The expression for the product of an even integer x and the next even integer can be written as 2x(x+1).
To find the product of an even integer x and the next even integer, we need to consider that consecutive even integers have a difference of 2.
Let's assume the even integer x is represented by 2k, where k is an integer.
The next even integer can be expressed as 2k+2.
Now, to find the product of x and the next even integer, we multiply 2k by (2k+2), resulting in 2k(2k+2).
Simplifying the expression, we can distribute the 2k across the terms inside the parentheses:
2k(2k+2) = 4[tex]k^2[/tex] + 4k.
Therefore, the expression for the product of an even integer x and the next even integer is 4[tex]k^2[/tex] + 4k, where x is represented by 2k.
This expression represents the multiplication of any even integer x by the next even integer.
The resulting expression is a quadratic polynomial in terms of k, which represents the product of the even integer x and the next even integer.
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Evaluate x^4.x^2 when x=5
Answer:
i got u fam!!!!!!!!!!!!!!!
its 15625
Step-by-step explanation:
An investor has decided to commit no more than $80,000 to the purchase of the common stocks of the companies, Company A and Company & He has also estimated that there is a chance of at most a 1% capital loss on his investment in Company A and a chance of at most a 4% loss on his investment in Company, and he has decided that these losses should not exceed $2000. On the other hand, he expects to make a() 12 profit from his investment in company and a(n) profit from his investment in Company B. Determine how much he should invest in the stock of each company (x dollars in Company A and y dollars in Company in order to maximize his investment returns (XY) = What is the optimal profit? Need Help? Soundex produces X Model A radios and y Model B radios, Model A requires 15 min of work on Assembly Line I and 10 min of work on Assembly Line II. Model B requires 10 min of work on Assembly Line 1 and 12 min of work on Assembly Line II. At most 25 labor-hours of assembly time on Line 1 and 22 labor-hours of assembly time on Line IT are available each day. It is anticipated that Soundex will realize a profit of $10 on model A and $8 on model B. How many clock radios of each model should be produced each day in order to maximize Soundex's profit? (x, y) - What is the optimal profit?
The investor should invest $40,000 in Company A and $40,000 in Company B to maximize their investment returns.
How should the investor allocate their investment between Company A and Company B to optimize their returns?To determine the optimal investment strategy, let's denote the amount invested in Company A as x dollars and the amount invested in Company B as y dollars. The investor has set a maximum capital loss of $2,000 for each company. Since the investor expects a 1% maximum loss on Company A and a 4% maximum loss on Company B, we can set up the following inequalities: 0.01x ≤ $2,000 and 0.04y ≤ $2,000. Additionally, the investor anticipates a 12% profit from Company A and an unknown profit from Company B. Let's denote the profit from Company B as p. Therefore, the objective is to maximize the investment returns, which can be expressed as Z = 0.12x + p. The total investment constraint is x + y = $80,000. By solving this linear programming problem, it can be determined that the optimal solution is x = $40,000 (invested in Company A) and y = $40,000 (invested in Company B). Consequently, the optimal profit will be 0.12($40,000) + p.
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Kiara made a square baby blanket she decided to add one foot of material to one side then cut 4 inches of material off from the bed to sit outside of area of the rest salting blankets in 960 square inches find the area of the original blanket
Answer:
Step-by-step explanation:
960 which is the area u divide by the one foot which is 144 square units to get your answer,6.66666666667
The length of a 1-inch paperclip is about 1.6 x 10−5 miles. It takes about 1.5 x 1010 paperclips linked together to reach the moon. What is the approximate distance to the moon?
Answer:
Distance from the moon is 2.4 × 10⁵ miles
Step-by-step explanation:
Length of 1-inch paperclip = [tex]1.6\times 10^{-5}[/tex] miles
Number of paperclips used to reach the moon = 1.5 × 10¹⁰
Total distance from the moon = Number of paper clips used × Length of one clip
= (1.5 × 10¹⁰) × (1.6 × 10⁻5)
= (1.5 × 1.6) (10¹⁰× 10⁻⁵)
= 2.4 × 10⁽¹⁰⁻⁵⁾ miles
= 2.4 × 10⁵ miles
Distance from the moon is 2.4 × 10⁵ miles.
consider the partially completed anova table shown. supposing all groups are of the same size, how many values are in each group of the data set this is based on?
The number of values in each group of the data set would be equal to the degrees of freedom for each group plus one.
In order to determine the number of values in each group of the data set based on the partially completed ANOVA table, we need to consider the total number of observations and the number of groups.
