The freezing point of water is 0^\circ\text{C}0

When f(n) = o(g(n)), it is not necessary that  log (f(n)) = o(log (g(n))) is proved using the limits.

Consider limits on both the sides,

lim(n→infinity) (log(f(n))/log(g(n)))

= lim n→infinity (1/(f(n)xln2)x f'(n)) / (1/(g(n) x(ln2)) x g'(n))

= lim n→infinity (g(n)/f(n) x f'(n)/g'(n))

lim n→infinity (g(n)/f(n)) = infinity

lim n→infinity (f'(n)/g'(n)) = 0

let f(n) = n, g(n) = n^2

lim n→ infinity (f(n)/g(n)) = lim n→ infinity (1/n)

                                      = 0

f(n) = o(g(n))​

In this condition

lim(n→infinity)(log(f(n))/log(g(n)))  = lim n→infinity (n x 1/2n)

                                                     = 1

so it is not necessary that  log (f(n)) = o(log (g(n))) when f(n) = o(g(n))

                                         Hence proved.

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The complete question is :

Given f(n) = o(g(n)), show that it is not necessary that  log (f(n)) = o(log (g(n)))

Based on the formula [tex]C (t) = 0.8 (1.2)^{t}[/tex] it is expected for the concentration of organisms to increase over time (third option).

To determine how the formula or function affects the number of organisms, let's use two different times in order to compare the results:

C (t) = 0.8 (1.2)^{t}

Let's find out the concentration of organisms with the times 2 and 5:

2:

C (t) = 0.8 (1.2)^{2}

C (t) = 0.8 x 1.44 = 1.152

5:

C (t) = 0.8 (1.2)^{5}

C (t) = 0.8 x 2.48 = 1.99

Based on this, it can be concluded that as time increases the concentration increases too.

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The powerline cable diagonal length is 61 feet.

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.

Given:

Hypotenuse = x

Base = 50 feet

Perpendicular = 35 feet

Using Pythagoras theorem

H² = P² + B²

H² = 35² + 50²

H² = 1225 + 2500

H² = 3725

H= 61 feet

Hence, the diagonal length is 61 feet.

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The actual temperature after the freezer was on for five minutes could be 18ºF or 12ºF

What is a graph ?

Graph can be defined as visual representation of the data through the given ordered pairs .

First, you must find the y-value returned by the line in the graph for a time (x) of five minutes, and then use the deviation of 3º to conclude the range of the real temperature.

1. y- value at x = 5 min.

The trend line (blue line in the graph) passes through the point (5,15), which means that it predicts a temperature of 15ºF when the time (x) is 5 minutes, or five minutes after the freezer was turned on.

2. Actual temperature value

It is stated that the temperature was actually three degrees from what the trend line shows, that means that the actual temperature could be 3ºF more or 3ºF less than the predicted temperature of 15ºF.

Mathematically:

Temperature = 15ºF ± 3ºF

Temperature = 15ºF + 3ºF or 15ºF - 3ºF

Temperature = 18ºF or 12ºF

Therefore, The actual temperature after the freezer was on for five minutes could be 18ºF or 12ºF

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The student's final grade in three significant digits is 247.

Significant Figures refer to the number of important single digits from 0 to 9 in the coefficient of the expression that conveys the message accurately. These significant figures help engineers or scientists in asserting the quantity of any measurement, length, volume, or mass. For example, 453 has three significant figures.

The two main applications to understand significant figures are - Precision and Accuracy. Let's learn how these two terms play an important role in real-life situations when it comes to the concept of significant figures.  

Precision - The closeness between two or more quantities to each other under the same condition is called precision. In precision, the level of measurement when repeated gives the same result and the individual measurement agrees to each other.

Accuracy - The closeness between the measurement and the accurate number is called accuracy. Accuracy helps in providing consistent results with no error along with accuracy in the result.

The final grade of the student would be equal to the sum of the grades obtained in the quizzes, exams, and final.

So, the total grades = 92 + 76+ 79 = 247

Calculating this to three significant digits, we get 247.

Thus, the student's final grade in three significant digits is 247.

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The value of g(f(-5)) is 5.

In mathematics, a function is represented as a rule that produces a distinct result for each input x. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.

Given:

f(x)= |2x+ 9|

Now, g(f(-5))

f(-5) = |2(-5)+9|

       = |-10 + 9|

       = |-1|

       = 1

and, g(f(-5)) = g(1)

So, from the graph we can see that the value of g(1) is 5.

