If u1,u2,u3 are linearly independent, show that v1,v2,v3 are also linearly independent if v1=u1+u2,v2=u1+u3,v3=u2+u3

Alice will receive £80 and Brian will receive £320

When dividing an amount of money in a given ratio, we need to first determine the total number of parts that the ratio is divided into.

In this case, the ratio between Alice and Brian is 1:4. This means that for every 1 unit of money that Alice receives, Brian receives 4 units of money. Therefore, the total number of parts in this ratio is 1 + 4 = 5.

To divide £400 in the ratio of 1:4, we need to find out how much of the total amount each part represents. To do this, we divide the total amount by the total number of parts:

£400 ÷ 5 = £80

So each part in this ratio represents £80.

Now that we know the value of each part, we can calculate how much Alice and Brian will receive.

Alice's share is one part, which represents £80. Therefore, Alice will receive

1 part x £80 = £80

Brian's share is four parts, which represents 4 x £80 = £320. Therefore, Brian will receive:

4 parts x £80 = £320

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The exponential growth equation is x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶ , where x ( t ) is the population in the year 2015 = 40,115,896

The exponential growth or decay formula is given by

x ( t ) = x₀ × ( 1 + r )ⁿ

x ( t ) is the value at time t

x₀ is the initial value at time t = 0.

r is the growth rate when r>0 or decay rate when r<0, in percent

t is the time in discrete intervals and selected time units'

Given data ,

Let the exponential growth equation be represented as x ( t )

Now , the average percentage change from the year 2000 - 2009 is calculated by

The total percentage = ( 1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93 )

The total number of years = 9 years

So , the average percentage change = total percentage / number of years

On simplifying , we get

The average percentage change = 12.21 / 9 = 1.35667 %

b)

The population in 2009 was 37,000,000

So , the population growth rate r = 1.35667 %

And , the population in the years 2015 is given by

The number of years n = 6 years

x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶

On simplifying , we get

x ( t ) = 37,000,000 ( 1.013567 )⁶

x ( t ) = 40,115,896

Hence , the population in the year 2015 is 40,115,896

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[tex]\huge\begin{array}{ccc}a_1=-2\\a_2=1\\a_3=6\\a_{10}=97\end{array}[/tex]

[tex]a_n=n^2-3[/tex]

Find the first 3 terms and the 10th term.

Substitute

n = 1, n = 2, n = 3 and n = 10:

[tex]a_1=1^2-3=1-2=-2\\\\a_2=2^2-3=4-3=1\\\\a_3=3^2-3=9-3=6\\\\a_{10}=10^2-3=100-3=97[/tex]

The measures of arcs and segments are

NK = 12, MLK = 41.5 degrees, JK = 24 and JPK = 277 degrees

Length NK

Given the triangle as the parameter.

Using the Pythagoras theorem, we have

NK = √(KL² - LN²)

So, we have

NK = √(15² - 9²)

Evaluate

NK = 12

The arc measure MK

Here, we start by calculating the central angle MLK

This is done using the following cosine ratio

cos(MLK) = NK/KL

cos(MLK) = 9/12

cos(MLK) = 0.75

Evaluate

MLK = 41.5 degrees

This means that

Arc MK = 41.5 degrees

Length JK

This is calculated as

JK = NK * 2

So, we have

JK = 12 * 2

JK = 24

The arc measure JPK

This is calculated as

JPK = 360 - 2 * MK

So, we have

JPK = 360 - 2 * 41.5 degrees

Evaluate

JPK = 277 degrees

Hence, the arc measure is 277 degrees

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Let's call the height of the building "h". We can use trigonometry to solve for "h" using the angle of elevation and the horizontal distance from the foot of the building to the point where the angle of elevation is measured.

In this case, we have a right triangle with the height of the building as one leg, the horizontal distance as the adjacent leg, and the angle of elevation as the angle opposite the height. So we can use the tangent function:

tan(60°) = h/55

Solving for "h", we get:

h = 55 tan(60°)

h ≈ 95.1

Rounded to the nearest tenth of a meter, the height of the building is approximately 95.1 meters.

Answer:

20%

Step-by-step explanation:

600÷3 of 100 = 20%

600/3 of 100 = 20%

The overall percentage increase is approximately 35.7%.

Given that, a company sales increased by 15% in year one and 18% in the following year.

