question 6
Match each number that is in scientific notation to its calculator shorthand representation

The proportional parts in triangles and parallel lines can be found by similarity and ratios.

The ratio of any related side lengths is always the same when two triangles are similar. The scale factor can be used to identify the triangle's proportionate components. The longer side of one triangle is twice as long as the comparable side in the other. In contrast, the alternate interior angles are identical when two parallel lines are cut by a transverse line. Additionally, the matching angle pairs are all equal. Trigonometry can be used to determine the length of the opposite side of a pair of parallel lines if we know the length of one side and the matching angle.

One can use similarity ratios to determine the proportional components of triangles and parallel lines. Comparable geometries have equivalent sides that are in proportion. To get the missing side length, one can arrange ratios of corresponding sides and utilise cross-multiplication. Further, similar triangles can also be employed. When two triangles are comparable, their corresponding sides and angles are proportionate and equal.

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Answer:

[tex]\large\boxed{\sf 0.133 }[/tex]

Step-by-step explanation:

The given equation to us is ,

[tex]\longrightarrow 3^{2-3x}= 4^{2x+1} \\[/tex]

Take log to base 10 on both sides,

[tex]\longrightarrow \log_{10} 3^{2-3x}= \log_{10} 4^{2x+1} \\[/tex]

Now recall that, [tex] \log a^m = m\ log \ a [/tex]

[tex]\longrightarrow (2-3x) \log_{10} 3 = (2x+1)(\log_{10} 4)\\[/tex]

We can write it as ,

[tex]\longrightarrow (2-3x) \log_{10}3=(2x+1)\log_{10}2^2 \\[/tex]

Again using the property mentioned above,

[tex]\longrightarrow (2-3x)\log_{10}3 = 2(2x+1)\log_{10} 2 \\[/tex]

[tex]\longrightarrow (2-3x)\log_{10}3 = (4x+2)\log_{10} 2 \\[/tex]

Now we know that ,

On substituting the values, we have ,

[tex]\longrightarrow (2-3x)0.477 = (4x+2)0.301 \\[/tex]

simplify by opening the brackets,

[tex]\longrightarrow 0.954 - 1.431x = 1.204x + 0.602 \\[/tex]

[tex]\longrightarrow 1.204x + 1.431x = 0.954 - 0.602 \\[/tex]

[tex]\longrightarrow 2.635x = 0.352 \\[/tex]

[tex]\longrightarrow x =\dfrac{0.352}{2.635} \\[/tex]

[tex]\longrightarrow \underline{\underline{ x = 0.133 }} \\[/tex]

and we are done!

The lines best approximate the directrices of the ellipse y = 10.6 or y = -6.6

The term called directrices of the ellipse is defined as a line parallel to the latus rectum of ellipse and is perpendicular to the major axis of the ellipse.

Here we know that first of all here the known form is an ellipse with a vertical major axis so it would have to be

=> d = a²/c      

Here d refers the distance between the center and the directrix.

When we take the value of a= 8, b=3, then we get the value of C as,

=> c² = a²- b²

Apply the value on it, then we get

=> c ² = 64-9

Then the value of is about 7.4

Then the value of d is calculated as,

=> d = 64/7.4 = 8.6

So the value of y is defined as,

=> 2 + 8.6 = 10.6

Or if it falls the negative direction, then we get

=> y = 2 - 8.6 = -6.6

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Answer:

Step-by-step explanation:

The transformation of triangle QRS that results in an image triangle DEF where angle Q is congruent to angle D, angle R is congruent to angle E, and angle S is congruent to angle F, and QR/DE = QS/DF = RS/EF ≠ 1 is a similarity transformation.

A similarity transformation is a type of transformation that preserves the angles between the corresponding sides of the original and the image figures. It means that the corresponding angles of the two figures are congruent, and the ratios of the corresponding sides are equal. In the case of a triangle, it preserves the angles and the ratios of the sides.

So, if a transformation of triangle QRS results in an image triangle DEF where angle Q is congruent to angle D, angle R is congruent to angle E, and angle S is congruent to angle F, and QR/DE = QS/DF = RS/EF ≠ 1, it is a similarity transformation.

