The probability of selecting the 8 tallest players randomly from a list of 27 is found by dividing the single way to choose the tallest players by the number of ways to choose any 8 players from 27, calculated using the combination formula C(n, k).
To determine the probability that the 8 tallest players will be selected from a list of 27 players, we need to consider the combinatorial aspect of the selection process. Since the selection is random, any group of 8 players can be chosen. The total number of ways to select 8 players out of 27 is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of players (27), and k is the number of players to be selected (8).
Firstly, the number of ways to choose the 8 tallest players is 1, since there is only one group of the 8 tallest players. Secondly, we calculate the total number of ways to choose any 8 players from the 27, which is C(27, 8). We can then find the probability by dividing the number of ways to choose the tallest players by the total number of ways to choose any group of 8 players.
Using the combination formula, C(27, 8) is calculated as:
27! / (8! * (27-8)!)
= 27! / (8! * 19!)
Factor out the common terms from the numerator and denominator
The remaining terms give us the total number of combinations
The probability is therefore: 1 / C(27, 8).
Humberto evaluates the expression 4t2 for t=3. He correctly substitutes 3 for t in the expression, but then says that the value is 144. However, he is incorrect!
Answer:
36
Step-by-step explanation:
He multiplied the four and the three first when he multiplied the equation together. However, according to the order of the operations, he was supposed to square three first. 3^2 = 9. Then multiplied 9*4 = 36.
Kairi spent $40.18 on CDs. Each CD cost the same amount. The sale tax was$2.33. Kairi also used a coupon for $1.00 off his purchase. How much did each CD cost?
Therefore the cost of each CD is =$2.30
Step-by-step explanation:
Given , Kairi spent $40 .18 no CDs. The sale tax was $2.33.Kairi also used a coupon for $1.00 0ff his purchase.
Total cost price of The CDs is = $(40+2.33-1.00)
=$41.33
Therefore the cost of each CD is =$(41.33÷18)
=$2.30
While visiting a pet store, you notice that there are only birds and cats in the cages. You can't help but wonder how many of each animal there is in the yard. But when you ask the store manager how many of each animal he has, he refuses to give you a direct answer. He says there are 16 animal heads and 42 animal feet. How many birds and cats are there in the pet store?
Answer: there are 11 birds and 5 cats
Step-by-step explanation:
Let x represent the number of birds in the pet store.
Let y represent the number of cats in the pet store.
A bird has one head and a cat also has one head. The store manager says that there are 16 animal heads in the store. It means that
x + y = 16
A bird has 2 feet and a cat has 4 feet. The store manager says that there are 42 animal feet in the store. It means that
2x + 4y = 42 - - - - - - - - - - - - 1
Substituting x = 16 - y into equation 1, it becomes
2(16 - y) + 4y = 42
32 - 2y + 4y = 42
- 2y + 4y = 42 - 32
2y = 10
y = 10/2 = 5
x = 16 - y = 16 - 5
x = 11
If Felicia lives 1 1/5 miles from the school and 9/10mile from the doctor field how much closer would she live if she lived 7/10 like from the doctor field
Mary buys p peaches at the farmer's market for d dollars each. She spends a total of tdollars on peaches. Create an equation that represents the relationship between t and p.
An equation that represents the relationship between t and p is [tex]\rm Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex].
Given
Mary buys p peaches at the farmer's market for d dollars each.
She spends a total of t dollars on peaches.
The cost of each peach is;
[tex]\rm Cost \ of \ each \ peaches = \Total \ number \ of \ peaches \times Amount \ of \ each \ peaches\\\\Cost \ of \ each \ peaches= p \times d[/tex]
Therefore;
An equation that represents the relationship between t and p is;
[tex]\rm Cost \ of \ each \ peaches = \dfrac{Total \ number \ of \ peaches \times Amount \ of \ each \ peaches} {Total \ money\ spend \ on \ peaches}\\\\Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex]
Hence, An equation that represents the relationship between t and p is [tex]\rm Cost \ of \ each \ peaches = \dfrac{p \times d }{t}[/tex].
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ABCD is a rhombus. If AC = 8 cm and BC = 6 cm, what is the area of the rhombus?
