Answer:
Option D) (k − 12)/m
Step-by-step explanation:
We have to write an expression:
the quotient of the quantity k minus 12 and m.
k minus 12 can be written as:
[tex]k - 12[/tex]
The quotient of (k-12) and m means (k-12) is divided by m.
This can be written as:
[tex]\dfrac{k-12}{m}[/tex]
Thus, the correct answer is
Option D) (k − 12)/m
Final answer:
The expression for the quotient of 'k minus 12' and 'm' is correctly written as (k - 12) ÷ m, making option D the accurate choice.
Explanation:
The expression for the quotient of the quantity k minus 12 and m should be written as a fraction where the numerator is k minus 12 and the denominator is m. This is mathematically represented by the fraction (k - 12) ÷ m, which indicates the division of the expression k minus 12 by m. Therefore, the correct expression from the given options is D) k - 12 ÷ m.
In circle A, ∠BAE ≅ ∠DAE. Circle A is shown. Line segments A B, A E, and A D are radii. Lines are drawn from point B to point E and from point E to point D to form secants B E and E D. Angles B A E and E A D are congruent. The length of B E is 3 x minus 24 and the length of E D is x + 10. What is the length of BE? 14 units 17 units 27 units 34 units
Answer:
The missing figure is attached down
The length of BE is 27 units ⇒ 3rd answer
Step-by-step explanation:
In circle A:
∠BAE ≅ ∠DAELine segments A B, A E, and A D are radiiLines are drawn from point B to point E and from point E to point D to form secants B E and E DThe length of B E is 3 x minus 24 and the length of E D is x + 10We need to find the length of BE
∵ AB and AD are radii in circle A
∴ AB ≅ AD
In Δs EAB and EAD
∵ ∠BAE ≅ ∠DAE ⇒ given
∵ AB = AD ⇒ proved
∵ EA = EA ⇒ common side in the two triangles
- Two triangles have two corresponding sides equal and the
including angles between them are equal, then the two
triangles are congruent by SAS postulate of congruence
∴ Δ EAB ≅ Δ EAD ⇒ SAS postulate of congruence
By using the result of congruence
∴ EB ≅ ED
∵ EB = 3 x - 24
∵ ED = x + 10
- Equate the two expressions to find x
∴ 3 x - 24 = x + 10
- Add 24 to both sides
∴ 3 x = x + 34
- Subtract x from both sides
∴ 2 x = 34
- Divide both sides by 2
∴ x = 17
Substitute the value of x in the expression of the length of BE to find its length
∵ BE = 3 x - 24
∵ x = 17
∴ BE = 3(17) - 24
∴ BE = 51 - 24
∴ BE = 27
The length of BE is 27 units
Answer:
27
Step-by-step explanation:
I TOOK THE QUIZ, AND GOT 100%
The ratio of boys to girls in a class is 2 to 3. There are 12 boys in the class. How many girls are in the class?
Answer:
18
you do 2/3=12/n. and find out n (im to lazy to explain sorry)
A certain television is advertised as a 29-inch TV (the diagonal length). If the width of
the TV is 20 inches, how many inches tall is the TV?
Answer:
21 inches
Step-by-step explanation:
refer to attached graphics
we can find the height by the Pythagorean theorem.
diagonal² = width² + height²
height² = diagonal² - width²
we are given that diagonal = 29" and width = 20", hence
height² = 29² - 20²
height² = 841 - 400
height² = 441
height = √441 = 21 inches
URGENT Consider the function f(x)=x3+6x2−20x+450.
What is the remainder if f(x) is divided by (x−12)? Report your answer as a number only. Do not include (x−12) in your answer.