The transformation is similarity transformation.

Similarity transformations are transformations where there occurs a dilation with one or more transformations like rotation, reflection or translation.

This is called so because the triangles so formed will be similar triangles.

We need two triangles QRS and DEF to be similar triangles.

Similar triangles are those triangles which has the same shape but different size and the length of sides are proportional.

This is the required transformation here.

Similar triangles can be formed by transforming a triangle first with dilation.

Then you can do any of the other rigid transformations.

The triangles will be similar.

Hence the transformation here is similarity transformation.

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The number line which shows all the values of x that make the provided inequality -3x+1<7 true is shown in number line D.

Now, According to the question:

The number line is the way of representation of number and digits in a arranged form to understand the data batter.

The inequality is :

- 3x + 1 < 7

Subtract 1 both side of the equation and solve it further,

-3x + 1 - 1 < 7 - 1

-3x < 6

Divide with number 3,on both sides

-x < 6/3

-x < 2

Change the sign of the inequality by filliping the inequality sign,

x > -2

Here, the inequality is greater than the number negative 2. The value of x is greater than, thus, the arrow will point towards left and there will be open circle. This is shown in the option D.

Thus, the number line which shows all the values of x that make the provided inequality -3x+1<7 true is shown in number line D.

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The tin with 250ml is better value for money. The solution has been obtained using the unitary method.

What is a unitary method?

The unitary method is used to calculate the value of a single unit from a given multiple, to put it simply. With the unitary method, you first determine the value of a single unit before determining the value of the necessary quantity of units.

We are given two tins.

First : 345ml tin of paint costing £4.80

Second: 250ml tin of paint costing £3.35

Now, using the unitary method, we get

First tin

For 350ml, cost is £4.80

So, for 1ml, cost will be

£4.80/350 = £0.014

Second tin

For 250ml, cost is £3.35

So, for 1ml, cost will be

£3.35/250 = £0.013

Hence, the tin with 250ml is better value for money.

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Answer:

d) 80°

Step-by-step explanation:

For isosceles triangle,

→ The other two angles are equal.

The given angle,

→ <A = 100°

Formula we use,

→ <A + <B + <C = 180°

Forming the equation,

→ 100° + x + x = 180°

Now the value of x will be,

→ 100° + x + x = 180°

→ 100° + 2x = 180°

→ 2x = 180° - 100°

→ 2x = 80°

→ x = 80°/2

→ [ x = 40° ]

Adding values of both angles,

→ x + x = 2x

→ 2 × 40° = 80°

Hence, option (d) is correct.

We will apply the percentage method to find that 48 students said red is their favourite colour.

Probability is a constituent part of computation that deals with the occurrence of a random event. The value is defined from zero to one. Probability is familiarised to predict how likely events are to happen. Probability is the degree to which something is possible to happen.

Total number of pupils in class 9 = 80

We need to calculate the number of students choosing red as their favourite colour.

We are given that 80 pupils are there in year 9.

Out of these, 3/5 of the pupils say that their favourite colour is red.

Let n be the number of pupils who say red is their favourite colour.

∴ n = (3/5) × 80

∴ n = 48

--The given question is incorrect; the correct question is

"3/5 of the pupils in a year 9 class say their favourite colour is red. There are 80 pupils in Year 9. How many students said red is their favourite colour?"--

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Using the arithmetic operation of addition the total thickness of granola bar and yogurt collectively is 1130 inches.

What is arithmetic operation?

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

The thickness of granola bar is = 1116 inches

The thickness of layer of yogurt is = 14 inches

To find the total thickness use the formula -

Total Thickness = Thickness of granola bar + Thickness of yogurt

The arithmetic operation of addition is used to find the total thickness.

Substitute the values into the equation -

Total Thickness = 1116 + 14

Total Thickness = 1130

Therefore, the total thickness is obtained as 1130 inches.

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The probability that two random people being born on the same day is thus 1/365 or about 0.3%

We apply our probability model above to the task at hand.

