Geometry: Unit 5: Relationships in Triangles. Homework 1: Triangle Midsegments

HELP!!!

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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Given:-

[tex] \: [/tex]

now, put the value in eqⁿ

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

hope it helps!:)

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 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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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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7 = a × 2 + b × 0 + c × 1, 7 = a × 0 + b × 4 + c × 1, 13 = a × 3 + b × 4 + c × 10 are the first vector is a linear combination of the other vectors.

In linear algebra, a linear combination is an expression formed by multiplying each element of a set of vectors by a scalar (a real or complex number) and then adding the results. The scalars are called coefficients and the vectors are called terms.

A linear combination can be written as a sum of the form:

a1 × v1 + a2 × v2 + ... + an × vn

where a1, a2, ..., an are the coefficients and v1, v2, ..., vn are the vectors. The result of the linear combination is also a vector, which can be thought of as a combination of the original vectors, weighted by the coefficients.

Linear combinations are used in many applications, including solving systems of linear equations, finding eigenvectors and eigenvalues, and representing a vector as a combination of basis vectors in a vector space.

(a) To determine whether the first vector is a linear combination of the other vectors, we can write a system of equations and solve for the coefficients. If the coefficients exist and are unique, then the first vector is a linear combination of the other vectors. If the coefficients do not exist or are not unique, then the first vector is not a linear combination of the other vectors.

-3 = a × 1 + b × 2 + c × -1

3 = a × -1 + b × 1 + c × 2

7 = a × 2 + b × 0 + c × 1

We can solve this system of equations using methods such as Gaussian elimination or matrix inversion to obtain the coefficients: a = -2, b = -1, c = 4

Therefore, the first vector is a linear combination of the other vectors.

(b) Following the same procedure, we can write a system of equations and solve for the coefficients:

-2 = a × 1 + b × 2 + c × -2

11 = a × -1 + b × 1 + c × 4

7 = a × 0 + b × 4 + c × 1

We can solve this system of equations to obtain the coefficients: a = 3, b = -2, c = 2

Therefore, the first vector is a linear combination of the other vectors.

(c) Using the same procedure, we can write a system of equations and solve for the coefficients:

2 = a × 1 + b × -1 + c × 1

7 = a × 2 + b × 2 + c × 6

13 = a × 3 + b × 4 + c × 10

We can solve this system of equations, but it will have no solution, meaning the coefficients do not exist.

Therefore, the first vector is not a linear combination of the other vectors.

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For all the given vectors the first vector is a linear combination of the other vectors.

In linear algebra, a linear combination is an expression formed by multiplying each element of a set of vectors by a scalar (a real or complex number) and then adding the results. The scalars are called coefficients and the vectors are called terms.

A linear combination can be written as a sum of the form:

a1 × v1 + a2 × v2 + ... + an × vn

where a1, a2, ..., an are the coefficients and v1, v2, ..., vn are the vectors. The result of the linear combination is also a vector, which can be thought of as a combination of the original vectors, weighted by the coefficients.

Linear combinations are used in many applications, including solving systems of linear equations, finding eigenvectors and eigenvalues, and representing a vector as a combination of basis vectors in a vector space.

(a) To determine whether the first vector is a linear combination of the other vectors, we can write a system of equations and solve for the coefficients. If the coefficients exist and are unique, then the first vector is a linear combination of the other vectors. If the coefficients do not exist or are not unique, then the first vector is not a linear combination of the other vectors.

-3 = a × 1 + b × 2 + c × -1

3 = a × -1 + b × 1 + c × 2

7 = a × 2 + b × 0 + c × 1

We can solve this system of equations using methods such as Gaussian elimination or matrix inversion to obtain the coefficients: a = -2, b = -1, c = 4

Therefore, the first vector is a linear combination of the other vectors.

(b) Following the same procedure, we can write a system of equations and solve for the coefficients:

-2 = a × 1 + b × 2 + c × -2

11 = a × -1 + b × 1 + c × 4

7 = a × 0 + b × 4 + c × 1

We can solve this system of equations to obtain the coefficients: a = 3, b = -2, c = 2

Therefore, the first vector is a linear combination of the other vectors.

(c) Using the same procedure, we can write a system of equations and solve for the coefficients:

2 = a × 1 + b × -1 + c × 1

7 = a × 2 + b × 2 + c × 6

13 = a × 3 + b × 4 + c × 10

We can solve this system of equations, but it will have no solution, meaning the coefficients do not exist.

Therefore, the first vector is not a linear combination of the other vectors.

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The correct solution is 24cm.

What is a property of similarity?

When two objects have the same shape but different sizes, they can be said to be comparable. This indicates that comparable shapes superimpose one another when amplified or demagnified. The term "Similarity" refers to this characteristic of like shapes.

Here, we have

Given: ∆ ABC ~ ∆ DEF

BC = 16cm

EF = 12cm

DE = 18cm

We have to find the correct solution.

By the property of similarity,

The corresponding sides are proportional.

So, AB/DE = BC/EF = AC/EF

AB/DE = BC/EF

x/18 = 16/12

x = 16×18/12

x = 24cm

Hence, the correct solution is 24cm.

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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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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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ysinx=x ² sinx+c where c is the constant of integration is the value of the differential equation.

