Find the standard matrix for the linear transformation t.t(x, y) = (3x 6y, 3x − 6y)

Answer:

[tex]\dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \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 indefinite integral:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x[/tex]

Take partial fractions of the given fraction by writing out the fraction as an identity:

[tex]\begin{aligned}\dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A}{x-4}+\dfrac{B}{x+2}\\\\\implies \dfrac{x+5}{(x-4)(x+2)} & \equiv \dfrac{A(x+2)}{(x-4)(x+2)}+\dfrac{B(x-4)}{(x-4)(x+2)}\\\\\implies x+5 & \equiv A(x+2)+B(x-4)\end{aligned}[/tex]

Calculate the values of A and B using substitution:

[tex]\textsf{when }x=4 \implies 9 = A(6)+B(0) \implies A=\dfrac{3}{2}[/tex]

[tex]\textsf{when }x=-2 \implies 3 = A(0)+B(-6) \implies B=-\dfrac{1}{2}[/tex]

Substitute the found values of A and B:

[tex]\displaystyle \int \dfrac{x+5}{(x-4)(x+2)}\:\:\text{d}x = \int \dfrac{3}{2(x-4)}-\dfrac{1}{2(x+2)}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ax^n\:\text{d}x=a \int x^n \:\text{d}x$\end{minipage}}[/tex]

If the terms are multiplied by constants, take them outside the integral:

[tex]\implies \displaystyle \dfrac{3}{2} \int \dfrac{1}{x-4}- \dfrac{1}{2} \int \dfrac{1}{x+2}\:\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating}\\\\$\displaystyle \int \dfrac{f'(x)}{f(x)}\:\text{d}x=\ln |f(x)| \:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\implies \dfrac{3}{2} \ln |x-4| - \dfrac{1}{2} \ln |x+2| + \text{C}[/tex]

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For an alternative approach, expand and complete the square in the denominator to write

[tex](x-4)(x+2) = x^2 - 2x - 8 = (x - 1)^2 - 9[/tex]

In the integral, substitute [tex]x - 1 = 3 \sin(u)[/tex] and [tex]dx=3\cos(u)\,du[/tex] to transform it to

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \int \frac{3\sin(u) + 6}{9 \sin^2(u) - 9} 3\cos(u) \, du \\\\ ~~~~~~~~~~~~ = - \int \frac{\sin(u) + 2}{\cos(u)} \, du \\\\ ~~~~~~~~~~~~ = - \int (\tan(u) + 2 \sec(u)) \, du[/tex]

Using the known antiderivatives

[tex]\displaystyle \int \tan(x) \, dx = - \ln|\cos(x)| + C[/tex]

[tex]\displaystyle \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C[/tex]

we get

[tex]\displaystyle \int \frac{x+5}{(x - 1)^2 - 9} \, dx = \ln|\cos(u)| - 2 \ln|\sec(u) + \tan(u)| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|\frac{(\sec(u) + \tan(u))^2}{\cos(u)}\right|[/tex]

Now, for [tex]n\in\Bbb Z[/tex],

[tex]\sin(u) = \dfrac{x-1}3 \implies u = \sin^{-1}\left(\dfrac{x-1}3\right) + 2n\pi[/tex]

so that

[tex]\cos(u) = \sqrt{1 - \dfrac{(x-1)^2}9} = \dfrac{\sqrt{-(x-4)(x+2)}}3 \implies \sec(u) = \dfrac3{\sqrt{-(x-4)(x+2)}}[/tex]

and

[tex]\tan(u) = \dfrac{\sin(u)}{\cos(u)} = -\dfrac{x-1}{\sqrt{-(x-4)(x+2)}}[/tex]

