What is the fastest way to find the distance between two points?.

Answer:

4/3

Step-by-step explanation:

The slope of line e

[tex]\frac{5-8}{8-4}[/tex] = [tex]\frac{-3}{4}[/tex]

Perpendicular slopes are opposite reciprocals.

So the slope of line f = 4/3

To find the missing length of a triangle using sine it is essential to know at least one side's length

One needs to know the lengths of at least one side and one angle of the triangle in order to use the sine function to get the missing length. In a right triangle, the sine function describes the relationship between the hypotenuse's length and the side length that is opposite the angle.

There are certain steps as to how the sine function determines a triangle's missing length. These steps are -

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The equation of the quadratic function y = x² + 6x + 3 in vertex form is y =  (x+ 3)² - 6

Quadratic function in math is defined as a polynomial function with one or more variables in which the highest exponent of the variable is two.

Here we have given the quadratic function y = x² + 6x + 3 and we need to find the vertex form of the given quadratic function.

As we all know that the vertex form of the equation is given by

=> f(x) = a(x - h)² + k

Here we have given the expression that can be written as,

=> y = x² + 6x + 3

Now, we have to convert it into vertex form, then we get,

=> x² + 6x + 3

Expand the expression as follows:

=> x² + 6x + 3 + 9 - 9

Now, we have rewritten the expression like the following,

=> y + 9 = (x + 3)² + 3

Here we have to subtract 9 on both sides, then we get

=> y =  (x+ 3)² + 3 - 9

Then the resulting equation is written as,

=> y =  (x+ 3)² - 6

Complete Question:

What is the equation of the quadratic function y = x² + 6x + 3 in vertex form?

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The set of -values that are appropriate in the given situation is known as the practical domain. Solution: F (x) = 2 x is the function's formula.

You are not accounting for or measuring any rainfall that occurred prior to the storm's onset, thus there is a total of 0 inches of precipitation at that time. Six and a half hours are spent in the storm.

Example :

Miami, Florida, experiences heavy rainfall due to a powerful hurricane. A 2 inch per hour downpour is occurring. In 6 and a half hours, the storm is over. The total amount of rain that has fallen throughout time should be modeled mathematically. The function is plotted. For this situation, identify the practical domain. The set of -values that are appropriate in the given situation is known as the practical domain.

Solution: Since there was no rain measured or included before the storm began, the function is as follows:

f(x)= 2x

There was 0 inches of rain at the moment the storm began. Six and a half hours are spent in the storm. f(6.5) = 2(6.5) = 13 inches of rain had been recorded at 6.5 hours, or so. It follows in words what the domain and range are:

Domain :

All real numbers between and including 0 and 6.5 hours.

Range :

All real numbers between and including 0 and 1.3 inches.

Domain and range can be expressed more easily with the help of some helpful notation, such as the following:

Domain { x∈R,0≤ x ≤ 6.5} [0,6.5]

Range { x∈R,0≤ x ≤ 6.13 [0,13].

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A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number.

A perfect number is a positive integer that is equal to the sum of its proper divisors, which are positive integers that divide the number without leaving a remainder. For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first four perfect numbers are 6, 28, 496, and 8128.

Perfect numbers have been studied by mathematicians for centuries, and continue to be an area of active research. They have connections to other areas of mathematics, such as number theory and algebraic number theory. Perfect numbers also have applications in computer science and cryptography.

It is also important to note that in number theory, a natural number is called abundant if it is less than the sum of its divisors and deficient if it is greater than the sum of its divisors. Perfect numbers are exactly equal to the sum of their divisors.