To calculate the overall percentage increase, we need to find the combined effect of the two increases.

One way to do this is to use the formula for compound interest, which is:

A = P(1 + r₁)(1 + r₂)

where:

In this case, let's assume that the initial sales amount is 100 (you could use any amount as long as you keep it consistent throughout the calculation).

Then, the first year's increase of 15% would result in a new amount of:

100(1 + 0.15) = 115

In the second year, the 18% increase would be applied to this new amount of 115, resulting in:

115(1 + 0.18) = 135.7

Hence, we have an overall increase of:

(135.7 - 100)/100

= 0.357 × 100%

= 35.7%

Therefore, the increase is approximately 35.7 percent.

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The Pythagorean Theorem is proven.

The reason that we can make the statement BC/AC = CD/AC is because corresponding sides of similar triangles are proportional. This is represented by option D. Corresponding sides of similar triangles are proportional.

To complete the proof of the Pythagorean Theorem, a² + b² = c², we can use the following steps:

As similar triangles have proportional sides, we can write the proportions AB/AC = AC/BC and AB/BC = BC/AC.
By the cross-product property, AB * BC = AC * AC and AB * AC = BC * BC.
By the addition property of equality, AB² + BC² = AC² + BC².
By the segment addition postulate, AC + BC = c.
By factoring, (AC + BC)² = c².
By the distributive property, AC² + 2*AC*BC + BC² = c².
By rearranging the terms, AC² + BC² = c² - 2*AC*BC.
By substituting the values of AC and BC from the proportions in step 1, we get a² + b² = c² - 2*a*b.
By simplifying, we get a² + b² = c².

Therefore, the Pythagorean Theorem is proven.

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Answer: 9

Step-by-step explanation:

When a=6, the value of b is equals to -43.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division, which can be simplified and evaluated.

In other words, an algebraic expression is a collection of terms and coefficients that are connected by mathematical operators.

Substitute the value of a=6 into the expression for b:

b = -1 - 7a

b = -1 - 7(6)

b = -1 - 42

b = -43

Therefore, when a=6, the value of b is -43.

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160 student tickets were sold.

Equations are used to model a wide range of phenomena, from physical laws in science to economic models in finance. They are also used in solving problems and making predictions in various fields.

Let's start by setting up a system of equations to represent the given information:

a + s = 200 (equation 1: total number of tickets sold)

7a + 5s - 210 = 870 (equation 2: profit after paying back Mr. Shum's investment)

We can simplify equation 2 by adding 210 to both sides:

7a + 5s = 1080 (equation 3)

Now we can use equation 1 to solve for one variable in terms of the other:

s = 200 - a (equation 4)

Substitute equation 4 into equation 3 and solve for a:

7a + 5(200 - a) = 1080

7a + 1000 - 5a = 1080

2a = 80

a = 40

So 40 adult tickets were sold. We can use equation 4 to find the number of student tickets sold:

s = 200 - a = 200 - 40 = 160

Therefore, 160 student tickets were sold.

To check our answer, we can verify that the total revenue from ticket sales is equal to the initial investment plus the profit:

7(40) + 5(160) = 210 + 870

280 + 800 = 1080

This checks out, so our answer is correct.

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Answer:

1/2

Step-by-step explanation:

Our answer options are: 0, 1, 1/2

0 = 0/100

1 = 100/100

1/2 = 50/100

Comparing the 3 options, we see that 1/2 is the closest to 48/100!

So, the answer is 1/2

Answer:

The given function is f(x) = √(x-2) - 2.

The graph that represents this function will have the following characteristics:

The domain of the function is x ≥ 2, since the expression inside the square root must be non-negative.

The function approaches the x-axis but never touches it as x approaches positive infinity, since the square root will always be positive.

The function has a vertical asymptote at x = 2, since the expression inside the square root becomes zero at x = 2.

The function has a horizontal asymptote at y = -2, since the square root term dominates the function as x gets large, and the -2 term remains constant.

Based on these characteristics, you can compare the given graphs to see which one matches the given function.

Answer:

6(x)-5=90

Step-by-step explanation:

Move all terms that don't contain x to the right side and solve.

Exact Form:

x=95/6

Decimal Form:

x=15.83

Mixed Number Form:

x= 15 5/6

tbh i don't know if this is right

Marcy spent 2.125 minutes of total time watching the millipede crawl.