Answer:24, it depend upon the picture
Step-by-step explanation:
if the ac and bc are diagonals the area of rhombus equal to 24.because area of the rhombuss = pq/2where p and q are the diagonals
A rectangular swimming pool measures 30 meters by 50 meters. Tope swims back and forth the complete length of the pool parallel to the long side of the pool. She swims at a constant speed of 50 meters per minute. Meanwhile, her sister Tomi walks clockwise along the edges of the pool at a constant positive speed in such a way that they meet every time Tope reaches her starting point at the shorter side of the pool. What is Tomi's slowest possible walking speed, in meters per minute?
Answer:
Tomi's slowest possible walking speed is 80 meters per minute
Step-by-step explanation:
Tope swims at a speed of 50 meters per minute
Distance back and forth the long side of the pool is 50+50 meters = 100 meters
Time required to complete one round (back and forth) is 100 / 50 = 2 minutes
Perimeter of the pool is 50+50+30+30 = 160 meters
To walk around the perimeter and meet Tope at the starting point, Tomi has to walk 160 meters in 2 minutes
Tomi’s speed = 160 meters / 2 minutes or 160 / 2 = 80 meters per minute
Tomi's slowest possible speed should be 80 meters per minute
A ship's mast is sighted just over the horizon at 4 nautical miles. How far is this in kilometers? There are 1.609 kilometers in a mile and there are 6,076 feet in a nautical mile.
Answer:
This is 7.408 kilometres far.
Step-by-step explanation:
Given:
A ship's mast is sighted just over the horizon at 4 nautical miles.
Now, to find the distance in kilometers.
As, given ship's mast is sighted just over the horizon at 4 nautical miles.
So, by using conversion factor we get the nautical mile into kilometers:
1 nautical mile = 1.852 kilometer.
Thus, 4 nautical miles
= [tex]4\times 1.852[/tex]
= [tex]7.408\ kilometers.[/tex]
Therefore, this is 7.408 kilometres far.
Answer:
7.4
Step-by-step explanation:
PLEASE HELP ASAP!!! I NEED CORRECT ANSWERS ONLY PLEASE!!!
Find m∠X.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠X = °
Answer:
45°
Step-by-step explanation:
From the figure;
Triangle WVX is a right-angled triangle
WV= 2
WX = 1
We are required to determine, m∠X
We need to determine the appropriate trigonometric ratio to use;
Therefore, since WV is the opposite and WX is the adjacent to m∠x, then the appropriate trigonometric ratio is tangent.
That is;
Tan m∠X = WV/WX
= 2/1
tan m∠X = 1
Thus,
angle m∠X = tan⁻¹ 1
= 45°
Thus, m∠X = 45°
9) Order the fractions from least to greatest. 2 4 , 4 5 , 7 10 , 2 3 A) 2 4 , 7 10 , 2 3 , 4 5 B) 7 10 , 2 4 , 2 3 , 4 5 C) 2 4 , 2 3 , 7 10 , 4 5 D) 2 4 , 7 10 , 4 5 , 2 3
Answer:
C) [tex]\frac{2}{4}, \frac{2}{3}, \frac{7}{10}, \frac{4}{5}[/tex]
Step-by-step explanation:
Given fractions:
[tex]\frac{2}{4}, \frac{4}{5}, \frac{7}{10},\frac{2}{3}[/tex]
To arrange the fractions from least to greatest.
Solution:
In order to arrange the fractions from least to greatest, we need to make the denominators common by taking LCD.
LCD of 4,5,10,3 can be found using their multiples.
4= 4,8,12,16,20,24,28,32,36,40,.........60
5= 5,10,15,20,25,30,.........60
10= 10,20,30,40,50,60
3= 3,6,9..........................60
So, 60 is the LCD.
Making the denominators common by multiplying same numbers to numerator and denominator.
[tex]\frac{2}{4}=\frac{2\times 15}{4\times 15}=\frac{30}{60}[/tex]
[tex]\frac{4}{5}=\frac{4\times12}{5\times 12}=\frac{48}{60}[/tex]
[tex]\frac{7}{10}=\frac{7\times 6}{10\times 6}=\frac{42}{60}[/tex]
[tex]\frac{2}{3}=\frac{2\times 20}{3\times 20}=\frac{40}{60}[/tex]
Comparing the numerators we can arrange the fractions.
