The function C(x) = 25.50x + 50 models the total cost for a cleaning company to clean a house, where x is the number of hours it takes to clean the house. What is the average rate of change of the function between 3 hours and 9 hours? A. $17.00 per hour B. $25.50 per hour C. $31.05 per hour D. $42.15 per hour

Answer:

[tex] 2b+f = 8[/tex]   (1)

[tex]3b+f=11[/tex]   (2)

[tex] b = 11-8=3[/tex]

[tex]f= 8-2(3) = 8-6 =2[/tex]

Step-by-step explanation:

For this case we can put some notation

Let b= dozen bagels and f=  delivery fee

And for this case we know that "On Monday, they delivered two dozen bagels, b, to an office at a total cost of $8", so then the total taling in count the delivery fee we have this:

[tex] 2b+f = 8[/tex]

And for the other part "On Tuesday, three dozen bagels were delivered at a total cost of $11", we can write the expression like this:

[tex]3b+f=11[/tex]

And our system of equations would be:

[tex] 2b+f = 8[/tex]   (1)

[tex]3b+f=11[/tex]   (2)

If we solve for f from equation (1) we got:

[tex]f= 8-2b[/tex]

And if w replace this into equation (2) we got:

[tex]3b+8-2b=11[/tex]  

[tex] b = 11-8=3[/tex]

And solving for f we got:

[tex]f= 8-2(3) = 8-6 =2[/tex]

Answer: 50%

Step-by-step explanation:

It is estimated that almost 50% of collisions involving a vehicle and a train were at crossings where warning devices such as lights, gates and bells were in working order. This high percentage may be as a result of malfunction of warning devices, because when warning devices malfunction they give wrong signals that could lead to accidents. It could also be as a result of violation or ignorance of traffic rules and warning signals which can also lead to collision.

The question discusses the number of accidents happening at railway crossings despite working warning systems, akin to a problematic intersection at Clay Street and Eagle Avenue that saw many accidents despite traffic rules. While specific statistics aren't given, such cases highlight the importance of adhering to traffic signals and safety measures for both drivers and pedestrians.

The question seems to be related to railroad crossing safety, specifically the effectiveness of warning devices like bells, lights, and gates. While the exact percentage is not provided, studies show that a significant proportion of vehicle-train collisions occur at crossings where these warning systems are fully functional. Like the traffic signal installed at the intersection of Clay Street and Eagle Avenue to prevent the high number of accidents, these warning devices aim to control vehicular and pedestrian traffic. Unfortunately, they may not always be heeded, and thus accidents occur.

Similar to the situation in the Clay Street and Eagle Avenue intersection, safety measures only work if they're respected. A traffic signal might reduce accidents, but only if drivers and pedestrians heed it. Unfortunately, despite railroad crossing warnings, accidents still occur, often due to negligence, recklessness, or distraction.

It emphasizes the importance of traffic rules and safety measures in our day to day life. Be it the students crossing the road or vehicles moving in traffic, the role of traffic signals and warning signs is critical. We must remember these are not there to restrict us, but to ensure everyone's safety.

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Answer:

He needs 23 containers

Step-by-step explanation:

1 container = 12 toy boxes

X container= 278 toy boxes

12x= 278

X= 278/12

X= 23 containers

Answer:

Step-by-step explanation:

The max and min values exist where the derivative of the function is equal to 0. So we find the derivative:

[tex]y'=9x^2+28x-11[/tex]

Setting this equal to 0 and solving for x gives you the 2 values

x = .352 and -3.464

Now we need to find where the function is increasing and decreasing.  I teach ,my students to make a table.  The interval "starts" at negative infinity and goes up to positive infinity.  So the intervals are

-∞ < x < -3.464           -3.464 < x < .352             .352 < x < ∞

Now choose any value within the interval and evaluate the derivative at that value.  I chose -5 for the first test number, 0 for the second, and 1 for the third.  Evaluating the derivative at -5 gives you a positive number, so the function is increasing from negative infinity to -3.464.  Evaluating the derivative at 0 gives you a negative number, so the function is decreasing from -3.464 to .352.  Evaluating the derivative at 1 gives you a positive number, so the function is increasing from .352 to positive infinity.  That means that there is a min at the x value of .352.  I guess we could round that to the tenths place and use .4 as our x value.  Plug .4 into the function to get the y value at the min point.

f(.4) = -48.0

So the relative min of the function is located at (.4, -48.0)

The amount of each of the 6 payments will be $41.50

Step-by-step explanation:

Given,

Cost of bike including taxes = $349

Down payment = $100

Amount left to pay = Cost of bike - Down payment

Amount left to pay = 349 - 100

Amount left to pay = $249

She agrees to pay in 6 equal payments, therefore, we will divide the total amount to pay by 6.

