Answers
We find the following conclusions concerning the telegraph wire:
Case 1: The perimeter of the circle is equal to 20π centimeters.
Case 2: The extra length for the telegraph wire is equal to 10π meters.
How to determine the perimeter of a circleIn this problem we find two cases where the perimeter of a circle must be computed for a telegraph wire, this can be done by using the following formula:
s = 2π · r
Where:
s - Perimeterr - RadiusNow we proceed to determine the perimeter for each case.
Case 1: (Please notice that diameter is twice the radius)
s = 2π · (20 / 2)
s = 20π cm
The circle has a perimeter of 20π centimeters.
Case 2: (The Earth has a radius of 6371000 meters)
Δs = 2π · (6371005 - 6371000)
Δs = 2π · 5
Δs = 10π m
An extra length of 10π meters is required for the telegraph wire.
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Related Questions
Carmen's swimming pool has the shape of a rectangular prism
with a length of 20 feet, a width of 10 feet, and a depth of 5
feet. The pool will be filled with water until the water level is 1 1/2 feet from the top of the pool. Which volume of water will be filled into the pool?
A. 1000 cubic feet
B. 700 cubic feet
C. 500 cubic feet
D. 300 cubic feet
Answers
[tex]1\dfrac{1}{2} = \dfrac{3}{2} = 1.5[/tex]
[tex]20 (10) (5-1.5) = 200(3.5) = 700 \ cubic \ feet[/tex]
Answer:
(B) 700 cubic feet
Step-by-step explanation:
The height of water in the pool is 5 - 1 1/2 = 3 1/2 feet.
The volume of water needed to fill the pool up to 3 1/2 feet is:
20 feet x 10 feet x 3 1/2 feet = 700 cubic feet
Therefore, the answer is (B) 700 cubic feet.
Translate the triangle then enter the new coordinates (1,4) (-3,2) (2,-2)
Answers
Step-by-step explanation:
the given pair (3, 2) is the length of the translation ?
it is really totally simple : you need to add the x value of the translation to the x-cordinate of each point. the same for the y value and the y- coordinates.
A' = (1 + 3, 4 + 2) = (4, 6)
B' = (2 + 3, -2 + 2) = (5, 0)
C' = (-3 + 3, 2 + 2) = (0, 4)
A dilation has center (0,0). Find the image of the point L(-4,3) for the scale
factor 3.
Answers
Answer: To find the image of point L(-4,3) after a dilation with a scale factor of 3 and center at the origin, we need to multiply the coordinates of point L by the scale factor.
The formula for dilation is:
(x', y') = (kx, ky)
where (x, y) are the original coordinates, (x', y') are the coordinates of the image, and k is the scale factor.
In this case, k = 3 and the coordinates of point L are (-4, 3).
So,
x' = kx = 3(-4) = -12
y' = ky = 3(3) = 9
Therefore, the image of point L(-4, 3) after the dilation with a scale factor of 3 and center at the origin is (-12, 9).
Step-by-step explanation:
Given the polynomial f(x) = x^3 + 3x^2 − x − 3, which of the following is true?
(x + 3) is a factor since f(3) = 0.
(x + 3) is a factor since f(−3) = 0.
(x − 3) is a factor since f(3) = 0.
(x − 3) is a factor since f(−3) = 0.
Answers
The true statement is that (x + 3) is a factor of f(x) = x³ + 3x² - x - 3 since f(-3) = 0.
How to find the factor of a polynomial ?A polynomial is an expression that consists of variables, terms, exponents and constants.
The factors are the polynomials which are multiplied to produce the original polynomial.
The factorisation of a polynomial is breaking the polynomial as a products.
Therefore,
f(x) = x³ + 3x² - x - 3
Hence, let's use the factor (x + 3)
Therefore,
f(-3) = (-3)³ + 3(-3)² - (-3) - 3
f(-3) = -27 + 27 + 3 - 3
f(-3) = 0
Therefore, the factor is (x + 3) since f(-3) = 0
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Help me right one gets brainiest
Answers
Answer:
7x = 7
with solution of x = 1
This is the first choice
Step-by-step explanation:
Choice 1
7x = 7 => x = 7/7 = 1
Choice 2
7x = 7x provides no additional info and therefore there are an infinite number of solutions; any value of will satisfy
Choice 3
x + 1 = x + 1; infinite solutions for reasons cited under choice 2
Choice 4
x + 1 = x + 2
Eliminating x from both sides gives 1 = 2
Such an equation will have no solution; no value of x can satisfy the equation
Correct answer: Choice 1: 7x = 7
Consider the following theorem. Theorem: For any integer , is odd if and only if + 1 is even.