Then the antiderivative we found is equivalent to

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = - \ln\left|-\frac{3(x+2)}{(x-4) \sqrt{-(x-4)(x+2)}}\right| + C[/tex]

and can be expanded as

[tex]\displaystyle - \int \frac{x+5}{(x - 1)^2 - 9} \, dx = -\ln\left| \frac{3(x+2)^{1/2}}{(x-4)^{3/2}}\right| + C \\\\ ~~~~~~~~~~~~ = - \ln\left|(x+2)^{1/2}\right| + \ln\left|(x-4)^{3/2}\right| + C \\\\ ~~~~~~~~~~~~ = \boxed{\frac32 \ln|x-4| - \frac12 \ln|x+2| + C}[/tex]

Answer:

4

Step-by-step explanation:

revenue has to be greater than cost for a profit

revenue minus cost = profit

2x^2 – 20x + 70 - (–x^2 + 5x + 18) =

2x^2 – 20x + 70 + x^2 - 5x - 18 =

3x^2 – 25x + 52 ≤ 0

(x - 4) (3x - 13)  ≤ 0

x ≤ 4

x ≤ 13/3

since its multiple choice you can enter one of the answers where the x is to get the answer

if you don't know what the question is and are short on time this may help

again:

revenue has to be greater than cost for a profit

revenue minus cost = profit

cost is 2x^2 – 20x + 70

revenue is –x^2 + 5x + 18

cost is

2(4)^2 – 20(4) + 70 =

32-80+70 =

102-80 =

22

revenue is

–x^2 + 5x + 18

–(4)^2 + 5(4) + 18

-16+20+18

-16+38

22

--------------------------------

cost is

2(5)^2 – 20(5) + 70 =

20

revenue is

–(5)^2 + 5(5) + 18

18

-------------------------------------

cost is

2(20)^2 – 20(20) + 70 =

470

revenue is

–(20)^2 + 5(20) + 18

-282

------------------------------

cost is

2(22)^2 – 20(22) + 70 =

598

revenue is

–(22)^2 + 5(22) + 18

-356

-----------------------------------------

cost is

2(3)^2 – 20(3) + 70 =

28

revenue is

–(3)^2 + 5(3) + 18

24

-------------------------------------

cost is

2(2)^2 – 20(2) + 70 =

38

revenue is

–(2)^2 + 5(2) + 18

24

Answer:

∠JKL = 98

Step-by-step explanation:

the two small vertical lines on JK & KL means that

JK & KL are the same length which means that

same length means same degrees

∠KJL & ∠KLJ are the same degrees

∠KJL & ∠KLJ are both 41 degrees

triangles are 180

41 + 41 = 82

180 - 82 = 98

This is reverse percentage

8.4% = 420

÷8.4 both side

1% is now 50

1% =50

x100

100% is 5000

Check:

8.4/100 = 0.084 our decimal multiplier

5000 x 0.084 = 420

Thus, x is 5000

Hope this helps!

Answer:

C

Step-by-step explanation:

2/5 x1/5 = 2/25  If you divide 2 by 25, you get.08.  To change a decimal into a percent, you move the decimal two places to the right to get 8%

Answer:

x = 1; y = 2

Step-by-step explanation:

y = 3x − 1

y = -4x + 6

In the first equation, y = 3x - 1.

Substitute 3x - 1 for y in the second equation.

3x - 1 = -4x + 6

7x = 7

x = 1

Substitute 1 for x in the first original equation.

y = 3x - 1

y = 3(1) - 1

y = 3 - 1

y = 2

Solution: x = 1; y = 2

Answer:

Twice

Step-by-step explanation:

In a convex hexagon the sum of all interior angle is equal to twice of exterior angles.

PLEASE MARK ME AS BRAINLIST

Answer: c. a-15= 20

Step-by-step explanation:

The answer is c because a represents her age and 15 less than that would mean that you have to subtract 15 from her age which is represented by a, and then 15 less than a would equal 20, so a-15=20

Answer:

Step-by-step explanation:

assuming that the figure is a triangle with sides (5y + 9) and (3y + 15) and (7y + 3) we find the perimeter by adding everything

5y + 9 + 3y + 15 + 7y + 3 =

15y + 27

or

3(5y+9)

next time put all the data and figures, we are doing you a pleasure, at least put ALL the data.

