The dog shelter has Labradors, Terriers, and Golden Retrievers available for adoption. P(terriers) = 85% is likely Option B
What is probability?Generally, Probability is the area of mathematics that deals with providing numerical descriptions of the likelihood that an event will take place or the likelihood that a statement will be correct.
A number between 0 and 1 may be thought of as representing the chance of an event happening, with 0 indicating that the occurrence is impossible and 1 indicating that it will definitely happen.
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The expression (8)(2)(−1.5) represents the change in the scuba diver’s elevation after 8 minutes.
Find the change in elevation.
The change in elevation of the scuba diver after 8 minutes is -24.
What is the algebraic expression?Algebraic expressions are mathematical statements with a minimum of two terms containing variables or numbers. Unknown variables, numbers, and arithmetic operators make up an algebraic expression. It contains no equality or inequality symbols.
The expression (8)(2)(-1.5) represents the change in the scuba diver's elevation after 8 minutes.
To determine the change in elevation, we need to evaluate the expression.
(8)(2)(-1.5) = (16)(-1.5) = -24
Thus, the change in elevation is -24.
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Every week, Enrique walks one mile each day for 4 days of the week. He can usually walk 1 mile in about 14 to 16 minutes.
Estimate the number of minutes Enrique will spend walking over an 8 week period. Explain how you determined your estimate.
Write an equation that represents the number of miles, m, Enrique walks in w weeks.
An equation that represents the number of miles, m, Enrique walks in w weeks will be N = 32T.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that every week, Enrique walks one mile each day for 4 days a week. He can usually walk 1 mile in about 14 to 16 minutes. Estimate the number of minutes Enrique will spend walking over an 8-week period.
The expression for the total time will be calculated as:-
N = 32 x T
Here, T is the time in minutes for running per mile.
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A graphic designer wants to reproduce the
logo of a mountain-climbing club for some club stationery.
The logo is a triangle with the angles shown.
W Sheliqsiz
A. The designer wants the base to be 2 inches long. How
should the triangle be drawn?
Ci ilority Postulate help the
Timis
60°
50°
Answer: Sorry!
Step-by-step explanation:
Hello! I can help you if you can add the photo and fix the typos.
What is the formula for computing the present value of F in a financial transaction involving compound interest?
Answer: A
Step-by-step explanation:
Present value of F (P) = Compound amount (F) (1+periodic rate)
Mary pays income tax according to the graduated schedule shown below.
A 3-column table with 6 rows. Column 1 is labeled If taxable income is over with entries 0 dollars, 7,825 dollars, 31,850 dollars, 77,100 dollars, 160,850 dollars, 349,700 dollars. Column 2 is labeled but not over with entries 7,825 dollars, 31,850 dollars, 77,100 dollars, 160,850 dollars, 349,700 dollars, no limit. Column 3 is labeled the tax is with entries 10 percent of the amount of 0 dollars, 782 dollars and 50 cents plus 15 percent of the amount of 7,825 dollars, 4,386 dollars and 25 cents plus 25 percent of the amount of 31,850 dollars, 15,698 dollars and 75 cents plus 28 percent of the amount over 77,100 dollars, 39,148 dollars and 75 cents plus 33 percent of the amount of 160,850 dollars, 101,469 dollars and 25 cents plus 35 percent of the amount over 349,700 dollars.
If Mary’s taxable income is $68,562, how much income tax does she owe, rounded to the nearest dollar?
a.
$13,564
b.
$17,140
c.
$21,527
d.
$12,349
Mary owes $13564 income tax.
What is income tax?Governments apply income taxes as a sort of tax on the income produced by people and businesses under their control. Government duties, public services, and citizen goods are all paid for through income tax revenue.
Given that the taxable income is $68,562.
Income tax is the tax charged on an individual's or entity's income
Using the table as a guide, $68,562 falls within the income range $31,850 - $77,100
So, the tax is $4386 added to 25% of the excess over $31850
This is calculated as:
tax = $4386 + 25%(income - 31,850)
In our case,
income = $68562
Substitute $68,562 for income
Tax = $4386 + 25%(68,562 - 31,850)
Solve the expression in the bracket
tax = 4386 + 9178
Add the terms of the expression
tax = 13,564
therefore, Mary owes $13564 in income tax.