The ANOVA table consists of three main components: "Source of Variation," "Sum of Squares (SS)," and "Degrees of Freedom (df)." The "Source of Variation" represents the different factors or groups in the data set, while the "Sum of Squares" measures the variability within each group. The "Degrees of Freedom" represents the number of independent pieces of information available for estimating the population parameters.
In this case, since all groups are of the same size, we can determine the number of values in each group by examining the "Degrees of Freedom" column. The degrees of freedom for each group is the group size minus one (df = group size - 1). By adding one to the degrees of freedom for each group, we obtain the number of values in each group.
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what is the fraction for 0.875?
Answer:
7/8
Step-by-step explanation:
Answer:
7/8
Step-by-step explanation:
simplify 875/1000
Jonathan has a bag that contains exactly one red marble (r), one yellow marble (y), and one green marble (g). He chooses a marble from the bag without looking. Without replacing that marble, he chooses a second marble from the bag without looking. Which outcomes would be included in the sample space for Jonathan’s experiment? Select three options.
yy
gr
gg
rg
yr
Answer: The answers B,D,E are correct
Step-by-step explanation:
Answer: gr, rg, yr
Step-by-step explanation:
PLZZZ HELP!
In 2002 Johan bought a collector car as an investment when its value was
$180,000. He sold the car in 2014. Over the time he owned it, its value
grew an average of 2.44% each year.
How much profit did Johan earn on
his investment?
bought for $180000 in 2002
sold it in 2014
180000*2.44\100
Mean Median Minimum Maximum 75th percentile 25th percentile Interquartile Range Variance Standard Deviation 1 Convert the data into an Excel Table. 2 3 Create the same analysis completed in A3 to post the 4 summary statistics above each table column. But reference 5 6 the table columns with structured references (use the "Black Downward Arrow"pointing to the column header to reference the data table column) rather than highlighting the range of 3 8 cells within the table. 9 LO 11 SALE TYPE HOME TYPE ADDRESS 12 MLS Listing 13 MLS Listing 4 MLS Listing 5 MLS Listing 6 MLS Listing 17 MLS Listing 18 MLS Listing 19 MLS Listing 20 MLS Listing Mobile/Manufactured Home Single Family Residential Single Family Residential Single Family Residential Single Family Residential Single Family Residential Single Family Residential Single Family Residential Mobile/Manufactured Home 300 NW MAIN St 111 SE RONALD St 824 NW ORCHARD Dr 651 NE CHRISTIAN St 1787 UPPER CAMAS Rd 12661 LOOKINGGLASS Rd 100 KENYA Ct 1 MLS Listing 22 MLS Listing 23 MLS Listing 24 MLS Listing 25 MLS Listing 26 MLS Listing Single Family Residential Single Family Residential Mobile/Manufactured Home Mobile/Manufactured Home Mobile/Manufactured Home Mobile/Manufactured Home Single Family Residential 7 MLS Listing 1160 BROCKWAY Rd 401 SE GREGORY Dr 1068 RICE CREEK Rd 4546 MELODY Ln 2205 SE BOOTH Ave 4690 COOS BAY WAGON Rd 119 RUBY MAY Way 282 RIVER PLACE Dr Unit SP 62 1178 SE MYRTLE VIEW Dr 524 NE BROADWAY St 1740 RIVERSIDE Dr 170 SE WOODY Ct 330 NE BROADWAY St 867 NE HOLLY St 417 NE BROADWAY St 152 NE DEBBIE Way 600 NW T St 28 MLS Listing Single Family Residential 29 MLS Listing 30 MLS Listing 31 MLS Listing 12 MLS Listing 13 MLS Listing 4 MLS Listing Single Family Residential Single Family Residential Mobile/Manufactured Home Single Family Residential Single Family Residential Single Family Residential Single Family Residential Single Family Residential Multi-Family (2-4 Unit) Single Family Residential 5 MLS Listing 16 MLS Listing 17 MLS Listing 18 MLS Listing 237 HARMONY Dr 228 NW CIVIL BEND Ave 135 NE PLUM RIDGE Ct CITY Winston Winston Myrtle Creek Myrtle Creek Camas Valley Roseburg Winston Winston Winston Winston Roseburg Roseburg Roseburg Roseburg Roseburg Myrtle Creek Myrtle Creek Myrtle Creek Myrtle Creek Myrtle Creek Myrtle Creek Myrtle Creek Myrtle Creek Winston Roseburg Winston Winston i 1 Myrtle Creek 1.5 Camas Valley 3 1 Roseburg 5 3
By using structured references, the formulas will automatically refer to the data within the column of the Excel Table, even if the table expands or shrinks when you add or remove data.
