What is a graph theory in math?

Step-by-step explanation:

the slope-intercept form is

y = ax + b

with "a" being the slope, and "b" being the y-intercept (the y- value when x = 0).

so, we have

y = x - 8

⇒The quadratic equation has already been factorized in such a way that the factors are equal to 0 ∴ they won`t be an need to multiply out, since you would still have to come back to the factorizing part.

⇒Let us solve for x now.

[tex]\sqrt{x} +2 =0\\\sqrt{x} =-2\\No\\ solution[/tex]

⇒There is no way that a square root of a number can result in a negative number... that is why the answer was a no solution.

[tex]\sqrt{x} -2=0\\\sqrt{x} =2\\(\sqrt{x} )^{2} =(2)^{2} \\x=4[/tex]

⇒The solution to the quadratic equation is only and only 4.

Answer:36

Step-by-step explanation:

Money is anything generally accepted as final payment for goods, services, and debt.

First, let us understand the money supply:

The money supply is the total amount of currency and liquid assets in a country's economy on the day measured. Cash and deposits that can be utilized virtually as quickly as cash are roughly included in the money supply.

Money supply refers to the total amount of money held by the public for a particular point.

Governments print paper money and coins through a combination of central banks and treasuries. Bank regulators impact the money supply available to the public by requiring banks to retain reserves, determining how to grant credit, and other monetary matters.

We can say that we are using the money for goods purchasing, service requirements, and for the debt taken.

Thus, money is anything generally accepted as final payment for goods, services, and debt.

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Answer:

[tex] x =\dfrac{7+\sqrt{41}}{4},\dfrac{7-\sqrt{41}}{4} [/tex]

Step-by-step explanation:

The given equation is ,

[tex]\longrightarrow 7x = 2x^2+1[/tex]

We can rewrite it as ,

[tex]\longrightarrow 2x^2-7x + 1=0 [/tex]

On comparing with the standard form of quadratic equation [tex] ax^2+bx + c [/tex] , we have ;

On using the quadratic formula , we have;

[tex]\longrightarrow x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\\[/tex]

[tex]\longrightarrow x =\dfrac{-(-7)\pm\sqrt{(-7)^2-4(2)(1)}}{2(2)} \\[/tex]

[tex]\longrightarrow x =\dfrac{7\pm\sqrt{49-8}}{4}\\ [/tex]

[tex]\longrightarrow x =\dfrac{7\pm\sqrt{41}}{4}[/tex]

separate the solutions ,

[tex]\longrightarrow x =\dfrac{7+\sqrt{41}}{4},\dfrac{7-\sqrt{41}}{4} [/tex]

And we are done !

Answer:

The length of the straw in the simplest radical form is [tex]2\sqrt{30}[/tex] inches

Step-by-step explanation:

The length of the straw can be calculated as follow

The first step is to calculate the diagonal of the base of the rectangular box  by using Pythagorean theorem

[tex]c^{2}=a^{2}+b^{2}[/tex]

Where [tex]c[/tex] is the base diagonal or the hypotenuse, [tex]a[/tex] is the short leg, [tex]b[/tex] is the long leg

Substitute [tex]2[/tex] for [tex]a[/tex], [tex]4[/tex] for [tex]b[/tex] in the Pythagorean theorem

[tex]c^{2}=2^{2}+4^{2}[/tex]

Simplify the right side of the equation

[tex]c^{2}=4+16[/tex]

[tex]c^{2}=20[/tex]

Take the square root of the right side to get the value of [tex]c[/tex]

[tex]c=\sqrt{20}[/tex]

Factor [tex]20[/tex] by [tex]4[/tex]

[tex]c=\sqrt{4\times 5}[/tex]

Substitute [tex]2^{2}[/tex] for 4

[tex]c=\sqrt{2^{2}\times 5}[/tex]

Rewrite the value of [tex]c[/tex] in the simplest radical form

[tex]c=2\sqrt{5}[/tex]

Since the value of [tex]c[/tex] is [tex]2\sqrt{5}[/tex], then the length of the diagonal of the base is

[tex]2\sqrt{5}[/tex] inches

The second step is to calculate the length of the straw by using Pythagorean theorem