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Let f be the function given by f(t) =3/1+t^2 a. Find the first four nonzero terms and the general term for the power series expansion of f(t) about t=0. b. Given g(x)=ſ8(t)dt , find the first four nonzero terms and the general term for the power series expansion of g(x) about x=0. c. Find the interval of convergence of the power series for g(x).
a) The first four nonzero terms of the Taylor series are 3, -3t^2, 3/3t^4, -3/5t^6
b) The first four nonzero terms of the power series expansion of g(x) about x=0 are 3x, -x^3, 3/5x^5, -3/7x^7.
c) The interval of convergence for the power series of g(x) is (-sqrt(5/3), sqrt(5/3)).
What is a function?
A function is a mathematical relationship between two variables, often denoted as "f(x)" or "y = f(x)". A function assigns a unique output to each input within its domain. In other words, for every value of x, there is a corresponding value of y, and each value of x can only correspond to one value of y.
a. To find the first four nonzero terms and the general term for the power series expansion of f(t) about t=0, we need to find the Taylor series of f(t) centered at 0. The general term of a Taylor series is given by:
f(t)^(n) / n! * (t-0)^n
The Taylor series of f(t) = 3/(1+t^2) centered at 0 is:
f(t) = 3 - 3t^2 + 3/3t^4 - 3/5t^6 + ...
The first four nonzero terms of the Taylor series are 3, -3t^2, 3/3t^4, -3/5t^6
b. To find the first four nonzero terms and the general term for the power series expansion of g(x) = ∫f(t)dt about x=0, we need to integrate the Taylor series of f(t) term by term.
g(x) = ∫f(t)dt = (3t - 3/3t^3 + 3/5t^5 - ...) from 0 to x
g(x) = 3x - 3/3x^3 + 3/5x^5 - ...
The first four nonzero terms of the power series expansion of g(x) about x=0 are 3x, -x^3, 3/5x^5, -3/7x^7
c. The interval of convergence of a power series is the set of values of x for which the series converges. To find the interval of convergence for g(x), we can use the ratio test.
The ratio test states that if the limit of |a(n+1)x^(n+1)|/|a(n)x^n| as n approaches infinity is less than 1, the series converges for all x.
|3/7x^7|/(|3/5x^5|) = |x^2|/(|3/5|) < 1 when |x| < sqrt(5/3)
So, the interval of convergence for the power series of g(x) is (-sqrt(5/3), sqrt(5/3)).
Hence,
a) The first four nonzero terms of the Taylor series are 3, -3t^2, 3/3t^4, -3/5t^6
b) The first four nonzero terms of the power series expansion of g(x) about x=0 are 3x, -x^3, 3/5x^5, -3/7x^7.
c) The interval of convergence for the power series of g(x) is (-sqrt(5/3), sqrt(5/3)).
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a customer service representative must sped different amounts of time with each customer to resolve various concerns the amount of time spent with each customer can be modeled by the following distribution find the percentile
The value of [tex]P$$(2 < x < 10)$[/tex] is 0.5350
What is meant by percentile?A score at or below which a given percentage falls is known as the k-th percentile in statistics, as is a score below which a given percentage k of scores in the frequency distribution falls. A percentile is the difference between a given score and the scores of the other members of a group. The proportion of other scores that a given score outperformed is displayed. In the 85th percentile, for instance, signifies that your test result of 75 is higher than the scores of 85% of other students.The exponential distribution is the probability distribution of the interval between events in a Poisson point process, that is, an event-producing process where events happen continuously and independently at a fixed average rate. It is an example of the gamma distribution.As per the Function, [tex]$P(x < x)=1-e^{-m x}$[/tex]
Where [tex]$x \sim E x p(0.2)$[/tex]
Therefore, [tex]$m=0.2$[/tex]
[tex]$$P(2 < x < 10)=P(x < 10)-P(x < 2)$$[/tex]
Applying all data
[tex]& \mathrm{P}\left(2 < \mathrm{x} < 10=\left(1-e^{0.2 \times 10}\right)-\left(1-e^{0.2 \times 2}\right)\right. \\[/tex]
=0.6703-0.1353=0.5350
The complete question is,
A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be
modelled by the following distribution: X-Exp(0.2)
Find [tex]$P(2 < x < 10)$[/tex].
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For the function ƒ(x) = 2 (x − 10)² + 6, find ƒ−¹(x).
For the function ƒ(x) = 2 (x − 10)² + 6 then F⁻¹ = [tex]\frac{x- 6}{2}[/tex] + 10 is the inverse of the given functions.
Calculating the problem:y = [tex]\frac{x - 6}{2}[/tex] + 10
F⁻¹ = [tex]\frac{x -6 }{2}[/tex] + 10
Inverse functions:A function that is capable of reversing into another function is referred to as an inverse function or anti function. Simply put, if a function is denoted by "f" or "F," then the inverse function is denoted by "f-1" or "F-1." The inverse function returns the original value for which a function provided the output.
If any function "f" takes x to y, then the inverse of "f" will take y to x. A function that consists of its inverse retrieves the original value when considering functions, where f and g are inverses, and f(g(x)) = g(f(x)) = x. Then, the opposite of f(x) is g(y) = (y-5)/2 = x.
How is the inverse of a function or relationship determined?The equation of the inverse relation can be obtained by substituting every y in the equation for every x in the equation that describes the relation between the variables x and y.
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which of the following refers to a statement that says that a change in one variable tends to follow or be associated with a change in another variable?
Correlation: A correlation is a statement that suggests there is a relationship between two variables, where a change in one variable tends to follow or be associated with a change in another variable.
A correlation is a statement that suggests there is a relationship between two variables. It does not necessarily mean that a change in one variable will cause a change in the other variable, but that a change in one tends to follow or be associated with a change in the other. For example, a correlation could be made between the amount of rainfall a region receives and the number of plants that are able to grow there. If the rainfall increases, then the number of plants that can grow in the region is likely to increase as well. A correlation can be either positive or negative, with a positive correlation meaning that an increase in one variable will likely result in an increase in the other, and a negative correlation meaning that an increase in one variable will likely result in a decrease in the other. Correlation can be used to make predictions, but should not be confused with causation, as correlation does not necessarily imply causation.
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The complete question: which of the following refers to a statement that there is a relationship between two variables that can change in one variable tends to follow or be associated with a change in another variable?
You invest $1,200 that is compounded continuously at 9% for 10 years. What is the equation you use?
Answer:
FV = $1,200 * e^(0.09*10)
Step-by-step explanation:
The equation used to calculate the future value of an investment compounded continuously is:
FV = PV * e^(r*t)
where:
FV = future value
PV = present value (initial investment) = $1,200
r = annual interest rate (9% = 0.09)
t = number of years (10 years)
So, the future value of the investment would be:
FV = $1,200 * e^(0.09*10)
The sports bar owner runs a regression to test whether there is a relationship between Red Sox away games and daily revenue. Which of the following statements about the regression output is true? SELECT ALL THAT APPLY.
A. The average daily revenue for days when the Red Sox do not play away is $1,768.32.
B. The average daily revenue for days when the Red Sox play away is $1,768.32.
C. The average daily revenue for days when the Red Sox play away is $2,264.57.
D. The average daily revenue for days when the Red Sox do not play away is $1,272.07.
E. On average, the bar’s revenue is $496.25 higher on days when the Red Sox play away than on days when they do not.
B and E are true.The regression output can be used to determine whether there is a relationship between Red Sox away games and daily revenue.
The output can also be used to calculate the average daily revenue for each scenario.
For the scenario where the Red Sox do not play away, the output will give the average daily revenue, which is $1,768.32. So, statement A is true.
For the scenario where the Red Sox play away, the output will give the average daily revenue, which is $2,264.57. So, statement B is true.
For the scenario where the Red Sox do not play away, the output will also give the average daily revenue, which is $1,272.07. So, statement D is false.
Finally, the output can be used to calculate the difference between the average daily revenue when the Red Sox play away and when they do not, which is $496.25. So, statement E is true.
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A 300g ball collides with a wall. The figure(Figure 1)shows the ball's velocity and the force exerted on the ball by the wall. What is vfx, the ball's rebound velocity?
The rebound velocity of the ball, vfx, is 10.0 m/s, calculated using the equation vfx = 2vix - viy, where vix is the initial velocity along the x-axis (4.0 m/s) and viy is the initial velocity along the y-axis (-2.0 m/s).
Using the equation vfx = 2vix - viy, the rebound velocity of the ball is calculated as follows:
vfx = 2(4.0 m/s) - (-2.0 m/s) = 10.0 m/s
1. Identify the initial velocity, vix, of the ball along the x-axis, which is 4.0 m/s.
2. Identify the initial velocity, viy, of the ball along the y-axis, which is -2.0 m/s.
3. Use the equation vfx = 2vix - viy to calculate the rebound velocity of the ball, vfx.
4. Calculate vfx = 2(4.0 m/s) - (-2.0 m/s) = 10.0 m/s.
The rebound velocity of the ball is 10.0 m/s.
The rebound velocity of the ball, vfx, is 10.0 m/s, calculated using the equation vfx = 2vix - viy, where vix is the initial velocity along the x-axis (4.0 m/s) and viy is the initial velocity along the y-axis (-2.0 m/s).
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Question:
(a) Find the equation for the family of linear functions with slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such that f(2)=1 and sketch several members of the family.
(c) Which functions belong to both families?
Slope-intercept form
Equation of slope-intercept form is given as y=mx+b where m is the slope of the line and c is the y-intercept. m is used to measure the steepness of the graph. This form is simply the method of writing the equation of a line so that the slope and y-intercept are rapidly apparent.
For a family of linear functions with slope 2, the equation is y = 2x + b, where b is the y-intercept. For a family of linear functions such that f(2) = 1, the equation is y = x + b, where b is the y-intercept. Both families of linear functions have the same equation, y = x + b, and all functions belonging to both families have a slope of 1 and a y-intercept of b.
(a) The equation for the family of linear functions with slope 2 is y = 2x + b, where b is the y-intercept. To sketch several members of the family, plot several points (x, y) with the equation, then connect the points to form a line.
(b) The equation for the family of linear functions such that f(2) = 1 is y = x + b, where b is the y-intercept. To sketch several members of the family, plot several points (x, y) with the equation, then connect the points to form a line.
(c) The equation for both families of linear functions is y = x + b, where b is the y-intercept. All functions belonging to both families have a slope of 1 and a y-intercept of b.
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16 years ago, Thomas invested $3,000 into a bank account with a 5% return compounded annually. Which of the following equations indicates the total amount of money Thomas has after t years of his original investment? A=3,000(.5)^tA=3,000(1.05)^t A = 3,000(.5)^(1/16)A = 3,000(1.05)^(1/16)
Thomas invested $3,000 into a bank account with a 5% interest return compounded annually. then the following equations indicates the total amount of money 3000(1.05)^1/16
Thomas invested $3,000 into a bank account with a 5% return compounded annually.
A: represents the current balance or compound amount in the account.
P: represents the principal (the original amount deposited).
R: annual interest rate (in decimal form).N: the number of times per year that interest is compounded .T: time in years the money has been invested.
The formula of interest is A=P (1+r)ⁿ
A=Final amount
P= Principal ( deposit invested)
r= interest rate
n= time
So, in this case
P=$3,000
r= 5% or 0.05
n=1/ 16
Replacing
A=3000(1+0.05)^1/16
A=3000( 1.05)^1/16
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a student sets up the following equation to convert a measurement. (the stands for a number the student is going to calculate.) fill in the missing part of this equation. note: your answer should be in the form of one or more fractions multiplied together
The missing part of the equation should be like 1000 g / 1 kg × 100 cm / 1 m
Given,
The set of units used in the final response is different. In particular, kilogram (kg) and meters (m) are converted to gramm(g) and centimeter (cm), respectively (cm). You must multiply the initial value by proportions to achieve this adjustment.
It is crucial to organize these proportions in a way that permits unit cancellation when writing them down. As an illustration, since kg and m are both in the numerator, they must be in the denominators of the conversions.
Proportions:
1 kg = 1,000 g
1 m = 100 cm
Here,
The complete equation is like;
-4.3 × 10⁴ (kg × m / s) × (1000 g / 1 kg) × (100 cm / 1 m) = ? (g × cm / s)
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Complete question is attached
72 is 120% of what number?
Answer:60 percent
Step-by-step explanation:
Step 1) 72 = 120% × Y
Step 2) 72 = 120/100× Y
Multiplying both sides by 100 and dividing both sides of the equation by 120 we will arrive at:
Step 3) Y = 72 × 100/120
Step 4) Y = 72 × 100 ÷ 120
Step 5) Y = 60
Which word is used incorrectly?
The mailperson's job is to put the envelope's in every mailbox on the block.
OÀ
A
O
O
B
C
D
every
mailperson's
block
envelope's
The word used incorrectly in the sentence is "envelopes".
The word used incorrectly in the sentence is "envelopes". The comment should be "envelopes" instead. This is because "envelopes" is a plural noun, referring to more than one envelope. The apostrophe in "envelopes" implies possession, indicating that the envelopes belong to someone or something, but it is not necessary for this context.
Hence, the correct sentence should be "The mailperson's job is to put the envelopes in every mailbox on the block."
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Suppose that sin A = 0.622. What is the value of cos A?
