(9L/min)

= 0.27kg/min

Rate Out

The solution is mixed and drains from the tank at the same rate.

Concentration, C(t) = Amount/Volume = y(t)/1000

[tex]R_{out}[/tex] =(concentration of salt in outflow)(output rate of solution)

= {y(t)/1000}kg/L*9L/min

= 0.009y(t) kg/min

Therefore, the differential equation for the amount of Salt in the Tank  at any time t:

dy/dt = 0.27 - 0.009y(t)

So the amount of salt in the tank after 2. 5 hours (2.5*60 min = 150 min) is  44.45 kg.

(c) The concentration of salt in the solution in the tank as time approaches infinity

As t -> -∞, e^-∞ = 0

so, the concentration of salt in the solution in the tank as time approaches infinity is 0.030 kg/L.

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A graph of ΔABC with vertices A(0, 0), B (3, 3), and C (4, 0) and its image after a 90° rotation about the origin is shown in the image attached below.

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

By applying a rotation of 90° counterclockwise to the vertices of triangle ABC (∆ABC), the coordinates of the vertices of the image (∆A′B′C′) are as follows:

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

Ordered pair A  = (0, 0) → Ordered pair A' = (0, 0)

Ordered pair B = (3, 3) → Ordered pair B' = (-3, 3)

Ordered pair C = (4, 0) → Ordered pair C' = (0, 4)

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The answer to the given question is  x = -2, x = 2. The given problem can be solved by the following steps :

Step 1: Rewrite the equation in standard form: x2 + 7x - 4 = 0

Step 2: Find the factors of the coefficient of x and the constant term: -4 = -2 x -2, 7 = 7

Step 3: Set up the factors to solve the equation: (x + 2)(x - 2) = 0

Step 4: Solve each factor: x + 2 = 0, x - 2 = 0

Step 5: Find the solutions: x = -2, x = 2

Therefore, the final answer is x = -2, x = 2.

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Answer:

47.07, 47.16, 47.3, 47.4, 48.1

Step-by-step explanation:

lol

The process of simplify the equation is explained below.

The term equation is known as a mathematical statement that defines the equality between two expressions.

Here we have to define the process of simplifying the equation.

The first and foremost step is to remove parentheses and brackets by multiplying factors.

And then we have to use the exponent rule to remove grouping if the terms contain exponents.

Then we have to combine like terms by adding or subtracting coefficients.

Finally, we have to Combine the constants.

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Answer:

  ln(x²y⁵/(5z⁷))

Step-by-step explanation:

You want to condense the logarithmic expression 2 ln(x) + 5 ln(y) - 7 ln(z) - In (5).

The relevant rules of logarithms are ...

  a·ln(b) = ln(b^a)

  ln(ab) = ln(a) +ln(b)

  ln(a/b) = ln(a) -ln(b)

  [tex]2\ln{(x)}+5\ln{(y)}-7\ln{(z)}-\ln{(5)}=\boxed{\ln\left(\dfrac{x^2y^5}{5z^7}\right)}[/tex]

Answer:

Step-by-step explanation:

I dont understand the question.

Wind direction [Mode] is South west

The [Mean] temperature is  

The  [Bimodal] pavement is  Dry Wet

From the question the first question is to determine the wind direction

Generally from the wind direction would be the data with the highest frequency on the wind column i.e the mode of the data, this because this variable(wind direction ) is a nominal level type of level of measurement where the value assigned to the variable are for classification of the data (i.e telling what the data is all about )

So

The mode is Southwest

This means that the wind direction is Southwest

From the question second question is to determine the temperature

Generally given that their are are no outlier that will skew the data we can obtain the temperature by calculating the mean for the temperature column

This is mathematically represented as

T = 89+86+....+93+93/8

T = 91

From the question, the first question is to determine the  pavement

Generally from the pavement would be the data with the highest frequency on the pavement i.e the mode of the data, this because this variable(pavement ) is a nominal level type of level of measurement where the value assigned to the variable are for classification of the data (i.e telling what the data is all about )

From the table we see that the pavement that the pavement column is bimodal

So the mode is Dry Wet

This mean that the pavement is Dry Wet

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Answer:

24d^5

Step-by-step explanation:

For an expression containing two variables, a and b, the expression with the largest value is a+b, the expression with the smallest value is a-b, and the expression closest to zero is either a+b or a-b, depending on the values of a and b.

