Factorise 10xy-12+15x-8y

Answer:

[tex]\displaystyle \int \dfrac{\ln x}{x^8}\:\:\text{d}x=-\dfrac{\ln x}{7x^7}- \dfrac{1}{49x^7}+\text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given integral:

[tex]\displaystyle \int \dfrac{\ln x}{x^8}\:\:\text{d}x[/tex]

[tex]\textsf{Apply exponent rule} \quad \dfrac{1}{a^n}=a^{-n}[/tex]

[tex]\implies \displaystyle \int x^{-8}\ln x\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Integration by parts} \\\\$\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x=uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x$ \\ \end{minipage}}[/tex]

Using Integration by parts:

[tex]\textsf{Let }u=\ln x \implies \dfrac{\text{d}u}{\text{d}x}=\dfrac{1}{x}[/tex]

[tex]\textsf{Let }\dfrac{\text{d}v}{\text{d}x}=x^{-8} \implies v=-\dfrac{1}{7}x^{-8+1}=-\dfrac{1}{7}x^{-7}=-\dfrac{1}{7x^7}[/tex]

Therefore:

[tex]\begin{aligned}\displaystyle \int u \dfrac{dv}{dx}\:dx & =uv-\int v\: \dfrac{du}{dx}\:dx\\\\\implies \displaystyle \int x^{-8}\ln x\:\:\text{d}x & = -\dfrac{1}{7x^7}\ln x- \int -\dfrac{1}{7x^7} \cdot \dfrac{1}{x}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}+ \int \dfrac{1}{7x^8}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}+ \int \dfrac{1}{7}x^{-8}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}-\dfrac{1}{7 \cdot 7}x^{-8+1}+\text{C}\\\\& = -\dfrac{\ln x}{7x^7}- \dfrac{1}{49x^7}+\text{C}\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $\ln x$}\\\\If $y=\ln x$, then $\dfrac{\text{d}y}{\text{d}x}=\dfrac{1}{x}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

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Substitute [tex]y=\ln(x) \iff e^y = x[/tex] and [tex]dy=\frac{dx}x[/tex] to get

[tex]\displaystyle \int \frac{\ln(x)}{x^8} \, dx = \int ye^{-7y} \, dy[/tex]

Integrate by parts with

[tex]u = y \implies du = dy[/tex]

[tex]dv = e^{-7y} \, dy \implies v = -\dfrac17 e^{-7y}[/tex]

Then

[tex]\displaystyle \int u\,dv = uv - \int v\,du \\\\ \implies \int y e^{-7y} \, dy = -\dfrac17 ye^{-7y} + \frac17 \int e^{-7y} \, dy + C \\\\ ~~~~~~~~~~~~ = -\frac17 ye^{-7y} - \frac1{49} e^{-7y} + C \\\\ ~~~~~~~~~~~~ = \boxed{-\frac{\ln(x)}{7x^7} - \frac1{49x^7} + C}[/tex]

19) The sum of the arithmetic series is - 397. (Correct choice: E)

33) The sum of the geometric series is 1700. (Correct choice: E)

19) Arithmetic series are sets of elements generated by a linear expression of the form:

aₙ = a₁ + (n - 1) · d      (1)

Where:

If we know that a₁ = 27, n = 20 and d = - 5, then sum of the first 20 terms of the series is:

x = 27 + 22 + 17 + 12 + 7 + 2 + (- 3) + (- 8) + (- 13) + (- 18) + (- 23) + (- 28) + (- 33) + (- 38) + (- 43) + (- 48) + (- 53) + (- 58) + (- 63) + (- 68)

x = - 397

The sum of the arithmetic series is - 397. (Correct choice: E)

33) Geometric series are sets of elements generated by a exponential expression of the form:

aₙ = a₁ · rⁿ     (2)

Where:

If we know that a₁ = 1458 and r = 1 / 3, then the sum of the first 6 terms of the geometric series is:

x = 1458 + 162 + 54 + 18 + 6 + 2

x = 1700

The sum of the geometric series is 1700. (Correct choice: E)

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Answer:

Step-by-step explanation:

ggggggg

Answer:

x= -4 + radical 13

Step-by-step explanation:

Answer: [tex]\Large\boxed{f(x-1)=x}[/tex]

Step-by-step explanation:

Given function

f(x + 1) = x + 2

Isolate [ x + 1 ] on the right-hand side

f(x + 1) = x + 1 + 1

f(x + 1) = (x + 1) + 1

Substitute [ x + 1 ] with x

f(x) = x + 1

Substitute [ x ] with [ x - 1 ]

f(x - 1) = (x - 1) + 1

[tex]\Large\boxed{f(x-1)=x}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Answer: 42 and 31

Step-by-step explanation:

       Let x be one number and y be the other.

