Homework 5: parts of similar triangles??

The given pair of linear equations is consistent.

What are Linear Equations?

An equation is said to be linear if the maximum power of the variable is consistent. Another name for it is a one-degree equation.

Equations are declarations of equality between two mathematical expressions containing one or more variables. A linear equation is one in which the maximum power of the variable is one. a real number equation of the form ax+by+c = 0, with real numbers a, b, and c with the highest power one variable x and y.

In our question, we can see:

a1 = -3 and a2 = 4

b1 = 3 and b2 = 4

c1 = 21 and c2 = 60

a1/a2 = -¾

b1/b2 = ¾

c1/c2 = 7/20

Since a1/a2 is not equal to b1/b2, therefore our pair of liner equation is consistent.

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A set S spans R^2 if any element of R^2 can be written as linear combination of elements of S the set S spans R2.

(a) Given that S={ (1,-1),(2,1) }

Let (x,y) be the any element in R^2.

(x,y) = C1(1,-1)+C2(2,1)

(x,y) = (c1+2c2,-c1+c2)

x  =  c1+2c2

y  = -c1+c2

x+y = 3c2

c2= {x+y}/{3}

c1 = c2-y

= {x+y}/{3}-y

={x-2y}/{3}

So for any element (x,y) in R^{2} we have the constants c_1,c_2 so that (x,y) can be written as,

(x,y)= {(x-2y)}/{3}(1,-1) + {(x+y)}/{3}(2,1)

So S= { (1,-1),(2,1) } spans R^2.

(b) Given that S = { (1,1) }

Let (x,y) be the any element in R^2.

(x,y)=c_1(1,1)

(x,y)=(c_1,c_1)

x=c_1,y=c_1

If x,y are distinct then do not get a value of c_1 so that (x,y) = c_1(1,1).

So set S= { (1,1) } does not span R^2.

(c) Given that S= { (0,2),(1,4) }

Let (x,y) be the any element in R^2.

(x,y)=c_1(0,2)+c_2(1,4)

(x,y)=(c_2,2c_1+4c_2)

x=c_2

y=2c_1+4c_2

c_1={y-4c_2} / {2}

={y-4x} / {2}

So (x,y)= {(y-4x)}/{2}(0,2)+x(1,4)

So any element of R^2 can be written as written as linear combination of elements of Set S.

Hence set S={ (0,2),(1,4) } spans R^2.

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Yes, the formula z = -b± √(b²-4ac)/2a is used to find the solutions (or roots) of a quadratic equation of the form az² + bz + c = 0. This is known as the Quadratic Formula.

What is the Quadratic Equation?

A quadratic equation is a type of polynomial equation of degree 2 in one variable (x), which can be written in the form of ax² + bx + c = 0, where a, b, and c are real numbers, and a is not equal to zero. The solutions or roots of this equation can be found by using the Quadratic Formula which is z = -b ± √(b²-4ac) / 2a.

The Quadratic formula is valid for any quadratic equation in standard form (ax² + bx + c = 0), where a, b, and c are real numbers, and a is not equal to zero. The two solutions are given by the formula are the roots of the equation, and they are always real and distinct, unless the discriminant b² - 4ac is negative, in which case the roots are complex numbers.

Hence, Yes, the formula z = -b± √(b²-4ac)/2a is used to find the solutions (or roots) of a quadratic equation of the form az² + bz + c = 0. This is known as the Quadratic Formula.

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As per the given confidence interval, the proportion of all city residents who would respond very supportive or somewhat supportive of the proposal is 336 ± 0.197

The term confidence interval is defined as the statistical measure used to indicate the range of estimates within which an unknown statistical parameter is likely to fall.

Here we have given that the random sample of 1,018 city residents were asked to rate their level of support for a proposal being considered by the city council.

Here we have also know that the percent of confidence interval is 95%.

Here we also identified the following values,

Total population = 1,018

Very supportive population = 336

Somewhat supportive population = 387

Not supportive population= 295

Then the value of confidence interval is calculated as,

Proportion of positive results = P = x/N = 0.3301

Lower bound = 0.3012

Upper bound = 0.3599

Therefore, it can be written as,

=> 336 ± 0.197

Complete Question:

A random sample of 1,018 city residents were asked to rate their level of support for a proposal being considered by the city council. Based on the responses, which of the following is a 95% confidence interval for the proportion of all city residents who would respond very supportive of somewhat supportive of the proposal?