Answer:
[tex]remainder=2802[/tex]
Step-by-step explanation:
Remainder Theorem: When a polynomial [tex]f(x)[/tex] is divided by [tex](x-a)[/tex] the remainder will be [tex]f(a)[/tex]
[tex]Here \ \ f(x)=x^3+6x^2-20x+450[/tex]
[tex]It\ is\ divided\ by\ (x-12)[/tex]
[tex]Then\ remainder =f(12)\\\\remainder=(12)^3+6(12)^2-20\times12+450\\\\remainder=1728+6\times144-240+450\\\\remainder=1728+864-240+450\\\\remainder=3042-240\\\\remainder=2802[/tex]
Final answer:
To find the remainder of the polynomial f(x) = x³ + 6x² - 20x + 450 when divided by (x - 12), we evaluate f(12) using the Remainder Theorem, which yields a remainder of 2802.
Explanation:
To find the remainder when the function f(x) = x³ + 6x² - 20x + 450 is divided by (x - 12), we use the Remainder Theorem. This theorem states that the remainder of a polynomial f(x) divided by (x - a) is f(a). Therefore, to find the remainder of f(x) divided by (x - 12), we evaluate f(12).
Substitute x with 12 into the function:
f(12) = (12)³ + 6(12)² - 20(12) + 450
f(12) = 1728 + 864 - 240 + 450
f(12) = 2802
Thus, the remainder is 2802.
A bag contains 7 red marbles, 5 yellow marbles, 6 blue marbles, 4 green marbles, and 3 orange marbles. What is the probability of randomly selecting a yellow marble out of the bag?
The probability of randomly selecting a yellow marble out of the bag is [tex]\frac{1}{5}[/tex]
Step-by-step explanation:
A bag contains:
7 red marbles5 yellow marbles6 blue marbles4 green marbles 3 orange marblesWe need to find the probability of randomly selecting a yellow marble out of the bag
Probability is the ratio of number of favorable outcomes to the total number of possible outcomes P(A) = [tex]\frac{n(A)}{n(outcoms)}[/tex]
∵ A bag contains 7 red marbles, 5 yellow marbles, 6 blue
marbles, 4 green marbles, and 3 orange marbles
- Add all the color to find the number of total marbles
∴ n(all) = 7 + 5 + 6 + 4 + 3 = 25
∵ There are 5 yellow marbles
∵ P(yellow) = [tex]\frac{n(yellow)}{n(all)}[/tex]
∴ P(yellow) = [tex]\frac{5}{25}[/tex]
- Divide up and down by 5 to simplify the fraction
∴ P(yellow) = [tex]\frac{1}{5}[/tex]
The probability of randomly selecting a yellow marble out of the bag is [tex]\frac{1}{5}[/tex]
Learn more:
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Answer:
1/5
Step-by-step explanation:
Write the numbers in order from least to greatest.
-3, 3 1⁄3, -3 3⁄4, 3 1⁄10
Answer:
Step-by-step explanation:
-3 3/4, -3, 3 1/10, and then 3 1/3 i think this is correct...sorry if it is wrong though.
FLAGPOLE Julie is 6 feet tall. If she stands 15 feet from the flagpole and holds a cardboard square, the edges of the square line up with the top and bottom of the flagpole. Approximate the height of the flagpole
Answer:
44 ft
Step-by-step explanation:
Given: Julie is 6 feet tall
She stands 15 feet from the flagpole.
The edges of the square line up with the top and bottom of the flagpole.
Lets assume the height of flagpole be "h".
As given, the edges of the square line up with the top and bottom of the flagpole.
∴ Angle and base of triangle are same then ratio of corresponding sides are also equal.
Now, finding the height of flagpole by using tangent rule.
we know, [tex]tan\theta= \frac{Opposite}{adjacent}[/tex]
Remember, both the angle are equal.