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By Solving by factorization method of equation 5x² - 3x - 2 = 0

x = 1 , [tex]\frac{-2}{5}[/tex]  

Given data :

Given equation is, 5x² - 3x - 2 = 0

Solving by factorization method,

5x² - 3x - 2 = 0

5x² - 5x + 2x - 2 = 0

( 5x + 2 ) ( x -*1 ) = 0

x = 1 , [tex]\frac{-2}{5}[/tex]

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The slope for the function f(t) = 3 - 3/5t at point (3/5, 2) is 5/3.

What is slope?

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.

The given function is - f(t) = 3 - 3/5t.

The data points are - (3/5, 2)

Rewrite the function -

f(t) = 3 - 3/5 t^-1

Differentiate with respect to t. Use the difference rule.

d/dt (3) - d/dt (3/5 t^-1)

Use the constant rule and the constant multiple rule.

0 - 3/5 d/dt (t^-1)

Use the power rule.

-3/5(-1)t^(-1 - 1)

3/5t^-2

Now plug in 3/5 for t on the derivative in order to find the slope at point (3/5, 2).

3/5(3/5)^-2

3/5(5/3)^2

3/5 × 25/9

5/3

Therefore, the slope value is 5/3.

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Find the slope of the graph of the function at the given point.

Function Point f(t) = 3 - 3/5t (3/5, 2)

The arc length of the circle will be 0.35 units.

Using the formula Length of an Arc = r x θ, where is in radians, one can determine the arc length of a circle given its radius and central angle. Arc length is equal to Arc = θ × (π/180) × r, where r is the radius in degrees.

Given that a circle has a radius of 3. An Arc in this circle has a central angle of 20 degrees.

The length of the arc of the circle will be calculated as:-

Arc = θ × (π/180) × r

Arc = 20 x  (π/180) × 3

Arc = 0.35 units

Therefore, the arc length of the circle will be 0.35 units.

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-pi is the  phase shift of y = -cos ( 1/2 x + pi/2 )

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

The given function is y = -cos ( 1/2 x + pi/2 )

Standard form of the cosine function is y=acos(bx+c)+d

Phase shift is -c/b

b=1/2 and c is pi/2

Now phase shift=-(pi/2)/(1/2)

=-pi

Hence, -pi is the  phase shift of y = -cos ( 1/2 x + pi/2 )

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The probability that someone who tries to buy tickets both through the internet and by telephone will obtain at least one ticket is equal to 1 minus the probability that they won't obtain a ticket through either method.

The probability that they won't obtain a ticket through either method is equal to the product of the probabilities of not obtaining a ticket through each method, since the events are independent.

Therefore, the probability that someone who tries to buy tickets both through the internet and by the telephone will obtain at least one ticket is:

1 - 0.006 = 0.994 = 0.994 (rounded to four decimal places)

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Coefficient of x on 4(3x+5) is 12

A coefficient refers to a number or quantity placed with a variable. It is usually an integer that is multiplied by the variable and written next to it.

To find the co-efficient of x in 4(3x+5)

Expand the bracket

4(3x+5) = 12x + 20

From 12x + 20

The co-efficient of x in 12, That is the number with the variable x

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Answer: I don't know if this is what you wanted, but two ways to represent 4/5 are 0.8 and 4 divided by 5.

Step-by-step explanation:

4/5 = 8/10

8/10 = 0.8

fraction symbol means to divide.

I hope this helps!

it is often useful to find the Factors of a polynomial in order to find its roots.

The opposite of multiplying, factoring is the process of expressing a polynomial as the product of two or more factors. To factor polynomials, we often combine the following identities or characteristics with various techniques. Ab + Ac = a(b + c) in the statistical distribution.

Because it is derived from the words poly-, which means "many," and -nominal, which in this context means "term," polynomial means "many terms." In mathematics, a polynomial is an expression with variables and coefficients that employs only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

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The multiples of 5 are completely divisible by 5. The multiples of 5 are 5, 10, 15, 20, etc.

What is multiple?

Manifold is the fundamental definition of multiple. A multiple in mathematics refers to the outcome of multiplying two numbers together.

The first 5 multiples of 5 are 5, 10, 15, 20, and 25. These can be written as:

5 × 1 = 5

5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

5 × 5 = 25

5 × 6 = 30

5 × 7 = 35

5 × 8 = 40

5 × 9 = 45

5 × 10 = 50

The number whose unit digit is either 0 or 5 is a multiple of 5.

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The slope of lines a, b, c, and d are 1/3, 1/4, 1/3, and -4 respectively. Line a is parallel to line c. Line b and line d are perpendicular to each other.

What is a line?

A line is an object in geometry that is infinitely long and has neither width nor depth nor curvature. Since lines can exist in two, three, or higher-dimensional spaces, they are one-dimensional objects. The term "line" can also be used to describe a line segment in daily life that has two points that serve as its ends.

The slope of a line that passes through the points (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁)/(x₂ - x₁).