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A matrix is in the row-echelon form if:

1. The first nonzero element of each nonzero row (called the leading entry) is to the right of the leading entry of the row above it.

2. Each leading entry is 1 (also called a pivot).

3. Each element above and below a pivot is 0.

A matrix is in the reduced row-echelon form if it is in row-echelon form and also:

4. Each pivot is the only nonzero element in its column.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The numbers, symbols, or expressions in a matrix are called its entries or elements. Matrices are often used in linear algebra, a branch of mathematics that deals with vector spaces and linear transformations.

They can be used to represent systems of linear equations, perform operations such as matrix addition and multiplication, and solve systems of linear equations.

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Answer:

A, C

Step-by-step explanation:

A)

(120 ∙ 8) ∙ 5 = 120 ∙ (8 ∙ 5)

Multiply 8 and 5 first to get 40.

Then 40 × 120 = 4800

C)

(17 + 75) + 25 = 17 + (75 + 25)

Add 75 and 25 first to get 100.

then 17 + 100 = 117

This is a system of equations problem. We can set up the following equations:

x = number of children admitted

y = number of adults admitted

1.25x + 7.20y = 1060 (the total amount of money collected)

x + y = 253 (the total number of people admitted)

To find the number of children admitted, we can use the second equation to solve for one of the variables in terms of the other.

x + y = 253

x = 253 - y

Now we can substitute this expression into the first equation:

1.25(253 - y) + 7.20y = 1060

Solving for y we get

y = 120

So there were 120 adults admitted.

We can use the second equation to find the number of children admitted:

x + y = 253

x = 253 - y

x = 253 - 120

x = 133

So there were 133 children admitted.

Therefore,

There were 133 children admitted.

There were 120 adults admitted.

Answer:

128 children, 128 adults

Step-by-step explanation:

1.25  per child (c)

7.2 per adult (a)

a+c = 253 so c = 253-a

1.25c+7.2a=1060

2 equations 2 variables, solve

1.25 (253-a) + 7.2*a = 1.25*253 -1.25a + 7.2a =1060

sp 5.95a =1060- 1.25*253

so a =125

c = 253-125 = 128

Step-by-Step Explanation:

3rd term (a3) = -6

12th term (a12) = -3

We know, nth term of an AP = a1 + (n - 1)d, where d = common difference

a3 = a + (3-1)d = a + 2d = -6...(I)

a12 = a + (12-1)d = a + 11d = -3..(II)

Subtracting (II) from (I), we have

2d - 11d = -6 + 3

=> -9d = -3

=> d = 1/3

Therefore, a + 2×1/3 = -6

=> a + 2/3 = -6

=> a = -6-2/3 = -20/3

Therefore, 18th term

= a + (18-1)d

= -20/3 + 17/3

= -3/3

= -1

Answer:

-1

Hope it helps.

If you have any query, feel free to ask.

Answer:

We can start solving the problem by using algebraic equations. Let x be the total number of stickers in the box at first.

According to the problem, Wenxuan took 8 more than of the total number of stickers, so he took x + 8 stickers.

The remaining stickers after Wenxuan took his share is x - (x+8) = x - x - 8 = -8 stickers.

Charlene took 7 more than of the remaining stickers, which is -8 + 7 = -1 stickers.

So, Charlene took x - (x+8) + 7 = -1 stickers.

Siti took the last 23 stickers, so the total number of stickers remaining is -1 - 23 = -24.

Now we can use this information to set up the equation:

x + (x+8) + (x-x-8+7) + (-1-23) = x

Solving for x we get x = 32

Therefore, there were 32 stickers in the box at first.

The value of a is 12. Therefore, the equation of the function is f(x) = 12|x|.

In algebra, an objective function function is one where the variables is inside the bars denoting absolute values. The absolute value is also referred to as the modulus function, and its most popular representation is f(x) = |x|, wherein x is a real number.

