A regular octagon rotates 360° about its center. How many times does the image of the octagon coincide with the preimage during the rotation?

The evaluation of[tex]\( \oint_C \mathbf{F} \cdot d\mathbf{r} \)[/tex]  using Stokes' Theorem is [tex]\( \pi e - \pi \).[/tex]

To evaluate[tex]\( \oint_C \mathbf{F} \cdot d\mathbf{r} \)[/tex]  using Stokes' Theorem, we first need to find the curl of the vector field [tex]\( \mathbf{F}(x, y, z) = yz\mathbf{i} + 9xz\mathbf{j} + e^xy\mathbf{k} \)[/tex] . The curl is given by[tex]\( \nabla \times \mathbf{F} \)[/tex] , which, for this vector field, is [tex]\( (e^x - 9z)\mathbf{k} \).[/tex]

Now, we compute the surface integral over the surface ( S) bounded by the circle  C . The given circle is[tex]\( x^2 + y^2 = 1 \)[/tex] with ( z = 3 ). The surface is the disk in the plane (z = 3) with radius ( r = 1 ). The surface integral is then [tex]\( \iint_S \nabla \times \mathbf{F} \cdot d\mathbf{S} = \iint_S (e^x - 9z) \, dS \)[/tex] , where  dS is the area element. The projection of  S  onto the xy-plane is the circle  C , so [tex]( dS = dA = \pi \).[/tex]

The final step is to evaluate[tex]\( \iint_S (e^x - 9z) \, dS \)[/tex] over the disk. Since  z = 3 over the entire surface, the integral simplifies to [tex]\( \iint_S (e^x - 9 \cdot 3) \, dS = \pi e - \pi \).[/tex]  Therefore, the final answer is[tex]\( \pi e - \pi \)[/tex] after applying Stokes' Theorem to evaluate the circulation of [tex]\( \mathbf{F} \)[/tex] around the circle C .

"The correct answer is [tex]\(52/7\)[/tex] written as a mixed number is [tex]\(7\frac{3}{7}\).[/tex]

To convert an improper fraction to a mixed number, one must divide the numerator by the denominator and express the result as a whole number plus a proper fraction.

For the fraction [tex]\(52/7\)[/tex] we divide 52 by 7:

[tex]\(52 \div 7 = 7\)[/tex] with a remainder of 3.

 This means that is equal to 7 whole units and [tex]\(3/7\)[/tex] of another unit. Therefore, as a mixed number,[tex]\(52/7\)[/tex] is written as [tex]\(7\frac{3}{7}\)."[/tex]

To divide 198 by 9 using the partial quotients method, you estimate how many times 9 fits into 198, subtract, and sum up the partial results to get the final answer of 22.

To solve 198 ÷ 9 using the partial quotients method, follow these steps:

Estimate how many times 9 can go into 198. We start with a rough estimate that 9 can go into 198 around 20 times.

Calculate 9 x 20 = 180. Subtract this from 198, yielding 198 - 180 = 18.

Next, determine how many times 9 can fit into the remaining 18. This is 2 times since 9 x 2 = 18.

Subtract 18 from 18, resulting in a remainder of 0.

Add the partial quotients: 20 + 2 = 22.

So, 198 ÷ 9 = 22 using the partial quotients method.

Answer:

D.

false; Kim Li could have received a 20% sale discount

The logarithmic equation log2(8) = 3 convert to exponential form as 2^3 = 8.

Converting the logarithmic equation log2(8) = 3 to exponential form involves recognizing the base and the exponent.Logarithms state that if logb(A) = C, then the equivalent exponential form would be bC = A. In this case, since 2 is the base and 3 is the exponent to which 2 must be raised to get 8, we can write 2^3 = 8.

To further clarify, logarithms are simply a way to represent exponents, and they are widely utilized in mathematics to simplify the multiplication and division of exponents. A logarithm with a base of 10 is known as the common logarithm and is often written without a base as log(A). In contrast, when another base is used, it is indicated as logb(A). The relationship between logarithms and exponents is leveraged in various scientific and engineering fields to represent and calculate large numbers conveniently.

Answer: Option B 3-4i is the correct option

Explanation:

we have formula for discriminant [tex]D=b^{2}-4ac[/tex]

after that we find the required variable that is to be find according to the quadratic equation given suppose we have to find x

then we have formula [tex]x=\frac{-b\pm\sqrt{D}} {2a}[/tex]

Here we have [tex]\pm[/tex] of roots if we have one root 3+41 other would be of opposite sign and hence,definitely be 3-4i.

Therefore Option B 3-4i is the correct option.

The equivalent ratios to 4/28 are obtained by dividing both the numerator and the denominator by 4 to get 1/7, and then by multiplying the resulting ratio to get the other two equivalent ratios, 2/14 and 3/21, which correspond to answer choice b.

The question is asking to write three ratios that are equivalent to the ratio 4/28. To find equivalent ratios, you can either multiply or divide both the numerator (top number) and the denominator (bottom number) of a ratio by the same non-zero number.

