Complete the left-hand column of the table below following the steps indicated in the right-hand column to show that "sin a sin b = 1/2[cos(a-b)-cos(a+b)]" is an identity. Use the definitions of sum and difference formulas for cosine.

When [tex]n=0[/tex], we have

[tex]5^{3^0} + 1 = 5^1 + 1 = 6[/tex]

[tex]3^{0 + 1} = 3^1 = 3[/tex]

and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)

Suppose this is true for [tex]n=k[/tex], that

[tex]3^{k + 1} \mid 5^{3^k} + 1[/tex]

Now for [tex]n=k+1[/tex], we have

[tex]5^{3^{k+1}} + 1 = 5^{3^k \times 3} + 1 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k}\right)^3 + 1^3 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k} + 1\right) \left(\left(5^{3^k}\right)^2 - 5^{3^k} + 1\right)[/tex]

so we know the left side is at least divisible by [tex]3^{k+1}[/tex] by our assumption.

It remains to show that

[tex]3 \mid \left(5^{3^k}\right)^2 - 5^{3^k} + 1[/tex]

which is easily done with Fermat's little theorem. It says

[tex]a^p \equiv a \pmod p[/tex]

where [tex]p[/tex] is prime and [tex]a[/tex] is any integer. Then for any positive integer [tex]x[/tex],

[tex]5^3 \equiv 5 \pmod 3 \implies (5^3)^x \equiv 5^x \pmod 3[/tex]

Furthermore,

[tex]5^{3^k} \equiv 5^{3\times3^{k-1}} \equiv \left(5^{3^{k-1}}\right)^3 \equiv 5^{3^{k-1}} \pmod 3[/tex]

which goes all the way down to

[tex]5^{3^k} \equiv 5 \pmod 3[/tex]

So, we find that

[tex]\left(5^{3^k}\right)^2 - 5^{3^k} + 1 \equiv 5^2 - 5 + 1 \equiv 21 \equiv 0 \pmod3[/tex]

QED

Answer:

a. 13/26 = 1/2

b. 3/13

c. 2/26 = 1/13

Step-by-step explanation:

A. 26 red cards and 13 of them are hearts so 13/26 which can be simplified to 1/2

B. 13 hearts but only jack, queen king are higher than 10 so 3/13

C. 26 red cards altogether and we have a total of 2 red jacks so 2/26 which can be simplified to 1/13

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Probability someone chosen at random will not have brown hair = 1/6

The number of girls = 15

The number of boys = 15

The total number of children = 15 + 15 = 30

Probability that someone chosen at random will have a brown hair, P(Brown hair) = 5/6

P(Brown hair) + P(Not Brown hair) = 1

5/6  +  P(Not brown hair) = 1

P(not brown hair) = 1 - 5/6

P(not brown hair) = 1/6

Therefore, if someone is chosen at random, probability that they will not have brown hair = 1/6

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The large cone can hold more ice cream than the four smaller cones combined.

The large cone is a better deal.

A cone is a three-dimensional structure with a round base and a point at the top, known as the vertex.

A cone is a three-dimensional solid geometric object with a point at the top and a circular base. A cone has a vertex and one face. For a cone, there are no edges.

The radius, height, and slant height of the cone are its three constituent parts.

For the supersized cone,

Volume = π(5)(5)(12)/3 = 314.29 cubic inches

For a small cone,

Volume= π(2.5)(2.5)(6)/3 = 39.29 cubic inches

Combined volume of 4 such cones = 4*39.29 = 157.16 cubic inches

As the volume of the larger cone is more, thus it holds more ice cream. Also it is a better deal, as it costs less than 4 small cones.

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The quantity of product that would result in the same profit is 1 thousand. Option A

Given the polynomial function as;

P(t) = –2t³ + 8t² + 2t

Where;

Let's equate the values

–2t³ + 8t² + 2t = 8

–2t³ + 8t² + 2t - 8 = 0

Factorize the expression

(-2t³ + 8t²) + ( 2t -8) = 0

- t² ( 2t - 8) + (2t -8) = 0

We take the similar factor

2t - 8 = 0

2t = 8

t = 8/2

t = 4

But we have that the company produced 4 thousand products for $8 million profit, so we have

= 4/ 4

= 1 thousand

Thus, the quantity of product that would result in the same profit is 1 thousand. Option A

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Answer:

[tex]x \approx 1.16[/tex]

Step-by-step explanation:

a.