∴ Ratio of opposite and adjacent leg for both right angle triangle= [tex]\frac{6}{15} : \frac{h-6}{15}[/tex]
We can put it; [tex]\frac{6}{15} = \frac{15}{h-6}[/tex]
Solving the equation now
⇒ [tex]\frac{6}{15} = \frac{15}{h-6}[/tex]
Multiplying both side by 15
⇒[tex]6 = \frac{15\times 15}{h-6}[/tex]
Multiplying both side by (h-6)
⇒ [tex]6\times (h-6) = 15\times 15[/tex]
Distributive property of multiplication
⇒ [tex]6h-36= 225[/tex]
Adding both side by 36
⇒[tex]6h= 225+36[/tex]
Dividing both side by 6
⇒[tex]h= \frac{261}{6}[/tex]
∴ [tex]h= 43.5\ feet[/tex] [tex]\approx 44 feet[/tex]
Hence, the height of flagpole is 44 feet.
Final answer:
To approximate the height of a flagpole given that Julie, who is 6 feet tall, lines up a cardboard square with the top and bottom of the flagpole while standing 15 feet away, we can use the principles of similar triangles. This results in a calculation showing that the flagpole is approximately 6 feet tall, the same as Julie's height.
Explanation:
The height of the flagpole can be approximated using similar triangles. Julie is 6 feet tall and stands 15 feet from the flagpole. Using the cardboard square, we understand that the triangle formed by Julie and her shadow is similar to the triangle formed by the flagpole and its shadow. Therefore, we can set up a proportion:
Julie's height / Julie's distance from flagpole = Flagpole's height / Flagpole's distance from cardboard.
If we assume that the cardboard square is held adjacent to Julie, the flagpole's distance from the cardboard is also 15 feet. The proportion simplifies to:
6 feet / 15 feet = Flagpole's height / 15 feet
Cross-multiplying to solve for the flagpole's height gives us:
Flagpole's height = 6 feet × (15 feet / 15 feet) = 6 feet
Therefore, the flagpole is approximately 6 feet tall.
3. (3x + 4) - (x + 2)
I don't understand plz help
Step-by-step explanation:
Perhaps you want to simplify the given expression: Let's do it.
[tex](3x + 4) - (x + 2) \\ \\ = 3x + 4 - x - 2 \\ \\ = 3x - x + 4 - 2 \\ \\ = 2x + 2 \\ this \: is \: the \: simplest \: form \: of \: the \: \\ given \: expression.[/tex]
Answer:
2(x + 1)
Step-by-step explanation:
(3x + 4) - (x + 2)
3x + 4 - x - 2
3x - x + 4 - 2
2x + 2
2(x + 1)
Which of the following equations can be a harmonic on a string that is 10 cm long? Select all that apply. (three correct answers)
A.) y=2sin(pi/5 x)
B.) y=2sin(2pi/7 x)
C.) y=2sin(pi/10 x)
D.) y=2sin(10pi x)
E.) y=2sin (5/2pi x)
Answer:
The options are: A, C and D
Step-by-step explanation:
The sine wave has a general form : y = A sin (BX)
Where A is the amplitude and B = 2π/period
So, we will check which of the options will be a harmonic on a string that is 10 cm long.
A.) y=2sin(pi/5 x)
B = π/5 ⇒ period = 2π/B = 2π ÷ π/5 = 2π * 5/π = 10
So, one cycle of y=2sin(pi/5 x) will be a harmonic on a string that is 10 cm long.
B.) y=2sin(2pi/7 x)
B = 2π/7 ⇒ period = 2π/B = 2π ÷ 2π/7 = 7
C.) y=2sin(pi/10 x)
B = π/10 ⇒ period = 2π/B = 2π ÷ π/10 = 20 = 2 * 10
So, half a cycle of y=2sin(pi/10 x) will be a harmonic on a string that is 10 cm long.
D.) y=2sin(10pi x)
B = 10π ⇒ period = 2π/B = 2π ÷ 10π = 1/5 = 10/50
So, 50 cycles of y=2sin(10pi x) will be a harmonic on a string that is 10 cm long.
E.) y=2sin (5/2pi x)
B = 5/2π ⇒ period = 2π/B = 2π ÷ (5/2π) = 4π²/5
So, options A, C and D can be a harmonic on a string that is 10 cm long.