Construct a proof for the theorem by selecting sentences from the following scrambled list and putting them in the correct order. Note that each statement will be used at most once (or not at all).
Since 2 | 2( + 1), 2 | ( + 1).
Suppose is odd.
Therefore, + 1 is even.
Suppose + 1 is even.
Therefore, + 1 is odd.
Thus, for some integer , + 1 = 2 + 1.
Thus, for some integer , = 2.
Thus, for some integer , = 2 + 1.
Therefore, is odd.
Thus, for some integer , + 1 = 2.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Now = (2 - 1) = 2( - 1) + 1 .
Proof:
(⇒)
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(⇐)
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Answers
The answer of the given question based on the theorem For any integer , is odd if and only if + 1 is even the answer is for any integer , is odd if and only if + 1 is even.
What is Theorem?A theorem is statement in mathematics that has been proven to be the true based on logical reasoning and the mathematical principles. In other words, theorem is a proposition that can be demonstrated or proved to be true through logical argument.
Proof:
(⇒) Suppose is odd.
Thus, for some integer , = 2 + 1.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Therefore, + 1 is even.
(⇐) Suppose + 1 is even.
Then, for some integer , + 1 = 2.
Thus, for some integer , = 2 - 1 = 2( - 1) + 1 .
Since 2 | 2( + 1), 2 | ( + 1).
Therefore, is odd.
Thus, we have shown that for any integer , is odd if and only if + 1 is even.
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What I still need to know is, what's Jonothan FLAW
Answers
As portrayed in the show, his biggest flaw is not knowing how to be a man. He had spent his life avoiding her responsibility, but she came anyway. He unnecessarily bows to Daenerys after she agrees to fight the Night King.
Perhaps Jon's biggest lingering flaw can be seen in book five: he's not a very good communicator. He struggled to sell his long-term ideas to his subordinates, and he also struggled to balance being overly friendly with them and alienating them, and he chose the latter. Other characters portray him as gruff, hard to read, and lacking in articulation, which isn't great when you're trying to bring about sweeping social reform. This is especially serious when you are dealing with people who are ready to kill you to prevent the implementation of the aforementioned social reforms.
I always say that Robb and Jon have the opposite problem: Robb is very good at selling himself, selling his ideas and thinking short term.
Jon is better at long-term strategic thinking and big-picture thinking, but lacks Robb's ability to really bring people to him.
Then of course, perhaps paradoxically, you have the underlying temper that he sometimes explodes (think of him heaving Thorne's throat with one hand, strangling Iron Emmett, etc.). He also felt very self-pity and defensive about his half-blood identity, both before and after joining the Night's Watch, while considering himself socially and militarily superior to his brothers in the Night guard. He needs someone to line him up (Donal Nye in the book, Tyrion in the show).
Then you have his flaws, which others might interpret as strengths. Lots of people criticized Jon for getting too involved in politics, like helping Stannis, throwing Creeker Stark in an ice prison, and getting Aryskar Stark to marry Saigon. Effective and ruthless efficiency, in the eyes of some, could on the other hand be described as a senseless (insidious?) intervention, the Pink Letter is Jon's intervention, which catches up with him. I also don't think the fictional character Harry Potter is that exciting.
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Using the bag of marbles below, find the probability of pulling a blue marble two times in a row. After pulling the first blue marble, you do NOT replace it in the bag before pulling the second one. Each marble has an equally likely chance of being pulled. Answer as a simplified fraction.
Answers
The probability of picking two blue marbles is 1/21.
What is the probability?Probability is used to determine how likely it is that a random event would occur. The probability of a random event happening lies between 0 and 1. If it is certain the event would happen, the probability value is 1 and if it is certain that the event would not happen, the probability value is 0.
Probability of picking two blue marbles = (number of blue marbles / total number of marbles) x (number of blue marbles -1 / total number of marbles - 1)
Total number of marbles = 7
(2/7) x (2-1/7-1)
2/7 x 1/6 = 1/21
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A rectangular piece of carpet has dimensions of 6 feet by 8 feet. A larger rectangular piece of carpet has dimensions that are times longer. What area, in square feet, does the larger piece of carpet cover?
Answers
Answer:
45
Step-by-step explanation:
Suppose that 20% of all copies of grade 7 Science text book fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.
b. What is the probability that at least 8 fail the test?