To find the values of m and n, we have to first expand  [tex](x + m)^2 + n[/tex] :

[tex](x + m)^2 + n[/tex]

⇒ [tex]x^2 + 2mx + m^2 + n[/tex]

[tex]f(x) = x^2 + 4x - 6[/tex]

Now  compare the coefficients of the x-terms:

• [tex]2mx = 4x[/tex]

⇒ [tex]\bf m = 2[/tex]

• [tex]m^2 + n = -6[/tex]

⇒  [tex]2^2 + n = -6[/tex]

⇒ [tex]n = -10[/tex]

b.

[tex]x^2 + 4x - 6 = 0[/tex]

∴ [tex](x + 2)^2 - 10 = 0[/tex]

⇒ [tex](x + 2)^2 = 10[/tex]

⇒ [tex]x + 2 = \sqrt{10}[/tex]

⇒ [tex]x = \sqrt{10} -2[/tex]

⇒ [tex]x \approx 1.16[/tex]

Answer:

Yes because putting value of x and y as 1 and -1 in equation :

y=3x-4

-1 = 3 * 1 - 4

-1 = -1 (True)

Thermal energy can be used in the heating systems either at home or in the industries

However, when solar energy is converted to thermal energy, the radiant energy from the sunlight is converted into heat which can then be used as space and hot water heating, power generation and industrial process heat

Energy simply refers to the capacity of doing work. Some forms of energy are:

So therefore, thermal energy can be used in the heating systems either at home or in the industries.

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Using a system of equations, the translated expression is:

P - 32.

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

In this problem, the variable is the number of dollars Paul owes, which is P. That amount minus 32 is represented by:

P - 32.

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Answer:

  h > 5

  h < 0

Step-by-step explanation:

We suppose that a "Special Solution" inequality is one that is technically incorrect, but is written using a compact "shorthand" form.

As written, the inequality claims that 17 < 2, which is false. There can be no value of the variable that will make this true. We presume the intended meaning is ...

  17 < 3h +2   or   3h +2 < 2

Subtracting 2 from the inequality gives ...

  15 < 3h < 0

Dividing by 3 gives ...

  5 < h < 0

There is no value of h that is both greater than 5 and less than 0, so we presume this means the OR (union) of the solution sets ...

  h > 5

  h < 0

The trigonometric form of the complex number given in the task content is; 18(isin(π/2)).

If follows from the task content that the complex number whose trigonometric representation is to be determined is; 18i.

Hence, It follows that the trigonometric form is;

= 18(cos(π/2) + isin(π/2)). where; cos(π/2) = 0.

Hence, we have;

= 18(isin(π/2)).

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Answer:

8

Step-by-step explanation:

The cos of 30 is [tex]\sqrt{x}[/tex] / 2.  Since the adjacent angle is 4 times that.  We would multiply 2 x 4 and get 8

Answer:

for what lol

Step-by-step explanation:

Answer:

It is an open circle and you would shade to the right on a number line.

Step-by-step explanation:

The circle is open because you are not including 3.  Put numbers in for t and you will see that the numbers greater than 3 are all on the right side of the number line.

Answer: equation: y=-4x+1

slope: -4

y intercept:1

the table complete is:

x      y

1       4

2      8

3      12

4      16

Where x is the ribbon length in yards and y is the number of bows she can make.

We know that Marina needs 1/4 yards of ribbon for each bow.

Then, with one yard of ribbon, she can make 4 bows, then the relation between y, the number of bows she can make, and x, the yards of ribbon that she has, is:

y = 4*x

Now we want to complete the table:

x           y

1

2

3

4

To do so, we just need to evaluate the above function.

when x = 1.

y = 4*1 = 4

When x = 2:

y = 4*2 = 8

when x = 3

y = 4*3 = 12

when x = 4

y = 4*4 = 16

Then the table complete is:

x      y

1       4

2      8

3      12

4      16

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Answer:

A

Step-by-step explanation:

the sum to n terms of a geometric series is

[tex]S_{n}[/tex] = [tex]\frac{a_{1}(r^{n}-1) }{r-1}[/tex] , then for n = 7

S₇ = [tex]\frac{-1(2^{7}-1) }{2-1}[/tex] = [tex]\frac{-1(128-1)}{1}[/tex] = - 1 (127) = - 127

Answer:

Undefined (DNE)

Step-by-step explanation:

Any vertical line has an undefined slope.