Answer:

x = 6

Step-by-step explanation:

First we divide both sides by 3 :

x² = 108 ÷ 3

x² = 36

Now we square root both sides :

x = √36

x = 6

Hope this helped and have a good day

Answer:

     x = ± 6

Step-by-step explanation:

Quick steps

[tex]3x^{2} =108[/tex]

[tex]x=\sqrt{\frac{108}{3} } =\sqrt{36}[/tex]

[tex]x=[/tex] ±6

A (positive) number has two square roots; one positive and one negative

Hope this helps

The dimensions of a rectangle with area 1728 square meter whose perimeter is as small(minimum) as possible are 41.57m and 41.56m.

Let the dimension of the rectangle be x and y m.

According to the given question.

The area of the rectangle is 1728 square meter.

⇒ x × y = 1728

⇒ x = 1728/y

Since, the perimeter of the rectangle is the sum of the length of its boundary.

Therefore,

Perimeter of the recatngle with dimensions x and y is given by

Perimeter of the rectangle, P = 2(x + y)

⇒ P = 2(1728/y + y)

⇒ P = 3456/y + 2y

Differentiate the above equation with respect to y

⇒ [tex]P^{'} = -\frac{3456}{y^{2} } + 2[/tex]

For the minimum(small) perimeter equate the above equation to 0.

⇒ [tex]-\frac{3456}{y^{2} } + 2 = 0[/tex]

⇒ [tex]2y^{2} =3456[/tex]

⇒ [tex]y^{2} = \frac{3456}{2}[/tex]

⇒ y^2 = 1728

⇒ y = √1728

⇒ y = 41.56 m

Therefore,

x = 1728/41.56

⇒ x = 41.57 m

Hence, the dimensions of a rectangle with area 1728 square meter whose perimeter is as small(minimum) as possible are 41.57m and 41.56m.

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Answer:

-2

Step-by-step explanation:

Every point would flip over the y axis the same distance it is from the y axis now, but just on the other side of the y axis.

In the original equation, when y = 0, x is 2, so when we flip it over the y axis, it is now at -2 when y = 0.

Answer:

Jeffrey has 225 % chance to go to Cafe Shirley.

Step-by-step explanation:

given,

chance of going to Cafe

Chris = 3/4

Nina =3/4

iana = 3/4

To find the total probability you should add all of them:

Total probability= 3/4 + 3/4 + 3/4

= 9/4 = 2.25

to convert to percent you should times 100

2.25 x 100 = 225%//

The height of the tree and the building are 7.6 m and 28.6 m respectively.

An equation is an expression that shows the relationship between two or more variables and numbers.

Two triangles are similar if they have the same shape and the ratio of their corresponding sides are in the same proportion.

Triangle BGD, CHD and AFD are similar triangles.

CH = 1.6, DH = 8, GH = 30,  GF = 105

FH = GF + GH = 105 + 30 = 135

FD = FH + DH = 135 + 8 = 143; GD = 30 + 8 = 38

Hence, using similar triangles:

BG/CH = GD / DH

BG / 1.6 = 38/8

BG = 7.6 m

Also:

building/CH = DF / DH

building / 1.6 = 143/8

building = 28.6 m

The height of the tree and the building are 7.6 m and 28.6 m respectively.

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The Biased samples that are most likely to be biased are as follows;

The residents of every other house on her street.

The people at the computer store.

At the mall every 100th person from the town’s phone registry.

Every 10th person who walks down the main street in her town.

According to the statement

we have given that the:

Maria wants to take a survey to predict who will win the election for her town’s mayor.

So, For this purpose

Biased sample In epidemiology, a sample of a group that does not equally represent the members of the group.

The above people are selected randomly with no specific characteristics, so it is less likely to be biased.

These people are selected according to specific characteristics; Subscribing to the golf magazine (who tends to be rich and biased in politics).

So, it is more likely to be biased.