Any equation containing one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation (i.e., independent variable)

dy/dx +ycotx=2x+x ² cotx  is of the form  dy/dx+Py=Q where P=cotx and Q=2x+x ² cotx

The solution is y×I.F=∫Q×I.Fdx+c

⇒ysinx=2∫xsinxdx+∫x² cosxdx

Consider ∫x² cosxdx

Take u=x ² ⇒du=2xdx and dv=cosxdx⇒v=sinx

∫x² cosxdx = x² sinx−2∫xsinxdx

∴ysinx=x ² sinx+c where c is the constant of integration.

Thus, ysinx=x ² sinx+c where c is the constant of integration is the value of the differential equation.

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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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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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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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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 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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On solving the exponential function y = 49 (0.81)⁹, the number of Bison left after 9 years is obtained as 7.35 million.

What is an exponential function?

The formula for an exponential function is f(x) = aˣ, where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

The number of Bison that roamed were a = 49 million

The survival rate dropped to b = 0.81.

The time period is t = 9 years

The exponential function is -

y = abˣ

Substitute the values into the equation -

y = 49 (0.81)⁹

y = 49 × 0.1500

y = 7.35

Therefore, the number of Bison left is 7.35 million.

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[tex]w^8[/tex] is the expression using positive exponents.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

w³ x [tex]w^5[/tex]

= [tex]w^{3 + 5}[/tex]

= [tex]w^8[/tex]

Thus,

[tex]w^8[/tex] is the final expression.

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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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If the probability distribution of x is known, this probability can be calculated as the probability mass or density at x = 1. If the probability distribution is unknown, then p(x = 1) cannot be determined.

1. The quantity p(x = 1) is equal to the probability of x occurring when x = 1.

This statement is true. It means that the probability of an event x occurring is equal to the probability of x having the value of 1.

2. If the probability distribution of x is known, this probability can be calculated as the probability mass or density at x = 1.

This statement is also true. If the probability distribution of x is known, then the probability of x having the value of 1 can be calculated using the probability mass or density at x = 1.

3. If the probability distribution is unknown, then p(x = 1) cannot be determined.

This statement is also true. If the probability distribution of x is unknown, then the probability of x having the value of 1 cannot be determined.

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List the frequency's percentage. To do this, multiply your answer by 100 after dividing the frequency by the total number of results.

What is the relative frequency?

The proportion of outcomes where a specific event occurs to all trials, not in a hypothetical sample space but in a real experiment, is known as the empirical probability, relative frequency, or experimental probability of an event.

The ratio of the total number of events occurring in a scenario to the number of times an event occurs is known as relative frequency.

Two facts must be known in order to calculate the relative frequency:

1) the total number of occasions/trials

2) frequency count for a subgroup or category

Relative frequency = f/n

f = the number of times the data occurred in an observation.

n = total frequency.

The relative frequency percentage is f/n × 100%.

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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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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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The area of the region outside the doghouse that skipper can reach is 3(pi)^2 yards.

This problem is based on the area of sectors. Now, we know that a sector is a pie-shaped part of a circle made of the arc along with its two radii.

The image which correctly explains the situation in the problem is attached.

From the figure, we can see that the spot can be located anywhere in the two sectors of 120  and  60 degrees each respectively. Now, these radii are respectively of 2 and 1 yards.

Area of one sector of 120 degrees = 120/360 π (2)^2 since radius is 2 yards.

So, area of two sectors is 2 x  120/360 π (2)^2 = 8π /3 square yards.

Similarly, Area of the two 60 degrees sectors is -

2 x  60/360 π (2)^2 = π /3 sq. yards.

Therefore , the total area that the skipper can reach outside the doghouse is given by the sum of all the four sectors i.e. π /3 + 8π /3 =

3π square yards.

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The area of the region outside the doghouse that the skipper can reach is [tex]3\pi ^2[/tex] yards.

This problem is based on the area of sectors. Now, we know that a sector is a pie-shaped part of a circle made of an arc along with its two radii.

The image which correctly explains the situation in the problem is attached.

From the figure, we can see that the spot can be located anywhere in the two sectors of 120  and  60 degrees each respectively. Now, these radii are respectively 2 and 1 yards.

Area of one sector of 120 degrees [tex]=\frac{(\pi*2^2*120) }{360}[/tex] since the radius is 2 yards.

So, the area of the two sectors is [tex]\frac{(2*120*2^2*\pi )}{360}=\frac{8\pi }{3}[/tex] square yards.

Similarly, the Area of the two 60 degrees sectors is -

[tex]\frac{(2*60*\pi *2^2)}{360} =\frac{\pi }{3}[/tex] sq. yards.

Therefore, the total area that the skipper can reach outside the doghouse is given by the sum of all the four sectors i.e. [tex]\frac{\pi }{3} +\frac{8\pi }{3} =3\pi[/tex]square yards.

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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 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.

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)))

The required coordinate of the point on the segment from (3,4) to (9,1)  is (7 , 7/3) which partitioned the segment into a ratio of 2 to 1.

The coordinates of the point on the directed line segment from (3,4) to (9,1)  that partitions the segment into a ratio of 2 to 1

Section formula.

x = mx₂ + nx₁ / m + n   ; y = my₂ + ny₁ / m + n

Where, m = 2 and n = 1
Substitute the value in the above equation,
x = 2 × 9 + 1 × 3 / 2 + 1 ; y = 3 × 1 + 1 × 4 / 2 +1
x = 21 / 3                     ; y = 7/3

x = 7                            ; y = 7/3

Thus, the required coordinate of the point on the segment from (3,4) to (9,1)  is (7 , 7/3) which partitioned the segment into a ratio of 2 to 1.

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A. 0. 0063 is the correct answer.

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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 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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