After conversion 98.2 °F=119.2°C

96.6 °F=116.3°C

100.1 °F==122.6°C

How can we convert temperature from Fahrenheit to Celsius?

We will convert using this formula

C= (F-32) *5/9

We will convert temperature from Fahrenheit to Celsius as shown below:

C= (F-32) *5/9

For Oral

Temperature=98.2°F

C= (98.2-32) *5/9

=119.16°C

Rounding to nearest tenth

=119.2°C

For Axillary

Temperature=96.6 °F

C= (96.6-32) *5/9

=116.28°C

Rounding to nearest tenth

=116.3°C

For Rectal

Temperature=100.1 °F

C= (100.1-32) *5/9

=122.58

Rounding to nearest tenth

=122.6°C

Hence, after conversion 98.2 °F=119.2°C

96.6 °F=116.3°C

100.1 °F=122.6°C

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Answer:

no

Step-by-step explanation:

because no

= 4×8 +2×3 -1+9

___________

20

= 8+6-9

_______

20

= 14 - 9

_______

20

= 5

____

20

The description of the corresponding partition is given as follows:
The infinite classes can be defined as:

At equal  distance of x from point [1,2] 

Where x can vary from o to infinity.

The analogous partition, commonly known as "the set of equivalence classes," is the set A/R, the elements of which are the subsets of A satisfying  the following:


For every a∈A there is some B∈A/R such that a∈B.

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electronic data interchange

Answer: B. 9 hours

Step-by-step explanation:

Question analysis (Optional if you can understand the question)

Since they are 1170 miles apart, with their speed, the question is asking how long would it take them to travel 1170 miles together.

In other words, the distance that Jo and Jan together add up to 1170 miles.

Given information

Total distance = 1170 miles

Jo's rate = 60 mph

Jan's rate = 70 mph

Set variables

Let x be the time they took to meet each other

Set equation

60x + 70x = 1170

Solve equation

130x = 1170

Divide 130 on both sides

130x / 130 = 1170 / 130

[tex]\boxed{x=9 \: hrs}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Using Green's Theorem to evaluate the given line integral with the given conditions and vertices gives us; -30

The line integral of a vector-valued function along the edges of a rectangle or any other closed curve can be found by converting the line integral into a double integral. We apply Green's theorem to do this. The resulting double integral is integrated over the two-dimensional region bounded by the same closed curve.

Green's Theorem can be applied as follows:

∮_c (P.dx + Q.dy) = ∫∫_R ((dQ/dx) - (dP/dy))dA

The vertices of the rectangle (1,1), (3,1), (1,4), and (3,4).

Applying Green's Theorem to the given function gives;

∮_c (ln x + y) dx - x² dy

= ∫14∫13 Dx(-x2) -Dy(ln(x) +y) dx dy

= ∫₁⁴∫₁³ (-2x - 1) dx dy

= -3·[x² +x]₁³

= -30

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The after-tax rate of return is 1.84%, the real return is 0.721%, the total monetary return is $1530 and the implications become very challenging to grow with such a return as the taxes

Given that money market account investment at 2.55 percent​ (APY), a ​1.11 % inflation​ rate, a 28 percent marginal tax​ bracket, and a constant ​$60000 ​balance.

Cash management is the backbone of any business. Cash management ensures that the company operates according to established standards and optimizes the use of funds to avoid wasting resources. Liquidity management decisions are therefore key to preserving a company's capital and profitability.

Firstly, we will find the after-tax return rate, we get

After-tax return rate= Before tax rate of return \times ( 1 - tax rate)

After-tax rate of return = 0.0255× ( 1 - 0.28)

After-tax rate of return = 0.0255×0.72

After-tax rate of return=1.84%

Now, we will find the real return, we get

Real  return=((1+After-tax Return)/(1+inflation rate))-1

Real return=((1+0.0184)/(1+0.0111))-1

Real return=(1.0184/1.0111)-1

Real return=1.00721-1

Real return=0.721%

Furthermore, we will find the total monetary return, we get

Total monetary rate=Balance×Before tax rate of return

Total monetary rate=60000×2.55%

Total monetary rate=$1530.

The implications of this result for cash management​ decisions is that it becomes very challenging to grow with such a return as the taxes and inflation is draining the returns and leaving only with negative returns.

Hence, the after-tax rate of return is 1.84%, the real return is 0.721%, the total monetary return is $1530 and the implications become very challenging to grow with such a return as the taxes and inflation is draining the returns and leaving only with negative returns.

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