To find the total time Marcy spent watching the millipede crawl, we need to add up the time it crawled during the first period and the time it crawled during the second period.

1 4/8 minutes is the same as 1 2/4 minutes, or 1.5 minutes.

2 1/8 minutes is the same as 2.125 minutes.

So the total time the millipede crawled is:

1.5 minutes + 2.125 minutes = 3.625 minutes

Therefore, Marcy spent a total of 3.625 minutes watching the millipede crawl.

We know that Marcy watched the millipede crawl for 1 4/8 minutes, which can be simplified to 1 2/4 minutes, or 1.5 minutes.

Next, the millipede stayed still for a while, and then crawled for 2 1/8 minutes. We can convert 2 1/8 minutes to a decimal by adding the whole number (2) to the fraction (1/8) converted to a decimal (0.125). So:

2 + 0.125 = 2.125 minutes

Marcy spent 2.125 minutes of total time watching the millipede crawl.

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To write an equation in standard form using integers, we need to eliminate any fractions by multiplying both sides of the equation by the least common multiple of the denominators.

In this case, the denominator is 5. So we can multiply both sides of the equation by 5 to get:-

5Y = X

Now, we can rearrange the equation so that the variables are on the left-hand side and the constants are on the right-hand side, in the form of Ax + By = C.

X - 5Y = 0 (subtracted X from both sides)

Therefore, the equation in standard form using integers is -X - 5Y = 0.

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The function [tex]g(x) = \frac{x}{x^2+3}[/tex] is neither even nor odd.

To determine if a function is even, we can use the following test:

f(x) = f(-x)

If this is true, then the function is even.

For the given function, [tex]g(x) = \frac{x}{x^2+3}[/tex], let's plug in -x for x and see if the function is equal to itself:

[tex]g(-x) = \frac{-x}{(-x)^2+3} = \frac{-x}{x^2+3}[/tex]

As we can see, g(x) does not equal g(-x), so the function is not even.

To determine if a function is odd, we can use the following test:

f(x) = -f(-x)

If this is true, then the function is odd.

For the given function, let's plug in -x for x and see if the function is equal to the negative of itself:

[tex]g(x) = \frac{-x}{(-x)^2+3} = \frac{-x}{x^2+3}\\\\ -g(-x) = \frac{x}{-x^2+3}[/tex]

As we can see, g(x) does not equal -g(-x), so the function is not odd.

After the algebraic analysis, we can conclude that the function g(x) is neither, i.e., it is neither even nor odd.

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IQR difference = 5

The interquartile range (IQR) is a measure of variability that describes the spread of a set of data. It is calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1) of the data.

The formula for calculating the IQR is:

IQR = Q3 - Q1

The Upper Quartile - Lower Quartile is the definition of the interquartile range.

The answer to the query is

Mount Vernon data set: 55, 60, 65, 70, 75, 80, 85 and 90

Lakewood data set = 65, 70, 75, 80 and 85

Mount Vernon median = 70+75/2 = 77.5

55+60+65+70/4 = 62.5 is the median for the lower half of the collection.

Upper half of set median: 82.5 (75, 80, 85, 90 divided by 4).

Mount Vernon's IQR is 82.5 minus 62.5, or 20.

Likewise, in the instance of Lakewood,

Average: 75

Lower half of the set median equals 65+70/2 = 67.5

Upper half of set median equals 80+85/2 = 82.5

IQR = 82.5 - 67.5 = 15

Hence, IQR difference = 20-15 = 5

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The end behavior is as x approaches negative infinity, f(x) approaches negative infinity as x approaches positive infinity, f(x) approaches negative infinity The only x-intercept of f(x) is x = 3.

The given function is f(x) = -x³ + 2x² + 3

1. End behavior:

As the leading coefficient of f(x) is negative, -x^3 dominates the behavior of the function as x approaches negative or positive infinity. Therefore, the end behavior of the function is:

2. X-intercepts:

To find the x-intercepts of f(x), we need to solve the equation f(x) = -x^3 + 2x^2 + 3 = 0. We can factor out a common factor of -1 to simplify the equation:

-x^3 + 2x^2 + 3 = 0

x^3 - 2x^2 - 3 = 0

Using synthetic division or long division, we can find that one of the roots of this equation is x = 3. We can then factor out (x - 3) using polynomial division to find the other two roots:

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

The quadratic factor x^2 + x + 1 has no real roots, so the only x-intercept of f(x) is x = 3.