[tex]c^{2}=a^{2}+b^{2}[/tex]

Where [tex]c[/tex] is the diagonal from the bottom left corner to the top right back corner or the hypotenuse, [tex]a[/tex] is the short leg, [tex]b[/tex] is the long leg

Substitute [tex]2\sqrt{5}[/tex] for [tex]a[/tex], [tex]10[/tex] for [tex]b[/tex] in the Pythagorean theorem

[tex]c^{2}=(2\sqrt{5})^{2}+10^{2}[/tex]

Simplify the right side of the equation

[tex]c^{2}=20+100[/tex]

[tex]c^{2}=120[/tex]

Take the square root of the right side to get the value of [tex]c[/tex]

[tex]c=\sqrt{120}[/tex]

Factor [tex]120[/tex] by [tex]4[/tex]

[tex]c=\sqrt{4\times 30}[/tex]

Substitute [tex]2^{2}[/tex] for 4

[tex]c=\sqrt{2^{2}\times 30}[/tex]

Rewrite the value of [tex]c[/tex] in the simplest radical form

[tex]c=2\sqrt{30}[/tex]

Since the value of [tex]c[/tex] is [tex]2\sqrt{30}[/tex], then the length of the diagonal from the bottom left corner to the top right back corner or the hypotenuse is

[tex]2\sqrt{30}[/tex] inches

Since the length of the diagonal from the bottom left corner to the top right back corner is [tex]2\sqrt{30}[/tex] inches, then the length of the straw is [tex]2\sqrt{30}[/tex] inches

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Answer:

  common denominator = 2·3·5

Step-by-step explanation:

You want the factorization of the common denominator of 2/3+7/10.

The denominators of the fractions in the given sum are 3 and 10. Their prime factors are ...

  3 = 3

  10 = 2·5

The common denominator factors will be the product of the unique factors in the above list:

  common denominator = 2·3·5

__

Additional comment

Using the common denominator, the sum is ...

  20/30 +21/30 = 41/30 = 1 11/30

Answer: [tex]\frac{11}{5}[/tex]

Step-by-step explanation

(y2-y1)/(x2-x1)

which means (5-(-6))/-4-(-9)=[tex]\frac{11}{5}[/tex]

Answer:

X-intercept: (6, 0)

Y-intercept: (0, -9)

Step-by-step explanation:

Required: Find the X and Y intercept of the standard from equation:

-3x + 2y = -18.

We shall write equation in the form [tex]\frac{x}{a} +\frac{y}{b} =1[/tex] and find the intercepts on X and Y axis:

-3x + 2y = -18

∴ -3x + 2y = -18

∴ -3x/-18 + 2y/-18 = 1

∴ x/6 + y/-9 = 1

Comparing this equation with [tex]\frac{x}{a} +\frac{y}{b} =1[/tex], we get:

intercept of X-axis = a = 6

intercept of Y-axis = b = -9

Thanks.

Answer:

The y-intercept for the line is ( 0, -9 ).

Hope this helps!

Step-by-step explanation:

The y-intercept is where the graph intercepts the y-axis and is the y-value of the point, ( x, y ).

Using implicit differentiation, the value of y in terms of x and y is; y = 0.5/(dy/dx)

In implicit differentiation, what we do is that we differentiate each side of an equation using the two variables (usually x and y) by treating one of the variables to be as a function of the other.

We are given the equation;

4x = 4y² + 1

Thus, first differentiation gives;

4 = 8y(dy/dx)

y(dy/dx) = 0.5

y = 0.5/(dy/dx)

This is the result when using implicit differentiation on the given function where y is the subject of the formula.

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The gradient of the function at the point (-1 -7) is 3a - 6.

What is the gradient of the function?

A function's gradient is also known as its slope, and the slope (of a tangent) at a given point on a function is known as its derivative.

We have,

g(x) = ax³ + 3x² + bx + c

Take the derivative of the function with respect to x, then substitute the x-coordinate of the point of interest in for the x values in the derivative to find the gradient.

to know the gradient of the function ax³ + 3x² + bx + c at the point (-1, -7) we would do the following:

Take the derivative with respect to x:

g'(x) = 3ax² + 6x + b

Substitute the x-coordinate (x = -1)in for x:

gradient = 3a(-1)² + 6(-1) + b = 3a - 6

So the gradient of the function at the point (-1 -7) is 3a - 6.