[tex]\frac{2}{4}, \frac{2}{3}, \frac{7}{10}, \frac{4}{5}[/tex]
In your class, you have scores of 66, 74, 71, and 81 on the first four of five tests. To get a grade of Upper C, the average of the first five tests scores must be greater than or equal to 70 and less than 80. a. Solve an inequality to find the least score you can get on the last test and still earn a Upper C. b. What score do you need if the fifth test counts as two tests?
Answer:
A score of greater than equal to 128 and less than 188 will get a grade of Upper C
Step-by-step explanation:
We are given the following in the question:
Scores:
66, 74, 71, 81
Let x be the score on fifth test.
[tex]\text{Average} = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
To get a grade of Upper C, the average of the first five tests scores must be greater than or equal to 70 and less than 80.
[tex]70 \leq \text{Average} < 80[/tex]
The fifth test counts as two tests.
Putting the values we get:
[tex]70 \leq \dfrac{66 +74+71 + 81+ x}{6} < 80\\\\420 \leq 292 + x < 480\\420 -292 \leq x < 480 - 292\\128 \leq x < 188[/tex]
Thus, a score of greater than equal to 128 and less than 188 will get a grade of Upper C
To earn an Upper C, solve an inequality with the average test scores. If the fifth test counts as two tests, the formula changes. Solve the new inequality to find the score needed.
Explanation:To find the least score you can get on the last test and still earn an Upper C, you need to solve the inequality:
(66 + 74 + 71 + 81 + x)/5 ≥ 70
(66 + 74 + 71 + 81 + x)/5 < 80
Solving this inequality will give you the range of scores you can get on the last test. If the fifth test counts as two tests, you can use the weighted average formula:
((66 + 74 + 71 + 81) + 2x)/7 ≥ 70
((66 + 74 + 71 + 81) + 2x)/7 < 80
Solving these inequalities will give you the score you need on the last test.
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You have just used the network planning model and found the critical path length is 30 days and the variance of the critical path is 25 days. The probability that the project will be completed in 33 days or less is equal to:_______.
Answer:
0.726 is the probability that the project will be completed in 33 days or less.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 30 days
Variance = 25 days
Standard Deviation,
[tex]\sigma = \sqrt{\text{Variance}} = \sqrt{25} = 5[/tex]
We assume that the distribution of path length is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(completed in 33 days or less)
[tex]P( x \leq 33) = P( z \leq \displaystyle\frac{33 - 30}{5}) = P(z \leq 0.6)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x \leq 33) = 0.726 = 72.6\%[/tex]
0.726 is the probability that the project will be completed in 33 days or less.
Beth gets on the elevator at the sixth floor. She rides up three floors to meet Doris. They ride down seven floors to meet Julio. How many floors have is Beth from where she started
Answer:
4
Step-by-step explanation:
Firstly, she gets on the elevator at the 6th floor. This means she was originally at the 6th floor.
She then stopped in the next three floors. This means she got off at floor 9 where we had Doris.
From Doris, she has to take 7 floors down to meet Julio. This means she went back to floor 2.
Now since her starting position was 6 and she is presently at the 2nd floor, this means she is four floors away from her starting position on the sixth floor.
The price-to-earning ratio for firms in a given industry is distributed according to normal distribution. In this industry, a firm with a standard normal variable value of Z=1:_________
a. Has an above average price-to-earning ratio
b. Has a below average price-to-earning ratio
c. Has an average price-to-earning ratio
d. May have an average or below average price-to-earnings ratio
Answer:
Option a) Has an above average price-to-earning ratio
Step-by-step explanation:
We are given the following in the question:
The price-to-earning ratio for firms in a given industry is distributed according to normal distribution.
For a particular firm the ratio x has a standard normal variable has a value,
z = 1
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
[tex]1 = \dfrac{x - \mu}{\sigma}\\\\\sigma = x - \mu\\x = \mu + \sigma[/tex]
Thus, the firm has an above average price-to-earning ratio as the ratio is one standard deviation above the mean.
Option a) Has an above average price-to-earning ratio
If a firm in a given industry has a standard normal variable value of Z=1, it signifies that the firm has an above average price-to-earning ratio.