3. Positive/negative:

To determine the sign of f(x) for different values of x, we can look at the sign of each factor in the polynomial:

Putting these together, we can make the following sign chart for f(x):

x                         f(x)

x < 0                   -

0 < x < 3                  +

x > 3                  -

4. Multiplicity:

The only root of f(x) is x = 3, which has multiplicity 1. This means that the graph of f(x) crosses the x-axis at x = 3 and changes sign at that point.

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a) A(t) = 1100(1 + 0.08/4)^(4t)
b) After 8 years, there will be $2203.99 in the account.

c) It will take approximately 8.03 years for there to be $2200 in the account.

d) With continuous compounding interest, there would be $2219.54 in the account after 8 years.

A(t) = P(1 + r/n)^(nt)

Where:
A(t) = the amount of money in the account after t years
P = the initial deposit
r = the annual interest rate
n = the number of times interest is compounded per year
t = the number of years

Find an equation that gives the amount of money in the account after t years:

A(t) = 1100(1 + 0.08/4)^(4t)

Find the amount of money in the account after 8 years:

A(8) = 1100(1 + 0.08/4)^(4*8)
A(8) = 1100(1.02)^32
A(8) = 2203.99

After 8 years, there will be $2203.99 in the account.

How many years will it take for the account to contain $2200?

2200 = 1100(1 + 0.08/4)^(4t)
2 = (1.02)^4t
log(2) = 4t*log(1.02)
t = log(2)/(4*log(1.02))
t = 8.03

It will take approximately 8.03 years for there to be $2200 in the account.

If the same account and interest were compounded continuously, how much money would the account contain after 8 years?

A(t) = Pe^(rt)

A(8) = 1100*e^(0.08*8)
A(8) = 1100*e^0.64
A(8) = 2219.54

With continuous compounding interest, there would be $2219.54 in the account after 8 years.

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Answer:

Step-by-step explanation:

sum of all angles in quadrilateral = 360

So, x+48+48+132 =360

x =132

The range of allowable values 24; can vary by 3.5% is 0.9.

The difference between the highest and lowest values for a given data set is the range in statistics. For instance, the range will be 10 - 2 = 8 if the given data set is 2, 5, 8, 10, and 3.

As a result, the range can also be thought of as the distance between the highest and lowest observation. The range of observation is the name given to the outcome. Statistics' range reflects the variety of observations.

In statistics, we must put the given values, set of data, or set of observations in ascending order in order to determine the range. That indicates that you should start by writing the observations from lowest to highest value. The formula must now be used to determine the range of observations.

24 × 3.5% =0.84

so round to nearest tenth is 0.9.

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Complete question:

Find the range of allowable values based on the given information. Round to the nearest tenth.

24; can vary by 3.5%

When BD is perpendicular tο AC, the value οf x is fοund tο be 19°.

When twο lines crοss at a straight angle οr 90 degrees, perpendicular lines are created. Perpendicularity is the name given tο this characteristic οf lines. A straight line that fοrms a right angle (90 degrees) with anοther line is referred tο as being perpendicular.

In οther wοrds, twο lines are perpendicular tο οne anοther if they cοnnect at a right angle. The term "perpendicular tο all pοints in the plane" οr "perpendicular tο each line it intersects" is used tο describe a line that is perpendicular tο the plane.  Perpendicular lines always intersect at 90 degrees but nοt all intersecting lines are perpendicular.

Given a vectοr diagram.

It is given that BD is perpendicular tο AC.

m∠CBE = (5x - 42)°

m∠CBE = (2x - 1)°

Since BD is perpendicular tο AC,

m∠DBC = m∠DBA = 90°

Nοw,

m∠DBC = 90°

m∠CBE + m∠CBE = m∠DBC

Then we can write,

m∠CBE + m∠CBE =  90°

(5x - 42)° +  (2x - 1)° = 90°

7x - 43 = 90

7x = 133

x = 133 / 7 = 19°

Therefοre when BD is perpendicular tο AC, the value οf x is fοund tο be  19°.

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The simplified form is 3/(x+7) + 14/(x)(x+7).

Simplified form of a fraction is the form of a fraction in which the numerator and denominator are in their lowest form, or the greatest common factor has been taken out of both the numerator and denominator. For example, the fraction 24/36 can be simplified to 2/3.