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Over 48.6 hours are worked by 2.28 percent of faculty members each week.

Given that,

In a college, a full-time faculty member typically puts in 43 hours each week.

We have to find how many faculty members work more than 48.6 hours each week, assuming the standard deviation is 2.8 hours.

The Z-score determines the measure's standard deviations away from the mean. After calculating the Z-score, we look at the p-value associated with it in the z-score table. This p-value is the probability that the measure's value is smaller than X, or the percentile of X. By subtracting 1 from the p-value, one can determine the likelihood that the value of the measure is greater than X.

In a college, a full-time faculty member typically puts in 43 hours each week. 2.8 hour standard deviation.

This means μ=43 and σ=2.8

What proportion of academic staff employees put in more than 58.6 hours each week.

When X = 48.6, the proportion is calculated by deducting 1 from the p-value of Z.

So,

Z=X-μ/σ

Z=48.6-43/2.8

Z=5.6/2.8

Z=2

Z=2 has a p-value of 0.9772

1 - 0.9772 = 0.0228

0.0228*100% = 2.28%

Therefore, Over 48.6 hours are worked by 2.28 percent of faculty members each week.

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The EF is the diameter. Then the area of the circle is  113.04  square meters.

circle :

A circle is a circular shape without corners or line segments. In terms of geometry, it has the shape of a closed curve.

EF = diameter

EF = 12 meter

then, radius  = d/2 = 12/2 = 6 meter

Area of the circle is calculated  as:

area = πr²

area = 3.14 x 6 x 6

area = 113.04 square meter

The given que is incomplete . The correct question is given as :

The EF is the diameter. If EF measures 12 meters, the approximate area of circle B is square meters.

The EF is the diameter. Then the area of the circle is  113.04  square meters.

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The area between z = -1.71 and z = 0.08 is 0.4883, in median .

What is median ?

The value of the middle observation found after the data are arranged in ascending order is referred to as the median of the data. In many cases, it is challenging to take all of the facts into account when representing something, and in these cases, median is helpful. The median is one of the simple to compute statistical summary metrics. The median is also known as the place average since it uses the data that is in the centre of a sequence to determine its value.

=> P(-1.71 < Z < 0.08 ) = P(Z < 0.08 ) - P(Z < -1.71)

= 0.5319 - [1 - P(Z < 1.71)]

= 0.5319- [1 - 0.9564]

= 0.4883

Hence, the area between z = -1.71 and z = 0.08 is 0.4883.

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Answer:

Step-by-step explanation: formula is (y2 - y1) / (x2 - x1)

so in this case it is: (4-(-4) / (-8 - 2) -> 8/-10 -> -4/5(final solution)

The graph of the function is increasing when x>0.

What is graph?

Graph a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

What is cube root parent?

The basic parent cube root function is of the form f (x) = ∛x. It increases on the interval (-∞, ∞). It increases on the interval [0, ∞). It is positive on (0, ∞) and negative on (-∞, 0). It is a non-negative function always (on [0, ∞)). Its absolute min is 0 but no absolute max. Its domain and range is the set of all real numbers.

Given,

The function f(x) = ∛x

Let x = 1, we get f(x) = 1,

x =2, f(x) = 8

if x =3, f(x) = 27.

As we know, with increase in the value of x, the value of f(x) is also increasing.

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The first term [tex]a_{1}[/tex] = 2 and the sum of first 30 terms [tex]s_{30}[/tex] = 930

Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value.