       We will write mathematical equations for these two numbers.

The sum of 2 numbers is 73.

x + y = 73

Their difference is 11.

x - y = 11

       See attached for the graph.

       The point of intersection is our solution.

a) The coordinates of point A that meet the linear equation y = 2 · x + 7 are A(x, y) = (0, 7).

b) The equation of the line parallel to the line y = 2 · x + 7 is the line y = 2 · x + 3.

c) The coordinates of the midpoint of line segment BC is M(x, y) = (1, 7).

Herein we find three cases related to the linear equation y = 2 · x + 7. a) We need to determine the coordinates of point A, the y-coordinate can be determined by evaluating the function:

y = 2 · 0 + 7

y = 0 + 7

y = 7

The coordinates of point A that meet the linear equation y = 2 · x + 7 are A(x, y) = (0, 7).

b) Two lines are parallel to each other whether both have the same slope but different intercept. Then, the new line is of the form:

y = 2 · x + b

Now we proceed to find the intercept of the line:

b = y - 2 · x

b = 3 - 2 · 0

b = 3

The equation of the line parallel to the line y = 2 · x + 7 is the line y = 2 · x + 3.

c) The coordinates of point C is:

y = 2 · 2 + 7

y = 11

If we know that B(x, y) = (0, 3) and C(x, y) = (2, 11), then the coordinates of the midpoint of BC is:

M(x, y) = 0.5 · B(x, y) + 0.5 · C(x, y)

M(x, y) = 0.5 · (0, 3) + 0.5 · (2, 11)

M(x, y) = (1, 7)

The coordinates of the midpoint of line segment BC is M(x, y) = (1, 7).

The statement presents typing mistakes and omissions, complete form is shown below:

a) The line of y = 2x + 7 meets the y-axis at A, where x = 0. Write down the co-ordinates of A.

b) A line parralel to y = 2x + 7 passes through B(x, y) = (0,3) Find the equation of this line.

c) C is the point on the line y = 2x + 7 where x = 2. Find the the co-ordinates of the midpoint of BC.

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Isolate y

Compare to slope intercept form y=mx+b

Slope=m=5/3

Answer:

[tex]\dfrac{5}{3}[/tex]

Step-by-step explanation:

To find the gradient (slope) of the given linear relation, use arithmetic operations to isolate y:

Given relation:

[tex]3y-5x=10[/tex]

Add 5x to both sides:

[tex]\implies 3y-5x+5x=10+5x[/tex]

[tex]\implies 3y=5x+10[/tex]

Divide both sides by 3:

[tex]\implies \dfrac{3y}{3}=\dfrac{5}{3}x+\dfrac{10}{3}[/tex]

[tex]\implies y=\dfrac{5}{3}x+\dfrac{10}{3}[/tex]

Slope-intercept form of a linear equation

[tex]y = mx+b[/tex]

where:

Comparing the rewritten equation with the slope-intercept formula, the gradient (slope) of the given linear relation is ⁵/₃.

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Answer:

Solution:

here,

6 cans cost = 9 dollars

1 can costs = 9÷6 dollars

= 1.5 dollars

Now,

8 cans cost = 8×1.5 dollars

= 12 dollars

Given

[tex]7 - px - x^2 = 16 - (q + x)^2[/tex]

expanding the right side gives

[tex]7 - px - x^2 = 16 - (q^2 + 2qx + x^2)[/tex]

[tex]7 - px - x^2 = 16 - q^2 - 2qx - x^2[/tex]

Two polynomials of equal degree are the same if their coefficients are identical. This means

[tex]\begin{cases}16 - q^2 = 7 \\ -2q = -p\end{cases}[/tex]

[tex]16 - q^2 = 7 \implies q^2 = 9 \implies q=\pm3[/tex]

[tex]-2q = -p \implies p = 2q = \pm6[/tex]

Both [tex]p[/tex] and [tex]q[/tex] are positive, so [tex]\boxed{p=6}[/tex] and [tex]\boxed{q=3}[/tex].

