legend has it that the great mathematician carl friedrich gauss ​(1777–​1855) at a very young age was told by his teacher to find the sum of the first 100 counting numbers. while his classmates toiled at the​ problem, carl simply wrote down a single number and handed the correct answer in to his teacher. the young carl explained that he observed that there were 50 pairs of numbers that each added up to 101. so the sum of all the numbers must be 50•101

I think the answer is x=4

Answer: x=4

Step-by-step explanation:

2x+3(5x-7)=47

2x+15x-21=47

17x-21=47

  +21  +21

17x= 68

17     17

x=4

Answer:

-18k

Step-by-step explanation:

-> -8k+ 10k

-> minus and plus is = minus

-> 8k+10k

-> -18k(ans)

The value of x is 12.

What is triangle proportion?

According to the Triangle proportion rule, When a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

In the diagram;

The side with x is a reflexive side of both triangles and as such from triangle proportionality theorem we can say that;

⇒6/x = 3/6

⇒ 36 = 3x

⇒ x = 12

Hence, The value of x is 12.

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The slope of the line that passes through  (8, 0) and (-4, -8) is calculated as: m = 2/3.

The slope (m) formula is given as:

Slope (m) = change in y/change in x = y2 - y1/x2 - x1.

Given the following points on the line:

(8, 0) = (x1, y1)

(-4, -8) = (x2, y2)

Plug in the values into the slope formula, x = y2 - y1/x2 - x1:

Slope (m) of the line = (-8 - 0) / (-4 - 8)

Slope (m) of the line = (-8) / (-12)

Slope (m) of the line = 2/3

Thus, the slope of the line that passes through  (8, 0) and (-4, -8) is calculated as: m = 2/3.

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The value of the expression log 10,000 is 4.

The logarithmic function is the inverse of the exponential function. This means that the logarithm of a given number, say x, is the exponent to which another fixed number that is the base b, must be raised, so as to produce that number x.

Log a to the base b can be written as- log a/ log b.

Moreover, log [tex]a^{k}[/tex] = k log a, for all non negative values of a

Here, we are given a logarithmic expression- log 10,000

As the base is not given, we always assume it to be 10.

We can expand this as-

log 10,000 = log 10,000/ log 10

= log [tex]10^{4}[/tex]/ log 10

= 4 log 10/ log 10

= 4

Thus, the value of log 10,000 is 4.

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Answer:

[tex]1.93475*10^{-10}[/tex]

Step-by-step explanation:

To multiply the decimal numbers in scientific notation, multiply the decimals and add the powers of the 10.

  [tex]\sf 5.45 * 10^{-4}* 3.55*10^{-7}=5.45*3.55 * 10^{(-4)+(-7)}[/tex]

                                         [tex]\sf = 19.3475*10^{-11}[/tex]

Now, write the decimal 19.3475 in scientific notation.

As we have to move the decimal one count towards left, add 1 to the power.

                                         [tex]\sf = 1.93475*10^{-11+1}\\\\ = 1.93475*10^{-10}[/tex]

Answer:

The dimensions of the rectangle are length = 22 meters and width = 13 meters.

Step-by-step explanation:

It is given that the length of a rectangle is 4 meters less than twice the width and if the area of the rectangle is 286 square meters we need to calculate the dimensions.

Firstly, let us look at the dimensions of the rectangle;

=> Let 'l' be the length and 'b' be the width of the rectangle.

Now, let us look at the area of a rectangle;

=> l × b

So, we get an equation from the above condition;

=> l = 2b - 4

Now, the area of the rectangle is given that is 286 square​ meters;

=> A = 286 sq. m.

Using the above-mentioned formula for the area of the rectangle we get;

=> A = l × b

=> 286 = (2b - 4) × b

=> 286 = 2b² - 4b

=> 2b² - 4b - 286 = 0

After solving it further we get;

=> 2b² - 4b - 286 = 0

=> b² - 2b - 143 = 0

=> b² - 13b + 11b - 143 = 0

=> (b - 13) + (b + 11) = 0

Now, after equating both the sides we get;

=> (b - 13) = 0

=> b = 13

and

=> (b + 11) = 0

=> b = -11

But, b = -11 is not possible so, the width of the rectangle is 13 meters.

And the length of the rectangle would be,

=> l = 2b - 4

=> l = 2 × 13 - 4

=> l = 22

Therefore, the dimensions of the rectangle are length = 22 meters and width = 13 meters.

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The value of the variable x in the equation 2log₂(x) - log₂(2x) = 3 is x =16

The equation is given as

2log₂(x) - log₂(2x) = 3

Apply the power rule of logarithm

So, we have

log₂(x²) - log₂(2x) = 3

Apply the quotient rule of logarithm

So, we have

log₂(x²/2x) = 3

Apply the change of base of logarithm

So, we have

x²/2x = 2^3

Evaluate the exponent

So, we have

x²/2x = 8

Evaluate the quotient

So, we have

x/2 = 8

Multiply both sides by 2

So, we have

x =16

Hence, the value of the variable x in the equation 2log₂(x) - log₂(2x) = 3 is x =16

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No extraneous solutions to this equation.

