Answer:
A. 20
Step-by-step explanation:
300 ÷ 2.5= 120
So
50÷2.5 = 20
Given f(x) = 8 - 5x, find f(2).
Answer: 320 i hope it works..
Here are four equations of absolute value functions and three coordinate pairs. Each coordinate pair represents the vertex of the graph of an absolute value function.
A. p(x)=[x-9] B. q(x)=[x]+9 C. r(x)=[x+9] D. t(x)=[x]-9
The vertex of p(x) is (9, 0).
Vertex of q(x) is (0, 9).
Vertex of r(x) is (-9, 0).
Vertex of t(x) is (0, -9).
What is absolute value function?
The non-negative value of a real number x without regard to its sign is known as its absolute value or modulus in mathematics and is denoted by the symbol |x|. Specifically, |x|=x and |x|=-x, respectively, if x is a positive number, and |0|=0 otherwise.
The vertex form of an absolute value function is f(x) = a |x - h| + k where
(h, k) is the vertex from which the graph starts and a = m is the slope.
So, the absolute value function p(x) = |x - 9|, means h =-(-9) = 9 and
k = 0, will result in the vertex at (9, 0).
The absolute value function q(x) = |x| +9 , means h = 0 h=0 and k =9, will result in a vertex (0, 9).
The absolute value function r(x) = |x + 9|, means, h = -9 and, k=0, will result in the vertex at (-9, 0).
The absolute value function t(x) = |x| -9 , means h = 0 and k = -9, will result in the vertex at (0, -9)
Hence,
The vertex of p(x) is (9, 0).
Vertex of q(x) is (0, 9).
Vertex of r(x) is (-9, 0).
Vertex of t(x) is (0, -9).
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i only need 8 someone pls i’m desperate i will mark you brainliest when i can
Answer:
[tex]y=-1/4x+3[/tex]
Step-by-step explanation:
Slope-intercept form is of the form y=mx+b where m is the slope and b is they y-intercept. To solve for the slope, we take the difference of two y-coordinates divided by the x-coordinates difference. Doing this, we get
[tex](3-1)/(0-4)[/tex]
[tex]=-1/4[/tex]
This is our slope. The y-intercept is 3, because the line hits the y-axis when y=3.
Thus, our equation is y=-1/4x + 3.
Feel free to contact me with any questions. I hope this helps!
Please help, i cannot figure out how to solve for 60%
(a) As time goes on and approaches infinity, the level of oxygen in the pond approaches normal, therefore;
The oxygen level will eventually approach its normal level in the long-run
(b) The number of weeks after which the oxygen level becomes 60% of its normal level are 0.5, 2 weeks
How can the rational polynomial be evaluated?The rational polynomial function in which the numerator and denominator have the same power and the coefficient of the highest power are the same indicates that the value of f(t) approaches 1 (normal) as t approaches infinity.
The function with which the oxygen level in the pond can be found is presented as follows;
[tex]f(t) = \dfrac{t^2-t + 1}{t^2+ 1}[/tex]
f(0) = 1, therefore
The end behavior of the function as the value ot t approaches infinity is therefore the asymptote;
y = 1/1 = 1
The value of the function f(t) as the value of t approaches infinity is that the f(t) approaches 1, which is the normal level of oxygen in the pond
(b) When the oxygen level is 60%, we get;
[tex]f(t) = \dfrac{t^2-t + 1}{t^2+ 1} = 60\% = 0.6[/tex]
0.6·(t² + 1) = t² - t + 1
0.6·t² + 0.6 = t² - t + 1
0.4·t² - t + 0.4 = 0
t² - 2.5·t + 1 = 0
Therefore;
(t - 2)·(t - 0.5) = 0
t = 2 or t = 0.5
The times when the oxygen level is 60% is 0.5 and 2 weeks after the organic waste is dumped in the pond.