Step-by-step explanation:

we have

B = J + 2

as the number of laps Bill ran.

to get the number of laps Jill ran we need to transform this equation, so that we get

J = ...

or

... = J

J has to be on one side, and the rest on the other (just as it is with B = ...).

so,

B = J + 2

B - 2 = J

or

J = B - 2

hey ! we are finished already !

The distance between the bird (B) and plane (P) exists at 10,779 feet.

Given the figure attached,

PT = OB = 7000 ft

PB = OT = x feet

and angle of elevation (∠POT) = 30°

By applying the Tan rule in the triangle POT,

tan 33° = Opposite side/Adjacent side = PT/OT

tan 33° = 7000/x

x = 7000/tan33°

x = 10779 feet

The distance between the bird (B) and plane (P) exists at 10,779 feet.

Therefore, the correct answer is option D. 10779 feet.

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The value of x from the given diagram is 20

According to theorem, the right triangle altitude theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse.

Using the theorem above;

RT^2 = 9 * 16

RT^2 = 144

R = 12 units

Determine the value of x using the Pythagoras theorem;

x² =12² + 16²

x² = 144 + 256

x² = 400

Take the square root of both sides

x = √400

x = 20

Hence the value of x from the given diagram is 20

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Answer:

b/a

Step-by-step explanation:

Perpendicular lines have slopes that are opposite sign and reciprocals (flipped over).

In the equation

ax + by = c,

the slope of the line is

-a/b

If you haven't memorized this pattern yet, you can calculate it by solving ax+by=c for y.

by = -ax +c

y = -a/b x + c/b

The slope is -a/b

So a perpendicular line would be opposite sign and flipped, b/a

Using it's concept, the probability of choosing a number that is a multiple of 9 is of 0.111.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

From 1 to 81, there are 81 numbers, of which 81/9 = 9 are multiples of 9, hence, considering that 81 is the number of total outcomes and that 9 is the number of desired outcomes, the probability is given the following division:

p = 9/81 = 1/9 = 0.111.

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Using proportions, it is found that the value of x is of 5.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

In this problem, since Figure A is a scale of Figure B, the side lengths are proportional, hence the following rule of three is established:

x/12.5 = 2/5

Applying cross multiplication:

5x = 25

x = 5.

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Answer:

nope

Step-by-step explanation:

simplify the first equation

Answer:

  yes

Step-by-step explanation:

The associative and commutative properties of multiplication allow you to rearrange the product to the form shown in the question.

Division by 2 is effectively multiplication by 1/2.

  [tex](n\times(n+1))/2\qquad\text{given}\\\\=\dfrac{n(n+1)}{2}=\dfrac{n}{2}(n+1)\\\\=\boxed{(n/2)(n+1)}[/tex]

__

Additional comment

Evaluation of the expressions according to the order of operations will proceed differently for the two expressions.

For the first expression, evaluation steps are ...

For the second expression, evaluation steps are ...

As we said above, the properties of multiplication ensure the results are the same either way.

As a practical matter, for integer values of n, one of n and (n+1) will be even. It is usually convenient to divide the even number by 2. This means the evaluation might be ...

For certain computer representations of the numbers, results may differ depending on the specific numbers and the order of evaluation.

Step-by-step explanation:

1. Entrance of Head of State and IOC President

2. Playing of the national anthem

3. The parade of the athletes

4. The symbolic release of doves

5. Olympic Laurel Award

6. Official Speeches

7. Opening of the Games

8. Raising the Olympic flag and playing the Olympic Anthem

9. Athletes, judges and coaches’ oath

10. Lighting of the Olympic flame

11. The artistic programme

Modern numbers are the ten numerical digits used in constructing other numbers.

Modern numbers are also called Arabic numerals. They are the ten numerical digits:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Uses of modern numbers;

Hence, modern numbers are the ten numerical digits used in constructing other numbers.

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Answer:

2.1 litres

Step-by-step explanation:

the 5 part of the ratio refers to 3.5 litres of red paint

divide this amount by 5 to find the value of one part of the ratio

3.5 litres ÷ 5 = 0.7 litres ← value of 1 part of the ratio , then

3 parts = 3 × 0.7 litres = 2.1 litres ← amount of blue paint used

Answer: 24+12=36

Step-by-step explanation:

36= _+_

2x(2 times)

x

X+2x=36

3x=36

12

12+2(12)=36

12+24=36

Answer:

Step-by-step explanation:

From, the given word explanation try to create your own equation like

From, equation (1) solve for x

Substitute, equation (3) to equation (2).