Answer:
A. y=2sin(pi/5x)
C. y=2sin(pi/10x)
D. y=2sin(10pix)
Step-by-step explanation:
How many different ways can 5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players
1507 are the different ways can 5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players
Solution:
Given that,
5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players
This is a combination problem
Combinations are a way to calculate the total outcomes of an event where order of the outcomes does not matter
The formula is given as:
[tex]n C_{r}=\frac{n !}{r !(n-r) !}[/tex]
Where n represents the total number of items, and r represents the number of items being chosen at a time
Let us first calculate 5 baseball players from 12 baseball players
Here, n = 12 and r = 5
[tex]\begin{array}{l}{12 C_{5}=\frac{12 !}{5 !(12-5) !}} \\\\{12 C_{5}=\frac{12 !}{5 ! \times 7 !}}\end{array}[/tex]
For a number n, the factorial of n can be written as:
[tex]n !=n \times(n-1) \times(n-2) \times \ldots . \times 2 \times 1[/tex]
Therefore,
[tex]\begin{aligned}12 C_{5} &=\frac{12 \times 11 \times 10 \times \ldots \ldots \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\\\12 C_{5} &=\frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2} \\\\12 C_{5} &=792\end{aligned}[/tex]
Similarly, 4 basketball players be selected 13 basketball players
n = 13 and r = 4
Similarly we get,
[tex]\begin{aligned}&13 C_{4}=\frac{13 !}{4 !(13-4) !}\\\\&13 C_{4}=\frac{13 !}{4 ! \times 9 !}\end{aligned}[/tex]
[tex]13C_4 = 715[/tex]
Thus total number of ways are:
[tex]12C_5 + 13C_4 = 792 + 715 = 1507[/tex]
Thus there are 1507 different ways
To determine the number of ways to select 5 baseball players from 12, and 4 basketball players from 13, we use the combination formula for both and multiply the results, applying the Counting Principle.
Explanation:The question asks how many different ways can 5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players. This is a problem of combinatorics, specifically the use of combinations, since the order of selection does not matter.
To find the number of ways to select the baseball players, we use the combination formula C(n, k) = n! / (k! * (n-k)!), where 'n' is the total number to choose from, 'k' is the number to choose, and '!' denotes factorial. For the 5 baseball players from 12, it is C(12, 5).
For the basketball players, it's C(13, 4), as we are choosing 4 out of 13. To find the total number of ways to form the group, we multiply these two values together, because each combination of baseball players can be paired with each combination of basketball players, which is an example of the Counting Principle.
So, the calculation is C(12, 5) * C(13, 4).
Find the midpoint of the line segment
with end coordinates of:
(2,0) and (8,8)
Show working out pls
Answer:
midpoint (5,4)
Step-by-step explanation:
The midpoint(M) of a segment with endpoints (x₁ , y₁) and ( x₂, y₂) is
where x₁ = 2 and x₂ = 8
y₁ = 0 and y₂ = 8
M = [tex]\frac{x_1 + x_2}{2} ,\frac{y_1 + y_2}{2}[/tex]
M = [tex]\frac{2 + 8}{2} ,\frac{0 + 8}{2}[/tex]
M = 5 , 4
One positive number is three larger than another positive number. If sixteen times the reciprocal of the smaller number is added to nine times the reciprocal of the larger number, the sum is one. Find the two number.
Answer: = [tex]\frac{25+\sqrt{949} }{6}[/tex] and y = \frac{25+\sqrt{949} }{6} - 3.
Step-by-step explanation:
Take x as the larger number and y as the smaller number.
x + 3 = y
[tex]\frac{16}{y}[/tex]+ [tex]\frac{9}{x}[/tex] = 1
Substitute x + 3 for y in the second equation.
[tex]\frac{16}{x+3}[/tex]+ [tex]\frac{9}{x}[/tex] = 1
Make a common denominator.