Amount to pay for one month = [tex]\frac{Amount\ left\ to\ pay}{No.\ of\ months}[/tex]

Amount to pay for one month = [tex]\frac{249}{6} = \$41.50[/tex]

The amount of each of the 6 payments will be $41.50

Keywords: division, subtraction

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Answer:

At the end of 6 years, he would have paid $13985.6

Step-by-step explanation:

Initial amount taken as load is $10,000 This means that the principal

P = 10000

It was compounded annually. This means that it was cam pounded once in a year. So

n = 1

The rate at which the principal was compounded is 5.75%. So

r = 5.75/100 = 0.0575

it takes you six years to pay off the loan. So

t = 6

The formula for compound interest is

A = P(1+r/n)^nt

A = total amount of money that you would have paid back by the end of the six years. Therefore

A = 10000 (1+0.0575/1)^1×6

A = 10000(1.0575)^6 = $13985.6

Answer: the maximum number of visits would be 38

Step-by-step explanation:

A person can pay $7 for a membership to the history museum and then go to the museum for just $1 per visit.

Let x represent the number of visits that the person makes to the museum after paying for the membership. This means that if the person makes x visits to the history museum, the total cost would be

7 + x

If a member of the history museum pays a total of $45. It means that the number of visits that he can make to the history museum would be

7 + x = 45

x = 45 - 7 = 38

Answer:

The opportunity cost will be 3 cars

So option (d) will be correct answer

Step-by-step explanation:

The opportunity is cost is best characterized as the following best action done without, it is something we give up to gain another.

For instance the opportunity cost of going to class can be the pleasure from watching movie, or spending time with companions or something different. The opportunity  cost can be determined by isolating what you forgone by what you gain.

So opportunity cost [tex]=\frac{12}{4}=3cars[/tex]

So option (d) will be the correct answer

Final answer:

Pete's opportunity cost of producing one truck is 3 cars, which is found by dividing Pete's car production rate by his truck production rate (12 cars per hour / 4 trucks per hour).

Explanation:

Pete's opportunity cost of producing one truck can be determined by looking at the alternative production of cars he could have achieved in the same amount of time. Since Pete can produce 12 cars per hour or 4 trucks per hour, we can set up a ratio of cars to trucks based on his production abilities. To find the opportunity cost of one truck, we divide the number of cars Pete can make by the number of trucks, which is 12/4, giving us 3. Hence, Pete's opportunity cost of one truck is 3 cars. This calculation is similar to how opportunity cost is determined in examples such as the production possibility frontier, where the slope shows the trade-off between two different products.

Answer:

The dimensions of the drawing are 10 units by 8 units

Step-by-step explanation:                        

we know that

The scale drawing is [tex]\frac{1}{15}\ \frac{unit}{feet}[/tex]

That means ---> 1 unit in the drawing represent 15 feet in the actual

To find out the dimensions of the drawing, multiply the actual dimensions by the scale drawing

so

[tex]150\ ft=150(\frac{1}{15})=10\ units[/tex]

[tex]120\ ft=120(\frac{1}{15})=8\ units[/tex]

therefore

The dimensions of the drawing are 10 units by 8 units

Answer:

The price of each hamburger is $3

The price of each hot dogs is $2  .

Step-by-step explanation:

Given as :

The total price of 3 hamburger and 2 hot dogs = $13

The total price of 2 hamburger and 5 hot dogs = $16

Let The price of each hamburger = $x

Let The price of each hot dogs = $y

Now, According to question

3 x + 2 y = 13              .........A

2 x + 5 y = 16              .......B

Now, Solving to eq A and B

3 × (2 x + 5 y ) - 2 × (3 x + 2 y ) = 3 × 16 - 2 × 13

Or, (6 x + 15 y) - (6 x + 4 y) = 48 - 26

Or, (6 x - 6 x) + (15 y - 4 y) = 22

Or, 0 + 11 y = 22

∴  y = [tex]\dfrac{22}{11}[/tex]

i.e y = $2

so, The price of each hot dogs = y = $2

Now, Put the value of y into eq B

i.e 2 x + 5 y = 16

or, 2 x + 5 × 2 = 16

or, 2 x = 16 - 10

or, 2 x = 6

∴  x = [tex]\dfrac{6}{2}[/tex]

i.e x = $3

So, The price of each hamburger = x = $3

Hence, The price of each hamburger is $3 and  The price of each hot dogs is $2  . Answer

Answer

Domain=[80, 100] ie. both of them inclusive.