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Brooke wanted to convert some money from pounds (£) into euros (€) so that she would receive €5382. Brooke converted her money on Tuesday when the exchange rate was £1 = €1.15. On Wednesday, the exchange rate was £1 = €1.17. How much more did Brooke spend on Tuesday than she would have if she had converted her money on Wednesday?
Answer:
she spent £80 more on Tuesday than on Wednesday
Step-by-step explanation:
Tuesday she spent = £4680
1 = €1.15
x = €5382
cross multiply
x = £4680
Wednesday she spent = £4600
£1 = €1.17
x = €5382
cross multiply
x = £4600
the difference is £80
(5382/1.15) - (5382/1.17) = 80
Answer: for £80 more
Step-by-step explanation:
To receive 5382€ on Tuesday Brooke needs:
5382/1.15=
£4680
To receive 5382€ оn Wednesday Brooke needs:
5382/1.17=
£4600
Thus,
4680-4600=
£80
What are the domain and range of f(x) = |x + 6|?
Answer:
Domain: [tex](-\infty, \infty)[/tex]
Range: [tex]f(x)\ge 0[/tex]
Step-by-step explanation:
Interval Notation:Interval Notation is one way to express a range of values. This is usually denoted with two values, let's just say "a" and "b" as: [tex](a, b)[/tex] where "a" is less than "b", so the lower value goes on the left the higher values goes on the right.
The next thing to note is the difference between square brackets and parenthesis. Whenever you use a square bracket next to a value in interval notation, that means the value is included. If you use a parenthesis, that means the value is not included.
So for example: [tex](2, 4][/tex] means all values between 2 (excluding 2) and 4 (including 4)
Domain of a Function:The domain of a function, or a relation, can be defined as all x-values that are on the graph. In most cases this is all real numbers which can also be denoted as: [tex](-\infty, \infty)[/tex] which is saying all values between negative infinity and positive infinity.. which is any real number. The only time the domain doesn't include x-values, is because the line isn't defined at that x-value OR based on the context, only certain values are in the domain e.g (time after an event; only positive numbers)
In the function provided: [tex]f(x)=|x+6|[/tex] there is no x-value in which this function is undefined. So the range is all real numbers or [tex](-\infty, \infty)[/tex]
Range of a Function:The range of a function, or a relation, similar to the domain, can be defined as all y-values that are on the graph.
Now to determine the range of the function, we first need to understand what the absolute value is doing to the (x+6) inside of it.
The absolute value of any real number can be defined as the distance from zero. When applied to any positive number, the value stays the same (since thd distance from zero IS that number). When applied to any negative number, the value becomes the positive value of that number since distance is positive. For example -6 has a distance of 6 units to the left of zero, but distance is a scalar it doesn't care about the direction, it's just 6 units away from zero, which is why it becomes positive when taking the absolute value. When applied to zero... it just stays zero, since 0 is 0 units away from zero...
So the minimum value of |x+6| is going to be zero, which occurs when x+6 is equal to zero. If it's a negative number or positive number inside it's always going ot output a positive number. And for positive numbers it actually remains positive, so once it reaches the turning point (where it goes from increasing to decreasing) it's going to keep increasing to positive infinity.
This means the range is all values greater than or equal to zero, which can be denoted as: [tex][0, \infty)[/tex] OR [tex]f(x)\ge 0[/tex] since the f(x) represents the "y" value, all the inequality is saying, is that the y-value will always be greater than or equal to zero.
Estimate the total amount of time Jessie practices by rounding to the nearest 100 minutes
Using simple mathematical operations, we know that Jessie's total time of practice is 410 minutes.
What are mathematical operations?A mathematical "operation" is the process of calculating a value utilizing operands and a math operator.
There are predefined rules associated with the math operator's symbol that must be applied to the supplied operands or numbers.
PEMDAS stands for Parentheses, Exponents, Multiplication, Division, and Addition Subtraction (from left to right).
So, the total time Jessie practices:
We must include the time spent practicing during the first week and the second week in order to determine the overall amount of time Jesse practices.
