A clothesline is tied between two poles, 10 m apart. The line is quite taut and has negligible sag. When a wet shirt with a weight of 0.5 kg is hung at the middle of the line, the midpoint is pulled down 10 cm. Find the tension in each half of the clothesline. (Use g = 9.8 m/s2 for the acceleration due to gravity. Enter your answers from smallest to largest value of the coefficient of i.)

Answer:

  x = 0

Step-by-step explanation:

You know the answer to this because you know the identity element for addition is 0: 5 + 0 = 5.

__

Or, you can make use of the addition property of equality and add -5 to both sides of the equation:

  5 - 5 + x = 5 - 5

  x = 0 . . . . . . . . . . simplify

Answer:x=0

Step-by-step explanation:

The student should watch for 32.135 on the dial pressure gauge when inflating the tire since the specified tire pressure is 32 27/200 psi.

The student's question involves converting a fraction to a decimal. The situation deals with the air pressure in a tire being 32 27/200 pounds per square inch (psi). To express this as a decimal, we perform the division operation of 27 divided by 200 which equals 0.135. This means the air pressure should be 32.135 psi on the dial pressure gauge.

For example, if a tire gauge reads 34 psi, this represents the pressure inside the tire without considering the atmospheric pressure. However, the absolute pressure within the tire will be gauge pressure plus atmospheric pressure. Therefore, if you're reading the gauge, you should aim for it to display 32.135 psi as it is in this context that we're discussing tire pressure.

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Answer: 0.9962

Step-by-step explanation:

Given : According to a 2009 Reader's Digest article, people throw away approximately 10% of what they buy at the grocery store.

i.e. the proportion of the people throw away what they buy at the grocery store [tex]p=0.10[/tex]

Test statistic for population proportion : -

[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]

For [tex]\hat{p}=0.02[/tex]

[tex]z=\dfrac{0.02-0.1}{\sqrt{\dfrac{0.1(1-0.1)}{100}}}\approx-2.67[/tex]

Now by using the standard normal distribution table , the probability that the sample proportion exceeds 0.02 will be :

[tex]P(p>0.02)=P(z>-2.67)=1-P(z<-2.67)=1-0.0037925\\\\=0.9962075\approx0.9962[/tex]

Hence, the probability that the sample proportion exceeds 0.02 =0.9962

From a statistical point of view, considering a normal sampling distribution with the known population proportion (10% or 0.10), the probability that the sample proportion of grocery shoppers throwing away groceries exceeds 0.02 or 2% is almost certain (0.996). This is calculated considering the Z-score for 0.02 using standard deviation calculated using the Central Limit Theorem.

This question is about the calculation of probability in relation to sampling distributions. In this case, we want to find out the probability that the sample proportion (the percentage of people who throw away groceries) exceeds 0.02 or 2%. Since the proportion of people who throw away groceries in the population (according to the Reader’s Digest article) is 10% or 0.10, the probability that the sample proportion exceeds 0.02 is basically 1, because 0.02 is significantly less than 0.10.

However, to apply this concept accurately, we need to consider the distribution for the sample proportion, which is approximately normal with a mean equal to the population proportion (0.10) and a standard deviation calculated as sqrt[(0.10*(1-0.10))/100] = 0.03, according to the Central Limit Theorem. Given this, the Z-score for 0.02 was calculated using Z = (sample proportion - population proportion)/standard deviation = (0.02-0.10)/0.03 = -2.67.

Looking up this Z-score in a standard normal table or using a probability calculator shows that the probability of getting a score this extreme or more (Z <= -2.67) is close to 0.004. Therefore, the probability that the sample proportion exceeds 0.02, in other words that Z > -2.67, is 1 - 0.004 = 0.996. So, it is almost certain (with a probability of 0.996) that the sample proportion will exceed 0.02.

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Answer:

  40.56 ft

Step-by-step explanation:

The perimeter is the sum of the lengths of the "sides" of this figure. Starting from the left side and working clockwise, the sum is ...

