5a. find the value of a.​

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:

  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

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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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:

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 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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The decimal equivalent of 13/10 is 1.3

Decimal:

Decimal is referred as the numbers, which consist of two parts namely, a whole number part and a fractional part separated by a decimal point. For example, 12.5 is a decimal number.

Give,

Here we have the fraction 13/10

And we need to find the decimal equivalent to the fraction.

In the given fraction, we have to use the long division method in order to get the decimal for,

The steps for the long division is as follows:

1) Set up the problem with long division bracket. Put dividend inside bracket and divisor on outside left.

2) 10 goes into 13 (1-times). Put a 1 in the next place of quotient and multiply 10 by 1 to get 10.

Subtract 10 from 13 to get remainder (13-10=3).

3) Now, bring down 0 from the dividend, to make 30. 10 goes into 30 (3-times). Put a 3 in the next place of quotient and multiply 10 by 3 to get 30.

Subtract 30 from 30 to get remainder (30-30=0).

Therefore, the decimal equivalent to 13/10 is 1.3

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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]

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

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]

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)

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:

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:

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.

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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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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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 ).

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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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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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] 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:

g = -5

z = -4

z = 2.1

t = -36

Step-by-step explanation:

-4 = g +1  Subtract 1 from both sides

-5 = g

z - 4 = -8  Add 4 to both sides

z = -4

10.5 = 5z  Divide both sides by 5

2.1 = z

-9 = t/4  Multiply both side sby 4

-36 = t

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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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 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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Answer: (0,0),  (-5,0),  (2,0)

Step-by-step explanation:

c(t)=4t⁴+12t³-40t²

1) t=0

c(t)= 4(0⁴)+12(0³)-40(0²)

c(t)=4(0)+12(0)-40(0²)

c(t)=0+0+0

c(t)=0

Thus (0,0)

2) c(t)=0

4t⁴+12t³-40t²=0

Divide both parts of the equation by 4:

t⁴+3t³-10t²=0

t²(t²+3t-10)=0

t²=0

t=0

Thus (0,0)

t²+3t-10=0

t²+(5t-2t)-10=0

(t²+5t)-(2t+10)=0

t(t+5)-2(t+5)=0

(t+5)(t-2)=0

t+5=0

t=-5

Thus (-5,0)

t-2=0

t=2

Thus (2,0)

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]