Total time:
165 + 245
410 minutes
Rounding off: 410 minutes
Therefore, using simple mathematical operations, we know that Jessie's total time of practice is 410 minutes.
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Complete question:
Jesse practices the trumpet for a total of 165 minutes during the first week of school. He practices for 245 minutes during the second week. A. Estimate the total amount of time Jesse practices by rounding to the nearest 10 minutes.
A group of friends wants to go to the amusement park. They have no more than $80 to spend on parking and admission. Parking is $14.75, and tickets cost $11.25 per person, including tax. Write and solve an inequality which can be used to determine
x
x, the number of people who can go to the amusement park.
The inequality which represents the number of people who can go to the amusement park will be 14.75 + 11.25x ≤ 80.
What is inequality?A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.
A mathematical phrase in which the sides are not equal is referred to as being unequal. In essence, a comparison of any two values reveals whether one is less than, larger than, or equal to the value on the opposite side of the equation.
As per the given,
Total budget = $80
Tickets cost of x number of tickets = 11.25x
Parking cost + tickets cost ≤ total budget
14.75 + 11.25x ≤ 80
Hence"The inequality which represents the number of people who can go to the amusement park will be 14.75 + 11.25x ≤ 80".
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Two systems of equations are shown: System A System B 6x + y = 2 2x − 3y = −10 −x − y = −3 −x − y = −3 Which of the following statements is correct about the two systems of equations? (1 point) The value of x for System B will be 4 less than the value of x for System A because the coefficient of x in the first equation of System B is 4 less than the coefficient of x in the first equation of System A. They will have the same solution because the first equations of both the systems have the same graph. They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 4 times the second equation of System A. The value of x for System A will be equal to the value of y for System B because the first equation of System B is obtained by adding −4 to the first equation of System A and the second equations are identical.
Since the first equation of system B is created by multiplying system A's second equation by four, both equations will have the same answer. option c is correct
What is equations?Equations are mathematical expressions that have two algebraic expressions on either side of an equals (=) sign. . For an unknown quantity represented by an unknown variable, equations can be solved to determine its value. A statement is not an equation if it contains no references to "equal to" symbols.
Here,
you have a two-equation system.
System A: 6x + y = 2, and -x - y = 3.
System B: -10 = 2x - 3y, -3 = x - y
Choose the statement about each of the two systems of equations that is true from the list provided.
System A is an example.
6x + y = 2 ......(1) (1)
-x -y = -3 ....(2) (2)
Equation (2) becomes when we double it by 4.
-4x -4y = -12 ......(3) (3)
We now have, after adding (1) and (3),
6x + y +(-4x - 4y) = 2 - 12
solving, we arrive at,
6x + y -4x - 4y = 2 - 12
Equation (1) of system B is 2x – 3y = – 10.
The answer to system A is now discovered.
Equation (2) is created by adding equation (1).
6x + y - x -y = 2 -3
5x = -1
Value of x in (2) yields,
The answer to system B is now discovered.
System B: 2x - 3y = -10... (4)
-x -y = -3 .....(5) (5)
By dividing equation (5) by 3, we get y=16/5
-3x -3y = -9 .......(6) (6)
Equation (6) is obtained by deducting equation (4).
2x - 3y-(-3x - 3y) = -10 +9
5x = -1
Using (5) as the value of x, we obtain x=-1/5
Therefore, the values of systems A and B are equal.
Thus, choice (C) is the right one.
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Answer:
They will have the same solution because the first equation of System B is obtained by adding the first equation of System A to 4 times the second equation of System A
PLEASE HELP ASAP!!!! FULL ANSWERS ONLY!!!!
Blake wants to build a rectangular enclosure for his animals. One side of the pen will be against the barn, so he needs no fence on that side. The other three sides will be enclosed with wire fencing. If Blake has 950 feet of fencing, you can find the dimensions that maximize the area of the enclosure.
a) Let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). Write a function for the area A of the enclosure in terms of w . (HINT: First write one equation with w and l and one equation with w and l and A. Solve for l in the first equation and substitute for l in the other). A ( w ) = ____________
b) What width w would maximize the area? w = __________________ ft
c) What is the maximum area? A = ______________ square feet
Answer: The width that would maximize the area would be 475, and the square feet would be 475.