Answer:

Step-by-step explanation:

Convert 2y=-3x+2 into y=mx+b form

[tex]\frac{2y}{2} =\frac{-3x+2}{2}[/tex]

[tex]y=\frac{-3x}{2} +1[/tex]

Y intercept is 1, slope is -3/2

Graph looks like this:

The perimeter of the triangle is 56.4 units.

The perimeter of the triangle can be found as follows:

The perimeter of a figure is the sum of the whole sides. Therefore, the perimeter of a triangle is the sum of the three sides.

perimeter of a triangle = x + y + z

where

Therefore,

perimeter of the triangle = 19.8 + (9.8 + 8.4) + (8.4 + (19.8 - 9.8))

perimeter of the triangle = 19.8 + 18.2 + 18.4

perimeter of the triangle = 56.4

perimeter of the triangle = 56.4 units

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The correct option regarding the data-sets represented by the box and whisker plots is:

B. The distribution for town A is positively skewed, but the distribution for town B is symmetric.

A box and whisker plot shows three things:

For data-set A, we have that:

Since 30 - 20 > 20 - 15, the distribution is positively skewed.

For data-set B, we have that:

Since 40 - 30 = 30 - 20, the distribution is symmetric.

Hence the correct option is:

B. The distribution for town A is positively skewed, but the distribution for town B is symmetric.

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Answer:3 3/4 meters

Step-by-step explanation:

6/(1 3/5)=3.75 meters or 3 3/4 meters

The correctly matched quadratic functions are given as follows:

Quadratics are polynomial equations of the second degree, which means that they contain at least one squared term.

Quadratic equations are another name for it. The quadratic equation has the following general form: ax2 + bx + c = 0.

Recall that:
f(x) - a (x -c) (x - d)

where

Part A) First graph

We are given to know

The solutions or zeros of the first graph are

x=-1 and x=3

The parabola open up, so the leading coefficient a is positive, the function therefore, is equal to

The vertex is equal to the point (1, -4)

Substitute and solve for a

-4 = a(1-3)(1+1)

-4 = a(-2)(2)

a = 1

Hence, the function is equal to f(x) = (x-3) (x + 1)

Part B) Second Graph

We are also given to know that

The solutions or zeros of the first graph are

x=-3 and x=1

The parabola open down, so the leading coefficient a is negative

The function is equal to f(x) = a (x-1)(x+3)

Find the value of the coefficient a

The vertex is equal to the point (-1,8)

to solve for a, we must substitute:

8 = a(-1-1) (-1+3)

8 = a(-2)(2)

a = -2

Hence, the function is equal to: f(x) = -2 (x - 1) ( x + 3)

Part C)

We are given to know that the solutions or zeros of the first graph are

x=-2 and x=6

The parabola open up, so the leading coefficient a is positive

The function is equal to f(x)  = a(x-6) (x+2).

Find the value of the coefficient a

The vertex is equal to the point (2,-8)

We substitute and solve for a:

-8 = a(2-6) (2+2)

-8 = a(-4)(4)

a = 0.5

Hence, the function is equal to f(x) = 0.5(x - 6) (x + 2)

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The assumption that is necessary before conducting a test of hypothesis is that there is a significant amount of complaints for use of e-tickets.

The assumption is that there is a significant amount of complaints for use of e-tickets.

H0: there is a significant amount of complaints for use of e-tickets.

H1: Number of complaints for use of e-tickets is not significant.

H1: parameter > value

Notice the inequality points to the right

Decision Rule: Reject H0 if test statistic > critical value

Decision Rule: Reject H0 if p -value = 0.05

The hypothetical mean is 15.00 and The actual mean is 13.50

The difference between these two values is -1.50.

The sample size is 20

Mean 13.50

SD 1.50

The test statistic will be -4.459.

Lastly z it's important to reject the null hypothesis since the test statistic is greater than the critical value of 1.7291.

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The behavior sees the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

The behavior of the graph of the polynomial function f(x) as the variable x approaches either positive or negative infinity represents the end behavior of the function. The final behavior of a polynomial function's graph is determined by the degree of the function as well as the leading coefficient.

Generally, the equation for is zeros  mathematically given as

y=x^4−2x^3−11x^2+12x+36

Therefore

(x+2)^2(x-3)^2=0

x=-2

x=3

since the graph attached has a positive leading coefficient and a positive degree

In conclusion, If the leading coefficient is already positive, then increasing the size of the inputs will simply result in the leading term being even more positively skewed. The line on the graph will start to slope to the right.

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