First, we suppose Person 1. This person is born on a certain date of the year.

Since this is a random person, we can fix this person's birthday to a specific date (a number between 1 and 365) without loss of generality for the problem.

Let's say this person is born on 10. (i.e. January 10th)

Now, we take Person 2. In order to achieve the event that Person 2 is born on the same date as Person 1,

we need them to be born on 10. In other words, we seek the probability of Person 2 being born on 10.

Based on our model, we have that this probability is 1/365.

The probability that two random people being born on the same day is thus 1/365 or about 0.3%

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Hi There! Your Answer would be 20. Why you might ask? allow me to tell you the reason!

you first make a "House" Around the Dividend, the biggest number goes outside of the box, then you just do basic division until you get a zero!

We would be left with 20.

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Where in the figure, ∠RQS Is-congruent to ∠QLK. 3 lines are shown. Lines SP and KN are parallel. Line R M intersects line SP at point Q, and intersects line KN at point L. ∠RQS is x degrees. ∠ KLM is (x minus 36) degrees. Note that the value of x will be 108°.

With regard to the problem description and the image attached,  we know that

m∠RQS≅m∠QLK ⇒ by corresponding angles

m∠KLM + m∠QLK=180° ⇒  by supplementary angles (consecutive interior angles)

Since

m∠RQS=x° ⇒  given in the problem

Therefore,

m∠QLX = x°

m∠KLM=(x-36)° ⇒  given problem

executing a substitution we have:

(x - 36)° + x° = 180°

2x = 180 + 36

2x = 216

x = 216/2

x = 108°

Thus, it is correct to state that the value of x in the diagram is 108°

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The cost of each pen is $25. This can be solved using unitary method.

In the unitary technique, the value of a single unit is determined first, and then the value of the required number of units is determined.

When employing ratio and proportion to solve a problem, it is essential to comprehend the units and values.

To make things simpler, always place the known facts on the left and the things that need to be calculated on the right. The number of apples in the aforementioned situation is known, but it is unclear how much the fruit is worth on the market.

It should be stressed that the unitary technique applies the notions of ratio and proportion to problems.

Let the cost of each pen be x .

then according to the question, the cost of book will be x-10

now

90 = 3x + x - 10

x = 25

hence the cost of the book is x - 10 = 15

and the cost of each pen is 25

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The ratios 3/11 and 17/6 is not proportional.

Proportions are defined as the concept where two or more ratios are set to be equal to each other.

Suppose we have two ratio p : q and r : s. If both these ratios are proportional, then we can write it as p: q : : r : s.

This is same as p : q = r : s or p/q = r/s

The given ratios are 3/11 and 17/6.

For two ratios a/b and c/d to be proportional, a/b must be equal to c/d.

If 3/11 and 17/6 are proportional,

3/11 must be equal to 17/6.

Or doing the cross multiplication,

3 × 6 must be equal to 11 × 17.

18 must be equal to 187.

But 18 ≠ 187

So the ratios are not proportional.

Hence 3/11 and 17/6 are not proportional.

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Answer:

Step-by-step explanation:Seriously you probably going to need a ruler because you don't like you can't just remember measurement inside your head you have to really like a ruler so actually go get a ruler and then take a picture of it doing with the ruler and then you might like get it.

The domain of the function will be All Real Numbers except x = -9. The correct option is B.

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the set of values it can take as input. After we enter an x value, the function outputs this set of values.

Given that the function is y=−9x−26/x+3. As the domain is defined as all the input values going into the function. The domain will be all the real numbers except -9. Because the function is undefined on -9.

The graph of the function is attached with the answer below.

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The constant of proportionality of the proportional relationship, given the values of x and y is D.  5 / 12

The ratio between two proportional quantities' constant values is known as the constant of proportionality. When either their ratio or their product gives a constant, two changing values are said to be in a proportionate relationship.