  P = left side (8 ft) + top side (10 ft) + semicircle (1/2×8 ft×π) + bottom side (10 ft)

  = 28 ft + 4π ft

  = (28 +12.56) ft

  P = 40.56 ft

Answer:

I am trying on this but I can solve you the 10th question

Answer:

12.4 miles, N84.4°E

Step-by-step explanation:

Split the translation over the components parallel to the direction S>N and W>E, then calculate the sum of both components, and get magnitude and direction of the movement. Here's my calculation, double check them regardless.

For the first hour, it travels [tex] 8.5 cos 37.5 [/tex] north and [tex] 8.5 sin 37.5 [/tex] east. Once the wind changes, it flies [tex] 6*1.5 cos (180-52.5) = 9 cos 127.5 [/tex] "north" ( the actual movement is southbound, which will appear calculating the cosine and getting a negative number) and [tex]6*1.5 sin (180-52.5) = 9 sin 127.5 [/tex]. The complete movement is 1.2 miles N and 12.3 miles E. The total movement is, with the Pythagorean theorem, 12.4 miles total, and the angle it forms with the north direction is the [tex]tan^{-1} \frac{12.3}{1.2} = 84.4°[/tex].

Answer:

The answer to your question is : 80 minutes

Step-by-step explanation:

Data

Mark = 120 minutes

Alexis = 240 minutes

Together = ??

We need to write an equation, we consider that in 1 minute, Mark loads 1/120 and Alexis 1/240. Then the equation is:

                   1 = x/120 + x/240             1 = truck uploaded ;   x = time in minutes

   Solve it     1 = (2x + x) /240

           240 = 3x

             x = 240/3

             x = 80 minutes              

The Mark and Alexis work together will take to load all of the packages is 80 minutes.

A word problem is a verbal description of a problem situation. It consists of few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

For the given situation,

Mark can load a truck with packages = 120 minutes

Mark's rate of work = [tex]\frac{1}{120}[/tex]

Alexis can load a truck with packages = 240 minutes

Alexis's rate of work = [tex]\frac{1}{240}[/tex]

Rate of work together by Mark and Alexis is

⇒ [tex]\frac{1}{120}+\frac{1}{240}[/tex]

⇒ [tex]\frac{2}{240}+\frac{1}{240}[/tex]

⇒ [tex]\frac{3}{240}[/tex]

⇒ [tex]\frac{1}{80}[/tex]

Thus Mark and Alexis work together will take to load all of the packages is [tex]\frac{1}{\frac{1}{80} }[/tex]

⇒ [tex]80[/tex]

Hence we can conclude that the Mark and Alexis work together will take to load all of the packages is 80 minutes.

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Final answer:

To find the function representing the total number of bacteria Q(t) in the petri dish at any time t, we integrate the growth rate function q(t)=12t and add the initial amount. The resulting function is Q(t) = 6t^2 + 89 million. After 3 hours, there will be 143 million bacteria in the dish.

Explanation:

The student is given the rate of bacterial growth in a petri dish as q(t)=12t, which represents the number of millions of bacteria per hour, and the initial number of bacteria is 89,000,000 (or 89 million). To find the function that gives the total number of bacteria at any time t, we integrate the rate of growth and add the initial amount. Since the rate of growth is linear, the integral of q(t)=12t with respect to t is 6t2, and adding the initial amount we get the total number of bacteria Q(t) = 6t2 + 89 million.

After 3 hours, we substitute t=3 into this function to get the number of bacteria:

Q(3) = 6(3)2 + 89 = 6(9) + 89 = 54 + 89 = 143 million bacteria.

Final answer:

A function q(t) representing the number of bacteria over time can be determined by integrating the growth rate function and adding the initial amount of bacteria. After 3 hours, there will be 89,000,054 million bacteria in the dish.