For a=-1/2 and b=6, the expression with the largest value is a+b, which is equal to 5.5. The expression with the smallest value is a-b, which is equal to -7.5. The expression closest to zero is a+b, which is equal to 5.5.

For a=12 and b=-6, the expression with the largest value is a+b, which is equal to 6. The expression with the smallest value is a-b, which is equal to 18. The expression closest to zero is a-b, which is equal to 18.

For a=-6 and b=-12, the expression with the largest value is a+b, which is equal to -18. The expression with the smallest value is a-b, which is equal to 6. The expression closest to zero is a+b, which is equal to -18.

For an expression containing two variables, a and b, the expression with the largest value is a+b, the expression with the smallest value is a-b, and the expression closest to zero is either a+b or a-b, depending on the values of a and b.

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Option D is correct, there are 16 possibilities are in the sample space at the end of the tree diagram

It is a branch of mathematics that deals with the occurrence of a random event.

Given that Cornelio creates a tree diagram to show all possibilities for flipping a coin four times

We need to find the number of  possibilities are in the sample space at the end of the tree diagram

The number of times coin is flipped=4

We can use 2ⁿ to figure out the number of possible outcomes where n is the number of tosses.

Now n=4, so the number of possibilities are 2⁴

2×2×2×2

16

Hence, option D is correct, there are 16 possibilities are in the sample space at the end of the tree diagram

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[tex]~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] ~\dotfill\\\\ (-243)^{\frac{1}{5}}\implies \sqrt[5]{(-243)^1}\implies \sqrt[5]{(-3)(-3)(-3)(-3)(-3)} \\\\\\ \sqrt[5]{(-3)^5}\implies -3[/tex]

The next operation that should be done when evaluating this expression 9-12+-2+3(-3) is A. 3(-3)=-9

An expression is simply used to show the relationship between the variables that are provided or the data given regarding an information. In this case, it is vital to note that they have at least two terms which have to be related by through an operator. Some of the mathematical operations that are illustrated in this case include addition, subtraction, etc.

In this case, it should be noted that the expression given is written as 9-12+-2+3(-3). In PEDMAS, the first operation is parentheses. Therefore, the number on parentheses will be solved first.

In this case, the number in parentheses will be 3(-3). Therefore, the correct option is A.

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The fraction that is bigger than 2/5 and smaller than 2/3 is 8/15.

A fraction is written in the form of a numerator and a denominator where the denominator is greater that the numerator.

Example: 1/2, 1/3 is a fraction.

We have,

Let the fraction be a/b.

2/5 < a/b < 2/3

Now,

a/b

= (2/3 + 2/5) / 2

= (10 + 6) / 30

= 16/30

= 8/15

= 0.533

Thus,

The fraction is 8/15.

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The equation of the line will be y = -2/3x + 5.

An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.

The general form of the equation of the line:-

y = mx + c

m = slope

c = y-intercept

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given that the line passes through points (9, -1) and (6, 1).

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope = ( 1 - (-1) / ( 6 - 9 )

Sloe = - 2 / 3

The y-intercept will be calculated as,

y = mx + c

-1 = (-2/3) x 9 + c

c = 5

The equation of the line,

y = mx + c

y = -2/3x + 5

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Inequality in its most basic form explains this circumstance is 5x + 4y ≤ 21

Jamal, a camp counsellor, purchased meal tickets at an arts festival for his ravenous kids. The campers may use them to purchase burgers and hot dogs, both of which cost four tickets each.

A camp counsellor purchased meal tickets at an arts festival for her ravenous kids.