The Constant of proportionality can therefore be found by picking a random point on the graph which is ( 2, 5 / 6 )

The constant of proportionality is:

= 5 / 6 ÷ 2

= 5 / 6 x 1 / 2

= 5 / 12

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Answer:

To find the coordinates of P so that P partitions segment AB in the part-to-whole ratio of 1 to 5 with A(-9, 3) and B(1, 8), we can use the formula:

P = (1 - t)A + tB

where t is a scalar value that represents the position of P on the segment AB, and A and B are the coordinates of the two endpoints of the segment.

In this case, we know that the part-to-whole ratio is 1:5, so t = 1/6.

So we substitute the values for A, B and t into the formula:

P = (1 - 1/6)A + (1/6)B

P = (5/6)A + (1/6)B

P = (-95/6, 35/6) + (1/6, 8/6)

P = (-15/2, 5/2) + (1/6, 4/3)

P = (-15/2 + 1/6, 5/2 + 4/3)

P = (-30/3, 23/6)

The coordinates of P are (-10,3.83)

For the second problem,

To find the coordinates of P so that P partitions the segment AB in the ratio 3 to 1 if A(7, -4) and B(-5, 4), we can use the same formula as above:

P = (1 - t)A + tB

In this case, we know that the part-to-whole ratio is 3:1, so t = 3/4.

So we substitute the values for A, B and t into the formula:

P = (1 - 3/4)A + (3/4)B

P = (1/4)A + (3/4)B

P = (7/4, -4/4) + (-53/4, 43/4)

P = (7/4 + -15/4, -4/4 + 3)

P = (-8/4, -1/4)

The coordinates of P are (-2,-0.25)

So in the first problem point P is (-10,3.83) and in the second problem point P is (-2,-0.25)

The best predicted weight for an adult male who is 185 cm tall is approximately 164.65 kg.

A linear pattern with no other non-linear patterns or outliers is indicated by a scatter plot.

To find the best predicted value of weight (y) for an adult male who is 185 cm tall, you can use the equation:

y = -101 + (1.09 * x)

where x is the height in centimeters.

Substituting 185 for x, we get:

y = -101 + (1.09 * 185) = 164.65 kg

So the best predicted weight for an adult male who is 185 cm tall is approximately 164.65 kg.

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The coefficients of the terms on the right side of a binomial equation are the numbers in Pascal's Triangle.

Pascal’s Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)n, where n can be any positive integer and x, and y are real numbers. Pascal's Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement. It begins with 1 on the top and with 1 running down on the two sides of the triangle. In the pascal triangle, each new number is between two numbers and below them and its value is the sum of two numbers above. This triangle is used in different types of probability conditions. Here each row represents the coefficient of expansion of (x + y)ⁿ.

Properties of Pascal’s Triangle:

Therefore, the coefficients of the terms on the right side of a binomial equation are the numbers in Pascal's Triangle.

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Answer: Let "t" be the fare of the train.

The fare of the bus is "b".

We know that t + b = 4000 (the total fare of the journey)

On the return trip, the fare of the bus was hiked by half of what was paid, so b*1.5 = b + (b/2) = new fare of the bus.

The total fare of the return journey is 4800, so t + (b + (b/2)) = 4800

Now we have two equations:

t + b = 4000

t + b + (b/2) = 4800

We can substitute the first equation into the second equation:

t + b + (b/2) = 4800

t + (4000 - t) + (b/2) = 4800

t + (b/2) = 800

t = 800 - (b/2)

Now we have an equation that only involves one variable "t" (the fare of the train)

We can substitute the first equation back in:

t + b = 4000

800 - (b/2) + b = 4000

800 = (3/2)b

b = 1200

So the fare of the bus is 1200 and the fare of the train is 800

Step-by-step explanation:

It will take Chantel approximately 2.028 hours or 122.68 minutes to complete the triathlon.

What is average ?

An average, also known as a mean, is a measure of central tendency that represents the typical value of a set of data. To calculate the average of a set of numbers, you add them up and then divide by the number of items in the set.