[tex]\frac{16(x) + 9(x+3)}{(x+3)(x)} =1[/tex]
Simplify and get rid of that fraction.
[tex]16x + 9x + 27 = x^{2} + 3x[/tex]
[tex]x^{2} + 3x - 25x - 27 = 0[/tex]
[tex]x^{2} -22x - 27 = 0[/tex]
By quadratic formula (and because they must be positive), x = [tex]\frac{25+\sqrt{949} }{6}[/tex] and then y = \frac{25+\sqrt{949} }{6} - 3.
To solve for the two positive numbers, we can set up an equation and solve for x.
Explanation:Let's call the smaller number x and the larger number x + 3.
From the given information, we can write the following equation:
16(1/x) + 9(1/(x + 3)) = 1
To solve this equation, we can find a common denominator and then simplify:
16(x + 3)/(x(x + 3)) + 9x/(x(x + 3)) = 1
After simplifying and solving for x, we find that the smaller number is 4 and the larger number is 7.
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When Sabine set off to climb Mt. Marcy, she had 18 gummi bears in her bag.
When she returned to the lodge, she had 6 gummi bears left. How many
gummi bears did she eat during her hike?
A square plot of land has a side length of 50 meters.
It is surrounded by a footpath that is 3 meters wide.
What is the area of the footpath?
50 m
3 m
Your answer
Answer:
636
Step-by-step explanation:
the area of the footpath=
(50m+3m+3m)^2-50^2
Eva and her children went into a restaurant and where they sell hotdogs for $5 each and tacos for $2.50 each. Eva has $30 to spend and must buy at least 7 hotdogs and tacos altogether. If Eva decided to buy 2 hotdogs, determine the maximum number of tacos that she could buy.
Answer: 8 tacos
Step-by-step explanation: 2 hotdogs are $10 as they are $5 each. Tacos are $2.50 each. $2.50 x 8 equals $20. $20 + 10 = $30. Eva can buy 8 tacos.
After buying 2 hotdogs with $10, Eva will have $20 left. With the remaining $20, she can buy a maximum of 8 tacos at $2.50 each.
Explanation:Since Eva is determined to buy 2 hotdogs at $5 each, she will spend $10 on hotdogs. She has a total of $30 to spend, meaning she will have $20 left after purchasing the hotdogs. Tacos cost $2.50 each. Therefore, with the remaining $20, Eva can afford to buy a maximum of 8 tacos (since $20 divided by $2.50 equals 8). This will also meet the condition of purchasing at least 7 hotdogs and tacos in total.
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Seven divided by four hundred ninety three
Seven divided by four hundred ninety three
Answer:
0.0141987829615
Step-by-step explanation:
The volume of a box is 39.375 inches³. What is the volume of the box if it is scaled down by a factor of 1/10?
The volume of scaled object is 0.039375 cubic inches
Solution:
Given that,
Volume of box = 39.375 cubic inches
Scaled down by a factor = [tex]\frac{1}{10}[/tex]
The volume of a scaled object will be equal to the volume of object times scale factor cubed
Therefore,
Volume of scaled object = Volume of box x scale factor cubed
[tex]Volume\ of\ scaled\ object = 39.375 \times (\frac{1}{10})^3\\\\Volume\ of\ scaled\ object = 39.375 \times \frac{1}{1000}\\\\Volume\ of\ scaled\ object = 0.039375[/tex]
Thus volume of scaled object is 0.039375 cubic inches
What is 2 to the power of 3 over 2 equal to? (5 points) squre root of 8 cube root of 8 cube root of 16 square root of 16
Answer:
Square root of 8.
Step-by-step explanation:
Given:
Number given : 2 to the power of 3 over 2 equal ...