Step-by-step explanation: 950 divided by 2 is 475 which would divide each of the two sides of the fence, to acquire the amount needed.
Input (a): a number
Output (y): 8 less than 3 times x
Which equation represents the rule?
y=-8x-3 y=-3x+8 y=-8x+3 y=3x-8
Ans: 3x-8
now, as per the question,
LHS=y .......(i)
and it is given in the question that 3 times x is greater than y by 8 .
so , by reframing the equation numerically,
we get,
y=3x-8.
Pina filled out the following information on the back of her monthly statement:
Ending balance from statement $1,115.5
Deposits outstanding(+) $409.03
Total of checks outstanding(-) $605.42
What is her revised bank statement?
The revised bank statement of Pina is $919.11 if ending balance from statement $1,115.5 and Deposits outstanding is $409.03.
What is meant by deposit outstanding?Unpaid deposits are receipts that appear in your accounting records but not on your bank statement. Money that you have received, such as cash and checks, is included in receipts. A receipt may occasionally be entered in your books before it shows up on your bank statement. Your books' line item corresponds to the unpaid deposit.
An outstanding deposit is a term used to describe a business's receipts (cash, customer checks, etc.) that have been entered into the general ledger accounts but have not yet been shown on the bank statement. Deposits in transit are another name for outstanding deposits.
Revised statement=End balance+ deposits-check outstand
=1115.5+409.03-605.42
=919.11
Therefore, the revised bank statement of Pina is $919.11
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The table shows the age and grade level of a sample of children in a club.
Create a scatter plot that represents the data that is shown in the table. The x-axis represents the child's age, and the y-axis represents the child's grade level.
Scatter plots exists essential because they can demonstrate the degree of correlation, if any, between the values of observed quantities or occurrences.
What is meant by scatter plot?Several dots are placed on a horizontal and vertical axis to form a scatter plot. In statistics, scatter plots are crucial because they can demonstrate the degree of correlation, if any, between the values of observed quantities or occurrences (called variables).
Using data visualization, a scatter plot may be used to show how several variables relate to one another. Put different data points between an x- and y-axis to display this data. The name of this form of data visualization comes from the way that each of these data points appears to be "scattered" over the graph.
The graphs known as scatter plots show the relationship between two variables in a data set. On a two-dimensional plane or in a Cartesian system, it represents data points. The dependent variable is represented on the Y-axis, and the independent variable or attribute is plotted on the X-axis.
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Walt made an extra $7000 last year from a part-time job. He invested part of the money at 7% and the rest
at 8%. He made a total of $530 in interest. How much was invested at 8%?
The amount invested at the rate of 8% is; $3000
How to calculate the amount invested?We are told that Walt made an extra $7000 last year from a part-time job. He invested part of the money at 7% and the rest at 8%.
If the amount invested is denoted as X , then we have;
0.07X + 0.08(7000 - X)
Now, since he made a total of $530, the we can equate that equation to $530 to get;
0.07X + 0.08(7000 - X) = 530
Expanding this equation gives us;
0.07X + 560 - 0.08X = 530
0.01X = 560 - 530
0.01X = 30
X = 30/0.01
X = $3000
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A random group of 114 adults and 136 teenagers completed a maze. Their times were recorded, and a prize was given if they
finished the maze in under 10 minutes.
There were 20 more teenagers than adults that received a prize for completing the maze in under 10 minutes.
There were 62 adults who finished the maze in 10 minutes or longer.
Create a two-way frequency table to represent this data. Type the correct answer in the box. Use numerals instead of words.
The adults less than 10 minutes are 52, the teenager less than 10 minutes 72, the teenager more than 10 minutes are 64, and the total is 250.
What is mathematical operations in reasoning?