Explanation:

The question asks us to find a function q(t) that gives the number of bacteria in a petri dish at any time t, given that it starts with 89,000,000 bacteria, and the growth rate is 12t million bacteria per hour. To find q(t), we need to integrate the growth rate function and then add the initial amount:

[tex]q(t) = \int (12t) dt + 89,000,000[/tex]

Upon integrating we get:

[tex]q(t) = 6t^2 + 89,000,000[/tex]

To find the number of bacteria after 3 hours, we plug in t = 3:

[tex]q(3) = 6(3)^2 + 89,000,000[/tex] = [tex]6(9) + 89,000,000[/tex] = [tex]54 + 89,000,000[/tex]

Therefore, the number of bacteria after 3 hours is 89,000,054 million.

Answer:

A.Cause and effect

Step-by-step explanation:

6^-2x • 6^-x = 1/216

x=1 is the answer

Answer: x=1

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

Step-by-step explanation: Negative power rule, quotient rule, then you'll simplify. After that use the negative power rule and multiply both sides by 6^3x. Then simplify 1/216 and multiply both sides by 216. Simplify 1*216=216, then finally you'll put both sides on the same base and cancel the base of six on both sides. Divide both sides by 3 and you'll get 1=x. Now just switch places so the x will be first (x=1).

Negative power rule x^-a=1/x^a

Quotient rule x^a/x^b=x^a-b

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

P.S. if you want me to right out on paper how I did it if it would be easier for you to visually see the text to learn it then it would be my pleasure! Math is hard so I'm happy to help more with this problem!

HOPE THIS HELPS, HAVE A BLESSED DAY! :-) ;-)

Answer:

4 boys, 3 girls

Step-by-step explanation:

brothers of John ⇒ x

sisters of John ⇒ y

John has as many brothers as sisters:

brothers of Emily ⇒ x + 1 (Emily have the same amount of brother as John, plus one (John))

sisters of Emily ⇒ y -1 (Emily have the same amount of sister as John, minus one (herself))

Emily has twice as many brothers as sisters:

Now we have a system of 2 equations and 2 variables

x=y (I)

2y - 2 = x + 1 (II)

____________

Replace x in equation II

2y - 2 = y + 1

2y - y = 2 + 1

y = 3

____________

Replace y in equation I

x = y = 3

That mean that John have 3 brothers plus himself, there is 4 boys in the family and John have 3 sister, so there is 3 girls in the family

John and Emily are both girls as they have 0 brothers and 0 sisters in the family.

Let's use variables to represent the number of brothers and sisters John and Emily have. Let b represent the number of brothers and s represent the number of sisters.

From the given information, we know that John has as many brothers as sisters. So, b = s.

We also know that Emily has twice as many brothers as sisters. So, b = 2s.

We can solve this system of equations to find the values of b and s. Substituting the value of b from the second equation into the first equation, we get 2s = s. Therefore, s = 0.

Since John and Emily are siblings, and John has as many brothers as sisters (0 sisters), it means John has 0 brothers as well. So, the family consists of only John and Emily, who are both girls.

Answer:

11,36 pounds

Step-by-step explanation:

Trayvon's weight on earth= 142 pounds

Trayvon's weight on Saturn= (0,92)*Trayvon's weight on earth=

(0,92)*142=130,64 pounds

The difference between Trayvon's weight on earth and Saturn:

Trayvon's weight on earth- Trayvon's weight on Saturn

142-130,64=11,36 pounds.