The equation is

[tex](5x- 4)^{1/2}[/tex] - x = 0

To find the value of x

[tex](5x- 4)^{1/2}[/tex] = x

Now square on both sides of the equation.

We get

5x - 4 =[tex]x^{2}[/tex]

[tex]x^{2}[/tex] - 5x + 4= 0

[tex]x^{2}[/tex] - 4x - x + 4 =0

x( x -4 ) - 1 ( x - 4) =0

( x- 4 ) ( x - 1)= 0

x = 4, 1

We get the value of x as 4 and 1

Now we have to put the value of x in the equation

First, put 4

[tex](5x - 4)^{1/2}[/tex] -x =0

[tex](5 * 4 - 4)^{1/2}[/tex] - 4= 0

[tex](20 - 4)^{1/2}[/tex] - 4 =0

[tex]16^{1/2}[/tex] - 4 = 0

4-4 =0

0 = 0

Now put 1

[tex](5 * 1 - 4)^{1/2}[/tex] - 1 = 0

[tex](5 -4)^{1/2}[/tex] - 1=0

1 - 1=0

0=0

Hence we get that both the value of x is not an extraneous solution.

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If the inverse of f exists, it is represented by f⁻¹ and exists only if f is a bijective function.

The value of this function is h⁻¹(4) = 2.

The inverse function of a function f is a function that reverses the operation of f. The inverse of f exists if and only if f is bijective, and it is denoted by f⁻¹ if it exists.

Because they allow mathematical operations to be reversed, inverse procedures are essential for solving equations. When introducing a mathematical procedure, one of the most important questions is how to invert it.

Inverse operations are mathematical manipulations in which one operation reverses the action of the other, such as addition and subtraction, multiplication and division. In most cases, the inverse of a number is its reciprocal, i.e. x - 1 = 1 / x. A number and its inverse (reciprocal) product equals one.

Let the function be h(x)= 1/x+2  

h(x) = 1/x + 2 then

y = 1/x + 2

x = 1/y + 2

simplifying the equation, we get

x-2 = 1/y

x-2 = y⁻¹

h⁻¹(x) = x - 2

substitute the value of x = 4,

h⁻¹(4) = 4 - 2

h⁻¹(4) = 2

The value of this function is h⁻¹(4) = 2.

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The number of ants that are more than people will be 9800000.

It should be noted that based on the information, it was stated that on earth, there are about 1 quadrillion ants and about 8 billion people.

Therefore, the difference between the values illustrated will be:

= 1 quadrillion - 8 billion

= 1 × 10^15 - 8 × 10^9.

= 9.2 × 10^5

Therefore, the number of ants that are more than people will be 9800000

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Use the following information to fill in the blanks. On earth, there are about 1 quadrillion ants and about 8 billion people.

On Earth, there are about ...... more ants than people.

Answer: A quadrilateral that has 4 sides is known as a rectangle.

Answer:

Step-by-step explanation:

This is either a rectangle or a square.

(A square is a special type of rectangle, with 4 equal sides).

ANSWER- X is  2.52083333 or 2.52083 rounded

265-23 is 242. 242 divided into 96 is 2.52083333

(96 x 2.52083333)+23 = 265

This means our X is 2.52083...

HOPE THIS HELPS & GOOD LUCK

Answer:

[tex]\displaystyle \frac{121}{48}[/tex]

or

[tex]2.5208\bar3[/tex]

To begin, subtract 23 from both sides of the equation. This will move the constants to the right side.

[tex]96x=242[/tex]

Then, divide both sides of the equation by 96. This will isolate the unknown, x, and allow you to solve the equation.

[tex]\displaystyle \frac{96x}{96}=\frac{242}{96}[/tex]

[tex]\displaystyle x=\frac{121}{48}[/tex]

There are two different answers that are equivalent but will depend on what you are looking for.

For a fraction answer, your final answer is [tex]\displaystyle \frac{121}{48}[/tex].

For a decimal answer, your final answer is [tex]2.5208\bar3[/tex].

Answer:

CD ≈ 10.8

Step-by-step explanation:

calculate the distance using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1} )^2 }[/tex]

with (x₁, y₁ ) = C (- 5, 4 ) and (x₂, y₂ ) = D (5, 8 )

CD = [tex]\sqrt{(5-(-5)^2+(8-4)^2}[/tex]

     = [tex]\sqrt{(5+5)^2+4^2}[/tex]

     = [tex]\sqrt{10^2+16}[/tex]

     = [tex]\sqrt{100+16}[/tex]

     = [tex]\sqrt{116}[/tex]

     ≈ 10.8 ( to the nearest tenth )

Answer:

8.25 Hours she worked last week

Step-by-step explanation:

The composite discount = 32%

The store offers 15% which  on all items and employees get a 20% discount.

