Select the correct answer.
If u =(1+i√3) and v=(1-i√3), what is uv?
Ο Α. 1
OB. -4
OC. 0
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The area of the rectangle is 1296 square feet.

Let's denote the width of the rectangle as "w" and its length as "4w" since the length is four times the width.

The formula for the perimeter of a rectangle is given by:

P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

Given that the perimeter of the rectangle is 180 ft, we can write the equation as:

180 = 2(4w + w)

Simplifying the equation:

180 = 2(5w)

180 = 10w

w = 18

So, the width of the rectangle is 18 ft.

Now, we can find the length of the rectangle:

Length = 4w = 4(18) = 72 ft

The area of a rectangle is given by the formula:

[tex]A = l \times w,[/tex]

where A is the area,

l is the length,

and w is the width.

Substituting the values we found:

Area [tex]= 72 ft \times 18 ft = 1296 ft^2[/tex]

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Answer:

D is the correct answer

Step-by-step explanation:

Acellus

The exact value of cos(-n/12) is equal to cos(n/12).

The cosine function (or cos function) in a triangle is the ratio of the adjacent side to that of the hypotenuse.

The value of cosine function is periodic with a period of 2π. Therefore, we can use the property cos(x) = cos(x + 2πk) for any integer value of k.

In this case, we have cos(-n/12). To find the exact value, we can use the fact that cos(x) = cos(-x) for any angle x. Therefore, we can rewrite cos(-n/12) as cos(n/12).

So, the exact value of cos(-n/12) is equal to cos(n/12).

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The simplified form of the rational expression is 1/(2[tex]x^5[/tex]).

To simplify the rational expression (7x)/(14[tex]x^6[/tex]), we can reduce the numerator and denominator by their greatest common factor.

In this case, the greatest common factor is 7x.

Dividing both the numerator and denominator by 7x, we get:

(7x)/(14[tex]x^6[/tex]) = (7x)/(7x × 2[tex]x^5[/tex])

Canceling out the common factor of 7x, we have:

= 1/(2[tex]x^5[/tex])

Therefore, the simplified form of the rational expression is 1/(2[tex]x^5[/tex]).

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Based on the given clues, the mystery number is 12.

Let's analyze the clues given to find the mystery number:

It is a 2-digit number: This means the number is greater than or equal to 10 and less than 100.

It is a factor of 48: The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

It is a multiple of 6: The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, and so on.

The sum of the digits is 3: From the clues, we know that the mystery number is a 2-digit number, so let's consider the possible numbers where the sum of the digits is 3:

12 (1 + 2 = 3)

21 (2 + 1 = 3)

Now let's check which of these numbers satisfy the first three clues:

12: This number is a factor of 48 (48 ÷ 12 = 4) and it is a multiple of 6 (12 ÷ 6 = 2). The sum of the digits is 3. Therefore, 12 satisfies all the clues.

21: This number is not a factor of 48 (48 ÷ 21 ≈ 2.29), so it doesn't satisfy the second clue.

Based on the given clues, the mystery number is 12.

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Manuel received $187.50, and George received $562.50 for the job.

Let's break down the information provided:

The job paid a total of $750.

George received $3 for every $2 that Manuel received.

To determine the amounts George and Manuel received, we can set up a ratio based on their earnings.

Let's assume Manuel's earnings as the base amount:

George's earnings : Manuel's earnings = $3 : $2

We can set up the following equation to solve for Manuel's earnings:

(Manuel's earnings / Manuel's earnings) = ($2 / $2)

Since the ratio is equivalent, we can say:

George's earnings / Manuel's earnings = $3 / $2

To find the amount each person received, we'll assign a variable to Manuel's earnings. Let's call it "x."

Therefore, George's earnings can be represented as "3x."

According to the given information, their total earnings should sum up to $750:

x + 3x = $750

Combining like terms, we have:

4x = $750

To solve for x, we divide both sides by 4:

x = $750 / 4

x = $187.50

Now, we can determine the amounts each person received:

Manuel's earnings = x = $187.50

George's earnings = 3x = 3 * $187.50 = $562.50

Therefore, Manuel received $187.50, and George received $562.50 for the job.

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Answer:

84

Step-by-step explanation:

a² + b² = c²

4² + b² = 10²

16 + b² = 100

b² = 100 - 16

b² = √84

Answer:

Hi

Please mark brainliest ❣️

Thanks

Step-by-step explanation:

Using Pythagorean theorem

hyp² = adj² + opp²

10² = 4² +b²

100-16 = b²

84= b²

√84 = b²

.°. b = 9.17

or b = √84

Fixed costs are expenses that do not change with the level of output or production. Examples of fixed costs in Manuel's case might include the permit cost, rent for the hot dog stand, or insurance premiums.