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The transformation will map ΔABC onto image ΔA'B'C' is, a reflection in the line y = x. So Option C is correct

In mathematics, the triangle is a type of polygon which has three sides and three vertices. the sum of all the interior angles of the triangle is 180°

If anyone of the triangle is 90° then it is called as right angle triangle

Given that,

Two triangles,

First,   ΔABC

Whose coordinates are A(1,3), B(1, 6), C(4, 6)

Second,  ΔA'B'C'

Whose coordinates are A'(3, 1), B'(6, 1), C'(6, 4)

There is only one transformation is  possible,

Which is a reflection in the line  y = x

Because,  the distance of the points will be equal from the line y = x

In comparison of corresponding points

Hence, the transformation is a reflection in the line y = x

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The function for the area is:

A = 0.5H² + 2H

And if the height is 6ft, then the area is 30ft²

We know that the area of a triangle is the product between the base and the height over two.

Here the base is 4ft more than the height H, then:

A = B*H/2

A = (H + 4)*H/2

A = 0.5H² + 2H

If the height is 6ft, then the area is:

A = 0.5*(6ft)² + 2*6ft = 30ft²

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There is no real number that, when multiplied by itself, produces a negative result. Therefore, the square root of a negative number is undefined in the context of real numbers.

A square root is defined as the value that, when multiplied by itself, equals a given number. The square of a real number is always positive, so the square root of a positive number is always positive or zero. However, there is no real number that, when multiplied by itself, produces a negative result. Therefore, the square root of a negative number is undefined in the context of real numbers.

Square root of a negative number is a complex number.

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В саду росте 500 дерев, з яких 24% є яблуні, що дає 120 яблунь. Груші становлять 115% кількості яблунь, тобто 135 груш. Вишні становлять 5/6 кількості груш, тобто 112,5 вишні. Решта дерев - сливи становлять 500 - 120 - 135 - 112.5 = 142.5 сливових дерев.

[tex]\it \dfrac{24}{100}\cdot500=24\cdot5=120\ apples\ trees\\ \\ \\ \dfrac{115}{100}\cdot120=\dfrac{115\cdot12}{10}=\dfrac{1380}{10}=138\ \ pears\ trees\\ \\ \\ \dfrac{5}{6}\cdot138=\dfrac{5\cdot138}{5}=\dfrac{690}{6}=115\ cherries\ trees\\ \\ \\ 500-(120+138+115)=500-373=127\ plums\ trees[/tex]

The inverse of the logarithmic function f(x) = log9x is f^-1(x) = 9^x.

The inverse of a function is a new function that "undoes" the original function. For a logarithmic function, such as f(x) = log9x, the inverse is an exponential function.

To find the inverse of f(x) = log9x, you can follow these steps:

Switch the roles of x and y in the original function to make y the subject of the formula.

log9x = y

Replace the logarithm function with its inverse function, the exponential function.

9^y = x

Switch x and y back to their original roles in the equation.

x = 9^y

It's worth noting that for this logarithmic function, the domain of the function is x > 0 and the range is y > 0, and the inverse function will have its domain as y > 0 and range as x > 0. Also, the base of the logarithm 9, is a positive real number, different than 1, this ensures that the logarithm function is defined for all x > 0, for b = 1 it would not be defined for x = 1.

So the inverse of the logarithmic function f(x) = log9x is f^-1(x) = 9^x

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Answer:14

Step-by-step explanation: This question is asked oddly but I believe it's still 14

3.5 x 4 = 14

Both coaches will charge $440 after 3 hours of coaching.

To find when the two coaches will charge the same amount, we need to set up the equation with help Compound Interest Formula:-

Where:-

Cost of Coach A = Cost of Coach B

200 + 80x = 230 + 70x

where x is the number of hours of coaching.

To solve for x, we can subtract 200 from both sides and 230 from both sides:

80x = 30 + 70x

10x = 30

x = 3

So, the two coaches will charge the same amount after 3 hours of coaching.

To find out how much it will cost, we can substitute 3 for x in the equation for either coach:

Cost of Coach A = 200 + 80x = 200 + 803 = $440

Cost of Coach B = 230 + 70x = 230 + 703 = $440

Therefore, both coaches will charge $440 after 3 hours of coaching.