Arithmetic sequence is gotten by adding a known term by a constant value.

if [tex]a_{33}[/tex] = 66 and d = 2 and [tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n -1)d and n = 33

Then

[tex]a_{33}[/tex] =  [tex]a_{1}[/tex] + 32d,

making  [tex]a_{1}[/tex] subject of the equation we have

[tex]a_{1}[/tex] =66 - 32 x 2

[tex]a_{1}[/tex] = 66 - 64

[tex]a_{1}[/tex] = 2

if n = 30,  [tex]a_{1}[/tex] =-2, d = 2

Then

[tex]a_{30}[/tex] = 2 + ( 30 - 1) x 2

[tex]a_{30}[/tex] = 2 + 29 x 2

[tex]a_{30}[/tex] = 60

if [tex]s_{n}[/tex] = n/2([tex]a_{1}[/tex] + [tex]a_{n}[/tex])

Then,

[tex]s_{30}[/tex] = n/2 ( [tex]a_{1}[/tex] + [tex]a_{30}[/tex]) where n = 30

[tex]s_{30}[/tex] = 30/2( 2 + 60)

[tex]s_{30}[/tex] = 930

In conclusion, the sum of he first 30 terms is 930 and the first term is 2

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The logarithm function can be used to convert large problem of multiplication and division into simple addition or subtraction. The value of x for the given equation is 2.

Logarithm function is defined as the inverse of the exponential function.

If a^(b) = c, then b = log(c) at the base of a.

The given equation is as below,

2^x=x^2

Take logarithm both sides,

x log(2) = 2 log(x)

=> x / log(x) = 2 / log(2)

Compare the denominator and numerator at both sides of the equation to get,

x = 2.

Hence the solution of the given equation is x = 2.

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Answer:

(32, 8)

Step-by-step explanation:

Substitute 4y for x in x + y = 40 and solve for y. Then find x.

[tex]4y + y = 40[/tex]

[tex]5y = 40[/tex]

[tex]y = 8[/tex]

[tex]x = 4(8) = 32[/tex]

Step-by-step explanation:

3x + 2y = -18

Set y = 0

3x + 2 × 0 = -18

3x + 0 = -18

3x = -18

x = -6

.: the x-intercept = -6

You can choose to put it in coordinate form (-6, 0).

[tex]Answer: \boxed {(-6,0)}[/tex]

Step-by-step explanation:

The x-intersection coordinate of the equation 3x+2y=-18 is (x,0)

Hence, y=0

3x+2(0)=-18

3x+0=-18

3x=-18

Divide both parts of the equation by 3:

x=-6

Thus, (-6,0)

Check the picture below.

[tex]\textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)\cos(C)\implies c = \sqrt{a^2+b^2-(2ab)\cos(C)} \\\\[-0.35em] ~\dotfill\\\\ b = \sqrt{7^2+11^2~-~2(7)(11)\cos(92^o)} \\\\\\ b = \sqrt{ 170 - 154 \cos(92^o) }\implies b\approx 13.24~km[/tex]

now, to find the bearing from C to A hmmm, let's use the Law of Cosines again, to find the angle at A, once we get that, we'll simply add 37 to it :)

[tex]\textit{Law of Cosines}\\\\ \cfrac{a^2+b^2-c^2}{2ab}=\cos(C)\implies \cos^{-1}\left(\cfrac{a^2+b^2-c^2}{2ab}\right)=\measuredangle C \\\\[-0.35em] ~\dotfill\\\\ \cos^{-1}\left(\cfrac{7^2+13.24^2-11^2}{2(7)(13)}\right)\approx\measuredangle A \implies \cos^{-1}\left(\cfrac{ 103.3 }{ 185.36}\right)\approx\measuredangle A \\\\\\ 56.13^o\approx \measuredangle A\hspace{15em} \underset{\textit{Bearing of C from A}}{\stackrel{56.13~~ + ~~37}{\boxed{\approx 93.13^o}}}[/tex]

An adjacency matrix is defined as the square matrix that is used to describe the finite graph

An adjacency matrix is defined as the square matrix that is used to describe the finite graph which is also an example of graph theory.

The adjacency matrix is made up of rows and columns which is used to represent a simple labelled graph. it is also called connection matrix.

The properties of an adjacency matrix include the following:

The adjacency matrix can be used to represent indirect or direct graph.

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An adjacency matrix is a square matrix used to describe the finite graph.

An adjacency matrix is a square matrix used to describe a finite graph, which is also a kind of graph theory.

The adjacency matrix, which consists of rows and columns, is used to depict a basic labeled graph. It is also known as a connection matrix. The adjacency matrix can represent either an indirect or direct graph.

An adjacency matrix has the following properties:

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The arc length of the given curve is L= 1/2([tex]e^{4} - \frac{1}{e^{4} }[/tex]).