In order to answer your question accurately, I would need the specific values for Manuel's profit and cost functions. The information you provided is incomplete, as you mentioned Manuel's initial profit hill is plotted in green on a graph, but the graph itself is not available for reference.

To determine how a change in fixed costs affects the profit-maximizing quantity, we typically analyze the cost and revenue functions. Without these functions or the corresponding data, it is not possible to provide an exact numerical answer.

However, I can explain the general concept. Fixed costs are expenses that do not change with the level of output or production. Examples of fixed costs in Manuel's case might include the permit cost, rent for the hot dog stand, or insurance premiums.

When fixed costs decrease, it reduces the overall cost of production for each level of output. This means that Manuel can achieve higher profits or reduce losses for any given level of sales. Consequently, the profit-maximizing quantity may change as a result.

If we assume that the decrease in the price of the permit is the only change in fixed costs and all other factors remain constant, the new profit hill can be expected to shift upward. This is because the reduction in fixed costs increases the potential for higher profits at each level of output.

Without more specific information about Manuel's profit and cost functions, it is not possible to determine the exact profit-maximizing levels before and after the decrease in the price of the permit.

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In a case whereby Darren says that more students hands are 4 2/4 inches longer than 4 and 5 1/4 inches combined, he is wrong

[tex]4\frac{2}{4}[/tex] that was given in the question can be seen as mixed fraction, we can expressed this as improper fraction so that it will be easier to handle.

[tex]4\frac{2}{4} = \frac{16+4}{2}[/tex]

=[tex]\frac{18}{4}[/tex]

Then [tex]4 + 5\frac{1}{4} = \frac{16}{4} +\frac{20+1}{4} \\\\=\frac{37}{4}[/tex]

Then after expressing the given fractions as improper fraction we can now compare them so that e will know may be Darren is right or wrong, then here we can see that [tex]\frac{18}{4}[/tex] is less than [tex]\frac{37}{4}[/tex] Hence, Darren is wrong.

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Answer:

Set your calculator to degree mode.

x^2 = 4^2 + 10^2 - 2(4)(10)(cos 29°)

x^2 = 46.0304

x = 6.78

The number that belongs in the green box is 6.78.

Answer:

Step-by-step explanation:

You have to use the quadratic formula

x = -b +/- [tex]\sqrt{(b^{2} -4ac)/2a[/tex]

and then you can simplify both answers so the first one would be -13/2 and the second would be -27/2

you usually have to reject an answer but since there is no equal sign in the question, you should be fine

The sentence with proper subject-verb agreement is the student as well as the teacher want to go to the museum. In this sentence, the subject is what we call a compound subject, meaning that the verb refers and agrees with more than just one singular word.

The compound subject is "student" and "teacher" and they are connected by "as well as", which functions as a coordinating conjunction would. That's why the verb should conjugate in its plural form.

Option A is incorrect because the structure inside parentheses is not related to the verb and does not influence its conjugation. Options C and D have a verb in the singular form for a compound subject - that would demand a plural conjugation.

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Susan uses 500 milliliters of solution b to obtain a resulting mixture with 102 milliliters of pure alcohol.

Let's set up the equation based on the given information.

Let x represent the amount of solution b in milliliters.

Since Susan uses 100 milliliters less of solution a than solution b, the amount of solution a is (x - 100) milliliters.

We know that solution a has a concentration of 13% alcohol, which means that for every 100 milliliters, it contains 13 milliliters of alcohol.

Similarly, solution b has a concentration of 10% alcohol, which means that for every 100 milliliters, it contains 10 milliliters of alcohol.

Given that the resulting mixture has 102 milliliters of pure alcohol, we can set up the equation:

0.13(x - 100) + 0.1x = 102

Simplifying the equation, we have:

0.13x - 13 + 0.1x = 102

0.23x - 13 = 102

0.23x = 115

x = 115 / 0.23

x = 500

Therefore, Susan uses 500 milliliters of solution b to obtain a resulting mixture with 102 milliliters of pure alcohol.

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Answer:

x=2

Step-by-step explanation:

I don’t really have an explanation, it was all mental math

Answer:

Step-by-step explanation:

[tex]\sf 2(4-x)-3(x+3)=-11[/tex]

Expand:-

Now, add 1 to both sides:-

Divide both sides by -5:-

Therefore, the value of x is 2!