Answer: a) Yes, there is enough fance

b) 58.1° and 47.9°

c) The city will not approve, because 1/3 of the area is just 2220.5ft²

Step-by-step explanation:

a) using law of cosines: x is the side we do not know.

x² = 126² + 110² - 2.126.110.cos74°

x² = 20335.3

x = 142.6 ft

So 150 > 142.6, there is enough fance

b) using law of sine:

sin 74/ 142.6 = sinα/126 = sinβ/110

sin 74/ 142.6 = sinα/126

0.006741 = sinα/126

sinα = 0.849

α = sin⁻¹(0.849)

α = 58.1°

sin 74/ 142.6 = sinβ/110

sin 74/ 142.6 = sinβ/110

0.006741 = sinβ/110

sinβ = 0.741

β = sin⁻¹(0.741)

β = 47.9°

Checking: 74+58.1+47.9 = 180° ok

c) Using Heron A² = p(p-a)(p-b)(p-c)

p = a+b+c/2

p=126+110+142.6/2

p=189.3

A² = 189.3(189.3-126)(189.3-110)(189.3-142.6)

A = 6661.5 ft²

1/3 A = 2220.5

So 2300 >  2220.5. The area you want to build is bigger than the area available.

The city will not approve

Answer:

  [tex]\sqrt[n]{x}=x^{\frac{1}{n}}[/tex]

Step-by-step explanation:

The index of a radical is the denominator of a fractional exponent, and vice versa. If you think about the rules of exponents, you know this must be so.

For example, consider the cube root:

[tex]\sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}=(\sqrt[3]{x})^3=x\\\\(x^{\frac{1}{3}})^3=x^{\frac{3}{3}}=x^1=x[/tex]

That is ...

[tex]\sqrt[3]{x}=x^{\frac{1}{3}} \quad\text{radical index = fraction denominator}[/tex]

Answer:

The answer to your question is below

Step-by-step explanation:

I think it will easy to understand if we graph this information but let's explain it without the graph.

According to the information given, we know that f(x) represents the number of baskets and x the number of players at the practice.

So, if we have the point (12, 36) we can conclude that during practice there were 12 players and there were 36 baskets.

For me, it makes sense.

Answer: Yes. The input and output are both possible

Step-by-step explanation:

The reason why is because f(x) stands for y and in parenthe this is how it looks like(x,y) and if you put the numbers in you have (12,36) 12 stands for the number of players and f(x) or y stands for the number of baskets made.

Answer:

A

Step-by-step explanation:

Let's first assume that the restriction doesn't hold.

So that way we can say that we can put ANY OF THE 9 DIGITS (1-9) on ANY OF THE 4 DIGIT COMBINATIONS.

Hence,

first digit can be any of 1 through 9

second digit can be any of 1 through 9

third digit can be any of 1 through 9

4th digit can be any of 1 through 9

So the total number of possibilities will be 9 * 9 * 9 * 9 = 6561

now, let's take into account the restriction. since all 4 digits cannot be the same, so we need to exclude:

1111

2222

3333

4444

5555

6666

7777

8888

9999

That's 9 numbers. So final count would be 6561 - 9 = 6552

Answer A is right.

The correct answer is a. 6552. There are 6561 total combinations for a 4-digit bike lock. After excluding 9 combinations where all digits are the same, 6552 combinations remain.

To determine the total number of possible combinations for a 4-digit bike lock with digits ranging from 1 to 9, we start with the total unrestricted possibilities. Each digit has 9 options (1 through 9), so we calculate:

However, the problem states that all 4 digits cannot be the same. This means we must subtract the 9 combinations where all four digits are identical (e.g., 1111, 2222, ..., 9999). Thus, we calculate:

The correct answer is a. 6552.

Answer:

10

Step-by-step explanation:

20/2=10

Answer:10

Step-by-step explanation:

Answer:

The answer to your question is: m∠CAT = 45;  m∠TAD = 135

Step-by-step explanation:

Data

∠CAT and ∠TAD are a linear pair

m∠CAT = 2x-5

m∠TAD = 5x+10

m∠CAT = ?

m∠TAD = ?