Line a passes through points (-1,4) and (5,6).

The slope of line a is (6-4)/(5-(-1)) = 2/6 = 1/3

Line b passes through points (-1,1) and (3,2).

The slope of line b is (2-1)/(3-(-1)) = 1/4

Line c passes through points (-3,-2) and (3,0).

The slope of line c is (0-(-2))/(3-(-3)) = 2/6 = 1/3

Line d passes through points (1,6) and (3,-2).

The slope of line d is (-2-6)/(3-1) = -8/2 = -4.

The slopes of line a and line c are the same, thus line a is parallel to line c.

If two lines are perpendicular to each other, then the product of therir slope is -1.

The product of slopes of line b and line d is

1/4 × (-4) = -1

Thus line b and line d are perpendicular to each other.

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The length of the curve is [5, 3].

A curve is a continuous and smooth function that can be represented in a three-dimensional space. The length of a curve is a measure of the distance between two points on the curve.

The curve r(t) = (-4t, -sin(t), -cos(t)) is a parameterized curve, where t is the parameter that defines the position of a point on the curve.

Here we need to find the length of this curve, we need to calculate the distance between two points on the curve for different values of t.

In order to find the length of the curve, we use the formula for the length of a vector, which is

=> √(x² + y² + z²),

where x, y, and z are the components of the vector.

In this case, the vector is the difference between two points on the curve. The length of the curve is then found by integrating the length of the vector over the interval [5, 3].

Complete Question:

Find the length of the given curve: r(t)=(−4t,−1sint,−1cost)where−5≤t≤3.

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The length of segment FB is given as follows:

FB = 4 in.

The theorem used to solve this problem is given as follows:

A line parallel to one side of a triangle divides the other two proportionately, as similar triangles are formed.

The parallel segments in this problem are given as follows:

DF, EF and AB.

Hence the proportional side lengths are given as follows:

Meaning that the length of segment FB is obtained as follows:

3/4 = 3/FB

FB = 4 in.

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To convert pennies to nickels, dimes, and quarters, you need to divide the number of pennies by the value of each coin.

The value of a nickel is 5 cents, a dime is 10 cents, and a quarter is 25 cents.

Here's an example of how to convert 100 pennies to nickels, dimes, and quarters:

Nickels: 100 pennies / 5 cents/nickel = 20 nickels

Dimes: 100 pennies / 10 cents/dime = 10 dimes

Quarters: 100 pennies / 25 cents/quarter = 4 quarters

So, 100 pennies can be converted into 20 nickels, 10 dimes, and 4 quarters.

Correct Question :

How to convert pennies to nickels dimes and quarters?

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The probability that a muffin picked at random contains raisins is 3/5.

The probability that a muffin picked at random contains raisins is calculated as follows:

The number of muffins that are not frozen = 30 - 14

The number of muffins that are not frozen = 16

The number of muffins that are not frozen but contain raisins = 16 - 4

The number of muffins that are not frozen but contain raisins = 12

The total number of muffins that contain raisins = 12 + 6

The total number of muffins that contain raisins = 18

The probability that a muffin contains raisins, P(contain raisins) = 18/30

P(contain raisins) = 3/5

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The requried amount of original money will we have after 8 quarters is $2,200,000.

The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.

Here,
The amount of money we lose every quarter is
= $5,000,000 × 7%
= $350,000.

After 8 quarters, the total amount of money we lose would be
=  $350,000 × 8
= $2,800,000.

So, after 8 quarters we would have
=  $5,000,000 - $2,800,000
= $2,200,000

Thus, the requried amount of original money will we have with after 8 quarters is $2,200,000.

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Vector is to scalar as displacement is to distance.

A vector quantity has both magnitude and direction. A scalar quantity has only magnitude and no direction. Displacement is the measure of distance between initial and final points in a particular direction. Distance is length between initial and final points no matter the direction.

here, we have,

Displacement is to distance is as vector is to scalar because displacement is a vector quantity and and distance is scalar version of displacement. Of the other options speed is a scalar quantity and velocity is a vector quantity and none of them arranged in the form of vector is to scalar.

Therefore, Vector is to scalar as displacement is to distance.

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Given:-

[tex] \: [/tex]

now, put the value in eqⁿ

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

hope it helps!:)

The value of x that makes triangles DEF ~ XYZ is -2. 25

It is important to note that perpendicular lines are equal to each other

From the triangles given, let us equate their perimeter one to another, we get;

8 + 10 + 3(x -1 ) = 7. 5 + 4 + x-1

expand the bracket

18 + 3x - 3 = 11. 5 + x - 1

collect like terms

3x - x = 10. 5 - 15

Add or subtract the collected like terms, we get;

2x = 4. 5

Now, make 'x' the subject from the equation

x = 4.5/2

Divide the values

x = -2. 25

Hence, the value is -2. 25

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Yes, that's correct. When a figure is dilated from the origin, each ordered pair of the figure is transformed according to the rule

What is dilation in maths ?