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

Hope this helps!

The probability that either event will occur is 0.67

From the question, we have the following parameters that can be used in our computation:

Event A = 3

Event B = 1

Other Events = 2

Using the above as a guide, we have the following:

Total = A + B + C

So, we have

Total = 3 + 1 + 2

Evaluate

Total = 6

So, we have

P(A) = 3/6

P(B) = 1/6

For either events, we have

P(A or B) = 3/6 + 1/6 = 0.67

Hence, the probability that either event will occur is 0.67

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Answer:$0.25

Step-by-step explanation:

To find the expected value of playing the game, we need to multiply the payoff for each possible outcome by its probability and then add up all the values.

Let's first find the probabilities of each outcome:

- Probability of spinning1:1/6

- Probability of spinning2:1/6

- Probability of spinning3:1/6

- Probability of spinning4:1/6

- Probability of spinning5 or6:2/6( or1/3)

Now let's calculate the expected value:

Expected value=(1/6*$1)+(1/6*$3)+(1/6*$5)+(1/6*$7)+(1/3*-$1.25)

Expected value=$0.25

So the expected value of playing the game is$0.25 per spin.

In the long run, Scott can expect to gain money since the expected value is positive. However, this doesn't guarantee that he will actually win money every time he plays the game, as there is still a certain degree of randomness involved in each spin.

The calculated value of the price per cup is $1.16

From the question, we have the following parameters that can be used in our computation:

A 3-quart jug of water costs $3.48

The price per cup is calculated as

Unit rate = Total cost/Size of the cup

Substitute the known values in the above equation, so, we have the following representation

Unit rate = 3.48/3

Evaluate

Unit rate = 1.16

Hence, the price per cup is $1.16

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Answer: 14 ounces

Step-by-step explanation:

         To find the mean, we add up all the values and divide by the number of values.

[tex]\displaystyle \frac{12+14+15+15+14}{5} =\frac{70}{5} =14\;ounces[/tex]

To find the common roots between two functions, we need to find the roots (or solutions) of each function individually and then identify the shared solutions.

For the function f(x) = x^2 + x - 6, we can find the roots by setting the function equal to zero and solving for x:

x^2 + x - 6 = 0

To factorize this quadratic equation, we need to find two numbers that multiply to -6 and add up to 1 (the coefficient of x). The numbers that satisfy these conditions are 3 and -2:

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

Setting each factor equal to zero:

x + 3 = 0 or x - 2 = 0

Solving for x in each equation:

x = -3 or x = 2

Therefore, the function f(x) = x^2 + x - 6 has two roots: x = -3 and x = 2.

To find the common roots between this function and another function, we would need to know the second function. If you provide the second function, I can help determine if there are any shared roots.

Given statement solution is :- The correct solutions to the quadratic equation  [tex]3(x-7)^2 = 27[/tex]  are x = 10 and x = 4.

Diego made an error in the step where he simplified [tex](x-7)^2 = 9 to x^2[/tex] - 72 = 9. The correct expansion of [tex](x-7)^2 is x^2 - 14x + 49[/tex], not [tex]x^2 - 72.[/tex]Therefore, his subsequent steps and solution are incorrect.

Mai made an error in the step where she simplified [tex](x-7)^2 = 9[/tex] to x - 7 = 9. The square root of 9 is ±3, so the correct equation should be x - 7 = ±3, not x - 7 = 9. Therefore, her solution is incorrect.

To solve the equation correctly, we start with the equation [tex]3(x-7)^2 = 27.[/tex]

Dividing both sides by 3, we have:

[tex](x-7)^2 = 9[/tex]

Taking the square root of both sides, we get:

x - 7 = ±3

Now, we need to solve for x by adding 7 to both sides:

x = 7 ± 3

This gives us two possible solutions:

x = 7 + 3 = 10

x = 7 - 3 = 4

So, the correct solutions to the quadratic equation [tex]3(x-7)^2 = 27[/tex] are x = 10 and x = 4.

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Answer:

Pin copied text snippets to stop them expiring after 1 hourText you copy will automatically show hereSlide clips to delete them

Step-by-step explanation:

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The length of each side of the original square paper is √812.25 = 28.5 inches.

Let us assume that the length and breadth of the square paper are 'a' and 'b' respectively. Now, Fabrizio cuts the paper vertically to make two rectangular pieces.

Since the paper is cut vertically, it can be assumed that the breadth of the square is equal to the height of the rectangle.