Answers
binomial distribution
Question: Suppose that 20% of all copies of grade 7 Science textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.b. What is the probability that at least 8 fail the test?To find the probability that at least 8 fail the test, we will use the binomial probability formula.The binomial distribution is a statistical distribution that describes the probability of success or failure in an experiment or survey with two possible outcomes.Suppose a trial consists of n independent experiments, each of which has two possible outcomes: either success with probability p or failure with probability q = 1 – p. If X is the number of successes in n trials, then X is said to follow a binomial distribution with parameters n and p. The probability that X = x is given by;P(X = x) = nCxpx(1-p)n-xwhere nCx = n! / x! (n-x)! is the binomial coefficient which counts the number of ways that x successes can be chosen from n trials.Using the formula, we have;P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)Here n = 15, p = 0.2 and q = 0.8P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)= 15C8 (0.2)^8 (0.8)^7 + 15C9 (0.2)^9 (0.8)^6 + 15C10 (0.2)^10 (0.8)^5 + 15C11 (0.2)^11 (0.8)^4 + 15C12 (0.2)^12 (0.8)^3 + 15C13 (0.2)^13 (0.8)^2 + 15C14 (0.2)^14 (0.8)^1 + 15C15 (0.2)^15 (0.8)^0= 0.0094 + 0.0267 + 0.0524 + 0.0864 + 0.1174 + 0.1312 + 0.1184 + 0.0352= 0.5771Therefore the probability that at least 8 fail the test is 0.5771 (rounded to four decimal places).Answer: 0.5771
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Lorie Reilly decides to go back to college. For transportation, she borrows money from her parents to buy a small car for $7,200. She plans to repay the loan in 7 months. What amount can she deposit today at 5.25% to have enough money to pay off the loan?
Answers
Lorie should deposit $6,986.05 today at 5.25% to have enough money to pay off the loan in 7 months.
What amount can Lorie Reilly deposit today at 5.25% to have enough money to pay off the loan?To determine the amount that Lorie should deposit today to have enough money to pay off the loan.
We use the simple interest formula for accrued amount.
A = P( 1 + rt )
Where A is total amount, P is initial amount, r is interest rate and t is elapsed time.
Given that;
Accrued amount A = $7,200Time = 7 monthsrate R = 5.25%Principal P = ?First, converting R percent to r a decimal
r = R/100
r = 5.25/100
r = 0.0525
Next, we put time into years for simplicity,
t = 7 months / 12 months
t = 7/12 year
Plug the values into the above formula.
A = P( 1 + rt )
P = A / ( 1 + rt )
P = $7,200 / ( 1 + ( 0.0525 × 7/12 ) )
P = $7,200 / ( 1 + 0.030625 )
P = $7,200 / 1.030625
P = $6,986.05
Therefore, the principal or initial amount is $6,986.05.
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The amount she should deposit today to have enough money to pay off the loan would be = $1,060
How to calculate the amount that should be deposited today?The cost of the small car that Lorie Reilly needs to buy for college = $7,200.(P)
The number of months that she plans to repay the money = 7 months = 0.58 yr.(T)
The rate at which the money needs to be deposited = 5.25%. (R)
The simple interest = Principal× Time × Rate /100
Simple interest = 7200× 0.58 × 5.25/100
= 21924/100
= $219.24
The amount she is expected to pay after 7 months = 7200+219.24 = 7419.24
The amount she should deposit today to have enough money to pay off the loan = 7419.24/7 = $1,060( approximately)
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5. Explain the properties of logarithms that you
would need to use to solve the following:
log₂2x + log₂4 = 5
Answers
Answer:
x = 4
Step-by-step explanation:
log a + log b = log ab
log(base b) x = y, then b^y = x
log₂2x + log₂4 = 5
log₂ (2x × 4) = 5
log₂ 8x = 5
8x = 2^5
8x = 32
x = 4
1. Explain what Marc did in steps 4 and 5.
2. Why did he do this?
3. Create your own radical equation and explain how to solve it.
4. Is there an extraneous solution to your equation?
Answers
2.He did this in order to simplify the cube root of five to the fourth
3. Sqrt 16
4. There are no extraneous solutions
Need urgently!! Fast asap
Answers
Answer:
1a) x=40°
2a) x=30°
Step-by-step explanation:
Use the fact that the sum of the angles of a triangle is always 180°.