Process

The sum of linear pairs angles equals 180°, so

                         m∠CAT + m ∠TAD = 180°

                         (2x - 5)  + (5x + 10) = 180°

                         2x - 5 + 5x + 10 = 180

Solve for x        7x + 5 = 180

                         7x = 180 - 5

                         7x = 175

                         x = 175 / 7

                         x = 25

m∠CAT = 2(25) - 5 =  50 -5 = 45

m∠TAD = 5(25) + 10 = 125 + 10 = 135

Answer:

[tex]m\angle CAT=45^{\circ}[/tex]

[tex]m\angle TAD=135^{\circ}[/tex]

Step-by-step explanation:

We are given that angle CAT and angle TAD are a linear pair.

[tex]m\angle CAT=2x-5[/tex]

[tex]m\angle TAD=5x+10[/tex]

We have to find the measure of angle CAT and angle TAD.

[tex]m\angle CAT+m\angle TAD=180^{\circ}[/tex]  (linear pair sum =180 degrees)

[tex]2x-5+5x+10=180[/tex]

[tex]7x+5=180[/tex]

[tex]7x=180-5=175[/tex]

[tex]x=\frac{175}{7}=25[/tex]

Substitute the values then we get

[tex]m\angle CAT=2(25)-5=45^{\circ}[/tex]

[tex]m\angle TAD=5(25)+10=125+10=135^{\circ}[/tex]

Answer:

  54 mph

Step-by-step explanation:

Let s represent the slower speed. The product of speed and time is distance, which did not change between the two trips. So, we have ...

  1.8(s +6) = 2(s)

  10.8 = 0.2s . . . . eliminate parentheses, subtract 1.8s

  54 = s . . . . . . . . divide by 0.2

Fran's average speed on Sunday was 54 miles per hour.

____

Her trip was 108 miles long.

To solve the problem, you can use the equation for speed which is distance divided by time. By substituting variables and solving the equation, you'll find that the average speed on Sunday was 54 mph when traffic was heavier.

To solve this, we need to use the formula for speed which is distance divided by time. Since the distance to her mother's house and back is the same for both trips, let's denote the distance as 'd'. We don't know the numerical distance, but we don't need to.

For Saturday, the formula is speed=d/1.8

For Sunday, the average speed is  d/2.

According to the problem, the average speed on Sunday was 6 mph slower than on Saturday. Therefore, the speed on Saturday minus 6 equals the speed on Sunday. So we have the equation: d/1.8 - 6 = d/2

To solve this equation, you first clear the fractions by multiplying each term by the common multiple of 2 and 1.8 which is 3.6. This gives us: 2d - 21.6 = 1.8d

Next, subtract 1.8d from 2d to get 0.2d = 21.6, then divide both sides by 0.2, yielding: d=108

Substitute d = 108 into the equation for Sunday to find the average speed: 108/2 = 54 mph. This is the answer, Fran's speed on Sunday was 54 mph when the traffic was heavier.

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Answer:

Top part lenght= 24 ft.

Middle part= 72 ft.

Bottom part = 144 ft.

Step-by-step explanation:

First we assign varibales to each rocket part

top part length= x;

middle part length= y;

bottom part length= z;

Then from the reading we can write the next equations:

X=1/6 Z; (1)

Y=1/2 Z; (2)

X + Y +Z = 240 (3)

Then solving, we replace x and y, in the equation (3)

1/6 z + 1/2 z + z = 240

Multiply by 6 both sides:

6/6 z + 6/2 z + 6 z = 1440

z + 3 z + 6 z = 1440

Then grouping similar terms

10 z = 1440

z= 144

Then replacing in (1) and (2)

Y=1/2 *144=72

X=1/6*144= 24

Answer:

see explanation

Step-by-step explanation:

Given

x + [tex]\frac{1}{2}[/tex] ≤ - 3 or x - 3 > - 2

Solve the left and right inequalities separately, that is

x + [tex]\frac{1}{2}[/tex] ≤ - 3 ( isolate x by subtracting [tex]\frac{1}{2}[/tex] from both sides )

x ≤ - 3 - [tex]\frac{1}{2}[/tex], that is

x ≤ - [tex]\frac{6}{2}[/tex] - [tex]\frac{1}{2}[/tex], thus

x ≤ - [tex]\frac{7}{2}[/tex]

OR

x - 3 > - 2 ( isolate x by adding 3 to both sides )

x > 1

Solution is

x ≤ - [tex]\frac{7}{2}[/tex] or x > 1

Answer:

  CD = 40

Step-by-step explanation:

Since D is the midpoint, the entire length CE is twice the length of CD, so we have ...