Finding x values that turn the equation inside the exact amount positive or negative into a constant is how absolute value equations are solved. Plot two for the negative and positive cases that intersect at the expression's 0 to graph exact value functions.

To find the value of a, we can use the point (1, 12) to set up the equation:

a = 12/|1| = 12

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Answer:

  -12,731

Step-by-step explanation:

You want to know what integer would be used to represent the value 12,731 feet below sea level.

Usually we measure elevation up from sea level. So, distances below sea level are usually represented with negative numbers. The integer corresponding to 12,731 feet below sea level would be ...

  -12,731 . . . . feet

Answer:

150 students in total

Step-by-step explanation:

30 students is 20%

20% x 5 = 100%

30 x 5 = 150 students

The first mistake Robbie made was, take improper lengths. AB is the diameter; its length is longer than the sides AX and XB. So, the triangle formed will not be an equilateral triangle.

An equilateral triangle in geometry is a triangle with equally long sides. The three angles opposing the three equal sides are equal in size because the three sides are equal.

The first mistake Robbie made was, take improper lengths. AB is the diameter; its length is longer than the sides AX and XB. So, the triangle formed will not be an equilateral triangle.

He should have, after drawing the diameter AB, construct a circle with point A as the center and AO as the radius. Mark the point of intersection as X. Then, AO, AX and OX would form an equilateral triangle.

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Answer:

1. constructing the perpendicular bisector of AB

2.

Step-by-step explanation:

Answer: c) 130

Step-by-step explanation:

x+8+42=180

x=180-50

x=130

The set {1,2,3} is not a group under multiplication modulo 4, as it does not contain an identity element. However, the set {1,2,3,4} is a group under multiplication modulo 5.

A group is a set of elements which is closed under an operation and contains an identity element. In order to be a group, the set must satisfy the four group axioms: closure, associativity, identity, and invertibility. The set {1,2,3} does not satisfy the group axioms under multiplication modulo 4, since it does not contain an identity element, and thus is not a group.

On the other hand, the set {1,2,3,4} is a group under multiplication modulo 5, since it contains an identity element (1) and is closed under multiplication. The other group axioms are also satisfied, as the operation is associative, and each element has an inverse (for example, the inverse of 1 is 4). Therefore, {1,2,3,4} is a valid group under multiplication modulo 5.

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Answer:

30 + 4n

Step-by-step explanation:

Company X = 30 + 4n

Answer:

75,000

Step-by-step explanation:

We have to round up to nearest thousand ,so we need to check hundred place ...

If hundred place is equal or greater than 5 ,we need to add +1 to the number which is in thousand place , If it is less than 5 we don't need to do so ....

The value of x that makes the equation true is 12°. The solution has been obtained using angle sum property of triangle.

What is Angle Sum Property?

The Angle Sum Property states that the sum of the measures of the angles of a triangle is 180°. This means that if you add up all three angles of a triangle, the total measure will be 180°. This property is true for all triangles, regardless of the size or shape. It is one of the most important properties in geometry and is used to solve problems involving angles in triangles.

We are given two angles as (2x)° and (5x+18)°

Also, we are given two sides of the triangle are same.

This means that the angles opposite to the sides are also same.

Thus, the angles of the triangle are (2x)°, (5x+18)° and (5x+18)°

Using angle sum property,

⇒ (2x)° + (5x+18)° + (5x+18)° = 180°

⇒ (2x)° + 2 * (5x+18)° = 180°

⇒ (2x)° + (10x+36)° = 180°

⇒ (12x + 36)° = 180°

⇒ (12x)° = 144°

⇒ x = 12°

Hence, the value of x that makes the equation true is 12°.

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The distance between the Dog from the Home is 360.55 m and the direction of the Dog from Home is North-East.

In the inquiry,

The daring dog begins, let's say, at O.

He walks 300 meters east by 3 blocks.

then,

200 m / 2 blocks north

100 m East of 1 block

Block North = 100 meters.

200 m = 2 blocks west

So,

the separation between the Dog's initial location and ending position.

OE is supplied by,

By using Pythagoras' theorem to triangle OLE,

OE²=OL²+LE²

OE²=200²+300²

OE²=40000+90000

OE=360.55m

Therefore, the distance of the Dog from the Home is 360.55 m and the direction of the Dog from Home is North-East.