The study is based on the mathematical operations that are multiplication, addition, subtraction, and division. “Mathematical Operations reasoning” is the simple process of the expression of the containing numbers and various operations in the mathematical field.
Time to complete Maze
less than 10 minutes 10 minutes or more Total
Adults 52 62 114
Teenager 72 64 136
Total 124 126 250
114 - 62 = 52 {adults less than 10 minutes}
52 + 20 = 72 {teenager less than 10 minutes}
136 - 72 = 64 {teenager more than 10 minutes}
124 + 126 = 250 {total}
Hence, the adults less than 10 minutes are 52, the teenager less than 10 minutes 72, the teenager more than 10 minutes are 64, and the total is 250.
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You run 12 laps in 4 minutes. Your friend runs 18 laps in 8 minutes. Are the rates equivalent?
Answer: The rates are not equivalent.
Step-by-step explanation: You have to find the unit rate. This means the number of laps per minute. To find this out you have to divide 12 by 4 which equals 3 and divide 18 by 8 which equals 2.25. Now we know the rates are not equivalent because your rate is 3 laps/minute and your friend's rate is 2.25 laps/minute.
Answer: The rates are not equivalent
Step-by-step explanation:
Firstly,You must identitify the ratio.
The ratio can be 12:4 and 18:8.
Now we must convert 4 minutes and minutes into seconds to make it easier for us to work with.
4 * 60 = 240
18 * 60 = 1080
Now the question is 12: 240 equavialent to 18: 1080.
Method A:
For 12:240 to get 1 lap :
we can divided 240/12 = 20.
1 lap takes 1/3 of a minute or 20seconds.
For 18:1080:
We divide 1080/18 = 60= 1 minute
12:240=20 seconds
18:1080= 60 seconds.
They are not equivalent.
Method B:Compare 12:240 to 18:1080 and divide by 2
if the ratio split 50/50,they are equavalent.
Add the seconds together,becuase that is the total time
1080+240= 1320 second.
12+ 18=20
Now the ratio is 30:1320
1 lap takes 1320 / 30 = 44seconds
12 laps * 44 = 528s
18laps * 44 = 792s
528s is not 792s.
Question 3 Multiple Choice Worth 10 points)
(04.05 LC)
D
A hockey player made 88% of her goal attempts last season. The coach wants to estimate the number of goals the player will make out of 10 attempts in a game and decides to
run a simulation. Which of the following could be a representation of the events for the simulation?
O Assign the numbers 00-88 to represent a made goal and the numbers 89-99 to represent a missed goal.
O Assign the numbers 1-9 to represent a made goal and the number 10 to represent a missed goal.
O Assign the numbers 1-88 to represent a made goal and the numbers 89-100 to represent a missed goal.
O Assign the numbers 0-8 to represent a made goal and the number 9 to represent a missed goal.
Answer: Assign the numbers 1-9 to represent a made goal and the number 10 to represent a missed goal.
Step-by-step explanation: The coach wants to ESTIMATE the number of goals the player will make out of 10 attempts. Since someone can’t make 8.8 goals, we can round up to 9 goals.
Type the 4-letter code into the answer box. All CAPS, no spaces.
answer choices
A:
y>-2 x+5
B:
y ≤ -3x+5
C:
y 2 x+5
I:
y≤ x+4
Using inequality graphing, we get
1 = option A and E
2= none
3= option B and G
4= option C and I
How can graph inequalities?
1. Rewrite the equation with "y" on the left and the rest of the terms on the right.
2. Draw the "y=" line (with a solid line for y or y and a dotted line for y or Y>).
3. Use a shaded line to indicate larger than or less than (y> or y) or vice versa.
The graph corresponding to the options is as follows
graph 1 corresponds to equations A and E
Graph 3 corresponds to equations B and G
Graph 4 corresponds to equations C and I
The 4-letter word is CAGE
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John earns $22 per hour for a regular 40 hour work week. Any hours worked over 40 hours are paid at time and a half.
What is John's gross pay if he worked 44 hour this week?