  2×CD = CE

  2×(5x) = 9x +8

  x = 8 . . . . . . . . subtract 9x and simplify

Then the length of CD is ...

  CD = 5x = 5·8 = 40

Answer: Option c.

Step-by-step explanation:

We know that the first month's interest payment was $477.82, therefore, we can calculate the Annual interest multiplying this first month's interest payment by 12:

[tex]Annual\ interest=\$477.82*12\\\\Annual\ interest=\$5,733.84[/tex]

Dividing it by the interest rate (Remember that [tex]7.125\%=\frac{7.125\%}{100}=0.07125[/tex]), we get:

[tex]\frac{\$5,733.84}{0.07125}=\$80,474.94[/tex]

Finally, since Kate and Bill secured a loan with a 75% loan-to-value ratio, we get:

[tex]\frac{\$80,474.94}{0.75}=\$107,299.92 \approx\$107,300[/tex]

Answer:

Area = length * width

11.25 sq ft = 4.5 ft * width

width = 11.25 / 4.5

width = 2.5 feet

5/2 feet = 2.5 feet

answer is D

Step-by-step explanation:

Answer:

The answer to your question is:  4w + 50

Step-by-step explanation:

Data

w = wide ft

length = w + 25

perimeter = ?

Process

Find the perimeter

Perimeter = 2 wide + 2 length

Perimeter = 2w + 2(w + 25)

Perimeter = 2w + 2w + 50

Perimeter = 4w + 50

The length of a fence around a rectangular park with width 'w' feet and length 'w + 25' feet would be 4w + 50 feet according to the perimeter formula for rectangles.

The length of the rectangular park is given as w + 25 feet, where w represents the width of the park. The length of a fence surrounding the park, including the gates, would cover the entire perimeter of the park. The formula to find the perimeter of a rectangle is 2*(length + width).

Substitute the given dimensions into the formula, the fence length therefore would be 2*(w + (w + 25)). Simplifying this equation gives us 2*(2w + 25) which is equal to 4w + 50. Thus, the length of the fence, including the gates, that surrounds the rectangular park is 4w + 50 feet.

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[tex]\rightarrow z^4=-625\\\\\rightarrow z=(-625+0i)^{\frac{1}{4}}\\\\\rightarrow x+iy=(-625+0i)^{\frac{1}{4}}\\\\ x=r \cos A\\\\y=r \sin A\\\\r \cos A=-625\\\\ r \sin A=0\\\\x^2+y^2=625^{2}\\\\r^2=625^{2}\\\\|r|=625\\\\ \tan A=\frac{0}{-625}\\\\ \tan A=0\\\\ A=\pi\\\\\rightarrow z= [625(\cos (2k \pi+pi) +i \sin (2k\pi+ \pi)]^{\frac{1}{4}}\\\\k=0,1,2,3,4,....\\\\\rightarrow z=(625)^{\frac{1}{4}}[\cos \frac{(2k \pi+pi)}{4} +i \sin \frac{(2k\pi+ \pi)}{4}] [/tex]

[tex]\rightarrow z_{0}=(625)^{\frac{1}{4}}[\cos \frac{pi}{4} +i \sin \frac{\pi)}{4}]\\\\\rightarrow z_{1}=(625)^{\frac{1}{4}}[\cos \frac{3\pi}{4} +i \sin \frac{3\pi}{4}]\\\\ \rightarrow z_{2}=(625)^{\frac{1}{4}}[\cos \frac{5\pi}{4} +i \sin \frac{5\pi}{4}]\\\\ \rightarrow z_{3}=(625)^{\frac{1}{4}}[\cos \frac{7\pi}{4} +i \sin \frac{7\pi}{4}][/tex]

Argument of Complex number

Z=x+iy , is given by

If, x>0, y>0, Angle lies in first Quadrant.