It's important to note that this calculation assumes that the cost of the coaches is based solely on the hourly rate and initial fee, and does not take into account any other factors such as discounts, taxes, or additional services offered by the coaches.

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2(3b+6) =78

Expanding the left side of the equation:

6b + 12 = 78

Solving for b:

6b = 66

b = 11

Now we will check the left side and the right side of the equation:

2(3(11) + 6) = 2(33+6) = 2(39) = 78

On the other side of the equation:

-2(6c-3) = -72

Expanding the left side of the equation:

-12c + 6 = -72

Solving for c:

-12c = -78

c = 6.5

Now we will check the left side and the right side of the equation:

-2(6(6.5) - 3) = -2(39 -3) = -2(36) = -72

This equation is true, the left side of the equation equals to the right side of the equation.

The solution for the first equation is b = 11

The solution for the second equation is c = 6.5

The sum symbol is represented as

Sum is determined by putting the values all together and results is called "total" or sum. The plus symbol (+) is used as sum symbol in case of small numbers of digits for addition. For example: sum of 5 and 6 is represented as 41 + 15 = 56 .

The symbol Σ (sigma) is used to denote a sum of multiple terms. This symbol is generally used by an index that varies to represent all terms that must be considered in the sum. For example, the sum of first whole numbers can be represented in the following 1 + 2 + 3 + 4 + .......+ n = [tex] \sum_{i = 1}^{n} {i}[/tex]

where, ∑ is the symbol that is denoted as "sigma".

Summation is used in case of sum of arithmetic sequence, geometric sequence and other sequences.

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We can use the tangent function to solve for the height of the lighthouse.

Let h be the height of the lighthouse from the top of the cliff.

tan(50.5 degrees) = (h + x) / 1100

tan(48.1 degrees) = h / 1100

We can divide the first equation by the second equation to get:

(h + x) / h = tan(50.5 degrees) / tan(48.1 degrees)

then we can simplify it by cross multiplying and canceling out h

h + x = (h * tan(50.5 degrees)) / tan(48.1 degrees)

then we can substract h from both sides

x = (h * tan(50.5 degrees)) / tan(48.1 degrees) - h

then we can substitute x = 1100

1100 = (h * tan(50.5 degrees)) / tan(48.1 degrees) - h

then we can add h to both sides

1100 + h = (h * tan(50.5 degrees)) / tan(48.1 degrees)

then we can multiply both sides by tan(48.1 degrees)

1100tan(48.1 degrees) + htan(48.1 degrees) = h * tan(50.5 degrees)

then we can divide both sides by tan(48.1 degrees)

h = (1100 * tan(48.1 degrees) + h * tan(48.1 degrees)) / tan(50.5 degrees)

after substituting the values we get:

h = (1100 * tan(48.1) + h * tan(48.1)) / tan(50.5)

we can round it to the nearest tenth

h = (1100 * tan(48.1) + h * tan(48.1)) / tan(50.5)

then we can use a calculator to get the exact number

h ≈ 191.5

Therefore, the height of the lighthouse from the top of the cliff is 191.5 meters.

Answer:

31.5

Step-by-step explanation:

The  60.42 percentage of 8th graders believe they spend less than an hour per day on social media.

A percentage is a ratio expressed as a fraction of 100.

A random sample of students was asked in a survey if they estimated they spent more or less than an hour per day on social media.

To calculate the percentage of 8th graders who say they spend more than an hour a day on social media, divide the number of students who said "less" by the total number of 8th grade students surveyed, then multiply by 100.

29 students out of 48 students responded "less,".

To find the percentage,

(obtained value / total score ) x 100%

So here,

percentage = (29/48) x 100

accounting for 60.42% of the total.

As a result, 60.42% of 8th graders believe they spend less than an hour a day on social media.

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The requried area of the actual room is given as 264 foot².

The scale factor is defined as the ratio of the modified change in length to the original length.

