.For any positive integer n, prove that n^3 - n is divisible by 6. ​

The Laplace transform of f(t) = cos² t is equal to the following expression: [tex]\mathcal {L} \{\cos^{2} t\} = \frac{1}{2\cdot s} - \frac{s}{2\cdot (s^{2}+2)}[/tex].

In this we need to rewrite the given function in terms of sines and cosines, whose Laplace transforms are well known. There is the following trigonometric formula:

[tex]\cos ^{2} t = \frac{1-\cos 2t}{2}[/tex]   (1)

Now we proceed to apply the Laplace transforms:

[tex]\mathcal {L} \{f(t)\} = \frac{1}{2}\cdot \mathcal {L} \left\{1 \right\}-\frac{1}{2}\cdot \mathcal {L}\left\{\cos 2t\right\}[/tex]

[tex]\mathcal {L} \{\cos^{2} t\} = \frac{1}{2\cdot s} - \frac{s}{2\cdot (s^{2}+2)}[/tex]

The Laplace transform of f(t) = cos² t is equal to the following expression: [tex]\mathcal {L} \{\cos^{2} t\} = \frac{1}{2\cdot s} - \frac{s}{2\cdot (s^{2}+2)}[/tex]. [tex]\blacksquare[/tex]

The statement is poorly formatted, the correct form is shown below:

Find [tex]\mathcal {L}\{f(t)\}[/tex] by first using a trigonometric expression. (Write your answer as a function of s). f(t) = cos² t

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Answer: Option (3)

Step-by-step explanation:

You can get 2 pairs of corresponding angles from the parallel segments, meaning that they are similar by angle-angle.

Answer:

1. perimeter =2(l+w)

                    =2(17+10)

                   = 54 yards

Are of the perimeter=l times b

                                =17 x 10

                                =170 yd square

Just follow this concept for all of them. It the same thing. Good Luck!!

Step-by-step explanation:

Answer:

minimum value is: the left whisker

Step-by-step explanation:

box an whisker plot: left line is the minimum, first part of the box is Q1, where the box has a middle line it is called Median, the other end of the box is Q3, the right line is called the maximum whisker.

Hope this helps

Answer:

Jim travelled 246 Miles.

Step-by-step explanation:

Answer:

94%

Step-by-step explanation:

400-24=376

Find 376 out of 400 as a decimal:

376/400 = 0.94

0.94x100 = 94%

Answer:

x = 0

Step-by-step explanation:

5^0 is 1, meaning that x must equal to 0 for the answer to be 5^0.

Answer:

811

Step-by-step explanation:

i d k

Answer:

Step-by-step explanation:

area of A=15*18=270 cm^2

area of B=15*18=270 cm^2

area of C=1/2 ×x×18=9x cm²

area of D=1/2×15×18=135 cm²

Answer: D

Step-by-step explanation:

Answer:

(i) no correlation as one variable is not dependent on the other.

(ii)  The age of a car does affects its value.  Therefore, the age will be the independent variable (x-axis) and the value will be the dependent variable (y-axis).  Unless the car is a classic, sought-after car, as the age increases, its value decreases, so this will be a negative correlation.

(iii)  The time spent watching TV may affect exam score (since the more time spent watching TV equals less available time to spend studying).  Therefore, the time spent watching TV will be the independent variable (x-axis) and the exam score will be the dependent variable (y-axis).  As time spent watching TV increases, the exam score could potentially decrease, so this will be a negative correlation.

(iv)  no correlation, since one variable is not dependent on the other.

(v)  The cost of an apartment may affect its sale, but there are many factors that can affect this such as location of the apartment and the buyers it appeals to.  The cost will be the independent variable (x-axis) and the sale will be the dependent variable (y-axis).  In usual circumstances, we would expect to see as the cost of an apartment increases, the sale may decrease, as less and less buyers can afford to buy it, so this will be a negative correlation.

Answer:

15- 12.27 = 2.73

2.73 is how much she spent on the ribbon.

2.73/ 0.21 = 13

She purchased 13 yards.

Answer: (4,6)

Step-by-step explanation:

We are splitting the line segment into 6 parts, so we can start by finding the displacement in the x and y coordinates.

The change in x is 12, and the change in y is 12.

This means that for each of the 6 parts we are splitting the segment into, the value of x will change by 2, and so will y.

This means that to find K, we can subtract one of these displacements from point L.