Using the laws of exponent:
We know that:
⇒ [tex]\sqrt{a} = (a)^1^/^2[/tex] and
⇒ [tex]\sqrt[3]{a}=(a)^1^/^3[/tex]
So,
According to the question:
2 to the power of 3 over 2 = [tex](2) ^3^/^2[/tex]
Using fractional exponent concept where : [tex]\sqrt[y]{a^x} = (a)^x^/^y[/tex]
This can be re-written as:
⇒ [tex](2^3)^1^/^2[/tex] and [tex]\sqrt{2^3}[/tex] that is equivalent to [tex]\sqrt{8}[/tex] as ...[tex]2^3=2\times 2\times 2=8[/tex]
Square root of 8 is our final answer.
write the equation of the line with the two given points (-12,14) . (6,-1)
Answer: 6y + 5x = 24
Step-by-step explanation:
The formula for calculating equation of line with two points is given as :
[tex]\frac{y-y_{1}}{x-x_{1}}[/tex] = [tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
[tex]x_{1}[/tex] = -12
[tex]x_{2}[/tex] = 6
[tex]y_{1}[/tex] = 14
[tex]y_{2}[/tex] = -1
substituting the values into the formula , we have
[tex]\frac{y-14}{x-(-12)}[/tex] = [tex]\frac{-1-14}{6-(-12)}[/tex]
[tex]\frac{y-14}{x+12}[/tex] = [tex]\frac{-15}{18}[/tex]
[tex]\frac{y-14}{x+12}[/tex] = [tex]\frac{-5}{6}[/tex]
cross multiplying , we have :
6(y - 14 ) = -5 ( x + 12 )
6y - 84 = -5x - 60
6y +5x = -60 + 84
6y + 5x = 24
What is the radius of a circle whose equation is x2 + y2 + 8x - 6y + 21 = 0?
units
Answer:r=2
Step-by-step explanation:
Answer:
A) 2
Step-by-step explanation:
I got it right on edge 2020.
Hope it helps :)
I purple you have a good year, stay safe.
G DHS hehhsjshsggdshjsnehejysgwfwgwhahahavavav
Joe, John and Jason are each a year apart in age. If the
sum of their ages i 39, how old is Jason?
equality
Substitute and Check (Answer in a Complete Sentence
HELP PLEASE!!!
Answer:
Jason is 14 year old.
Step-by-step explanation:
Given: Joe, John and Jason are each a year apart in age.
Sum of their age is 39.
Lets assume the age of Joe be x
∴ Age of Jahn will be [tex](x+1)[/tex]
And age of Jason will be [tex](x+2)[/tex]
Now, putting up an equation for the sum of their age.
∴ [tex]x+(x+1)+(x+2)= 39[/tex]
Opening the parenthesis.
⇒[tex]x+x+1+x+2= 39[/tex]
⇒[tex]3x+3= 39[/tex]
Subtracting both side by 3
⇒ [tex]3x= 36[/tex]
Dividing both side by 3
⇒[tex]x= \frac{36}{3}[/tex]
∴ [tex]x= 12\ years[/tex]
Hence, Joe is 12 year old.
Next subtituting the value of x to find age of Jason and John.
Jason= [tex](x+2)= 12+2[/tex]
∴Jason= 14 years.
John age= [tex](x+1)= 12+1[/tex]
∴ John Age= 13 years.
Hence, Jason is 14 year of age.
Can you show me the steps when reducing 58/48 to 29/24?
Answer:
58/48 to 29/24
Step-by-step explanation:
An easy way that I always remember is that if both the numbers are even, divide by 2. Keep dividing until you get a odd number and then you know that you have your lowest form.
Which of the following decimal numbers is the greatest
Is it
0.206
2.06
0.026
0.26
Answer:
converting to fraction:
0.206 = 0.206/1000 =206//1000
2.06 = 2.06/100 = 206/100
0.026 = 0.026/1000 = 26/1000
0.26 = 0.26/100 = 26/100
answer = 2.06
Step-by-step explanation:
The fraction having the greatest value will be 2.06.