If, x<0, y>0, Angle lies in Second Quadrant.

If, x<0, y<0, Angle lies in third Quadrant.

If, x>0, y<0, Angle lies in fourth Quadrant.

We have to find those roots among four roots whose argument is between 270° and 360°.So, that root is

   [tex] \rightarrow z_{2}=(625)^{\frac{1}{4}}[\cos \frac{5\pi}{4} +i \sin \frac{5\pi}{4}][/tex]

The solutions are[tex]\( z = 5e^{i(\frac{9\pi}{4})} \), \( z = 5e^{i(\frac{17\pi}{4})} \), \( z = 5e^{i(\frac{25\pi}{4})} \), and \( z = 5e^{i(\frac{33\pi}{4})} \).[/tex]

To find the solutions of the equation [tex]\( z^4 = -625 \) in the given range of argument, we first rewrite the equation in polar form. Let \( z = re^{i\theta} \), where \( r \) is the magnitude of \( z \) and \( \theta \) is its argument.[/tex]

The equation becomes:

[tex]\[ (re^{i\theta})^4 = -625 \]\[ r^4e^{4i\theta} = -625 \]Now, since the right side is a negative real number, we can express it in polar form as \( -625 = 625e^{i\pi} \). So we have:\[ r^4e^{4i\theta} = 625e^{i\pi} \][/tex]

Comparing the magnitudes and arguments on both sides, we get:

[tex]\[ r^4 = 625 \]\[ 4\theta = \pi \]Solving for \( r \) and \( \theta \):\[ r = \sqrt[4]{625} = 5 \]\[ \theta = \frac{\pi}{4} \][/tex]

However, we need solutions in the given range of argument, which is between [tex]\( 270^\circ \) and \( 360^\circ \). Since \( \frac{\pi}{4} \) is approximately \( 45^\circ \) or \( \frac{\pi}{4} \), we need to add multiples of \( 2\pi \) to this angle to get solutions within the desired range.[/tex]

The solutions are:

[tex]\[ z_1 = 5e^{i(\frac{\pi}{4} + 2\pi)} \]\[ z_2 = 5e^{i(\frac{\pi}{4} + 4\pi)} \]\[ z_3 = 5e^{i(\frac{\pi}{4} + 6\pi)} \]\[ z_4 = 5e^{i(\frac{\pi}{4} + 8\pi)} \]Simplifying the angles:\[ z_1 = 5e^{i(\frac{9\pi}{4})} \]\[ z_2 = 5e^{i(\frac{17\pi}{4})} \]\[ z_3 = 5e^{i(\frac{25\pi}{4})} \]\[ z_4 = 5e^{i(\frac{33\pi}{4})} \][/tex]

Finally, we can convert these back to rectangular form if needed:

[tex]\[ z_1 = 5\left(\cos\frac{9\pi}{4} + i\sin\frac{9\pi}{4}\right) \]\[ z_2 = 5\left(\cos\frac{17\pi}{4} + i\sin\frac{17\pi}{4}\right) \]\[ z_3 = 5\left(\cos\frac{25\pi}{4} + i\sin\frac{25\pi}{4}\right) \]\[ z_4 = 5\left(\cos\frac{33\pi}{4} + i\sin\frac{33\pi}{4}\right) \][/tex]

You can compute the approximate values of these complex numbers and round them to the nearest thousandth if necessary.