What is arc length?

The distance between two places along a segment of a curve is known as the arc length.

Main Body:

So, we have [tex]y=\frac{1}{2} (e^{x} +e^{-x} )[/tex]

This is in the form y=f(x), so we know arc length L on [0,4] is given by,

L=[tex]\int\limits^4_0{\sqrt{1+(\frac{dy}{dx} })^{2} \, dx[/tex]

The derivative y=

dy/dx= [tex]\frac{1}{2} (e^{x} +e^{-x} )[/tex]= sinhx

([tex]\frac{dy}{dx}^{2}[/tex])= sinh²x

L=[tex]\int\limits^4_0{\sqrt{1+sinh^{2}x \, dx[/tex]

L= [tex]\int\limits^4_0{\sqrt{cosh^{2}x \, dx[/tex]

L=[tex]\int\limits^4_0{coshx \, dx[/tex]

Now putting the limits as given=

L= [tex]sinhx\left \{ {{y=4} \atop {x=0}} \right.[/tex]

L= [tex]\frac{1}{2} (e^{x} -e^{-x} )[/tex]|⁴₀

L=1/2([tex]e^{4} - \frac{1}{e^{4} }[/tex])

Hence the answer is  1/2( [tex]e^{4} - \frac{1}{e^{4} }[/tex]).

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The other three length are 9km,12km,14km as mode is 12km

What is mode?

The value that appears most frequently in a set of values is referred to as the mode. It is the value that shows up the most frequently. The value that consistently appears in a given set is known as the mode in statistics. The mode or modal value is the value or number that appears most frequently in a data set and has a high frequency. Along with mean and median, there are three other ways to measure central tendency.

The mean, median, and mode all have the same values for this distribution. This indicates that this value—the one that appears the most frequently in the data—is the average, the centre, and the mode.

Four lengths have a mode of 12 km and a range of 5 km.

The range is the largest number minus the smallest number.

As mode is 12 so one more number is 12 as well.

Now another number given 14km, so the last number is (14-5 = 9)

Hence the other three lengths are 9km,12km,14km.

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Answer:

∴ x = 6.40 units (to 3sf)

Step-by-step explanation:

Following the Pythagoras' Theorem,

 x² = 4² + 5²

∴ x = 6.40 units (to 3sf)

Answer: [tex]\frac{\ln(241/200)}{4}[/tex]

Step-by-step explanation:

If [tex]P(t)[/tex] represents the number of bacteria after [tex]t[/tex] hours, then [tex]P(t)=2400e^{kt}[/tex], where [tex]k[/tex] is the hourly growth rate parameter.

Using the fact that [tex]P(4)=2892[/tex],

[tex]2892=2400e^{4k}\\\\e^{4k}=\frac{241}{200}\\\\4k=\ln(241/200)\\\\k=\frac{\ln(241/200)}{4}[/tex]

Answer:

f(-7) = 18

f(-3) = 14

f(4) = 7

Step-by-step explanation:

f(-7)

l-7-8l + 3

l-15l + 3

15 + 3

18

F(-3)

l-3-8l + 3

l-11l + 3

11 + 3

14

F(4)

l4-8l + 3

l-4l + 3

4 + 3

7

Step-by-step explanation:

Using Triangle Intersection Theorem,

2(LM)=2+x

[tex]2(2x - 11) = 2 + x[/tex]

[tex]4x - 2 2 = 2 + x[/tex]

[tex]3x = 24[/tex]

[tex]x = 8[/tex]

Answer:

the interior angles of a triangle add to be 180 degrees.

the value of angle x = 36 degrees

the value of angle y = 84 degrees

The required implicit differentiation is given as dy/dx = -4x/3.

Given that,
To determine the implicit differentiation of the given function 4 = 2x² + 3y.

The method of determining the derivative of an implicit function by differentiating each term separately, expressing the derivative of the dependent variable as a symbol, and solving the resulting expression for the symbol

Here,
4 = 2x² + 3y
Differentiate with respect to x on both sides,
d4/dx = d[2x²]/dx + 3dy/dx
0 = 4x + 3dy/dx
dy/dx = -4x/3

Thus, the required implicit differentiation is given as dy/dx = -4x/3.

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