To subtract the two fractions, we need to find a common denominator. The common denominator of (x+7) and (x) is (x)(x+7). We can then rewrite the two fractions with this common denominator and subtract the numerators:

(x+8)/(x+7) - (x-2)/(x) = (x+8)(x)/(x)(x+7) - (x-2)(x+7)/(x)(x+7)

= (x^2+8x)/(x^2+7x) - (x^2-2x+7x-14)/(x^2+7x)

= (x^2+8x - x^2+2x-7x+14)/(x^2+7x)

= (3x+14)/(x^2+7x)

= 3x/(x^2+7x) + 14/(x^2+7x)

= 3/(x+7) + 14/(x)(x+7)

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The number that was excluded from the 10 consecutive numbers would be = 228.

Let's take the smallest number of the 10 consecutive number to be = X

Then, the 10 consecutive numbers are x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, and x+9.

If we exclude one of these numbers, then the sum of the remaining 9 numbers would be:

(x) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) + (x+8)

Collect the like terms and add up.

= 9x + 36

Recall that the sum of the remaining 9 numbers = 2023.

That is:

= 9X + 36 = 2023

=9x = 1987

x = 221

Therefore, the smallest number in the consecutive sequence is 221, and the 10 numbers are 221, 222, 223, 224, 225, 226, 227, 228, 229, and 230.

If we exclude the number 228, then the sum of the remaining 9 numbers is 2023.

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The equation of the line passing through the point (8,3) that is parallel to the line y=-3/4+4 is y = (-3/4)x + 9.

The slope of the given line is -3/4. The desired line will likewise have a slope of -3/4 because it is parallel to this line.

The equation of the line travelling through the point (8,3) with a slope of -3/4 may be found using the point-slope form of a line's equation:

y - y1 = m(x - x1)

where the provided point is (x1, y1), and m is the slope.

When we change the values, we obtain:

y - 3 = (-3/4)(x - 8)

Making the right side simpler:

y - 3 = (-3/4)x + 6

To each sides, add 3:

y = (-3/4)x + 9

Hence, y = (-3/4)x + 9 is the equation of the line going through the point (8,3) that is parallel to the line y=-3/4+4.

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When A U B = C & A ∩ B = φ, then A = C - B. The solution has been obtained by using properties of sets.

What is a set?

In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Curly brackets are typically used to represent sets.

We know that

A - B = (A ∪ B) - B

On substituting A U B = C in this, we get

A - B = C - B       ...(1)

Similarly,

A - B = A - (A ∩ B)

On substituting  A ∩ B = φ in this, we get

A - B = A        ...(2)

From (1) and (2), we get

A = C - B

Hence, when A U B = C & A ∩ B = φ, then A = C - B.

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When the smaller circle has a radius οf 6 cm, the area οf the shaded regiοn is 339.3 cm², rοunded tο the nearest tenth.

The cοllectiοn οf all pοints in a plane that are equally spaced frοm a specific pοint knοwn as the centre is referred tο as a circle in geοmetry. It is usually represented by a clοsed, rοunded curve in twο dimensiοns. The radius is the distance acrοss the centre οf the circle, and the diameter is the distance frοm the centre οf the circle tο any spοt οn the curve. The area encircled by a circle is determined by the fοrmula A = r2, where r is the radius. The circumference οf a circle is alsο referred tο as its edge. Circles are a key idea in geοmetry and have many uses in bοth mathematics and daily living.

given:

We must deduct the area οf the smaller circle frοm the area οf the bigger circle in οrder tο determine the area οf the shaded area.

The bigger circle has a 12 cm radius, sο the fοllοwing is its area:

A = πr² = π(12 cm) (12 cm)

A ≈ 452.39 cm²

The smaller circle has a radius οf 6 centimetres, sο the fοllοwing is its area:

A = πr² = π(6 cm) (6 cm)

A ≈ 113.1 cm²

As a result, the shaded regiοn's size is:

113.1 cm² - 339.3 cm² = 452.39 cm²

When the smaller circle has a radius οf 6 cm, the area οf the shaded regiοn is 339.3 cm², rοunded tο the nearest tenth.

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Answer:

The third one

Step-by-step explanation:

The only one which gives the last number of the sequence is the third one as subbing in 16 you get 3×16 + 3 = 48 + 3 = 51