Answer :
0.783
Step-by-step explanation:
We know that,
sin^2 (x) + cos^2 (x) = 1
Thus,
[tex] \sin^{2} (A) + cos^{2} (A) = 1 \\ cos^{2} A =1 - sin^{2} A \\ cos^{2} A = 1 - (0.622)^{2 } \\ cos^{2} A = 1 - 0.386884 \\ cos A = \sqrt{0.613116} \\ = 0.783[/tex]
Complete the missing values in the solution to the equation.
The missing value in the solution to the equation is 4. The value of x is 4 and y is -6 which is (4, -6).
What is solution to a equation?A solution to an equation is a value or values that make the equation true. For example, if the equation is x + 3 = 7, then the solution to the equation would be x = 4.
To solve an equation, you must use a combination of addition, subtraction, multiplication, division, and algebraic manipulations to isolate the unknown variable. For instance, to solve the equation x + 3 = 7, you would subtract 3 from both sides of the equation to find x = 4.
It is important to note that an equation may have more than one solution. For example, the equation x2 + 4 = 0 has two solutions, x = 2 and x = -2.
In some cases, equations may not have a solution. This happens when the equation is not true for any value of the unknown variable. For example, the equation 0 = 1 is not true for any value of x, so it does not have a solution.
In summary, a solution to an equation is a value or values that make the equation true. To solve an equation, you must use a combination of addition, subtraction, multiplication, division, and algebraic manipulations to isolate the unknown variable. An equation may have multiple solutions or no solution at all.
The given equation is 4x + y = 10.
We have the value of y is -6,
To find the value of x, using the equation 4x + y = 10
4x - 6 = 10
4x = 10 +6
4x = 16
x = 4
The missing value in solution is 4.
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use the following dataset to answer the question 11 to 14
History class final test scores
79, 82, 85, 91, 97, 73, 88, 87, 85, 92, 90, 85, 86, 91, 94, 85, 92
what is the average class score? Round your answer to the nearest whole number.
Answer:
82
Step-by-step explanation:
average class score=total number of test scores/number of test scores
sum up the scores 79+82+85+91+97+73+88+87+85+92+90+85+86+91+94+85+92
=1482÷18=82.33333
=82
Before women were allowed to vote, both men and women organized, protested, and marched. Finally, in 1920, the 19th Amendment to the Constitution gave women the right to vote.
How does this example support the author's main purpose in the book?
It shows how citizens' actions affect today's issues.
It analyzes how citizenship was different in the past.
It explains what an amendment to the Constitution is.
It shows that citizens' actions can change society.
Answer:
it analyses how citizenship was different in the past
P( – 3, – 5) after a reflection over the y-axis.
Answer: 3,-5
Step-by-step explanation:
Answer: (3, -5)
......................................
What is the solution to the equation? 3|x| − 1 = 8
x = 3 or 9
x = −9 or 9
x = −3 or 3
x = −3 or −9
please help
Answer:
x = 3
Step-by-step explanation:
3|x|- 1= 8
=3x = 8 + 1
=3x = 9
=3x/3 = 9/3
x = 3.
Answer:
The equation is 3|x| - 1 = 8. To solve this equation, we first need to consider the absolute value of x.
The absolute value of x is defined as |x| = x for x ≥ 0, and |x| = -x for x < 0.
We have 3|x| - 1 = 8, so we have to consider two cases:
x ≥ 0: in this case, |x| = x and we have 3x - 1 = 8, which gives x = 3 or x = 9.
x < 0: in this case, |x| = -x and we have 3(-x) - 1 = 8, which gives -3x - 1 = 8, and then x = -3 or x = -9.
So the solution to the equation is x = 3 or 9 or x = -3 or -9.
Step-by-step explanation:
A line that has a slope or 1/5 and passes through (-10,4)
The Equation of the line which has a slope of 1/5 and passes through (-10,4) will be [tex]y = \frac{1}{5} x + 6[/tex] .
Given, We have the slope of the line = 1/5 and a point (-10,4) though which the line passes. We know that the form of a linear line is y=mx + c (slope intercept form) where m is the gradient/slope of the line and c is the point where it intercepts the y axis on the graph. We can directly put the value slope in the place of m.
To find C , we can put the coordinates of the given point in the equation to solve for C. Therefore :
⇒ y = mx + c
⇒y = 1/5 x + c
⇒4 = 1/5 x (-10) + c
⇒4 = -2 + c
⇒ 4 +2 = c
⇒c = 6
Putting all the values in equation gives us the linear equation of the line : [tex]y = \frac{1}{5} x + 6[/tex]
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The complete question is :
What is the locus/ linear equation/ equation of the line that has a slope of 1/5 and passes through coordinates (-10,4)?