Explanation:A student has asked about the interpretation of a standard normal variable value, Z, in the context of a firm's price-to-earning (P/E) ratio in a given industry. If a firm has a Z-value of 1, it means that the firm's P/E ratio is one standard deviation above the mean for the industry. Since the standard normal distribution has a mean of 0 and a standard deviation of 1, a Z-value of 1 corresponds to performance that is better than average. Therefore, a correct interpretation of this scenario is a. Has an above average price-to-earning ratio.
The price-to-earning ratio for firms in a given industry is distributed according to a normal distribution. A firm with a standard normal variable value of Z=1 would have an above average price-to-earning ratio.
This is because the standard normal distribution has a mean of 0 and a standard deviation of 1. A Z-score of 1 corresponds to a value that is one standard deviation above the mean. Since the mean represents the average price-to-earning ratio in the industry, a firm with a Z-score of 1 would have a ratio that is above average.
Can someone help me with this Exponential Growth and Decay word problem? I have been able to get the rest of them done but this one stumps me...
Answer:
A) 61.68
Step-by-step explanation:
This is an exponential decay of 0.875. The way to know that is to get that is[tex]\frac{right}{left} \\[/tex] (Pick a random number and divide by the number to the left of it) From there, there is a 12 hour difference from 7 am to 7pm! Divide 306.25/0.875 and divide it by 0.875 by the answer of that each time 12 times. The exact answer is 61.68402914
Answer: option A is the correct answer.
Step-by-step explanation:
Looking at the table, the amount of medicine in her system is decreasing constantly with time. This constant amount by which this amount of medicine is decreasing is the decay constant.
Decay constant = 350/400 = 306.25/ 350 = 0.875
Let y represent the amount of medicine in her system after t hours.
Let a represent the initial amount of medicine in her system.
Let b represent the decay constant.
Let t represent the number of hours. Therefore, the function representing the exponential decay would be
y = 400b^t
When she wakes up at 7.00 am, the time from 5.00 pm would be 14 hours. Therefore
y = 400 × 0.875^14
y = 61.7 mg
If ABC ~ DEF and the scale factor from ABC to DEF is 1/5, what are the lengths of DE , EF and DF , respectively?
Answer:
option a (2,2,5)
Step-by-step explanation:
First we have to notice that it tells us that the scale is 1/5.
This means that the measures of the new triangle will be 5 times smaller than those of the other triangle.
Now we just have to take the measurements of the sides of the triangle and multiply them by 1/5 or divide them by 5
AB = 10 DE = 10/5 = 2
BC = 10 EF = 10/5 = 2
AC = 25 DF = 25/5 = 5
DE = 2
EF = 2
DF = 5
(2, 2, 5)
If 64.7% 2017 college students attended a 4year program and u took a random 500students in 2017 how many college students would you expect to attend 4year program
Answer:
Around 337 students
Step-by-step explanation:
We are given the following in the question:
Percentage of student who attended a 4 year programme in 2017 = 64.7%
Sample size, n = 500
We have to find he expected number of students who would attend 4 year programme.
Formula:
[tex]67.4\% \times n\\=67.4\% \times 500\\\\=\dfrac{67.4}{100}\times 500\\\\=337[/tex]
Thus, around 337 students are expected to attend 4 year program.
Sheila has a plan to save $45 a month for 18 months so that she has $810 to remodel her bathroom. After 13 months Sheila has saved $510. If the most Sheila can possibly save is $70 per month, which of the following statements is truea. Sheila will meet her goal and does not need to adjust her plan. b. Sheila must save 50permonthtoachievehergoal.c.Sheilamustsave60 per month to achieve her goal. d. Sheila will not be able to achieve her goal.
Answer:
The answer is C.
Step-by-step explanation:
810 (goal) - 510 (amount saved so far) = 300 (Balance left to achieve goal) divided by 5( months remaining to achieve goal) = $60 a month
The statement is Sheila needs to save $60 per month to achieve her goal, the option is C.
We are given that;
Monthly saving= $45
Bathroom remodel amount= $810
Now,
To find the answer, we need to calculate how much more Sheila needs to save and how many months she has left.