Here,
As given in the question,
1/2 inch = 1 foot
1 inch = 2 foot

Now,
A rectangular room measures 6 inches by 11 inches on the drawing.
= 6 inch × 11 inch
= 6 (2 foot) × 11 (2 foot)
= 12 × 22 foot²
= 264 foot²

Thus, the requried area of the actual room is given as 264 foot².

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The answer to (9a²b³) x (2a⁴b)/ 6b² is 3[tex]a^6[/tex]b².

Exponents and powers can be used to represent extremely big or extremely small numbers in a more straightforward manner.

To illustrate 3 x 3 x 3 x 3 in a straightforward manner, for instance, we could write it as 34, where 4 is the exponent and 3 is the base. It is claimed that the entire statement 34 has power.

Given:

(9a²b³) x (2a⁴b)/ 6b²

So, (9a²b³) x (2a⁴b)/ 6b²

= 3 x 3 x a x a x b x b x b x 2 x a x a x a x a x b / 2 x 3 x b x b

= 3 x a x a x a x a x a x a x b x b

= 3[tex]a^6[/tex]b²

Hence, the answer is 3[tex]a^6[/tex]b².

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Answer:

h= 13.6 ft

Step-by-step explanation:

Let,

a= 8 ft

b= 11 ft

The ladder against the top of the house is creating a right triangle

So, we can find the length of the ladder using

[tex]h^{2} = {a}^{2} + {b}^{2} \\ h^{2} = {8}^{2} + {11}^{2} \\ h^{2} = 64 + 121 \\ h = \sqrt{185} \\ = 13.6ft[/tex]

Therefore, the length of the ladder (h) = 13.6 ft ( rounded to the nearest tenth)

The factored form is when the equation is written as "a(x-r1)(x-r2) = 0", where r1 and r2 are the roots (or solutions) of the equation. The vertex form of a quadratic equation is "a(x-h)^2 + k" where (h,k) is the vertex of the parabola represented by the equation.

The standard form of a quadratic equation is "ax^2 + bx + c = 0", where a, b, and c are constants. The 'a' coefficient is called leading coefficient, it determines the direction of parabola (whether it opens upward or downward) and the degree of the equation. The factored form of the equation is obtained by solving the equation and factoring out the common factors of the roots. This form of the equation is useful for identifying the x-intercepts (or zeroes) of the parabola, which are the points where the graph of the equation crosses the x-axis.

The vertex form of a quadratic equation is "a(x-h)^2 + k" where (h,k) is the vertex of the parabola represented by the equation. The vertex is the lowest point on the parabola if the leading coefficient 'a' is positive and highest point if 'a' is negative. The 'h' value in the equation represents the x coordinate of the vertex and 'k' represents the y coordinate. This form of the equation is useful for identifying the vertex of the parabola and the axis of symmetry, which is the line that divides the parabola into two mirror images.

The vertex form can be obtained by completing the square on the quadratic equation in standard form. By adding and subtracting the square of half of the coefficient of x, we can convert the equation in the form of (x-h)^2 which is a square of binomial.

Hence, the factored form is when the equation is written as "a(x-r1)(x-r2) = 0", where r1 and r2 are the roots (or solutions) of the equation. The vertex form of a quadratic equation is "a(x-h)^2 + k" where (h,k) is the vertex of the parabola represented by the equation.

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The functions could represent g(x) is C. g(x) = f(x - 5)

Using the general equation as Witten which can be used to find the y intercept of the line

g(x) = f(x) + b

The g(x) has intercept of y=-2,

Then the  y-intercept of f(x) is at y=3

If we find the difference between both y-intercepts, we have

Recall that the y-intercept is the point where the graph intersects the y-axis

( -2 -3) = -5

This implies that the intercept of the line is -5

Then b=-5

Hence, g(x) = f(x - 5) is correct option for the function

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The ratio of the two legs of the triangle 30-90-60 will be 1 : √3.

The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.

The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.