By conducting mathematical operations, we know that John's gross pay will be $1,012 if he works 44 hours a week.
What are mathematical operations?A mathematical "operation" is the process of calculating a value utilizing operands and a math operator.
The supplied operands or integers must adhere to a set of predefined rules that are connected to the symbol of the math operator.
The order of operations refers to the rules that specify how to solve an expression including many operations.
Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction are all referred to as PEMDAS (from left to right).
So, we know that:
John earns $22 per hour for the first 40 hours of a week.
After every additional hour, he earns 1.5 times $22 each hour.
22 * 1.5 = $33
So, John's gross pay if he worked 44 hours will be:
22*40 + 33*4
880 + 132
$1,012
Therefore, by conducting mathematical operations, we know that John's gross pay will be $1,012 if he works 44 hours a week.
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Solve the given equation
2 x+3 y-6
x+y=6
Answer:
= 2x=6-3y
= 2x/2= 6-3y/2
= x= 6-3y/2
= x = -3y/2+3
6
4
2
Divide (16x - 12x + 4x) by 4x
Answer:2
Step-by-step explanation:
Combine like terms and divide
8x/4x
2
I’m in need of help
Answer:
(200, 10000)
Step-by-step explanation:
graph the function: f(x) = {x+1 if x<0}
{2 if 0<- x <-1 on the coordinate plane}
{x if x>1}
The graph of the function f(x) = { x + 1 } at different domain are plotted and attached
{ if x < 0 }
{ if 0 < x < 1 }
{ if x > 1 }
What is a graph?In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way.
The relationships between two or more items are frequently represented by the points on a graph.
The graph of f(x) = x + 1 for different domain is plotted and attached
{ if x < 0 } the domain includes { -∞....-3 -2 -1}
{ if 0 < x < 1 } the domain includes { 0...0.7 0.8 0.9}
{ if x > 1 } the domain includes { 0 1 2 3 4 5...∞ }
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7. How many possible combinations are possible if you choose exactly 4
toppings for a pizza from the choices below? You may use your formula sheet
or Desmos formulas to show your work.
pepperoni, pineapple, sausage, onions, frogs. peppers, ham, fish, olives.
mushrooms, tomatoes
The number of possible combinations is 330.
What is combination?
The definition of the combination is "An arrangement of objects where the order of the objects is irrelevant." Combining these two words indicates "Selection of things," where the order of the items is irrelevant.
The toppings for a pizza is pepperoni, pineapple, sausage, onions, frogs. peppers, ham, fish, olives, mushrooms, tomatoes.
The number of toppings is 11.
you choose exactly 4 toppings for a pizza.
Here r = 4
n = 11
The formula of combination rCn.
The number of possible combinations of a pizza is
₁₁C₄ = 11!/[4! (11-4)!]
₁₁C₄ = 11!/[4! 7!]
₁₁C₄ = 330
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17. Of the 4,325 grapefruits harvested so far,
Bob's has sold 1,250 of them at a farmer's
market. How many gift cartons can Bob's
fill with the remaining grapefruits? How
many grapefruits will be left?
Answer:
170 cartons ; 15 Grapefruits
Step-by-step explanation:
Given:-
Total grapefruits : 4325
Bob sold : 1250
Remaining is : 3075 (4325-1250)
How many gift cartons can Bob’s fill with the remaining grapefruits
Grapefruit per carton: 18 (given)
= 3075/18
= 170 cartons
How many grapefruits will be left
= 3075 - 3060
= 15 Grapefruits
I hope my answer helps you.
Cesar invested a total of $41000 in two accounts. The first account earned 7% after one year. However the second account suffered a 5% loss in the same time. At the end of one year the total amount of money gained was $1670. How much he invested in each account?
He invested in each account as follows:
First account -----> 31,000
Second account -----> 10,000
What is the percentage?