We can subtract the amount she has saved from the amount she needs to save:
810 - 510 = 300
So, Sheila needs to save $300 more. We can also subtract the number of months she has saved from the number of months in her plan:
18 - 13 = 5
So, Sheila has 5 months left. To find the monthly amount she needs to save, we can divide the remaining amount by the remaining months:
300 / 5 = 60
Therefore, by algebra the answer will be $60 per month.
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Jenny uses a roller to paint a wall. The roller has a radius of 1.75 inches and a height of 10 inches. In two rolls, what is the area of the wall that she will paint. Use 3.14 for pi
In two rolls, 219.8 square inches of wall she will paint
Solution:
Given that,
Jenny uses a roller to paint a wall
The roller has a radius of 1.75 inches and a height of 10 inches
We have to find the area of the wall that she will paint in two rolls
Find the lateral surface area of cylinder
[tex]A_L = 2 \pi r h[/tex]
Where, "r" is the radius and "h" is the height
From given,
r = 1.75 inches
h = 10 inches
Substituting the values we get,
[tex]A_L = 2 \times 3.14 \times 1.75 \times 10\\\\A_L = 6.28 \times 1.75 \times 10\\\\A_L = 109.9[/tex]
Thus lateral surafce area is 109.9 square inches
For two rolls,
Area = 2 x 109.9 = 219.8
Thus in two rolls, 219.8 square inches of wall she will paint
The length of a rectangle is 1 7/9 in., and its width is 3/4 of its length. Find the area of this rectangle.
Answer:
The answer to your question is Area = 1024/243 or 4 52/243
Step-by-step explanation:
Data
length = 1 7/9
width = 3/4 of its length
Area = ?
Formula
Area of a rectangle = length x width
Process
1.- Convert the mixed fraction to improper fraction
1 7/9 = (9 + 7) / 9 = 16/9
2.- Get the width
16/9 / 3/4 = (16 x 4) / (9 x 3)
= 64 / 27
3.- Get the area
Area = (16/9)(64/27)
= 1024/243
= 4 52/243
Answer:
2.37 square inches
Step-by-step explanation:
l = 16/9 = 1.78
b = 16/9 *3/4 = 1.33
Area = l * b
Area = 1.78 * 1.33
Area = 2.37 square inches
what is 7x-6=22
A 4
B 5
C 16
D 11
Find the least-squares regression line: ŷ =b0+b1x, through the points (−1,0),(0,9), (4,13), (8,20), (10,23). For x=5, what is ŷ? For x=9, what is ŷ?
The value of y =14.45689 when x=5 and y= 21.74137 when x= 9.
What is Regression line?An estimate of the line that depicts the actual, but unidentified, linear relationship between the two variables is called a regression line. When the value of the explanatory variable is known, the regression line's equation is used to predict (or estimate) the value of the response variable.
Because it is the line that fits the points the best when drawn through them, the regression line is occasionally referred to as the "line of best fit." It is a line that minimises the difference between the projected and actual scores.
Given Data:
(−1,0),(0,9), (4,13), (8,20), (10,23).
Sum of X = 21
Sum of Y = 65
Mean X = 4.2
Mean Y = 13
Sum of squares (S[tex]S_x[/tex]) = 92.8
Sum of products (SP) = 169
Regression Equation = ŷ = bx + a
b = SP/S[tex]S_x[/tex] = 169/92.8 = 1.82112
a = [tex]M_y[/tex]- [tex]bM_x[/tex]= 13 - (1.82x4.2) = 5.35129
ŷ = 1.82112x + 5.35129
For x= 5
ŷ = 1.82112(5) + 5.35129
ŷ = 14.45689
and, when x= 9
ŷ = 1.82112(9) + 5.35129
ŷ = 21.74137
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Final answer:
The least-squares regression line equation is ŷ = -1.72 + 2.18x. For x = 5, ŷ is approximately 9.88. For x = 9, ŷ is approximately 18.62.
Explanation:
To find the least-squares regression line, we need to use the formula ŷ = b0 + b1x, where b0 is the y-intercept and b1 is the slope. Using the coordinates of the given points, we can calculate the values of b0 and b1. Substituting these values into the formula, we get ŷ = -1.72 + 2.18x. For x = 5, we can plug this value into the equation: ŷ = -1.72 + 2.18(5) = 9.88. Similarly, for x = 9, we can substitute x into the equation: ŷ = -1.72 + 2.18(9) = 18.62.