So for 1a), x + x+40 + x+20 = 180
Simplify to
3x + 60 = 180
3x = 120
x = 40
apply the same logic to all the others.
For circles, the sum of the angles is 360°, so for 2b)
5x+3x+4x = 360 =>
12x = 360
x = 30°
Drag each label to the correct location on the image.
Dave, Frida, Natalie, and Robbie have four adjacent seats at a baseball park. They randomly chose an order to sit in. Make a list of all the possible ways in which they can sit in the four seats. Then match the events to their correct probabilities.
the probability of Dave and
Natalie sitting together
the probability of sitting
boy, girl, boy, girl or
girl, boy, girl, boy
the probability of the two
boys sitting in the middle
the probability of Frida
and Natalie sitting
together
the probability of Robbie
sitting between the girls
the probability of Natalie
sitting between Dave
and Robbie
Answers
The probability of Natalie sitting between Dave and Robbie: 1/6
What is Probability?Probability is a branch of mathematics that looks at the likelihood of a particular event occurring. It is used to determine the chance that a certain outcome will occur, such as the probability of rolling a particular number when a six-sided die is thrown. Probability can also be applied to more complex scenarios, such as determining the likelihood of a particular stock's price increasing over a certain period of time. The probability of an event is typically expressed as a number between 0 and 1, where 0 indicates that the event will not occur and 1 indicates that the event is certain.
Dave, Frida, Natalie, Robbie:
1. Dave, Frida, Natalie, Robbie
2. Dave, Natalie, Frida, Robbie
3. Frida, Dave, Natalie, Robbie
4. Frida, Natalie, Dave, Robbie
5. Natalie, Dave, Frida, Robbie
6. Natalie, Frida, Dave, Robbie
Probabilities:
1. the probability of Dave and Natalie sitting together: 1/6
2. the probability of sitting boy, girl, boy, girl or girl, boy, girl, boy: 1/2
3. the probability of the two boys sitting in the middle: 1/6
4. the probability of Frida and Natalie sitting together: 1/6
5. the probability of Robbie sitting between the girls: 1/6
6. the probability of Natalie sitting between Dave and Robbie: 1/6
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3n-6=-21 ieuhfuewigdhfu43w
Answers
Answer:
-5
Step-by-step explanation:
3n-6=-21
+6 +6
3n=-15 --> divide both sides by 3
n=-5
Find the principal.
Annual rate of interest
=
6. 5
%
=6. 5%equals, 6, point, 5, percent
Period
=
4
=4equals, 4 years
Total interest
=
1222
=1222equals, 1222 rupees
Principal
=
=equals
rupees
Answers
The principal is $467.69 rupees (rounded to two decimal places).
The concept of simple interest is a type of interest calculation based on a fixed percentage rate applied to an initial amount of money, called the principal, for a specific period of time
To find the principal, we can use the formula:
Total Interest = (Principal x Rate x Time) / 100
Where,
Rate = Annual Rate of Interest
Time = Period in years
Plugging in the given values, we get:
1222 = (Principal x 6.5 x 4) / 100
Simplifying the equation:
1222 = (26 x Principal) / 10
Multiplying both sides by 10/26:
Principal = $467.69
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what number is missing?
Answers
The number that is needed in the missing cell on the figure in the sequence of numbers, is 4.
How to find the missing figure ?The pattern of the numbers on the figure is such that the number on top of two numbers is acquired by subtracting the smaller of the two numbers below the number from the larger number.
This is why 2 is placed on top which was reached with :
= 5 - 3
= 2
The number that is missing can therefore be reached by subtracting 3 from 7 :
= 7 - 3
= 4
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If you place 37 foot ladder against the top of a 15 foot building
Answers
I you place the 37 foot ladder against the top of the 15 foot building, it will reach a height of 52 feet.
If you place a 37 foot ladder against the top of a 15 foot building, then the ladder will reach a height of 52 feet (37+15=52). To calculate this, you can use the formula for addition, which is x + y = z, where x is the height of the building, y is the height of the ladder, and z is the total height the ladder will reach. In this case, x = 15 and y = 37, so the total height the ladder will reach will be z = 15 + 37 = 52. This means that if you place the 37 foot ladder against the top of the 15 foot building, it will reach a height of 52 feet.
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For the equation y=8-2, complete the table and plot the ordered pairs on the graph.
Оо
Size
O
X
y
< Previous
-3
-2
-1
0
2
3
Answers
Step-by-step explanation:
For the equation y=8-2, complete the table and plot the ordered pairs on the graph.