Therefore, the perimeter of one rectangle can be calculated by the formula P = 2l + 2wwhere l = length and w = widthNow, P = 57 inches.

Therefore, 2l + 2w = 572(l + w) = 57l + 57w = 28.5(2l + 2w)Since the paper is cut into two rectangles, the length of the square paper is divided equally into the two pieces.

Therefore, the length of the side of the original square paper can be calculated as follows:Length of one rectangle = l = 28.5Width of one rectangle = w = 57 - 28.5 = 28.5

Therefore, the breadth of the original square paper = width of one rectangle = 28.5 inches.

The area of the square can be calculated as length * breadthTherefore, the area of the square paper is given by a * b = 28.5 * 28.5 = 812.25 square inches.

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The function notation of 9x + 3y = 12 is given as follows:

f(x) = 4 - 3x.

The function in the context of this problem is given as follows:

9x + 3y = 12.

The format for the function notation is given as follows:

Hence we must isolate the variable y, as follows:

3y = 12 - 9x

y = 4 - 3x (each term of the expression is divided by 3).

f(x) = 4 - 3x.

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Answer:

yes

Step-by-step explanation:

Approximately 37.5% of the data are greater than 28, 33.3% of the data are less than 23, 91.7% of the data are greater than 17.5, and 62.5% of the data are between 17.5 and 28.

To answer the questions, we can analyze the given data set.

First, let's count the number of data points that satisfy each condition:

Greater than 28:

There are 9 data points greater than 28 (29, 29, 29, 30, 29, 29, 28, 28, 28).

Less than 23:

There are 8 data points less than 23 (20, 18, 15, 17, 17, 15, 17, 19).

Greater than 17.5:

There are 22 data points greater than 17.5.

Between 17.5 and 28:

There are 15 data points between 17.5 and 28.

Now, let's calculate the approximate percentage for each condition:

Percent greater than 28:

The total number of data points is 24. Approximately, 9 out of 24 data points are greater than 28.

Percentage =[tex](9 / 24) \times 100 = 37.5[/tex]%.

Percent less than 23:

Approximately, 8 out of 24 data points are less than 23.

Percentage = [tex](8 / 24) \times 100 = 33.3[/tex]%.

Percent greater than 17.5:

Approximately, 22 out of 24 data points are greater than 17.5.

Percentage = [tex](22 / 24) \times 100 = 91.7[/tex]%.

Percent between 17.5 and 28:

Approximately, 15 out of 24 data points are between 17.5 and 28.

Percentage = [tex](15 / 24) \times 100 = 62.5[/tex]%.

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The given limit is 0.

To solve the given limit, we can recognize the sum as a Riemann sum and convert it into an integral.

The given sum can be rewritten as:

[tex]\lim_{n \to \infty} \sum_{i=1}^{n} \frac{3}{n} \sqrt{1+\frac{3i}{n}}[/tex]

Let's rewrite it in terms of integration:

[tex]\lim_{n \to \infty} \frac{3}{n} \sum_{i=1}^{n} \sqrt{1+\frac{3i}{n}}[/tex]

Since we are taking the limit as n approaches infinity, we can approximate the sum as an integral.

The integral that corresponds to the given sum is:

[tex]\lim_{n \to \infty} \frac{3}{n} \sum_{i=1}^{n} \sqrt{1+\frac{3i}{n}} \approx \lim_{n \to \infty} \frac{3}{n} \int_{0}^{1} \sqrt{1+3x} ,dx[/tex]

To solve this integral, we can use a change of variables.

Let u = 1 + 3x, then du = 3dx.

The integral becomes:

[tex]\lim_{n \to \infty} \frac{3}{n} \int_{0}^{1} \sqrt{1+3x} ,dx = \lim_{n \to \infty} \frac{1}{n} \int_{1}^{4} \sqrt{u} ,du[/tex]

Integrating [tex]\sqrt{u}[/tex], we get,

[tex]\lim_{n \to \infty} \frac{1}{n} \int_{1}^{4} \sqrt{u} ,du = \lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} u^{3/2}\right]_{1}^{4}[/tex]

Substituting the limits, we have:

[tex]\lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} (4)^{3/2} - \frac{2}{3} (1)^{3/2}\right][/tex]

Simplifying further:

[tex]\lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} (8 - 2)\right] = \lim_{n \to \infty} \frac{1}{n} \left[\frac{12}{3}\right] = \lim_{n \to \infty} \frac{4}{n} = 0[/tex]

Therefore, the given limit is 0.

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