A dilation is a function f from a metric space M into itself that fulfils the identity d=rd for all locations x, y in M, where d is the distance between x and y and r is some positive real number.

Yes, that's correct!

When a figure is dilated from the origin, each ordered pair of the figure is transformed according to the rule (x, y) → (kx, ky), where k is the scale factor. The scale factor determines the size of the dilation; if k > 1, the dilation increases the size of the figure, while if 0 < k < 1, the dilation decreases the size of the figure. If k = 1, the figure remains unchanged.

It's important to note that dilations preserve the orientation of the figure and can be thought of as a non-uniform scaling of the figure.

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this arithmetic sequence will be 10, 14, and 18.

What is arithmetic sequence?

There are two approaches to defining an arithmetic sequence. It is a "series where the differences between every two succeeding terms are identical" (or) In an arithmetic sequence, "each term is obtained by adding a fixed number (positive or negative or zero) to its preceding term." The series where the common difference between any two next terms stays constant is known as the arithmetic sequence. Let's go through what a sequence is once again. A collection of numbers that follow a pattern is referred to as a sequence.

The first term of an arithmetic sequence is 6

common difference 2/3.

a2 = a+d = 6+4 =10

a3= a+8 =14

a4=a+12 =18

Hence this arithmetic sequence will be 10, 14, and 18.

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The factor of the expression becomes:

π(R² - r²) = π(R + r) (R - r)

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It is possible to multiply, divide, add, or subtract in math. Any mathematical statement with variables, numbers, and an arithmetic operation between them is called an expression or an algebraic expression. For instance, 10m + 5 is an expression including the terms 10m and 5 as well as m, the variable of the supplied expression, all separated by the arithmetic sign "+".

An annulus is the region between two concentric circles. The area of the annulus shown in the figure is πR² - πr²

= πR² - πr²

= π(R² - r²)

(R² - r²)  is of the form (a² - b²)

where, a = R and b = r

As we know, (a² - b²) = a² - 2ab + b² = (a + b) (a - b)

so, (R² - r²) = (R + r) (R - r)

The polynomial factored completely is then:

π(R² - r²) = π(R + r) (R - r)

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According to the above, the fractions numbers would be: 0.8 = 4/5, 2.34 = 117/50, and 1.25 = 5/4.

To convert a decimal to a fraction, we must write the decimal number as a numerator and its place value as the denominator. Also, if there is a number to the left of the decimal point, we leave that number as an integer. According to the above, the fractional numbers would be:

Those fractions are simplified.

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The data have a mean of 200 and a standard deviation of 5, with 95% of the data falling between 190 and 210.

Mean = 200

5 is the standard deviation.

10 divided by two standard deviations

Lower Boundary = Mean – Two SDs = 200 – 10 = 190

Upper Boundary = Mean + Two SDs, which is 200 + 10 = 210.

As a result, 190 through 210 should contain 95% of the data.

The data provided indicate that the data set's mean is 200 and its standard deviation is 5. Accordingly, the range of the data for 95% of the samples should be between 190 and 210, with 200 serving as the average. Due to the fact that for a set of data with a normally distributed distribution, around 95% of the values should be within two standard deviations of the mean, this is the case. Two standard deviations from the mean of 200 would be 190 and 210 since 5 is the standard deviation. As a result, these two figures should encompass 95% of the data.

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The complete question is

a+given+data+set+is+normally+distributed+with+a+mean+of+200+and+a+standard+deviation+of+5.+++which+two+values+does+95%+of+the+data+fall+between 190 and 210.

The exact doubling time calculation provides a more accurate estimate of how long it will take a given value to double.

Approximate Doubling Time:

5% - 14 Years

10% - 7 Years

15% - 5 Years

20% - 4 Years

Exact Doubling Time:

5% - 14.2 Years

10% - 6.93 Years

15% - 4.77 Years

20% - 3.58 Years

Approximate doubling times are calculated by dividing 72 by the percentage rate of growth. This method assumes a consistent rate of growth, which is not always the case. The exact doubling time can be calculated by using the formula T=ln(2)/ln(1+r) where r is the rate of growth. This formula accounts for the compounding effect of the growth rate, which is not taken into account in the approximate calculation. For example, for a 5% annual growth rate, the approximate doubling time is 14 years, but the exact doubling time is 14.2 years. The exact doubling time calculation provides a more accurate estimate of how long it will take a given value to double.

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