The triangle is 30-90-60. The ratio of the two legs will be calculated as,

tan(30) = AB / BC

1 / √3 = AB / BC

Hence, the ratio of the two legs will be 1 : √3.

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The proof that ∠FCB≅∠DCE is shown below

Given that AECF is a rhombus, opposite angles of the rhombus are congruent.

This is represented as

∠EAF = ∠ECF

Also, given that EB ≅ FD, we can conclude that:

∠EBF = ∠DFC

This is by the definition of congruent segments having equal corresponding angles

Add the angles

∠CEB + ∠EBF = ∠CEB + ∠DFC = 180°

Recall that

Since ∠CEB = ∠AEF

So, we have:

∠AEF + ∠EBF = ∠AEF + ∠DFC = 180°

Hence, ∠FCB=∠DCE.

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We need to remove the decimal and multiply and divide by 10, resulting in 0.4/1 x 10/10 = 4/104/10. This can be further reduced to 2/5, so 0.4 as a fraction is 4/10 or 2/5.

A fraction shows part of a whole. This whole can be a region or a collection. The word fraction is derived from the Latin word "fraction" which means 'to break'. The Egyptians, being the earliest civilization to study fractions, used fractions to resolve their mathematical problems, which included the division of food, supplies, and the absence of a bullion currency.

In Ancient Rome, fractions were only written using words to describe a part of the whole. In India, the fractions were first written with one number above another (numerator and denominator), but without a line. It was the Arabs only, who added the line which is used to separate the numerator and the denominator.

First, we need to remove the decimal.

For that, we multiply and divide by 10 as there is only one digit after the decimal.

We get, 0.4/1 x 10/10 = 4/10

4/10 can further be reduced to 2/5

Hence, we can say that 0.4 as a fraction is 4/10 or 2/5.

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From probability of occurence of events the large box holds 180 hats.

The probability of an event is the proportion of favorable outcomes to all other potential outcomes.The number of positive results for an experiment with 'n' outcomes can be represented by the symbol x. The following formula can be used to determine how likely an event is to occur.

Here, probability can be used. When throwing coins, rolling dice, or drawing a card from a deck of cards, probability is employed to forecast the results.

The probability is classified into theoretical probability and experimental probability.

Probability(Event) = Positive Results/Total Results = x/n

Here, in the mall box, are 60 hats.

And a big box holds 18% more hats than the box provided by the mall.

So, 180/100 ×60 = 180

Therefore, from probability of occurence of events the large box holds 180 hats.

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Answer:

Step-by-step explanation:

You want to know the consecutive integers that bound the values of 30^(1/5) and 45^(1/3).

The fifth root of 30 will lie between the two integers whose 5th powers lie on either side of 30:

  1^5 = 1 < 30

  2^5 = 32 > 30

So, the fifth root of 30 lies between 1 and 2:

  [tex]\boxed{1 < \sqrt[5]{30} < 2}[/tex]

The cube root of 45 will lie between the two integers whose cubes lie on either side of 45:

  3^3 = 27 < 45

  4^3 = 64 > 45

So, the cube root of 45 lies between 3 and 4:

  [tex]\boxed{3 < \sqrt[3]{45} < 4}[/tex]

__

Additional comment

A calculator confirms this:

  [tex]\sqrt[5]{30}\approx 1.974\\\sqrt[3]{45}\approx3.557[/tex]

The statement "In the statement ~(M ⊃ X)~) governs the simple statement M" is false

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

In the statement ~(M ⊃ X)~) governs the simple statement M.

In logic and binary, the negation of an implication (M ⊃ X) is written as ~(M ⊃ X), which is read as "not (M implies X)".

This statement does not govern or control the simple statement M. The simple statement M can be true or false independently of the negation of the implication.

In other words, the truth value of ~(M ⊃ X)is dependent on the truth values of both M and X, not just M.

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Answer:

undefined

At x = 0 the function is undefined, because there is a zero denominator.

Step-by-step explanation: