To determine the number of different eight-note melodies that alternate between black and white keys within a single octave, you calculate the permutations starting with either type of key and add them together, resulting in 141,120 possible melodies.
Explanation:The question asks: How many different eight-note melodies within a single octave can be written if the black keys and white keys need to alternate? To solve this, we need to understand the structure of a piano octave, which consists of seven white keys and five black keys. Since melodies must alternate between black and white keys, starting with a white key will always result in a pattern of white-black-white-black, and so on, until eight notes are reached. Conversely, starting with a black key follows a black-white pattern.
If we start with a white key, we have 7 options for the first note. The next note (a black key) gives us 5 options. This alternating pattern continues, decreasing the number of options by 1 for each type of key used, until we have selected all eight notes. Mathematically, this calculates as 7 × 5 × 6 × 4 × 5 × 3 × 4 × 2. Similarly, starting with a black key would result in a calculation of 5 × 7 × 4 × 6 × 3 × 5 × 2 × 4.
However, since an eight-note melody can start with either a white or a black key, we calculate both scenarios and add them together for the total amount of possible melodies. The sum of the series for both starting options gives us 141,120 possible eight-note melodies that alternate between black and white keys within a single octave.
A man who has lost $37 playing roulette in Las Vegas wins $43 playing blackjack. He then loses $44 playing wheel of fortune. Write an expression using the numbers -37, 43 and 44 to describe this situation and then simplify it. Write your simplified result below.
_____
Answer:
(-37+43)-44=x
Step-by-step explanation:
i put the parentheses only because of PEMDAS. it's a pretty straightfoward question. To put it in an expression however, I'm not sure. Best of luck!
At a toy store the children's department has bicycles and tricycles. There are 20 wheels on a group of bicycles each bike has 2 wheels how many bikes are there
Answer:
10
Step-by-step explanation:
20/2=10
Answer:10
Step-by-step explanation:
ABC Auto Insurance classifies drivers as good, medium, or poor risks. Drivers who apply to them for insurance fall into these three groups in the proportions 30 percent, 50 percent, and 20 percent, respectively. The probability a "good" driver will have an accident is .01, the probability a "medium" risk driver will have an accident is .03, and the probability a "poor" driver will have an accident is .10. The company sells Mr. Brophy an insurance policy and he has an accident. What is the probability Mr. Brophy is: a. A "good" driver? b. A "medium" risk driver? c. A "poor" driver?
Answer:
a.[tex]P(E_1/A)=0.0789[/tex]
b.[tex]P(E_2/A)=0.395[/tex]\
c.[tex]P(E_3/A)=0.526[/tex]
Step-by-step explanation:
Let [tex]E_1,E_2,E_3[/tex] are the events that denotes the good drive, medium drive and poor risk driver.
[tex]P(E_1)=0.30,P(E_2)=0.50,P(E_3)=0.20[/tex]
Let A be the event that denotes an accident.
[tex]P(A/E_1)=0.01[/tex]
[tex]P(A/E_2=0.03[/tex]
[tex]P(A/E_3)=0.10[/tex]
The company sells Mr. Brophyan insurance policy and he has an accident.
a.We have to find the probability Mr.Brophy is a good driver
Bayes theorem,[tex]P(E_i/A)=\frac{P(A/E_i)\cdot P(E_1)}{\sum_{i=1}^{i=n}P(A/E_i)\cdot P(E_i)}[/tex]
We have to find [tex]P(E_1/A)[/tex]
Using the Bayes theorem
[tex]P(E_1/A)=\frac{P(A/E_1)\cdot P(E_1)}{P(E_1)\cdot P(A/E_1)+P(E_2)P(A/E_2)+P(E_3)P(A/E_3)}[/tex]
Substitute the values then we get
[tex]P(E_1/A)=\frac{0.30\times 0.01}{0.01\times 0.30+0.50\times 0.03+0.20\times 0.10}[/tex]
[tex]P(E_1/A)=0.0789[/tex]
b.We have to find the probability Mr.Brophy is a medium driver
[tex]P(E_2/A)=\frac{0.03\times 0.50}{0.038}=0.395[/tex]
c.We have to find the probability Mr.Brophy is a poor driver
[tex]P(E_3/A)=\frac{0.20\times 0.10}{0.038}=0.526[/tex]
Final answer:
The probability that Mr. Brophy is a good, medium, and poor risk driver given he has had an accident is approximately 0.079, 0.395, and 0.526 respectively, as calculated using Bayes' theorem.
Explanation:
Using Bayes' theorem, we can find out the probability that Mr. Brophy is a good, medium, or poor risk driver given that he has had an accident:
Let's denote A as the event of having an accident and Gi, Mi, Pi as the events of Mr. Brophy being a good, medium, or poor risk driver respectively.
P(Gi) = 0.30, P(Mi) = 0.50, P(Pi) = 0.20P(A|Gi) = 0.01, P(A|Mi) = 0.03, P(A|Pi) = 0.10The total probability of an accident, P(A), is given by:
P(A) = P(A|Gi)P(Gi) + P(A|Mi)P(Mi) + P(A|Pi)P(Pi)
P(A) = (0.01)(0.30) + (0.03)(0.50) + (0.10)(0.20) = 0.003 + 0.015 + 0.02 = 0.038
The probability of Mr. Brophy being a good driver given he had an accident, P(Gi|A), is:
P(Gi|A) = (P(A|Gi)P(Gi)) / P(A) = (0.01)(0.30) / 0.038 ≈ 0.079
The probability of Mr. Brophy being a medium risk driver given he had an accident, P(Mi|A), is:
P(Mi|A) = (P(A|Mi)P(Mi)) / P(A) = (0.03)(0.50) / 0.038 ≈ 0.395
The probability of Mr. Brophy being a poor risk driver given he had an accident, P(Pi|A), is:
P(Pi|A) = (P(A|Pi)P(Pi)) / P(A) = (0.10)(0.20) / 0.038 ≈ 0.526
In this year, the average cost of a traditional Thanksgiving dinner for 10, featuring turkey, stuffing, cranberries, pumpkin pie, and trimmings, was $48.05, a decrease of 1.76% from the cost in the last year. What was the average cost of a traditional Thanksgiving dinner in the last year?
Answer:
Last year, the average cost of a traditional Thanksgiving dinner was $48.91.
Step-by-step explanation:
If this year the cost of a traditional Thanksgiving dinner was decreased by 1.76% compared to last year, then
last year cosr - 100%
this year cost - 100%-1.76%=98.24%
So,
$x - 100%
$48.05 - 98.24%
Write a proportion:
[tex]\dfrac{x}{48.05}=\dfrac{100}{98.24}[/tex]
Cross multiply:
[tex]98.24x=48.05\cdot 100\\ \\98.24x=4,805\\ \\x\approx 48.91[/tex]
Last year, the average cost of a traditional Thanksgiving dinner was $48.91.
Final answer:
The average cost of a traditional Thanksgiving dinner in the last year was approximately $48.91, calculated by dividing this year's cost of $48.05 by the percentage decrease converted to a decimal (1 - 0.0176).
Explanation:
To calculate the average cost of a traditional Thanksgiving dinner in the last year, given this year's average cost and the percentage decrease, we can use the formula original price = discounted price / (1 - discount rate). The discount rate in this case is the percentage decrease in cost, expressed as a decimal. Given that this year's average cost is $48.05 and the decrease is 1.76%, we first convert the percentage to a decimal by dividing by 100, which gives us 0.0176.
The formula becomes:
original price = $48.05 / (1 - 0.0176)
original price = $48.05 / 0.9824
original price = $48.91 approximately
Therefore, the average cost of a traditional Thanksgiving dinner in the last year was about $48.91.
Select the graph that represents the given set. (Click on the graph until the correct one is showing.) B = {(1, 2), (2, 1), (3, 0), (4, -1)}
Answer:
3rd graph
Step-by-step explanation:
Hi, the answer would be the 3rd graph because each given set is shown on that coordinate relation.
Hope this helps, I attached the graph in case my explanation was confusing.
Third graph represents the set B = (1, 2), (2, 1), (3, 0), (4, -1).
Here, Set of points given as,
B = (1, 2), (2, 1), (3, 0), (4, -1).
We have to check the graph which represents the given set.
How to show points on graph?
Points are identified by stating their coordinates in the form of (x, y).
Where x is coordinate of x - axis and y is coordinate of y - axis.
Now,
Set of points given as,
B = (1, 2), (2, 1), (3, 0), (4, -1).
⇒ Third graph represents the set B = (1, 2), (2, 1), (3, 0), (4, -1).
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Can someone help me with number 6?
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].
A rectangular solar panel has a length that is 12 inches shorter than 3 times its width. If the perimeter of the panel is 160 inches, what are the dimensions of the panel?
Answer:
width = 23 inches
length = 57 inches
Step-by-step explanation:
Let x inches be the width of the rectangular solar panel.
So,
width = x inches
3 times the width = 3x inches
12 inches shorter than 3 times the width = 3x - 12 inches
length = 3x - 12 inches
The perimeter of the rectangle is
[tex]P=2(\text{Width}+\text{Length})[/tex]
Hence,
[tex]160=2(x+3x-12)\\ \\160=2(4x-12)\\ \\80=4x-12\ [\text{Divided by 2}]\\ \\4x=80+12\\ \\4x=92\\ \\x=23\ inches\\ \\3x-12=3\cdot 23-12=69-12=57\ inches[/tex]
walt averages 98 miles per hourin 4 races. if each race is 95 miles in lenght, how many miles did walt drive in the race?
Answer: 380
Step-by-step explanation: Did you mean to ask how much time it took him? If he’s doing 4 races each being 95 miles it would simply be 95 x 4 = 380
The total distance driven by Walt is calculated by multiplying the length of each race (95 miles) by the number of races (4), giving a total of 380 miles.
Explanation:Walt averages 98 miles per hour, each race is 95 miles long, and he completed 4 races. To find the total distance driven by Walt, we only need to know how long each race is and how many races Walt completed because the speed at which Walt drives does not affect the total distance he covered in the race. Therefore, to get the total distance, we simply multiply the length of each race by the number of races. In this case, Walt drove 95 miles x 4 = 380 miles in total.
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A rocket is divided into three sections: the payload and navigation section in the top, the fuel section in the middle, and the rocket engine section in the bottom. The top section is one-sixth the length of the bottom section. The middle section is one-half the length of the bottom section. The total length is 240 ft. Find the length of each section?
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
Samantha has cut a pastry into four parts. Suppose PS||RQand T is the midpoint of PR. Determine whether SPT = QRT brainly
Answer:
Here's ur answer
Step-by-step explanation:
find the area and circumference.
Answer:
area ≈ 176.7 cm²circumference ≈ 47.1 cmStep-by-step explanation:
The radius is half the diameter, so is ...
r = d/2 = 15 cm/2 = 7.5 cm
The area formula is ...
A = πr²
Filling in the radius, we have ...
A = π·(7.5 cm)² = 56.25π cm² ≈ 176.7 cm² . . . area
__
The circumference formula is ...
C = πd
Filling in the diameter, we have ...
C = π·(15 cm) = 15π cm ≈ 47.1 cm . . . circumference
Solve for x.
5+4/7(21+3x)=41
(note the 4/7 is a fraction not division)
Enter your answer in the box.
x =
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.
A rectangular park is w feet wide. Its length is 25 feet larger than its width. Record using algebra:
b
What is the length of a fence (including the gates) that was built around the park?
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.
Explanation: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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A triangle hss one angle mesuaring 3x degrees. A second angle mesures 2x + 20 degrees and the third angle measures 4x - 20 degrees. What is the value of x
Answer:
Step-by-step explan 3x+2x+20+4x-20=180. 9x=180 X=180÷9. X=20
Find the perimeter of this figure. Please show work.
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
Solve for n.
1/2n+3/4n=1/2
Answer:
n=2/5 or 0.4
Step-by-step explanation:
thx mate brainliest plzz
If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation (s+4)2=196 for s to find the length of a side of the patio.
Answer:
The answer is 10
Step-by-step explanation:
The side length cannot be negative, hence the side length of the patio will be 10 feet
Give the expression that represents the statement given as:
[tex](s+4)^2 = 196[/tex]
We need to get the length of the side of the patio "s"
[tex](s+4)^2 = 196\\s+4 = \pm\sqrt{196}\\s+4=\pm14[/tex]
Subtract 4 from both sides
[tex]s+4-4=14-4\\s=14-4\\s =10ft[/tex]
Since the side length cannot be negative, hence the side length of the patio will be 10 feet
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Kate and Bill secured a loan with a 75% loan-to-value ratio. The interest rate was 7.125% and the term was for 30 years. The first month's interest payment was $477.82. What was the appraised value of the property?a) $103,700b) $80,475c) $107,300d) $79,239
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]
Alexandra has 78 emails in her inbox. She deletes 47 emails.How many emails are left in het inbox? Draw jumps and level the number line to show your thinking.
Answer:
31 emails, but I can't do a # number line.
Step-by-step explanation:
Answer: There are 31 emails left in her inbox.
Step-by-step explanation:
Since we have given that
Number of emails in her inbox = 78
Number of emails she deleted = 47
We need to find the number of emails that are left in her inbox.
So, for left we would use "Subtraction operator"
So, Number of emails left in her inbox = Number of emails in her inbox - Number of emails she deleted.
[tex]=78-47\\\\=31[/tex]
Hence, there are 31 emails left in her inbox.
Explain the distance formula. Then use it to calculate the
distance between A(1, 1) and B(7, −7).
Answer:
[tex]dAB=10\ units[/tex]
Step-by-step explanation:
we know that
The distance formula is derived by creating a triangle and using the Pythagorean theorem to find the length of the hypotenuse. The hypotenuse of the triangle is the distance between the two points
The formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
A(1, 1) and B(7, −7)
Let
(x1,y1)=A(1, 1)
(x2,y2)=B(7, −7)
substitute the given values in the formula
[tex]dAB=\sqrt{(-7-1)^{2}+(7-1)^{2}}[/tex]
[tex]dAB=\sqrt{(-8)^{2}+(6)^{2}}[/tex]
[tex]dAB=\sqrt{64+36}[/tex]
[tex]dAB=\sqrt{100}[/tex]
[tex]dAB=10\ units[/tex]
Answer: The required distance between the points between A(1, 1) and B(7, −7) is 10 units.
Step-by-step explanation: We are given to explain the distance formula. Also, to calculate the distance between A(1, 1) and B(7, −7).
Distance formula : The distance between any two two points with co-ordinates (a, b) and (c, d) is given by
[tex]D=\sqrt{(c-a)^2+(d-b)^2}.[/tex]
Therefore, the distance between the points A(1, 1) and B(7, −7) is given by
[tex]D\\\\=\sqrt{(7-1)^2+(-7-1)^2}\\\\=\sqrt{36+64}\\\\=\sqrt{100}\\\\=10.[/tex]
Thus, the required distance between the points between A(1, 1) and B(7, −7) is 10 units.
John and Emily are brother and sister. John has as many brothers as sisters, and Emily has twice as many brothers as sisters. How many boys and girls are there in this family?
Answer:
4 boys, 3 girls
Step-by-step explanation:
brothers of John ⇒ x
sisters of John ⇒ y
John has as many brothers as sisters:
x = ybrothers 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:
2 (y-1) = x+ 1Now 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.
Explanation: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.
h(x)=(f o g)(x).
h(x)=3/x-2
g(x)=x−2
What is f(x)?
Sophie is buying fabric to make items for a craft fair. The table shows some combinations of how much of each color fabric she might buy. Which equations model the total yards of fabric Sophie will buy? Check all that apply.
A. X+Y=28
B. 28+X=Y
C. X-Y=28
D. 28-X=Y
E. 28-Y=X
Answer:
OPTION A.
OPTION D.
OPTION E.
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope and "b" is the y-intercept.
The Standard form of the equation of the line is:
[tex]Ax + By = C[/tex]
Where "A" is a positive integer, and "B" and "C" are integers.
Choose two points from the table and find the slope with this formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex].
Points:
[tex](1,27)\\\\(4,24)[/tex]
So we get that the slope is:
[tex]m=\frac{24-27}{4-1}=-1[/tex]
Let's substitute the slope and the coordinates of the point (1,27) into [tex]y=mx+b[/tex] and then solve for "b":
[tex]27=(-1)(1)+b\\\\27+1=b\\\\b=28[/tex]
Then, we get that the equation of the line in Slope-Intercept form is:
[tex]y=-x+28[/tex] or [tex]28-x=y[/tex]
In order to write it in Standard form, we can add "x" to both sides of the equation:
[tex]y+x=-x+28+x\\\\x+y=28[/tex]
We can solve for "x" by subtracting "y" from both sides of the equation:
[tex]x+y-y=28-y\\x=28-y\\\\28-y=x[/tex]
Answer:
A,D, and E
Step-by-step explanation:
What is the answer to this How wide is a poster that has a length of 9/2 feet and area of 45/4 square feet? A) 1/4 feet B) 5/8 feet C) 2/5 feet D) 5/2 feet
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:
The register that holds bit values (flags) that describe comparison operation results, control conditional BRANCH execution, or indicate actual or potential error conditions ____.
Answer:
program status word
Step-by-step explanation:
The acronym for PSW is Program Status Word. The Program Status Word or the PSW consists of status bits which shows the present Central Processing Unit state.
The PSW or Program Status Word is IBM System/360 architecture which means it is an independent model architecture for S/360 series of computers developed by IBM.
The status register contains all the information about the processor.
Thus the answer is program status word.
Find the solution of the following equation whose argument is strictly between 270^\circ270 ∘ 270, degree and 360^\circ360 ∘ 360, degree. Round your answer to the nearest thousandth. z^4=-625z 4 =−625
[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 temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean μ, the actual temperature of the medium, and standard deviation σ. What would the value of σ have to be to ensure that 95% of all readings are within 0.6° of μ?
Answer: 0.3061.
Step-by-step explanation:
Given : The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean [tex]\mu[/tex], the actual temperature of the medium, and standard deviation [tex]\sigma[/tex].
Let X be the random variable that represents the reading of the thermometer.
Confidence level : [tex]=95\%[/tex]
We know that the z-value for 95% confidence interval is 1.96.
Then, we have
[tex]-1.96<\dfrac{X-\mu}{\sigma}<1.96[/tex] [tex]z=\dfrac{X-\mu}{\sigma}[/tex]
[tex]\Rightarrow\ -1.96\sigma<X-\mu<1.96\sigma[/tex]
But all readings are within 0.6° of [tex]\mu[/tex].
So, [tex]1.96\sigma=0.6[/tex]
[tex]\Rightarrow\ \sigma=\dfrac{0.6}{1.96}=0.30612244898\approx0.3061[/tex]
Hence, the required standard deviation will be
The confidence level is 95% in normal distribution then The value of standard deviation is 0.3061.
What is a normal distribution?It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
Given
The temperature reading from a thermocouple placed in a constant temperature medium is normally distributed with mean μ, the actual temperature of the medium, and standard deviation σ.
Let x be the random variable that represents the reading of the thermometer.
The confidence level is 95%. Then the z-value for 95% confidence level interval is 1.96.
Then we have
[tex]-\ \ 1.96 \ < \dfrac{x- \mu}{\sigma} < 1.96\\-1.96 \sigma < x- \mu \ < 1.96 \sigma[/tex]
But all the readings are within 0.6° of μ. Then
[tex]1.96 \sigma = 0.6\\[/tex]
On solving
[tex]\sigma = \dfrac{0.6}{1.96}\\\\\sigma = 0.306122 \approx 3061[/tex]
Thus, the standard deviation is 0.3061.
More about the normal distribution link is given below.
https://brainly.com/question/12421652
D is the midpoint of CE . If CD = 5x and CE = 9x + 8, what is CD?
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
Fred is 4 times as old as Nathan and is also 27 years older than Nathan. Let f be Fred's age and let n be Nathan's age. Which system of equations represents this situation?
Answer:
The system of equations is
[tex]f=4n[/tex] and [tex]f=n+27[/tex]
Fred's age is 36 years old and Nathan's age is 9 years old
Step-by-step explanation:
Let
f -----> Fred's age
n ----> Nathan's age
we know that
[tex]f=4n[/tex]-----> equation A
[tex]f=n+27[/tex] ----> equation B
equate the equations and solve for n
[tex]4n=n+27\\4n-n=27\\3n=27\\n=9[/tex]
Find the value of f
[tex]f=4(9)=36[/tex]
therefore
Fred's age is 36 years old and Nathan's age is 9 years old
Answer:
The system of equations is f=4n and f=n+27
Step-by-step explanation:
A basketball team practices their shooting. The function f(x) represents the number of baskets made during practice, where x is the number of players at the practice. Does a possible solution of (12, 36) make sense for this function? Explain your answer.
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.