To convert the data into an Excel Table and perform the analysis using structured references, you can follow these steps:
Select the entire data range, including the headers and the values.
In the Excel menu, go to the "Insert" tab and click on "Table." Choose a table style that you prefer.
Excel will automatically detect the range of your data. Make sure to check the box that says "My table has headers" if your data has headers.
Click "OK" to create the Excel Table.
Once you have created the Excel Table, you can perform the analysis and display the summary statistics using structured references. Here's how you can do it:
To calculate the Mean, use the formula =AVERAGE(Table1[LO]) and place it above the "LO" column header.
To calculate the Median, use the formula =MEDIAN(Table1[LO]) and place it above the "LO" column header.
To calculate the Minimum, use the formula =MIN(Table1[LO]) and place it above the "LO" column header.
To calculate the Maximum, use the formula =MAX(Table1[LO]) and place it above the "LO" column header.
To calculate the 75th percentile, use the formula =PERCENTILE.INC(Table1[LO],0.75) and place it above the "LO" column header.
To calculate the 25th percentile, use the formula =PERCENTILE.INC(Table1[LO],0.25) and place it above the "LO" column header.
To calculate the Interquartile Range, use the formula =QUARTILE.INC(Table1[LO],0.75) - QUARTILE.INC(Table1[LO],0.25) and place it above the "LO" column header.
To calculate the Variance, use the formula =VAR(Table1[LO]) and place it above the "LO" column header.
To calculate the Standard Deviation, use the formula =STDEV(Table1[LO]) and place it above the "LO" column header.
Make sure to adjust the table name (Table1) and column reference (LO) in the formulas based on your actual table and column names.
By using structured references, the formulas will automatically refer to the data within the column of the Excel Table, even if the table expands or shrinks when you add or remove data.
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Can someone help me with this. Will Mark brainliest.
Answer:
(-6, -7)
Step-by-step explanation:
For the x coordinates:
-1 - 4 = -5
-1 + -5 = -6
For the y coordinates:
-5 + -3 = -2
-5 + -2 = -7
I don't know how to explain it well sorry
The Lewis family and the Pham family each used their sprinklers last summer. The water output rate for the Lewis family's sprinkler was 30 L per hour. The water output rate for the Pham family's sprinkler was 25 L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 1475 L. How long was each sprinkler used?
The Lewis family used their sprinkler for 30 hours, while the Pham family used theirs for 25 hours.
Let's assume x represents the number of hours the Lewis family used their sprinkler, and y represents the number of hours the Pham family used their sprinkler. We can set up a system of equations based on the given information.
Equation 1: x + y = 55 (The combined total of hours)
Equation 2: 30x + 25y = 1475 (The total water output in liters)
To solve this system of equations, we can use substitution or elimination methods. By solving Equation 1 for x and substituting it into Equation 2, we get:
30(55 - y) + 25y = 1475
1650 - 30y + 25y = 1475
-5y = -175
y = 35
Substituting the value of y into Equation 1, we find:
x + 35 = 55
x = 20
Therefore, the Lewis family used their sprinkler for 20 hours, while the Pham family used theirs for 35 hours.
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Why do banks offer higher interest on savings accounts than on checking accounts?
The benefit: Savings accounts typically have higher interest rates than checking, making it easy for you to grow your money faster. ... Going over that limit can result in a fee or, if you do it multiple times, your bank might convert the account to checking.
In 2020 Phoenix, AZ was the fastest growing cities in the United States. In 2020 the population was approximately 1,730,000. The city population grew by 25,000 people that year. Write a model for the population of Phoenix x years after 2020 assuming it continues to grow by 25,000 people per year.
Answer : P(x) = 25,000x + 1,730,000P(x) represents the population of Phoenix after x years since 2020.
Explanation:
Given information: The population of Phoenix in 2020 was approximately 1,730,000 and the city's population grew by 25,000 people in 2020.
Model for the population of Phoenix x years after 2020 if it continues to grow by 25,000 people per year:
To find the population of Phoenix after x years since 2020, we need to add the number of people that moved into Phoenix since 2020, i.e., 25,000 people per year.
If x represents the number of years since 2020, then the model is given as follows:
P(x) = 25,000x + 1,730,000P(x) represents the population of Phoenix after x years since 2020.