The cost of 3 boxes of envelopes and 4 boxes of notebook paper is $22.65. Two boxes of envelopes and 6 boxes of notebook paper cost $24.60. Find the cost of each.
Answer: the cost of one box of envelope is $3.75.
the cost of one box of notebook paper is $2.85
Step-by-step explanation:
Let x represent the cost of one box of envelope.
Let y represent the cost of one box of notebook paper.
The cost of 3 boxes of envelopes and 4 boxes of notebook paper is $22.65. This means that
3x + 4y = 22.65 - - - - - - - - - - -1
Two boxes of envelopes and 6 boxes of notebook paper cost $24.60. This means that
2x + 6y = 24.6 - - - - - - - - - - -2
Multiplying equation 1 by 2 and equation 2 by 3, it becomes
6x + 8y = 45.3
6x + 18y = 73.8
Subtracting, it becomes
- 10y = - 28.5
y = - 28.5/-10
y = 2.85
Substituting y = 2.85 into equation 1, it becomes
3x + 4 × 2.85 = 22.65
3x + 11.4 = 22.65
3x = 22.65 - 11.4 = 11.25
x = 11.25/3 = 3.75
Please help, will mark brainliest if correct!!!
Answer:8.5
Step-by-step explanation:
Pythagorean Theorem
A
a^2 + b^2 = c^2
3^2 + 8^2 = c^2
9 + 64 = c^2
73 = c^2
C= Sqrt (73)
C = 8.5
A card chosen at random from a deck of 52 cards. There are 4 queens and 4 kings in a deck of playing cards. What is the probability it is a queen or a king?
1. Find X
A. 16.45
B. 15.92
C. 12
D. 11.5
Applying the tangent ratio, the value of x in the given image is calculated as B. x = 15.92
How to find x using the tangent ratio?The tangent ratio is a trigonometric ratio that represents the ratio of the length of the side opposite a given angle to the length of the side adjacent to that angle in a right-angled triangle. It is denoted as tan(θ), where θ is the angle. Mathematically, tan(θ) = opposite/adjacent.
Thus, using the tangent ratio, we have:
tan 53 = x/12
x = tan 53 * 12
x ≈ 15.92
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Are these ratios equivalent?
18 professors : 5 students
36 professors : 10 students
yes/no
Answer:
yes
Step-by-step explanation:
Answer:
Yes
Step-by-step explanation:
18 : 5
36 : 10
36/2 : 10/2
18 : 5
A company collected data for the number of text messages sent and received using a text-message application since October 2011. The table shows the number of text messages sent and received in billions over time. The data can be modeled by a quadratic function. Which function best models the data?
A. n(t) = -0.002t^2 + 0.55t + 5.02
B. n(t) = 0.072t^2 - 0.15t + 2.73
C. n(t) = -0.002t^2 + 5.02
D. n(t) = 0.072t^2 + 2.73
Answer:
B. n(t) = 0.072t^2 - 0.15t + 2.73
Step-by-step explanation:
Plot data as pair of coordinates where t =x and n(t)=y
Use a graph tool to plot the coordinate points and join the points with a smooth curve
From the answers, test for the function that fits the points as plotted on the tool.
In this case, the function that fits the plot of the data is ;
n(t) = 0.072t^2 - 0.15t + 2.73
See attached;
The function that best used to model this situation is n(t) = 0.072t² - 0.299t + 2.57
Quadratic functionQuadratic function is in the form:
y = ax² + bx + cwhere a, b, c are constants.
Let n(t) represent the number of text at time t months.
It is given by:
n(t) = at² + bt + c
At point (5, 3):
3 = a(5)² + 5b + c25a + 5b + c = 3 (1)At point (20, 27):
27 = a(20)² + 20b + c400a + 20b + c = 27 (2)At point (40, 112):
112 = a(40)² + 40b + c1600a + 40b + c = 112 (3)a = 0.072, b = -0.29, c = 2.57
n(t) = 0.072t² - 0.299t + 2.57
The function that best used to model this situation is h(t) = -4.9t² + 295 + 2
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Show that the Fibonacci numbers satisfy the recurrence relation fn = 5fn−4 + 3fn−5 for n = 5, 6, 7, . . . , together with the initial conditions f0 = 0, f1 = 1, f2 = 1, f3 = 2, and f4 = 3. Use this recurrence relation to show that f5n is divisible by 5, for n = 1, 2, 3, . . . .