Step-by-step explanation:

Albert has total $105.

Let the cost of a product be c.

Therefore, the total cost of a product after the delivery and all is represented by a function f(c)

f(c)=c+[tex]\frac{c}{20}[/tex]

f(c)=[tex]\frac{21c}{20}[/tex]

The domain, of the function is the no. of possible values of c that can satisfy the equation.

Now, lets check the maximum price of the product that Albert can buy.

f(c)=105

[tex]\frac{21c}{20}[/tex]=105

c=$100

As, the minimum price of a product of is $80 , the domain needs to start from here. And it will go on till $100.

Domain=[80, 100]

Answer: 1/16,384 microbes

Step-by-step explanation:

Since the microbes divides every 30minutes, if the microbes divides by half every 30minutes, then after an hour it will divides by another half that is (1/2)of 1/2 to give 1/4.

Subsequently after another hour, the microbes will reduces to 1/4 of its remains (i.e 1/4) to give 1/16.

Since the denominator is increasing geometrically each hour.

Looking at the trend;

After 1hr = 1/4 of the original microbe = 1/4

After 2 hrs = 1/4 of the remaining microbe = 1/4 of 1/4 = 1/16

Generalizing, we will let the number of hours be n

According to the progression, 1, 1/4,1/16,...n

Since its a Geometric Progression, nth term = ar^n-1

Where a is the first term = 1

r is the common ratio = 1/4

n is the number of hours of decay

After 8hours, the microbes will have been divided by

(1)(1/4)^8-1

= (1/4)^7

= 1/16,384 microbes

Answer:

1/2^16 or 0.0000152588 microbes

Step-by-step explanation:

Assuming you start with just one microbe that divides every 30 minutes.

This means that the half life of the microbe is 30 minutes or 0.5hours

The interpretation of this half life is that after every half-life, which in this case is 30 minutes, half of the microbe will be gone, and half will remain.

It follows that after another half hour the amount remaining will be

1/2 of 1/2= 1/4 microbes

Thus after 8 hours, there would have been (8/0.5)=16 half lives.

Therefore the amount of microbes remaining will be 1/2^16 of 1 = 0.0625

Alternatively, we could solve the differential equation

dM/dt=kM, where dM/dt is the rate of decay, and M is the amount at any time t, k is the decay constant

Solution of this first order differential equation by separating the variables and integrating yields {dM/M={kt+c, lnM=kt+c, and ......

M=Moexp(-kt)

The initial value Mo=1, when t=0, and given value M=0.5, t=0.5h yields the value of k as follows

0.5=exp(-k*0.5)

ln(0.5)=-k*0.5

k=1.386

After any time time, thus the given expression holds

M=exp(-1.386t)

Thus after 8 hours, the microbes remaining will be

M=exp(-1.386t)=exp(-1.386*8)=0.000152588 microbes.

Answer: the sale price of the dress is $87.75

Step-by-step explanation:

Chanice and destiny went shopping at new fashion shop. Everything in the store was at a 30% discount. Chanice found a dress that was originally $65.00. This means that

the sale price would be the original price + 35% of the original price. It becomes

65 + 35/100×65

= 65 + 0.35×65

= 65 + 22.75

= 87.75

The equation of the least squares regression line (LSRL) predicting sodium level from the number of calories in beef hot dogs is Sodium Level (mg) = 37.77 mg + 4.077 mg/calories × Calories.

The equation of the least squares regression line (LSRL) predicting sodium level from the number of calories in beef hot dogs can be determined using the given statistics and the formula for the slope (b) of the LSRL:

b = r (sy/sx)

Where r is the correlation coefficient, sy is the standard deviation of the y-values (sodium levels), and sx is the standard deviation of the x-values (calories).

Given that r = 0.887, sy = 102.43 mg, and sx = 22.64 calories, we can calculate the slope (b):

b = 0.887 (102.43 mg / 22.64 calories) = 4.077 mg/calories

Then, we can use the y-intercept (a) in the equation y = a + bx, where a = y-bar - b*x-bar.