Оо
Size
O
X
y
< Previous
-3
-2
-1
0
2
3
$4, 700 is invested in an account earning 6.8% interest (APR), compounded continuously. Write a function showing the value of the account after t years, wher the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage o growth per year (APY), to the nearest hundredth of a percent.
Answers
The function for the value of the account after t years is:[tex]\rm V(t) = 4700 \times e^{(0.068t)[/tex].
The percentage growth per year (APY) is 7.03%.
What is formula for continuous compounding?The formula for the value of an account with continuous compounding is given by:
[tex]\rm V = Pe^{(rt)[/tex]
where:
V is the value of the account after t years
P is the initial principal amount invested (in this case, $4,700)
e is the mathematical constant (approximately equal to 2.71828)
r is the annual interest rate expressed as a decimal (in this case, 0.068)
t is the time period in years.
Therefore, the function for the value of the account after t years is:
[tex]\rm V(t) = 4700 \times e^{(0.068t)[/tex]
To find the annual percentage yield (APY), we use the formula:
[tex]APY = 100 \times (e^r - 1)[/tex]
where r is the annual interest rate expressed as a decimal (in this case, 0.068).
[tex]APY = 100 \times (e^{0.068} - 1)[/tex]
= 7.03%
Therefore, the percentage growth per year (APY) is 7.03%.
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Show your work plssss
Answers
As a result, the Moon orbits the Earth roughly 2.8 billion times while the Sun revolves once around the Milky Way's nucleus.
How is earths rotation calculated?
One cycle of the Sun around the Milky Way's nucleus is thought to take between 225 and 250 million years to complete. The synodic month, or the length of time it takes the Moon to complete one orbit of the Earth, lasts roughly 29.5 days.
We must divide the total time of one orbit of the Sun around the Milky Way by the amount of time it takes for the Moon to orbit the Earth once in order to determine how many times the Moon orbits the Earth during one orbit of the Sun around the Milky Way's centre.
The result is (Total time for one orbit of the Sun around the centre of the Milky Way)/(Time for one orbit of the Moon around the Earth) = (225 million years x 365.25 days per year)/(29.5 days per orbit of the Moon) = around 2.8 billion orbits of the Moon around the Earth.
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You invest $3000 in an account at 3.5% per year simple interest. How much will you have in the account at the beginning of the 7th year? Round your answer to the whole dollar.
Answers
Answer: the answer is that you will have $3630 in the account at the beginning of the 7th year.
Step-by-step explanation:
I = Prt
Using this formula:
I = Prt
I = 3000 x 0.035 x 6
I = $630
Therefore, the total amount in the account at the beginning of the 7th year will be
Total = Principal + Interest
Total = $3000 + $630
Total = $3630
Needddd the answerrrrr plssss
Answers
The expοnential functiοn fοr the given table is y = 5(2)ˣ.
What is a Functiοn?A functiοn is relatiοn between set οf inputs (called dοmain) and set οf οutputs (called range), where each input is assοciated with exactly οne οutput. A functiοn is usually denοted by symbοl like f, and is written as f(x), where x is element οf dοmain.
we can use the general fοrm οf an expοnential functiοn, which is:
y = abˣ
where a and b are cοnstants that we need tο determine.
Using the given values οf x and y in the table, we can fοrm fοur equatiοns as fοllοws:
2.5 = ab⁽⁻¹⁾
5 = ab⁰
10 = ab¹
20 = ab²
Dividing the secοnd equatiοn by the first equatiοn, we get:
5/2.5 = (ab⁰)/(ab⁽⁻¹⁾ )
2 = b
Substituting this value οf b in the οther equatiοns, we get:
2.5 = a(2)⁽⁻¹⁾
a = 5
10 = 5(2)¹
10 = 10
20 = 5(2)²
20 = 20
Therefοre, the expοnential functiοn fοr the given table is:
y = 5(2)ˣ.
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Function: f (x) = x + 2
Answers
Step-by-step explanation:
let f(x) be y
y=x+2
make x the subject of the formula
x=2-y
Answer:
Step-by-step explanation:
Write an explicit formula for an,the nth term of the sequence 23,19,15,
Answers
If sequence 23,19,15, the nth term of the sequence 23, 19, 15 is given by the formula aₙ = 23 - 4(n-1).