The Biased samples that are most likely to be biased are as follows

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Answer:

A: The residents of every house on her street

B: The subscribers to a golf magazine

C: The people at the computer store at the mall

Natasha got the power right. The power is 295 hp.

t = 5.825∛w / p

where,

Therefore,

t = 13.4 s

p = 3590

13.4 = 5.825∛ 3590 / p

cube both sides

13.4³ = 5.825³ × 3590 / p

2406.104 = 197.645890625 × 3590 / p

cross multiply

2406.104 p = 709548.747344

p = 709548.747344 / 2406.104

p = 294.895294361

p = 295 hp

Therefore, Natasha got the power right

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It will take the diver 1.5seconds for her to hit the water

Linear equations are equation that has a leading degree of 2. Given the expression that expresses the distance covered by the diver as a function of time as shown;

d(t)= -16(2t - 3)(t+1)

where;

d is the height of the diver above the water and;

t is the time in seconds.

Given the following

d = 0 (the distance on the ground)

Substitute into the formula below;

-16(2t - 3)(t+1)= 0

Divide through by -16

(2t - 3)(t+1) = 0

Determine the time

2t - 3 = 0

t = 3/2

Similarly;

t +1 = 0
t = -1

Hence it will take the diver 1.5seconds for her to hit the water

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Step-by-step explanation:

3. B

because the figure to the left of the y-axis yields the one to the right when you reflect it across y = x.

4. C

Flip either of the figures vertically across x = -1 which is their intersection point and you'll get the other.

5. D

it can't be clockwise because the x value of the vertices would have been negative. so it is 270° counterclockwise resulting in ( -y , x )

6. A

Similar to #4 which except this time, you flip it horizontally like how you lay a book or pick it up.

Answer:

B. Property right are assigned to the party who is being

[tex]\boldsymbol{\sf{Therefore \to \ 5.75\not{h}*\dfrac{60 \ min}{1\not{h}} =345 \ min }}[/tex]

5.75 hours is equal to 345 minutes.

Answer:  48 cm

Step-by-step explanation:

A perimeter(P) of a rectangle is equal to 2(L + W), or 2L + 2W, so we write the equation as:  

1st way:

W = 50%(16) = 0.5 x 16 = 8

   P = 2(L + W)

   P = 2(16 + 8)

   P = 2(24)

   P = 48 cm

2nd way:

   P = 2L + 2W

   P = 2(16) + 2[50%(16)]

   P = 2(16) + 2[0.5 X 16]

   P = 32 + 2(8)

   P = 32 + 16

   P = 48 cm

Answer:

x=-22

Step-by-step explanation:

Let's substitute the number you are thinking of with x

(x+4)*3=(x-5)*2

3x+12=2x-10

x+12=-10

x=-22

Answer:

40

Step-by-step explanation:

7 - 3 + 9 x 8 / 2

Multiplication and division left - right.

7 - 3 + 72 / 2

7 - 3 + 36

Add and subtract left to right.

4 + 36 = 40

The number of servings of pasta were in the dish Lois made is 31.67.

Number of servings of pasta were in the dish Lois made = Number of cups of pasta Lois made / A serving of the pasta

= 6 1/3 ÷ 0.2

= 19/3 ÷ 0.2

= 19/3 × 1/0.2

= (19×1) / (3 × 0.2)

= 19/0.6

= 31.6666666666666

Approximately,

Number of servings of pasta were in the dish Lois made is 31.67 servings

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The number of gigabytes of data that Christopher can use while staying within his budget exists 1.3 gigabytes.

The number of gigabytes of data that Christopher can use while staying within his budget is 1.8 gigabytes.

Based on the information, we get

59 + (3 × g) = 62.90

59 + 3g = 62.90

simplifying the equation, we get

3g = 62.90 - 59

3g = 3.9

Divide both sides by 3, and we get

3g/3 = 3.9/3

g = 1.3

The number of gigabytes of data that Christopher can use while staying within his budget exists 1.3 gigabytes.

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Answer:

Step-by-step explanation:

4x - 10 + x + 7 = 4x

5x - 3 = 4x

-3 = -x

x = 3

4(3) - 10 = 12 - 10 = 2

3 + 7 = 10

10+2 = 12

4(3)= 12