Given that the average sodium level y-bar = 401.15 mg and the average number of calories x-bar = 156.85 calories, the y-intercept (a) will be:

a = 401.15 mg - (4.077 mg/calories * 156.85 calories) = 37.77 mg

Therefore, the equation of the LSRL is:

Sodium Level (mg) = 37.77 mg + 4.077 mg/calories × Calories

Answer:

Even though 14 candies were gone there still is a very low change of getting pineapple. out of 12 candies 2 are pineapple.

Step-by-step explanation:

Answer:

[tex]\frac{2\sqrt3}{3}[/tex] [tex]cm^2[/tex]

Step-by-step explanation:

Since [tex]\triangle ABC[/tex] is equilateral triangle, then [tex]\angle A = \pi /3[/tex]

So,

[tex]S_{\triangle ABC} = \frac{2 * 2 * sin(\pi /3)}{2} = \frac{2 * 2 * \sqrt{3}/2}{2} = \sqrt{3}[/tex] [tex]cm^2[/tex]

Then we need to find [tex]S_{\triangle AEF}[/tex] which can be computed by finding length_AF.

Let's call x = length_AF.

By Menelao's Theorem,

[tex]\frac{BE*x*CD}{AE*(2-x)*BD} = \frac{1*x*2}{1*(2-x)*4}  = 1[/tex]

⇒ x = 4/3 cm

Thus,

[tex]S_{\triangle AEF} = \frac{1 * x * sin(\pi /3)}{2} = \frac{1 * 4/3 * \sqrt{3}/2}{2} = 1/\sqrt{3}[/tex] [tex]cm^2[/tex]

To find the area of quadrilateral [tex]BEFC[/tex], we have to subtract [tex]S_{\triangle AEF}[/tex] from [tex]S_{\triangle ABC}[/tex]

Hence,

[tex]S_{BEFC} = S_{\triangle ABC} - S_{\triangle AEF} = \sqrt3 -1/\sqrt3 = \frac{2\sqrt3}{3}[/tex] [tex]cm^2[/tex]

Answer:

160 basket of each fruit baskets we can make using all fruit.

Step-by-step explanation:

Given:

Oranges comes in bags = 16

Apples comes in bags = 20

bananas come in bags = 32

We need to find the greatest number of fruit baskets that you can make using all fruit.

Also Given:

You have one bag of each fruit basket must be identical.

So we will first find the least common multiple of all the numbers we get;

20 = 20,40,60,80,100,120,140,160

16 = 16,32,48,64,80,96,112,128,144,160

32 = 32,64,96,128,160

The least common multiple is 160.

Hence we can say that, 160 basket of each fruit baskets we can make using all fruit.

Answer:

0.03

Step-by-step explanation:

Anthony can expect to pull out a blue candy 30 times in 150 draws, as each draw has a 1/5 probability of being blue, and the total number of draws is 150.

The question can be answered by calculating the expected frequency of pulling a blue candy based on the probability of drawing a blue candy from the bag. Anthony has 4 blue candies, 6 green candies, and 10 yellow candies, making a total of 20 candies in the bag. Since each draw is independent and the candies are replaced each time, the probability of drawing a blue candy is 4/20 or 1/5.

To find the expected number of times Anthony could pull out a blue candy in 150 tries, we multiply the probability by the number of trials:

Expected blue candies = (Probability of blue candy) x (Number of trials)

Expected blue candies = (1/5) x 150

Expected blue candies = 30

Therefore, Anthony can expect to pull out a blue candy 30 times in 150 draws.

Answer:

  [tex]h_1(t)=-16t^2+154[/tex]

Step-by-step explanation:

Put the initial height of ball 1 into the given formula:

  [tex]h_1(t)=-16t^2+154[/tex]

Answer:

[tex]p_v =P(t_5>1.965)=0.0533[/tex]

We can use the following excel code to find it :"=1-T.DIST(1.965,5,TRUE) "

Step-by-step explanation:

Data given and notation  

[tex]\bar X[/tex] represent the average score for the sample  

[tex]s[/tex] represent the sample standard deviation  

[tex]n=6[/tex] sample size  

[tex]\mu_o =98.6[/tex] represent the value that we want to test  

[tex]\alpha[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to apply a one upper tailed test.  

What are H0 and Ha for this study?  

Null hypothesis: [tex]\mu \leq 98.6[/tex]  

Alternative hypothesis :[tex]\mu > 98.6[/tex]  

Compute the test statistic  

The statistic for this case is given by:  

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

The calculated value on this case is given t=1.965

Give the appropriate conclusion for the test  

First we need to calculat ethe degrees of freedom given by:

[tex]df=n-1=6-1=5[/tex]

Since is a one side right tailed test the p value would be:  

[tex]p_v =P(t_5>1.965)=0.0533[/tex]

We can use the following excel code to find it :"=1-T.DIST(1.965,5,TRUE) "

Conclusion  

If we compare the p value and a significance level assumed [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the true mean is not significantly higher than 98.6 at 5% of significance.  

Answer:

Step-by-step explanation:

6 flights from  New York to Denver

7 flights from Denver to San Francisco.

total number of combination

6*7= 42

Final answer:

For a trip from New York to San Francisco via Denver, with six airlines on the first segment and seven on the second segment, there are 42 different pairs of airlines that can be chosen.

Explanation:

The question asks how many different pairs of airlines can be chosen for a trip from New York to San Francisco via Denver, considering one airline for the leg from New York to Denver and another airline for the continuation from Denver to San Francisco. Since there are six different airlines that fly from New York to Denver and seven that fly from Denver to San Francisco, the total number of combinations of airlines for both segments of the journey is a simple multiplication of the two separate choices. Therefore, the number of unique airline pairs is 6 airlines from New York to Denver multiplied by 7 airlines from Denver to San Francisco.

To calculate this, we use the formula for the number of combinations for two independent choices, which is:

Total combinations = (Choices for first segment) × (Choices for second segment)

Total combinations = 6 × 7 = 42 different pairs of airlines.

Thus, a traveler could choose from 42 different combinations of airlines for their trip from New York to San Francisco via Denver.

Answer:

12 years.

Step-by-step explanation:

We have been given that the average 8 year old gets 30 questions right on an intelligence test, whereas the average 12 year old gets 50 questions right. We are asked to find the mental age of an 8 year old who gets 50 questions right.

Since the 8 year old boy gets 50 questions right and the average 12 year old also gets 50 questions right, so the mental age of 8 year old (getting 50 questions right) will be equal to the mental age of 12 years old.

Therefore, the mental age of an 8 year old who gets 50 questions right would be 12 years.

Answer:

The answer is either 9 or 12

Step-by-step explanation:

Answer: UTS is similar to UQR, second choice

The other choices refer to the same triangle but we have to have corresponding vertices in the same order.  U corresponds to itself, so has to be listed first in the similar triangle.   T corresponds to Q and S to R, so UQR is our answer.

Answer:

She would have to work 11 hours to get enough money to buy the sneakers

But if you want the hours included for the time she needs to work to pay off the loan she would have to work 8 hours, so all together 19 hours.

Answer:

It would be x ≥ 18

Step-by-step explanation:

Answer:

For the given circle, the center is at (-7,4)  and radius is 7

Step-by-step explanation:

Given equation of the circle is [tex](x+7)^2+(y-4)^2=49\hfill (1)[/tex]

To find the center and radius of given circle with the help of its equation:

Equation of the circle is of the form

(x-h)^2+(y-k)^2=r^2\hfill (2)[/tex] with center at (h,k) and radius is r.

Given [tex](x+7)^2+(y-4)^2=49[/tex]

The above equation can be written as,

[tex](x-(-7))^2+(y-4)^2=7^2\hfill (3)[/tex]

Now comparing the equations (2) and (3) we get

h=-7, k=4 and r=7

Therefore center at (h,k) is (-7,4) and radius is 7

For the given circle, the center is at (-7,4)  and radius is 7

Answer:

Marginal Product of Labour = 5K

Step-by-step explanation:

Marginal product of labor is the change in output when additional labor is added,

To calculate marginal product of labor you simply divide the change in total product by the change in labour.

Marginal product of labour (MPL) = Change in total products/ Change in labour

Here, change in labour = 5KL

Change in labour = L

MPL = 5KL/L

MPL = 5K

The marginal product of labor for Joe's coffee house, where the production function is q = 5KL, is found by differentiating q with respect to L, resulting in 5K, which indicates that each additional employee increases output by 5 times the number of coffee machines available.

The marginal product of labor (MP) is the additional output generated by employing one more unit of labor while holding other factors of production constant. In Joe’s coffee house, where the production function is given by q = 5KL (with q as the number of cups produced per hour, K as the number of coffee machines, and L as the number of employees), the marginal product of labor can be found by differentiating the production function with respect to labor. As capital (K) is held constant, the marginal product of labor will be calculated as follows:

MP = d(q)/d(L) = d(5KL)/d(L) = 5K

This equation indicates that the marginal product of labor is 5 times the number of coffee machines in use because K is constant in the short run. Therefore, every additional employee hired will increase the output (q) by an amount equal to 5 times the number of coffee machines that Joe has in his coffee house.

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18 kg of 15% copper and 72 kg of 60% copper should be combined by the metalworker to create 90 kg of 51% copper alloy.

Step-by-step explanation:

Let x = kg of 15% copper alloy

Let y = kg of 60% copper alloy

Since we need to create 90 kg of alloy we know:

x + y = 90

51% of 90 kg = 45.9 kg of copper

So we're interested in creating 45.9 kg of copper

We need some amount of 15% copper and some amount of 60% copper to create 45.9 kg of copper:

0.15x + 0.60y = 45.9

but

x + y = 90

x= 90 - y

substituting that value in for x

0.15(90 - y) + 0.60y = 45.9

13.5 - 0.15y + 0.60y = 45.9

0.45y = 32.4

y = 72

Substituting this y value to solve for x gives:

x + y = 90

x= 90-72

x=18

Therefore, in order to create 90kg of 51% alloy, we'd need 18 kg of 15% copper and 72 kg of 60% copper.

Answer:The number of minutes that Alexandra talked on her cell phone is 120

Step-by-step explanation:

A cell phone company charges a flat rate of 4.75 per month with an additional charge 0.19 per minute. Assuming the total number of minutes of call made for the month is represented by x and the total cost of x minutes of call is y, then

y = 0.19x + 4.75

To determine how many minutes that Alexandra talked on her cell phone if his monthly bill was 27.55, we would substitute y = 27.55 into the equation. It becomes

27.55 = 0.19x + 4.75

0.19x = 27.55 - 4.75 = 22.8

x = 22.8/0.19 = 120 minutes.

Answer:

It is an obtuse angle.

Step-by-step explanation:

It is not a 180° angle. Because 180° angle forms a straight line.

The given angle is an obtuse angle which means it is greater than 90° and less than 180°

Note that 90° angle is perpendicular angle(interior angles of a rectangle)

That angle can be measured using a protractor.

Final answer:

The student asked for the completion of a Java function to compute the nth Fibonacci number using recursion. The base case for the function is missing, which should return the index when it is either 0 or 1. Two alternatives to improve runtime are memoization and dynamic programming, with an example provided for the latter technique.

Explanation:

The student is asking for the missing code in a recursive Java function to compute the nth Fibonacci number. To complete the base case of the provided function, we should return the index itself because the 0th and 1st Fibonacci numbers are defined as 0 and 1, respectively.

The function should look as follows:

public static int fib(int index) {
   if (index == 0) return 0; // Base case for 0
   if (index == 1) return 1; // Base case for 1
   else // Reduction and recursive calls
       return fib(index - 1) + fib(index - 2);
}

Using this recursive algorithm is inefficient due to its exponential time complexity, represented by O(2^n). To illustrate the concept of reducing computation time, two alternative approaches can be used: memoization and dynamic programming. Memoization stores the results of expensive function calls and reuses them when the same inputs occur again, thus avoiding the need to recompute them. Dynamic programming approaches the problem from a bottom-up perspective, computing and storing the result of smaller subproblems first, which then are used to compute larger ones.

An example of a dynamic programming approach is to use a loop to calculate the sequence iteratively:

public static int fib_loop(int index) {
   if (index == 0) return 0;
   if (index == 1) return 1;
   int a = 0, b = 1, sum;
   for (int i = 2; i <= index; i++) {
       sum = a + b;
       a = b;
       b = sum;
   }
   return b;
}

The time complexity for the iterative solution is O(n), which is significantly faster than the recursive approach. Therefore, fib_loop() can compute Fibonacci numbers much quicker than the recursive fib(). For example, calculating fib(40) using the recursive method can take a significant amount of time, while fib_loop(40) will complete in a fraction of the time due to its linear time complexity.