To find the explicit formula for the nth term of the sequence 23, 19, 15, we need to look for a pattern in the sequence. We can observe that each term in the sequence is obtained by subtracting 4 from the previous term. Therefore, we can write the recursive formula for this sequence as:
a₁ = 23
aₙ = aₙ-1 - 4, for n ≥ 2
To find the explicit formula, we need to use the recursive formula to eliminate the dependence on the previous term. We can do this by repeatedly substituting the recursive formula into itself, until we get an expression for an in terms of a1:
a₂ = a1 - 4 = 23 - 4 = 19
a₃ = a₂ - 4 = (a₁ - 4) - 4 = a1 - 8 = 23 - 8 = 15
a₄ = a₃ - 4 = (a₁ - 8) - 4 = a1 - 12 = 23 - 12 = 11
From this pattern, we can see that each term can be written as:
aₙ = a₁ - 4(n-1)
Substituting a1 = 23, we get the explicit formula:
aₙ = 23 - 4(n-1)
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i need help on this question
Answers
Option C is correct. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by -3. The solution to system B will be the same as the solution to system A.
What is equations ?
An equation is a mathematical statement that shows the equality of two expressions. It usually contains one or more variables and is solved to find the value of the variable(s) that make the equation true. Equations are used extensively in mathematics, physics, engineering, and many other fields to describe relationships between quantities and to solve problems.
The given system of equations is:
System A:
x + y = 5
3x - 2y = -35
And the solution to this system is (-7, 2).
To obtain System B, we need to manipulate the equations in System A. Let's start by isolating y in the first equation:
y = 5 - x
Now we can substitute this expression for y into the second equation and simplify:
3x - 2(5 - x) = -35
3x - 10 + 2x = -35
5x = -25
x = -5
Now that we have x = -5, we can substitute it back into either of the original equations to solve for y:
x + y = 5
-5 + y = 5
y = 10
So the solution to System B is (-5, 10).
Comparing the two solutions, we see that they are not the same. Therefore, the correct answer is A. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by 3. The solution to System B will not be the same as the solution to System A.
Therefore, Option C is correct. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by -3. The solution to system B will be the same as the solution to system A.
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What is the factor of the equation
Answers
Answer:
(2x - 3) (x + 3)
Step-by-step explanation:
Let's check
(2x - 3) (x + 3)
2x² + 6x - 3x - 9
2x² + 3x - 9
So, (2x - 3) (x + 3) is the correct answer.
The cubic function has three roots. Verify
that x=2 is one of them and find the other two.
Answers
a) Verified
b) 4 + √7 and 4 - √7
Given a cubic function with three roots, one of the roots is given as x = 2. The other two roots can be found by using the factor theorem. Using factor theorem, we have(f(x) = x³ - 10x² + 27x - 18)(x - 2) is a factor of f(x)if and only if f(2) = 0.So, we have f(2) = 2³ - 10(2²) + 27(2) - 18 = 0. Therefore, (x - 2) is a factor of f(x) and we can find the other two roots by factorizing the cubic function. Using long division method, we get f(x) = (x - 2)(x² - 8x + 9) Now, to find the other two roots, we solve the quadratic equation x² - 8x + 9 = 0 using the quadratic formula. x = (-b ± √(b² - 4ac))/2a. We have a = 1, b = -8 and c = 9. Substituting the given values in the above formula, we get x = (8 ± √(8² - 4(1)(9)))/2= (8 ± √28)/2= 4 ± √7. Therefore, the three roots of the cubic function are 2, 4 + √7 and 4 - √7.
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O RATIONAL EXPRESSIONS Multiplication and division of 3 rational expressions Simplify. (x^(2)+9x+14)/(2x)*(x^(2)-4x)/(x^(2)-49)-:(x-4)/(x-7)
Answers
The expression (x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) when simplified is 2x + 4
How to simplify the expressionGiven that
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7)
To start with, we factorize the factors of the expression
The factored expressions are
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) = (x + 2)(x + 7)/(2x) * x(x - 4)/(x - 7)(x + 7) ÷ (x-4)/(x-7)
Cancel out the common factors to reduce the expression
So, we have
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) = (x + 2)/(2) * (x - 4)/(x - 7) ÷ (x-4)/(x-7)
Express division as product
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) = (x + 2)/(2) * (x - 4)/(x - 7) * (x - 7)/(x - 4)
Evaluate the quotient
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) = (x + 2)/(2)
So, we have
(x² + 9x + 14)/(2x) * (x² - 4x)/(x² - 49) ÷ (x-4)/(x-7) = 2x + 4
Hence, the solution is 2x + 4
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