Answer with step-by-step explanation:
We are given that the recurrence relation
[tex]f_n=5f_{n-4}+3f_{n-5}[/tex]
for n=5,6,7,..
Initial condition
[tex]f_0=0,f_1=1,f_2=1,f_3=2,f_4=3[/tex]
We have to show that Fibonacci numbers satisfies the recurrence relation.
The recurrence relation of Fibonacci numbers
[tex]f_n=f_{n-1}+f_{n-2}[/tex],[tex]f_0=0,f_1=1[/tex]
Apply this
[tex]f_n=(f_{n-2}+f_{n-3})+f_{n-2}=2f_{n-2}+f_{n-3}[/tex]
[tex]f_n=2(f_{n-3}+f_{n-4})+f_{n-3}=3f_{n-3}+2f_{n-4}[/tex]
[tex]f_n=3(f_{n-4}+f_{n-5})+2f_{n-4}=5f_{n-4}+3f_{n-5}[/tex]
Substitute n=2
[tex]f_2=f_1+f_0=1+0=1[/tex]
[tex]f_3=f_2+f_1=1+1=2[/tex]
[tex]f_4=f_3+f_2=2+1=3[/tex]
Hence, the Fibonacci numbers satisfied the given recurrence relation .
Now, we have to show that [tex]f_{5n}[/tex] is divisible by 5 for n=1,2,3,..
Now replace n by 5n
[tex]f_{5n}=5f_{5n-4}+3f_{5n-5}[/tex]
Apply induction
Substitute n=1
[tex]f_5=5f_1+3f_0=5+0=5[/tex]
It is true for n=1
Suppose it is true for n=k
[tex]f_{5k}=5f_{5k-4}+3f_{5k-5}[/tex] is divisible 5
Let [tex]f_{5k}=5q[/tex]
Now, we shall prove that for n=k+1 is true
[tex]f_{5k+5}=5f_{5k+5-4}+3f_{5k+5-5}=5f_{5k+1}+3f_{5k}=5f_{5k+1}+3(5q)[/tex]
[tex]f_{5k+5}=5(f_{5k+1}+3q)[/tex]
It is multiple of 5 .Therefore, it is divisible by 5.
It is true for n=k+1
Hence, the [tex]f_{5n}[/tex] is divisible by 5 for n=1,2,3,..
The Fibonacci sequence with its initial conditions does satisfy the given recurrence relation. With this, it can also be demonstrated that f5n is divisible by 5 for all positive integers n.
Explanation:The problem involves a specific series in mathematics known as the Fibonacci series. We are given the Fibonacci numbers initial conditions: f0 = 0, f1 = 1, f2 = 1, f3 = 2, and f4 = 3. From these initial conditions, we can calculate the next values of the Fibonacci numbers.
Using the given Fibonacci numbers recurrence relation: fn = 5fn−4 + 3fn−5, we get:
f5 = 5f1 + 3f0 = 5 + 0 = 5f6 = 5f2 + 3f1 = 5 + 3 = 8f7 = 5f3 + 3f2 = 10 + 3 = 13So, fn = 5fn−4 + 3fn−5 is valid for n = 5, 6, 7, and presumably, for greater integers as well.
We then use this recurrence relation to show that f5n is divisible by 5, for n = 1, 2, 3, etc. This is immediately obvious for n = 1, because f5 = 5. But for n greater than 1, we can see that f5n = f(5*(n-1)+5) = 5f(5*(n-1)) + 3f(5*(n-1)-1), which will be divisible by 5 because 5 is a factor in the first term and the second term is also divisible by 5 according to our earlier results. This confirms that f5n is divisible by 5 for n = 1, 2, 3, and so on.
Learn more about Fibonacci sequence here:https://brainly.com/question/34482134
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Evaluate the given expression. 5 P 2
Answer:
20
Step-by-step explanation:
nPr = n!/(n-r)!
5P2 = 5!/(5-2)!
= 5!/3!
= 5×4×3!/3!
= 5×4
= 20