Complete Question;

Find the solution of the following equation whose argument is strictly between [tex]$270^\circ$[/tex] , degree and [tex]$360^\circ$[/tex] , degree. Round your answer to the nearest thousandth.  [tex]\[z^4 = -625z\][/tex]

The probability of nobody winning in four rounds of a "nonconformity" game where players choose rock, paper, or scissors is [tex]\((\frac{1}{3})^4 = \frac{1}{81}\).[/tex]

To find the probability that nobody wins any of the 4 games, we need to consider under what conditions there is no winner in this game. There are two cases for no winner :

1. All three players choose the same symbol.

2. All three players choose different symbols.

Let's consider each case separately and calculate the probabilities.

Case 1: All three choose the same symbol

Each player has 3 choices (rock, paper, scissors).

For all three to choose the same symbol, the first player can choose any symbol, but the second and third must match.

Thus, there are 3 possible outcomes (RRR, PPP, SSS) out of a total of [tex]\(3 \times 3 \times 3 = 27\)[/tex] possible outcomes for each game.

Thus, the probability of this case is: [tex]\[\frac{3}{27} = \frac{1}{9}.\][/tex]

Case 2: All three choose different symbols

There are three possible different symbols (rock, paper, scissors). The possible outcomes for this case are RPS, RSP, PRS, PSR, SRP, SPR, giving a total of 6 distinct outcomes. Hence, the probability of this case is:

[tex]\[\frac{6}{27} = \frac{2}{9}.\][/tex]

Total Probability of No Winner

Combining the probabilities from Case 1 and Case 2, we get the total probability that there is no winner in a single game:

[tex]\[\frac{1}{9} + \frac{2}{9} = \frac{3}{9} = \frac{1}{3}.[/tex]

Since there are 4 games and each game is independent, the probability of no winner in any of the 4 games is: [tex]\[\left( \frac{1}{3} \right)^4 = \frac{1}{81}.\][/tex]

Thus, the probability that nobody wins any of the 4 games is[tex]\(\boxed{\frac{1}{81}}\).[/tex]

The complete question is : Mario, Yoshi, and Toadette play a game of "nonconformity": they each choose rock, paper, or scissors. If two of the three people choose the same symbol, and the third person chooses a different symbol, then the one who chose the different symbol wins. Otherwise, no one wins. If they play 4 rounds of this game, all choosing their symbols at random, what's the probability that nobody wins any of the 4 games? Express your answer as a common fraction.

Answer:

x=14

Step-by-step explanation:

Solve for x by simplifying both sides of the equation, then isolating the variable.

Answer:

The answer is x=14.

Step-by-step explanation:

In order to determine the answer, we have to solve for x, that is, we have to free the "x" variable in any side of the equation. We have to do the same procedure for any variable, independent of the amount of variables in the equation.

Solving the expression for x:

[tex]5+\frac{4}{7}*(21+3x)=41\\\frac{4}{7}*(21+3x)=41-5\\\\21+3x=\frac{36}{\frac{4}{7} } \\\\21+3x=\frac{36*7}{4}\\3x=63-21\\3x=42\\x=\frac{42}{3}= 14[/tex]

The solution for x is x=14.

Answer:

  16

Step-by-step explanation:

The constant in a perfect square trinomial is the square of half the coefficient of the linear term.

  (x +a)² = x² +2ax +a²

For 2a=-8, a=-4 and a² = 16. The constant 16 must be added:

  x² -8x +16 = 7+16 . . the left side is a perfect square trinomial

  (x -4)² = 23 . . . . . . . the left side is a perfect square

Consumer goods are goods produced with the help of Capital goods.

Capital Good=Machine and Machine Parts, Parts which are used in manufacture of a tool,

Consumer Good=Chocolate, Different Commodities, Car.

Capital goods last for longer period of time, whereas Consumer goods are manufactured again and again with the help of same Capital goods.

It is given that, we want to increase production of consumer goods to a total of 6 units.

Producing 7.5 ,units of capital goods is too high for producing 6 unit of Consumer good, as Capital good last for Longer duration of time.