What is a number system?A numeral system is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set in a consistent manner using digits or other symbols. In different numeral systems, the same sequence of symbols can represent different numbers.
The greatest value of the fraction will be calculated as below in the calculation:-
0.206 = 0.206/1000 =206//1000
2.06 = 2.06/100 = 206/100
0.026 = 0.026/1000 = 26/1000
0.26 = 0.26/100 = 26/100
Therefore, the fraction having the greatest value will be 2.06.
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Write a function that gives the car’s value, V(t), t years after it is sold.
5f + 3s +6
use f= 6 and s = 7
What is the decimal from of 12%
Answer:
0.12
Step-by-step explanation:
12/100
There can be many based on what the total amount of the number is.
But in this case I'll say 0.12
how would I find x?
Step-by-step explanation:
In the question figure,
∠ 1 = 115 °, ∠ 2 = 115 °, ∠ 3 = 120 °, ∠ 4 = 14x °, ∠ 5 = 133 °, ∠ 6 = 167 °, ∠ 7 = 138° and ∠ 8 = 18x °
To find, the value of x = ?
We know that,
The sum of all angles of heptagon = 1080°
∴ ∠ 1 + ∠ 2 + ∠ 3 + ∠ 4 + ∠ 5 + ∠ 6 + ∠ 7 + ∠ 8 = 1080°
⇒ 115 ° + 115 ° + 120 ° + 14x ° + 133 ° + 167 ° + 138° + 18x ° = 1080°
⇒ 32x ° + 788° = 1080°
⇒ 32x ° = 1080° - 788° = 292°
⇒ x ° = 9.125°
∴ x ° = 9.125°
the numerator of a fraction is 12 the gcf witch stand for great common factor of the numerator and denominator is 4. what is the denominator
Answer:
16.
Step-by-step explanation:
The denominator could be 16.
The GCF of 12 and 16 is 4.
Final answer:
The denominator of the fraction with a numerator of 12 and a GCF of 4 with the denominator is 12. You divide the numerator by the GCF and then multiply the result by the GCF to get the denominator.
Explanation:
The student is asking for the denominator of a fraction when the numerator is 12 and the greatest common factor (GCF) of the numerator and the denominator is 4. To find the denominator, you would divide the numerator (12) by the GCF (4). This gives us 12 ÷ 4, which equals 3. Therefore, the denominator of the fraction must be a number that when divided by the GCF (4) will give us a quotient of 3. Since the denominator is 4 times larger than this quotient, we multiply 3 by 4 to find the denominator. Therefore, the denominator is 3 × 4, which equals 12.
Convert the complex number z = -7 - 8i from rectangular form to polar form.
The polar form of z = -7 - 8i is:
z = 3√13 (cos 230.2° + i sin 230.2°)
Converting -7 - 8i to polar form:
The rectangular form of a complex number is given by z = a + bi, where a is the real part and b is the imaginary part. In this case, a = -7 and b = -8.
The polar form of a complex number is given by z = r(cos θ + i sin θ), where:
r is the modulus (or absolute value), which represents the distance of the complex number from the origin in the complex plane.
θ is the argument (or angle), which represents the direction of the complex number relative to the positive real axis.
1. Finding modulus (r):
r = √(a² + b²) = √((-7)² + (-8)²) = √(113) = √(13 * 9) = √13 * 3 (using factorization and perfect squares)
Therefore, r = 3√13.
2. Finding argument (θ):
θ = arctan(b/a) = arctan((-8)/(-7)) ≈ 50.2° (using the arctangent function on a calculator). However, this only gives one possible angle for the complex number.
Note: The arctangent function typically outputs values between -90° and 90°, which corresponds to Quadrant 1 or 4 in the complex plane. Since -7 - 8i lies in Quadrant 3, we need to add 180° to get the correct angle:
θ = 50.2° + 180° = 230.2°
Therefore, the polar form of z = -7 - 8i is:
z = 3√13 (cos 230.2° + i sin 230.2°)