The campers may use them to purchase burgers and hot dogs, both of which cost four tickets each.

Let x be the number of hot dogs and y be the number of burgers. In all, campers are allowed to utilise up to 21 tickets.

Thus, inequality in its most basic form explains this circumstance.

= 5x + 4y ≤ 21

Inequality in its most basic form explains this circumstance is 5x + 4y ≤ 21

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Christopher sold 20 cups of large 20-ounce cups of lemonade

outside his house.

A straight line on the coordinate plane is represented by a linear function. As an illustration, the equation y = 3x – 2 depicts a linear function because it is a straight line in the coordinate plane.

Any function whose graph is not a straight line is said to be nonlinear. Any curve other than a straight line can be a graph of it.

An example of a non-linear function is a quadratic function.

Given, Christopher has set up a lemonade stand outside his house and sells small cups and large cups of lemonade.

Small cups sold is x and the number of large cups sold is y.

Each small cup holds 10 ounces of lemonade and each large cup holds 20 ounces of lemonade and he sold 800 ounces.

Therefore,

10x + 20y = 800...(i)

He also sold twice as many small cups as large cups.

Therefore,

2y = x...(ii)

10(2y) + 20y = 800.

20y + 20y = 800.

40y = 800.

y = 20.

So, He sold 20 cups of large lemonade.

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The value 8 is a numerical value that represents the quantity of eight. It is the natural number following seven and preceding 9. It is an even number and can be written as the sum of two squares: 8 = 3^2 + 2^2 = 4^2. In binary, it is 1000.

Eight is in the first position and has a place value of 8. Understanding the place value of digits in numbers aids in number comparison. It also aids in the composition of numbers in their enlarged form. For example, the extended version of the previous number, 13,548, is 10,000 + 3,000 + 500 + 40 + 8.

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Answer:

2 pounds of cheese

Step-by-step explanation:

First, we have to find out how much did she pay by taking

$5.00 - $1.40 = $3.60

So, she paid $3.60 in total. Then we find how many pounds of cheese she bought by taking

3.60 / 1.80 = 2 pounds of cheese

So, she bought 2 pounds of cheese.

The theoretical probability that the coin lands on heads and roll the cube and the number are less than 5 is 1/3.

What is an event?

An event is a collection of experiment results (a subset of the sample space) to which a probability is ascribed in probability theory.

There are 6 faces of a cube.

The numbers less than 5 on a cube are 1, 2, 3, 4.

The number of favorable outcomes is 4.

The total number of outcomes is 6.

The probability of getting less than 5 is

favorable outcome/ total number of outcomes = 4/6 = 2/3.

The number of faces of a coin is 2.

The number of heads of a coin is 1.

The probability of getting head is 1/2.

The probability of getting less than 5 in a roll and head in the coin is

2/3 × 1/2 = 1/3.

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Answer:

Step-by-step explanation:

You want the percentage multiplier associated with the following changes:

To find the multiplier, add 1 to the change. When you want the multiplier in percent, then add 100% to the change expressed as a percent. (1 = 100%)

  100% -25% = 75% . . . . of last year

  100% +50% = 150% . . . . of sales last month

  100% -10% = 90% . . . . of last marathon's time

Jake and Hakeem's rocket's flight is approximately modeled by the equation h(t) = - 12t ^ 2 + 36t + 3. The rocket's highest point is 30m

Generally, Since we are not instructed on what to look for, we are able to determine the highest point that the rocket ever achieved.

Given that the rocket's flight can be approximated approximatively using the equation,

h(t) = - 12t ^ 2 + 36t + 3

At maximum height dh/dt = 0

dh/dt = --24t + 36

0 = -24t + 36

24t = 36

t= 36/24

t = 1.5s

Find out the maximum height.

h(1.5) = - 12 (1.5)^ 2 + 36(1.5) + 3

h(1.5) = - 12(2.25) + 54 + 3

h(1.5) = -27 + 57

h(1.5) = 30m

As a result, the rocket is only capable of reaching a height of 30 meters at its highest point.