2. question 2 fill in the blanks to make the print prime factors function print all the prime factors of a number. a prime factor is a number that is prime and divides another without a remainder.
A Prime factor is a natural number that has only two factors namely 1 and the number itself. One is not a prime number. Prime factors are odd in nature except for 2 because 2 is the only even prime factor. Example of prime factors are 2,3,5,7,11 etc
def print_prime_factors(number):
factor = 2
while factor <= number:
# Check if factor is a divisor of number
if number % factor == 0:
# If it is, print it and divide the original number
print(factor)
number = number / factor
else:
# If it's not, increment the factor by one
factor+=1
return "Done"
print_prime_factors(100)
Complete question- Above mentioned is the correct code with all blanks filled.
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solve:
g(16)=-1/2x+5
pls show work
The function g(16) = -1/2 x + 5 is solved by substitution to get -3
What is substitution as used in math?The process of changing an algebraic letter to it's value corresponds to the procedure known as substitution.
In the problem, the equation g(16) = -1/2 x + 5,
Considering the equation, the value of x is 16. The letter x will be replaced by 16 in the equation
g(16) = - 1/2 x + 5
g(16) = - 1/2 * 16 + 5
g(16) = - 8 + 5
g(16) = -3
Substituting x = 16 in the equation g(16) = -1/2 x + 5 results to a solution of -3
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Find the volume of a square based pyramid with a base length of 9 and a height of 11
Answer: It is 36cm the height of it and the length W times L
Step-by-step explanation:
a model of a car is 4 inches long. if the actual car is 10 feet long, find the scale of the model. a. 4 in
Scale of the model is found to be 1 : 30 using the ratio method.
A model of a car is 4 inches long.
The actual car is 10 feet long.
4 in. = 10 ft. ( since given )
Divide both sides by 4, we get
∴ 1 in. = 2.5 ft.
We use the ratio here to compare the lengths of the actual car and its model.
A ratio scale is a quantitative measurement scale that is used to compare numbers.
The model is 4 inches long while the actual car is 10 feet long.
One foot is equal to 12 inches. Therefore the car in terms of inches is:
10 feet × 12 inches = 120 inches
The ratio scale is the length of the model of the car : length of the actual car
= 4 : 120
= 1 : 30
Therefore the scale of the model is 1: 30.
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Which numbers lie between -2 and -3 on the number line? Select all that apply.
Answer:
-2.5, -2.02, -11/4
Step-by-step explanation:
Any number beginning with -2 will be the answer. -11/4 is -2.75 when converted to a decimal!
There are integers b, c for which both roots of the polynomial x^2-x-1 are also roots of the polynomial x^5-bx-c. Determine the product bc.
It is discovered that the phrase under the division symbol is 3x2 - (b - 2)x - c, meaning that b = 5 and c = 3.
find the roots of x2 - x - 1 = 0
---> Using the quadratic formula, the roots are [ 1 + sqrt(5) ] / 2 and [ 1 + sqrt(5) ] / 2.
Placing these values into the equation x5 - bx - c = 0, we get:
x = [ 1 + sqrt(5) ] / 2 ---> ( [ 1 + sqrt(5) ] / 2 )5 - b( [ 1 + sqrt(5) ] / 2 ) - c = 0
---> [ 176 + 80sqrt(5) ] / 32 - ( [ 1 + sqrt(5) ] / 2 )b - c = 0
multiplying by 32 ---> 176 + 80sqrt(5) - 16b - 16sqrt(5)b - 32c = 0
Using the same steps for x = [ 1 - sqrt(5) ] / 2 , we get: 176 - 80sqrt(5) - 16b + 16sqrt(5)b - 32c = 0
Now, let's combine these two results: 176 + 80sqrt(5) - 16b - 16sqrt(5)b - 32c = 0
176 - 80sqrt(5) - 16b + 16sqrt(5)b - 32c = 0
Subtract the lower equation from the upper: 160sqrt(5) - 32sqrt(5)b = 0 ---> 160sqrt(5) = 32sqrt(5)b ---> b = 5
Now, substitute this value for b into 176 + 80sqrt(5) - 16b - 16sqrt(5)b - 32c = 0
--->176 + 80sqrt(5) - 16(5) - 16sqrt(5)(5) - 32c = 0
--->176 + 80sqrt(5) - 80 - 80sqrt(5) - 32c = 0
--->96 - 32c = 0
--->c = 3
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