Time = Distance / Speed

For the swimming segment, the distance is 0.93 miles and the speed is 2.1 mph. So the time it will take Chantel to swim the 0.93 miles is:

Time = 0.93 / 2.1 = 0.445 hours or 26.7 minutes

For the biking segment, the distance is 24.8 miles and the speed is 18.2 mph. So the time it will take Chantel to bike the 24.8 miles is:

Time = 24.8 / 18.2 = 1.367 hours or 81.02 minutes

For the running segment, the distance is 6.2 miles and the speed is 5.1 mph. So the time it will take Chantel to run the 6.2 miles is:

Time = 6.2 / 5.1 = 1.216 hours or 72.96 minutes

To get the total time, you have to add the time for each segment:

Total time = 0.445 + 1.367 + 1.216 = 2.028 hours or 122.68 minutes

So , It will take Chantel approximately 2.028 hours or 122.68 minutes to complete the triathlon.

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The example for the y-intercept, slope, two points, and the ordered pair

are given below.

In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction

Let a line in the slope-intercept form,

y = x + 1.

And the line passes through the two ordered pairs (3, 4) and (5, 6).

To find the slope;

m =  (y₂- y₁) / (x₂ - x₁)

m = 1

If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,

then the equation of line is

y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)

y - 4 = x -3

y = x+1

Substitute x = 0,

then y = 1

the y-intercept is 1.

Therefore, all the required values are given above.

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The segment lengths in the diagram are given as follows:

Using the Pythagorean Theorem, the length of segment GE is given as follows:

(GE)² = 5² + 12²

(GE)² = 169

GE = 13.

Applying the geometric mean theorem, the length AM is obtained as follows:

(GM)² = ME x AM

25 = 12 AM

AM = 2.083.

Then, applying the Pythagorean Theorem, the length AG is found as follows:

(AG)² = 2.083² + 5²

AG = 5.42.

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The bolts that are destroyed due to being more than 2 standard deviations from the mean will have a length of 10 cm ± 0.15 cm.

This means that any bolts that are shorter than 9.85 cm or longer than 10.15 cm will be destroyed. Any bolts that fall within this range will pass the quality-control check and will not be destroyed.

The manufacturer should also have a policy in place to ensure that bolts that are found to be too short or too long are discarded and not put into circulation. This will help maintain the quality of the bolts that are produced.

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20-s is the number of pages written by Leo this week.

An expression is combination of variables, numbers and operators.

Given that Leo wrote s of his 20-page paper last week.

In last week she wrote s pages.

Leo completed his writing the paper this week.

20-s

Twenty minus s is the expression which gives the pages Leo write this week

Hence, 20-s is the number of pages written by Leo this week.

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From the data points given, the linear equation is obtained as y = 6x - 3.

What is a linear equation?

A linear equation is one that has a degree of 1 as its maximum value. No variable in a linear equation, thus, has an exponent greater than 1. A linear equation's graph will always be a straight line.

The data points given are -

X = 1, 2, 3, 4, 5

f(x) = 3, 9, 15, 21, 27

The slope of the line is given as m.

The slope can be deduced using the formula -

(y2 - y1) / (x2 - x1)

m = (9 - 3) / (2 - 1)

m = 6/1

m = 6

The slope-intercept form of a equation is y = mx + b

Substitute the values of x, y and m in the equation -

3 = 6(1) + b

3 = 6 + b

b = 3 - 6

b = -3

So, the equation becomes -

y = 6x - 3

This equation forms a straight line in the graph.

Therefore, y = 6x - 3 is a linear equation.

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The required algebraic expression is x-3

An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms.

Given here: The statement "A number subtracted from 3 "

Thus translating the given text we get the following:

Let the number be x

And we have have to subtract 3 from x

Thus the algebraic expression is given by x-3

Hence, The required algebraic expression is x-3

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The inverse equation of the function is y = 5x - 12

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

f(x) = 1/5x + 12

Express as an equation

So, we have

y = 1/5x + 12

Swap x and y

This gives

x = 1/5y + 12

Next, we make y the subject

1/5y = x - 12

This gives

y = 5x - 12

Hence, the inverse is y = 5y - 12

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