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An absolute value equation is an equation that contains an absolute value expression. An inequality is a mathematical statement that compares algebraic expressions using greater than (>), less than (<), and other inequality symbols.

The absolute number of a number a is written as  |a|

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation IxI = a

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

x+7|=14

You begin by making it into two separate equations and then solving them separately.

x+7=14

x+7−7=14−7

x=7

or

x+7=−14

x+7−7=−14−7

x=−21

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality   |x|<2  represents the distance between x and 0 that is less than 2 Whereas the inequality  |x|>2 represents the distance between x and 0 that is greater than 2.

You can write an absolute value inequality as a compound inequality.

$$\left | x \right |<2\: or

−2<x<2

This holds true for all absolute value inequalities.

|ax+ b| < c, where c>0

=−c < ax+ b < c

|ax+ b|> c, where c>0

=ax+ b <−corax+ b>c

You can replace > above with ≥ and < with ≤.

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.

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An absolute value equation is an equation that contains an absolute value expression. An inequality is a mathematical statement that compares algebraic expressions using greater than (>), less than (<), and other inequality symbols.

The absolute number of a number a is written as  |a|

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation IxI = a

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

x+7|=14

You begin by making it into two separate equations and then solving them separately.

x+7=14

x+7−7=14−7

x=7

or

x+7=−14

x+7−7=−14−7

x=−21

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality   |x|<2  represents the distance between x and 0 that is less than 2 Whereas the inequality  |x|>2 represents the distance between x and 0 that is greater than 2.

You can write an absolute value inequality as a compound inequality.

[tex]$$\left | x \right | < 2\[/tex]: or

−2<x<2

This holds true for all absolute value inequalities.

[tex]|ax+ b| < c, where c > 0\\=−c < ax+ b < c\\|ax+ b| > c, where c > 0\\=ax+ b < −corax+ b > c[/tex]

You can replace > above with ≥ and < with ≤.

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.

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The probability that a randomly selected customer will have to wait between 32 minutes and 37 minutes is 59.81%

How likely something is to occur is known as its probability. We can discuss the probabilities of various outcomes—how likely they are—when we aren't sure how a particular event will turn out. Statistics describes the examination of events subject to probability. Look up all the related responses.

The likelihood that an event will happen is quantified by probability. An event's likelihood is determined by how frequently it will occur over the long run. Probabilities are all digits that fall between 0 and 1, inclusive—that is, all digits that fall between 0 and 1.

Let the equation of Z score be:

z = (raw score - mean) / standard deviation

Given mean of 36 minutes and a standard deviation of 3 minutes.

If x = 32:

z = (32 - 36)/3 = -1.33

and x = 37:

z = (37 - 36)/3 = 0.33

P(-1.33 < z < 0.33) = P(z < 0.33) - P(z < -1.33)

= 0.6179 - 0.0198

= 0.5981

A consumer selected at random has a 59.81% chance of having to wait between 32 and 37 minutes.

The complete question is:

At a local restaurant, the amount of time that customers have to wait for their food is normally distributed with a mean of 36 minutes and a standard deviation of 3 minutes. What is the probability that a randomly selected customer will have to wait between 32 minutes and 37 minutes, to the nearest thousandth?

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Gracie poured out 3/10 gallons of the original mixture.

In mathematics →

Given is that Gracie has made 2 gallons o punch that contains 50% juice. After Gracie pours out some of her mixture and replaced with an equal amount of pure 100% juice, she has 2 gallons of punch that contains 65% juice.

Assume that she pour out {x} gallons of original mixture. We can write -

{50% of 2} - {x} = {(100 - 65)% of 2}

{50/100 x 2} - {x} = {35/100 x 2}

{1/2 x 2} - {x} = {35/100 x 2}

1 - x = 7/10

1 - 0.7 = x

x = 0.3

x = 3/10

Therefore, Gracie poured out 3/10 gallons of the original mixture.

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