For the ratio 4/28, if we divide both the numerator and the denominator by 4, we get 1/7. If we then multiply 1/7 by different numbers, we can find additional equivalent ratios. Therefore, multiplying by 2 gives us 2/14, and multiplying by 3 gives us 3/21. This means that the set of equivalent ratios is 1/7, 2/14, 3/21, which corresponds to answer choice b.

Answer:

Option B is correct.

Step-by-step explanation:

Given Geometric series : 3 , 1 , [tex]\frac{1}{3}\:,\:\frac{1}{9}\:,\:\frac{1}{27}[/tex]

To find: Sum of the series.

First term of the geometric series, a = 3

Common ration of the Geometric series, r = [tex]\frac{second\:term}{first\:term}=\frac{1}{3}[/tex]

Sum of the finite Geometric series , [tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]

Sum of the given 5 term term of given series , [tex]S_5=\frac{3(1-(\frac{1}{3})^5)}{1-\frac{1}{3}}=\frac{3(\frac{3^5-1}{3^5})}{\frac{3-1}{3}}[/tex]

=  [tex]\frac{\frac{3^5-1}{3^3}}{2}=\frac{243-1}{2\times3^3}=\frac{121}{27}[/tex]

Therefore, Option B is correct.

The domain of this function is  { - 4, - 1, 0, 3 } and range of the function is

{ - 4 }.

Suppose we have an ordered pair (x, y) then the domain of the function is the set of values of x and the range is the set of values of y for which x  is defined.

The domain of the function f(x) is the set of x values that are

{ - 4, - 1, 0, 3 }

and the range of the function f(x) is { - 4 }.

As for every input, the output of the function remains the same so it is a constant function.

learn more about the domain and range here :

https://brainly.com/question/28135761

#SPJ2

Answer:

Find out the what is the percent of increase .

To prove

As given

A store increases the price of a sweater from $20 to $22.

Increase in the price = Increase price - Initial price  

                                  = $22 - $20

                                  = $2

Formula

[tex]Percentage = \frac{increase\ in\ price\times 100}{Initial\ price}[/tex]

Here initial price = $20

increase in price = $2

put in the formula

[tex]Percentage = \frac{2\times 100}{20}[/tex]

[tex]Percentage = \frac{200}{20}[/tex]

Percentage = 10%

Therefore the increase in the price is 10% .

Option (e) is correct.

Final answer:

The percent of increase when a store raises the sweater's price from $20 to $22 is calculated as 10%, using the formula of difference over original price times 100, Option E is correct.

Explanation:

The question asks for the percent of increase when a store increases the price of a sweater from $20 to $22. To find the percentage increase, we take the increase in price ($2), divide it by the original price ($20), and then multiply by 100 to convert to a percentage. Thus, the calculation is (($22 - $20) / $20) * 100 = (2 / 20) * 100 = 0.10 * 100 = 10%.

Answer:

Step-by-step explanation:

Notice that the yellow area is a compound area. Its height is 10 inches.

We can divide this area in three figures, one rectangle and two squares, where the square sides are 2 inches long.

So, the area of the squares is

[tex]A_{squares}=(2in)^{2} +(2in)^{2}=4in^{2} +4in^{2} =8in^{2}[/tex]

On the other hand, the rectangle has a height of 10 inches, and its base is 2 inches long. So its area is

[tex]A_{rectangle}=(10in)(2in)=20in^{2}[/tex]

Therefore, the yellow area is

[tex]A_{yellow}=A_{squares} +A_{rectangle}=8in^{2} +20in^{2} =28in^{2}[/tex]

So, the answer is 28 square inches.

In a second it will go 90

In a minute it will go 5400

The Greek letter ω is often used to represent the SI unit of angular motion, the radian per second. Additionally, the SI unit for angular frequency is the radian per second. The shift in an object's orientation measured in radians per second is known as a radian per second.

Given

A computer’s hard disk is spinning at 90 revolutions per second.

In 1 second --- hard disk spins 90 revolutions

in 1 min = 60 sec = 90*60 = 5400 revolutions

To learn more about revolution per second refer to:

https://brainly.com/question/858645

#SPJ2

The computer's hard disk travels 4 degrees in a second and 240 degrees in a minute.

To find the number of degrees traveled by a computer's hard disk in a second, we need to know the time taken for one revolution. Since the disk is spinning at 90 revolutions per second, the time taken for one revolution is given by:

Time taken for one revolution = 1 / 90 seconds

Now, to find the number of degrees, we can use the formula:

Degrees = (Revolutions) x (360 degrees)

Therefore, the number of degrees traveled in a second is:

Degrees = (1 / 90) x (360) = 4 degrees

To find the number of degrees traveled in a minute, we multiply the number of degrees per second by the number of seconds in a minute:

Degrees per minute = (Degrees per second) x (60 seconds)

Degrees per minute = 4 x 60 = 240 degrees

Answer:

There is a common ratio of 2/3 between the height of the ball at each bounce. So, the bounce heights form a geometric sequence: 27, 18, 12. Two-thirds of 12 is 8, so on the fourth bounce, the ball will reach a height of 8 feet.

Step-by-step explanation:

Sample response :)

On dividing 646.1 by 34, we get 19.

Dividing 646.1 by 34:

Set up the division: 646.1 ÷ 34.

Perform the division to get the result: 646.1 ÷ 34 = 19.