The composite discount formula is:

 -x - y + ( x ×y / 100)

Here x = 15% and y = 20%

So by putting the value of x and y we get

- 15 - 20 + ( 15 × 20 /100)

- 15 -20 + 3

- 35+ 3

-32  

Therefore we get the composite discount i.e., 32%

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The quadratic function of graph given characteristics vertex(1,\9); passes through (4,\6) is y=1/3x2+2/3x+26/3.

A quadratic function is a polynomial function with one or more variables where the highest exponent of the variable is two. Due to the fact that the biggest degree term in a quadratic function is of second degree, it is sometimes referred to as the polynomial of degree 2. There must be a quadratic function with at least one second-degree term. It performs algebraic operations.

Assum:y=ax2+bx+c

Point (1,9)(4,6) into it:{9=a+b+c/6=16a+4b+c} (point 1and 2)

Therefore, vertex(1,9) –b/2a=1→-b=2a

Point 3 into 1 and 2={9=-a+c/b=8a+c}

a=1/3, b=2/3, c=26/3

y=1/3x2+2/3x+26/3

For: y=ax2+bx+c

The graph intersects the x axis at x1 and x2,

X1= -b+√b2-4ac/2a , X2= -b-√b2-4ac/2a

The x-coordinate of the vertex must be in the middle of x1 and x2.

Assume the x-coordinate of the vertex as x

So : x=x1+x2/2 = -b+√b2-4ac/2a + -b-√b2-4ac/2a / 2 = -b/2a

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Answer:

If the state is 0, then Alberto's is -3 and Rebecca's is +2 or 2

The distance between Alberto's and Rebecca's: 2 - (-3) = 2 + 3 = 5

Step-by-step explanation:

If the state is 0, then Alberto's is -3 and Rebecca's is +2 or 2

The distance between Alberto's and Rebecca's: 2 - (-3) = 2 + 3 = 5

After solving the given equation log₃x = 4, the value of x = 81.

So,

Then,

Therefore, after solving the given equation log₃x = 4, the value of x = 81.

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The new coordinates after transformation of the original coordinates using the transformation rule (x, y) → (x, -y) are;

P(6, -4) → P'(6, 4)

Q(2, -2) → Q(2, 2)

R(2, -6) →  R(2, 6)

S(6, -6) → S(6, 6)

The given reflection transformation of (x, y) → (x, -y) simply means a reflection across the x-axis.

Thus, the new coordinates after undergoing the above transformation are;

P(6, -4) → P'(6, 4)

Q(2, -2) → Q(2, 2)

R(2, -6) →  R(2, 6)

S(6, -6) → S(6, 6)

Thus, we conclude that the new coordinates after transformation of the original coordinates using the transformation rule (x, y) → (x, -y) are;

P(6, -4) → P'(6, 4)

Q(2, -2) → Q(2, 2)

R(2, -6) →  R(2, 6)

S(6, -6) → S(6, 6)

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The Magnitude of earthquake that hit Texas is 5.8; With the help of the logarithmic equation so found, we can get the value of M, with which we can get the magnitude of the Texas earthquake.

Given:

To find: Magnitude of earthquake that hit Texas.

Finding:

=>  [tex]\frac{I_2}{I_1}[/tex] =  [tex]30^M[/tex]

For  [tex]\frac{I_2}{I_1}[/tex] = 74900, the above expression becomes:

=> 74900 = [tex]30^M[/tex]

Taking logarithm on both sides, with base 10 gives:

=> log (74900) = log ([tex]30^M[/tex])

As [tex]log_b(a)^m = m(log_b(a))[/tex],

=> log (74900) = M ( log (30))

=> [tex]\frac{log(74900)}{log (30)}[/tex] = M

=> M ≈ 3.3, i.e., the increase in magnitude difference is approximately 3.3.

Thus, the magnitude of the earthquake that hit Texas = 9.1 - 3.3 = 5.8.

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The equation that shows the different and incorrect relationship is C. b = ✓(a² + c²)

It should be noted that from the information the Pythagoras theorem was illustrated and given as c² = a² + b²

Therefore, the equations that can be gotten from it include b² = c² - a².

It should be noted that the equation that shows the different and incorrect relationship is b = ✓(a² + c²)

In conclusion, the correct option is C.

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The unknown angle in the triangle is as follows:

m∠A = 52°

External angle theorem states that If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Therefore,

m∠D = m∠A + m∠B

m∠D = 116°

m∠A =  (2x)°

m∠B = (x + 38)°

Hence,

116 = 2x + x + 38

116 = 3x + 38

3x = 116 - 38

3x = 78

divide both sides by 3

x = 78 / 3

x = 26

Therefore,

m∠A = 2(26)

m∠A = 52°

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