It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
gained - lost = 1670
0.07x - 0.05(41000 - x) = 1670
Find x :
0.07x - 0.05(41000 - x) = 1670
0.07x - 2050 + 0.05x = 1670
0.12x = 1670 + 2050
x = 31,000
41000 - 31,000 = 10,000
He invested in each account as follows:
First account -----> 31,000
Second account -----> 10,000
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Solve the system of equations below by
graphing. Type in the ordered pair (x,y)
for your answer
y=-2
x+y=5
Answer:
(7, -2)
Step-by-step explanation:
You want a graphical solution to the equations ...
y = -2x + y = 5GraphThe graph is shown in the attachment. The solution is (x, y) = (7, -2).
let a 5 c 1 2 3 4 5 6 2 1 3 5 4 6 d and b c 1 2 3 4 5 6 6 1 2 4 3 5 d . compute each of the following. a. a21 b. ba c. ab
(A×B)∩(A×C)={(1,4),(2,4),(3,4)}
In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Sets are typically denoted by curly brackets; for instance, A = 1, 2, 3, 4 is a set. Check the set symbols here as well.
The various kinds of sets, symbols, and operations are described by the set theory.
Given,
A={1,2,3},B={3,4} and C={4,5,6}
A×B={(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}
A×C={(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}
∴ (A×B)∩(A×C)={(1,4),(2,4),(3,4)}
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A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 109-cm and a standard deviation of 1.6-cm. For shipment, 27 steel rods are bundled together.
Find the probability that the average length of a randomly selected bundle of steel rods is less than 108.7-cm. P(M < 108.7-cm) =
The probability that the average length of a randomly selected bundle of steel rods is 0.165
Given population mean(μ)=109cm
standard deviation(σ)=1.6cm
sample size(n)=27
Let x represents the lengths of an individual rod then the z-score for
x can be given as:
z=x-μ/σ----------(1)
Also, the z score for the sample mean, x:
z=(¯x−μ)/σx
The standard error:
σ¯x=σ/√n-------------(2)
=1.6/√27
=1.6/5.196
σ¯x =0.3079
For a bundle of 27rods, the probability that the average length is between 109 cm and 108.7-cm:
considering m=108.7cm
P(n-m/σx)------------(3)
[tex]P(109 < x < 108.7)=P(\frac{109-108.7}{0.3079} )\\\\ P(109 < x < 108.7)=P(z < -0.974)\\[/tex]
=0.165.
And also find using another method:
P(z < (M - population mean)/(standard deviation/square root(sample size))
P(z < (108.7-109)/(1.6/sqrt(27)) = P(z < -0.974) = 0.165.
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Find a polynomial P3 such that {po, P1, P2, P3} (see Ex- ercise 11) is an orthogonal basis for the subspace P3 of P4. Scale the polynomial p3 so that its vector of values is (-1,2,0, -2,1).
The equation of the polynomial P3 is p3(t) = 5/6t³− 17/6t
Equation:
In algebra, an equation consists of variable, number and constants.
Given,
Here we need to find the polynomial P3 such that {P0, P1, P2, P3} is an orthogonal basis for the subspace P3 of P4.
And the scale the polynomial p3 so that its vector of values is (-1,2,0, -2,1).
Here we know that the polynomial (before the scaling) p3 is just the difference between t³ and its orthogonal projection on the span of 1,t,t²−2.
Then we showed that the projection is , hence
p3(t) = t³− 17/5t
When here we have the scale this polynomial such that the vector of values at t=−2,−1,0,1,2 is (−1,2,0,−2,1).
Now, we need to find the scalar α, it can be written as,
=> α⋅p3(−2)=−1
=> α⋅p3(−1)=2
=> α⋅p3(0)=0
=> α⋅p3(1)=−2
=> α⋅p3(2)=1
Here from the 4-th equation we have the value of
=> α⋅(1- 17/5) = -2
=> α = 5/6
And it is easy to check that all the equations are satisfied with this α.
Therefore, the required polynomial p3 is
=> p3(t) = 5/6(t³− 17/5t)
=> p3(t) = 5/6t³− 17/6t
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