This means that K has coordinates (6-2, 8-2)=(4,6)

volume : 7.5 cm²

[tex]\hookrightarrow \sf Length \ * Width \ * \ Height[/tex]

[tex]\hookrightarrow \sf 2 \ * 1\dfrac{1}{4} \ * \ 3[/tex]

[tex]\hookrightarrow \sf 2 \ * \dfrac{5}{4} \ * \ 3[/tex]

[tex]\hookrightarrow \sf \dfrac{15}{2}[/tex]

[tex]\hookrightarrow \sf 7.5 \ cm^2[/tex]

Answer:

[tex]\sf volume=7 \frac12 \ cm^3[/tex]

Step-by-step explanation:

Formula

volume of cuboid = length × width × height

Given dimensions of the ice cube:

Substituting the given values into the formula:

[tex]\sf \implies volume=2\times 1 \frac14 \times 3[/tex]

Converting to improper fractions:

[tex]\sf \implies volume=\dfrac21\times \dfrac54 \times \dfrac31[/tex]

                  [tex]\sf =\dfrac{ 2\times 5 \times 3}{1 \times 4 \times 1}[/tex]

                  [tex]\sf =\dfrac{30}{4}[/tex]

Dividing the numerator and denominator by the highest common factor:

[tex]\sf \implies volume=\dfrac{30 \div 2}{4 \div 2}[/tex]

                  [tex]\sf =\dfrac{15}{2}[/tex]

Converting the improper fraction into a mixed number:

[tex]\sf \implies volume=7 \frac12 \ cm^3[/tex]

Answer:

5/32

Step-by-step explanation:

Answer:

70 dolls

Step-by-step explanation:

Hello!

The ratio is 5:2, which are division of  a whole. They can be represented as 5x and 2x.

Jacky has 5x dolls, and Peter has 2x dolls. They were equal after subtracting 15 dolls from 5x and adding them to 2x.

Equation:

Solve:

So Jacky has 5(10) dolls, or 50 dolls, and Peter has 2(10) or 20 dolls.

The total sum is 70 dolls.

Answer:

[tex]w=8[/tex]

Step-by-step explanation:

[tex]2(2w-16)=0\\4w-32=0\\4w=32\\w=8[/tex]

Answer:

w = 8

Step-by-step explanation:

2(2w − 16) = 0

Divide each side by 2

2w-16 = 0

Add 16 to each side

2w-16 +16 =0 +16

2w = 16

Divide each side by 2

2w/2 = 16/2

w = 8

Answer:

[tex]\frac{\sqrt{2} }{2}[/tex]

Step-by-step explanation:

Simplify the expression:

sin [tex]\frac{3pi}{4}[/tex]

Rewrite the angle as a sum or difference with its reference angle:

sin (pi - [tex]\frac{pi}{4}[/tex])

Simplify the expression

(sin [tex]\frac{pi}{4}[/tex])

Evaluate the expression:

[tex]\frac{\sqrt{2} }{2}[/tex]

hope this helps and is right!! p.s. i really need brainliest :)

Answer:

64

Step-by-step explanation:

[tex]\sqrt{256}[/tex] = 16 (length of each side)

A square (rug) has 4 sides.

Add to find perimeter:

16 + 16 + 16 + 16

= 64

Answer: 8ft

Step-by-step explanation:

4A= 256

4A/4     =    256/4

A = 64

A = a²

64 = 4²

a² = 64

a = ±√64

a = ±8

∴a = ±8

Hence, the length of a side of each rug is 8ft.

Answer:

-4.1-4.1212

Step-by-step explanation:

Log2^2-log 2^16

log2^2/2^16

log2^2-16

log2^-14

-14log2

-14×0.3010

-4.1-4.1214

Here the common ratio is -5

next terms will be 375 and -1875

Answer:

The answer is 2

-16/-8=2

Answer:

x is 19.5 degress

Step-by-step explanation:

hmmm so here oh i see now okay:

since a line is 180 degress lets write

2x + 82.5 + 3x = 180

now we umm add the ones with the same variable and power

5x + 82.5 = 180

5x = 180 - 82.5

5x = 97.5

x = 19.5

This is an angle on a straight line and It is equal to 180°

2x° + 3x° + 82.5° = 180°

5x° = 180° - 82.5°

5x = 97.5°

x = 19.5°

To indicate the angle

2x => 2(19.5°) => 39°

3x => 3(19.5°) => 58.5°

The angle are (39°, 82.5°, 58.5°

Answer:

14.6

Step-by-step explanation:

The surface area of the box is 1048 square inches

The formula for calculating the surface area of the prism is expressed as:

Surface area = 2(lw + wh +lh)

Given the following

l = 20in

w = 8in

h = 13in

Substitute the given parameters'

Surface area = 2(lw + wh +lh)

Surface area = 2(20(8) + 8(13) +20(13))

Surface area = 2(160+104+260)

Surface area = 1048 square inches

Hence surface area of the box is 1048 square inches

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Step-by-step explanation:

using cosine rule,

BC²= AC² + BC² - 2*AC*BC*COSc

= 13² + 6² - 2*13*6*-cos(180-91)

= 169 + 36 - (-2.72)

= 202.28

√BC² = √202.28

BC = 14.22

BC = 14.20cm to nearest tenth

Answer:

$31.05

Step-by-step explanation:

You can get this answer by changing percent (13.5%) to just 13.5, or in other words, a decimal. you also change 230. to a decimal by by moving the decimal point two times to the left to get 2.30. You can then get your answer by multiplying the values together (2.30 x 13.5)

Answer:

1. any number from 42 to 68

e.g 52

2. any number less than 42 or greater than 68

e.g 76