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The volume of the solid obtained by rotating the region bounded by the given curves about the specified line about x-axis is 3.93740 and y=-1 is 5.0740

We have to find the volume of given curve about x-axis

we have [tex]$\mathrm{y}=\mathrm{e}-\mathrm{x}^2,[/tex] y=0, x=-5, x=5

(a).

The volume of solid of revolution about the x-axis using disk method is evaluated as:

Volume V=[tex]\pi \int_{\mathrm{x}=-5}^{\mathrm{x}=5} \mathrm{y}^2 \mathrm{dx}$[/tex]

[tex]$$\begin{aligned}& \mathrm{V}=\pi \int_{\mathrm{x}=-5}^{x=5}\left(\mathrm{e}^{-\mathrm{x}^2}\right)^2 \mathrm{dx} \\& \mathrm{V}=\pi \int_{\mathrm{x}=-5}^{x=5} \mathrm{e}^{-2 \mathrm{x}^2} \mathrm{dx} \\& \mathrm{V}=\pi \sqrt{\frac{\pi}{2}}=3.93740 .\end{aligned}$$[/tex]

V=3.93740

we use the formula

(b). The volume of solid of revolution about the y= - using washer method is:

[tex]$$\begin{aligned}& \text { volume } \mathrm{V}=\pi \int_{\mathrm{x}=-5}^{\mathrm{x}=5} \mathrm{y}^2-{ }^2 \mathrm{dx} \\& \mathrm{V}=\pi \int_{\mathrm{x}=-5}^{\mathrm{x}=5} \mathrm{y}^2 \quad 2 \mathrm{y}-{ }^2 \mathrm{dx} \\& \mathrm{V}=\pi \int_{\mathrm{x}=-5}^{\mathrm{x}=5} \mathrm{y}^2 2 \mathrm{ydx} \\& \mathrm{V}=\pi \int_{-5}^5 \mathrm{e}^{-2 \mathrm{x}^2} 2 \mathrm{e}^{-\mathrm{x}^2} \mathrm{dx} \\& \mathrm{V}=\pi \sqrt{\frac{\pi}{2}} .77245 \\& \mathrm{~V}=5.0740\end{aligned}$$[/tex]

Therefore, the The volume of solid of revolution about the y= - using washer method is 5.0740

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The domain and the range of y = 12x - 36 are the set of all real numbers

From the question, we have the following parameters that can be used in our computation:

y = 12x - 36

There is no restriction on the input values, x

So, the domain of y = 12x - 35 is all real numbers, since there is no restriction on the input (x).

The range of y = 12x - 35 is also all real numbers, since the output (y) can be any real value, depending on the input (x).

This can be seen by considering that for any real number x, the output y = 12x - 35 will also be a real number.

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Complete question

What is the Domain of y = 12x - 36

x > 3 ,

-3 ≤ x ≤ 3,

all real numbers

Also, determine the range

The function in terms of x and y for the line that contains the points (2, 4) and (6, 16) would be; y=3x-2

The slope is the ratio of the vertical changes to the horizontal changes between two points of the line.

The slope of the line is calculated as follows:

m = Δy/Δx

Given that the line that passes through the points  (2, 4) and (6, 16)

To find the function of the line that passes through points  (2, 4) and (6, 16)

Therefore, the slope of the line is;

m = Δy/Δx

m = 3

The equation is;

y = mx + c

y = 3x - 2

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Projects that adopt agile methodology are not prone to the typical rapid changes that occur throughout the system development lifecycle, The statement is False.

The methodology is defined as to the overarching strategy and rationale of your research project.

It involves studying the methods used in your field and the theories or principles behind them, in order to develop an approach that matches your objectives.

Agile project management methodology is commonly used for software development projects. It has greater adaptability to frequently changing scopes. Adopt Agile because they want to increase speed to market, meet customer demand, or increase team productivity.

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The 1x8 vector resulting from multiplying each entry of x by 4.x4 = [4*x1, 4*x2, 4*x3, 4*x4, 4*x5, 4*x6, 4*x7, 4*x8]

For example, if x = [1, 2, 3, 4, 5, 6, 7, 8], x4 would be [4, 8, 12, 16, 20, 24, 28, 32]. This can be represented mathematically as x4 = [4x1, 4x2, 4x3, 4x4, 4x5, 4x6, 4x7, 4x8] which is the same as 4*x = [4x1, 4x2, 4x3, 4x4, 4x5, 4x6, 4x7, 4x8]. By multiplying each entry of x by 4, we can create a new vector x4 with every entry multiplied by 4.Multiplying each entry of a 1x8 vector x by 4 can be expressed as multiplying the vector by the scalar 4, resulting in a 1x8 vector x4. For example, if x = [1, 2, 3, 4, 5, 6, 7, 8], then x4 = [4, 8, 12, 16, 20, 24, 28, 32].

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Answer:

3x^2 + 4x + 3

Step-by-step explanation:

we can use this information to find g(x) in terms of f(x).

substitute f(x) = 3x-1 into f(g(x)) = 9x^2 + 12x + 8, we get:

3g(x) - 1 = 9x^2 + 12x + 8

Now, we can solve for g(x) by isolating it on one side of the equation:

3g(x) = 9x^2 + 12x + 9

g(x) = 3x^2 + 4x + 3

therefore g(x) is equal to 3x^2 + 4x + 3

The volume of the solid of revolution is  0

The volume of the solid of revolution obtained by rotating the region bounded by the curves x=0 and y=0 is zero since the area of the region is zero.

To calculate the volume of a solid of revolution, we use the formula V = ∫2dx, where r is the radius of the circle obtained by rotating the curve about the x-axis. Since the region bounded by the curves x=0 and y=0 has an area of 0, the radius r is also 0 and the integral becomes 0. Therefore, the volume of the solid obtained by rotating the region bounded by the curves x=0 and y=0 is 0.

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The total profit as the subject of formula is t = 3p

From the question, we have the following parameters that can be used in our computation:

p = t/3

Where

Individual Profit = p

total profit = t

Recall that, we have

p = t/3

Multiply both side of the equation by 3

So, we have

3p = t

Rewrite as

t = 3p

Hence, the solution is t = 3p

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Complete question

The equation p=t/3 is used to calculate the individual profit,p, made by each of three brothers operating a lemonade stand with total profit, t

make t the subject

The probability that the student got a a a given that the student was in the afternoon class is 25/57

In math the term probability is defined as based on the possible chances of something to happen

Here we have given that  a test to a group of students, the grades by the two different classes are summarized.

and it can be written like the following;

                                a             b               c                   Total

Morning class         4             20              8                    32

Evening Class         13              5               7                   25

Total                        17             25              15                  57

Here we know that the total number of students is obtained as

=> 57

And the number of students who have been taking the afternoon class is calculated as,

=> 25

Then the probability the student got a a a given that the student was in the afternoon class is written as,

=> 25/57

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The length of the ladder used in the question is; 6.974 ft

The 3 main trigonometric ratios are;

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

Now, since the ladder makes an angle of 35 degrees with the wall, it means the ladder will be the hypotenuse of the right angle triangle and the distance from the base of the wall to the base of the ladder will be labelled as h = 4 will be the opposite side of the triangle. Thus;

4/hypotenuse = sin 35

hypotenuse = 4/sin 35

hypotenuse = 4/0.5736

Hypotenuse = 6.974 ft

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The x-intercepts of this graph from least to greatest are x = -0.73, x = 1, and x = 2.73.

The local minimum of this graph is (0, -2).

The local maximum of this graph is (2, 2).

The intervals on which the function is decreasing are x < -0.73 and x > 1.

The intervals on which the function is increasing are 1 < x < 2.73.

The local minimum of a graph can be defined as the point on the graph of a function where it changes from a decreasing function to an increasing function, which is given by this ordered pair (0, -2).

Conversely, the local maximum of a graph can be defined as the point on the graph of a function where it changes from an increasing function to a decreasing function, which is given by this ordered pair (2, 2).

For any given function, y = f(x), if the output value is decreasing when the input value is increased, then, the function is referred to as a decreasing function.

In conclusion, we would use an online graphing calculator to plot the given function f(x) = x³ + 3x² - 2 in order to determine its key features.

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