We need to add 1,730,000 to 25,000x because 1,730,000 is the initial population in 2020.
Therefore the required model P(x) = 25,000x + 1,730,000P(x) represents the population of Phoenix after x years since 2020.
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Please help thanks!!!
For question 3, you don’t need to calculate it. Please explain the steps of how you would work it out.
Answer:
30 or 780 is your answer for the first one. i dont know how to work out the answer for the 2 one im sorry :(
Step-by-step explanation:
13+12+5 = 30
13 x 12 x 5 = 780
I would go for 780
it seems more like to be the answer
In a certain game of chance, a wheel consists of 40 slots numbered 00, 0, 1, 2,..., 38. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. Complete parts (a) through (c) below. OU. ne sample space is 00, 0, 1, 2,..., 38). (b) Determine the probability that the metal ball falls into the slot marked 4. Interpret this probability, The probability that the metal ball falls into the slot marked 4 is 0.025 (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box within your choice. (Type a whole number.) O A. If the wheel is spun 1000 times, it is expected that about of those times result in the ball landing in slot 4. O B. If the wheel is spun 1000 times, it is expected that exactly of those times result in the ball not landing in slot 4, (c) Determine the probability that the metal ball lands in an odd slot. Interpret this probability The probability that the metal ball lands in an odd slot is (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box within (Type a whole number.) your choice O A. If the wheel is spun 100 times, it is expected that exactly of those times result in the ball not landing on an odd number, B. If the wheel is spun 100 times, it is expected that about of those times result in the ball landing on an odd number
(a) The sample space for this game is the set of numbers on the wheel are 00, 0, 1, 2, ..., 38.
For this game of chance with a wheel consisting of 40 slots, the sample space is defined as the set of all possible numbers that the metal ball can fall into. The numbers range from 00 to 38, including both the double zero and single-digit numbers.
(b) The probability of the ball landing in the slot marked 4 is 1/40, which is equivalent to 0.025 when rounded to four decimal places.
To determine the probability that the metal ball falls into the slot marked 4, we need to calculate the ratio of the favorable outcomes (the ball landing in slot 4) to the total number of possible outcomes.
There is only one slot marked 4, so the number of favorable outcomes is 1. The total number of possible outcomes is 40 since there are 40 slots on the wheel.
This means that if the game is played many times, we can expect the ball to land in slot 4 approximately 0.025 (or 2.5%) of the time.
(c) The probability that the metal ball lands in an odd slot is 0.5.
To determine the probability that the metal ball lands in an odd slot, we count the number of odd slots on the wheel. Odd numbers occur every other slot starting from 1, so there are a total of 20 odd slots on the wheel.
The probability of the ball landing in an odd slot is given by the ratio of the number of odd slots to the total number of possible outcomes. Therefore, the probability is 20/40, which simplifies to 1/2 or 0.5 when rounded to four decimal places.
This means that if the game is played many times, we can expect the ball to land in an odd slot approximately 0.5 (or 50%) of the time.
The correct choices are:
(b) If the wheel is spun 1000 times, it is expected that about 25 of those times result in the ball landing in slot 4.
(c) If the wheel is spun 100 times, it is expected that exactly 50 of those times result in the ball landing on an odd number.
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please help me i don't understand this. the teacher said that she does not care and its due in 20 mins please help me :(
A galaxy is likely to be a collection of which of the following?
A :Universe and interstellar matter
B : Stars and interstellar matter
C: Clusters and constellations
D :Stars and clusters
Answer:
C
Step-by-step explanation:
47 students have a dog 76 students have a cat a) Represent this data in a Venn diagram in the box provided (3) b) How many students have both a dog and a cat?
Answer:
89 people
Step-by-step explanation:
using the fplot command in matlab graph the function f(x)=xsin(x) between x=0 and x=2.5
To graph the function f(x) = (x)sin(x) in MATLAB using the fplot command, you can follow the steps below:
matlab
Define the function
f = (x) (x.)sin(x);
Set the range of x values
x = linspace(0, 2.5, 100);
Plot the function
fplot(f, [0, 2.5])
Add labels and title
xlabel(x)
ylabel(f(x))
title(Graph of f(x) = (x)sin(x))
Display the grid
grid on
In this code, we first define the function f(x) = (x)sin(x) using an anonymous function (x). Next, we create a range of x values using linspace from 0 to 2.5 with 100 points. Then, we use the fplot command to plot the function f over the specified range. Finally, we add labels, title, and grid to the graph.
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Billy made 2 gallons of juice for a picnic. He said that he made
2
4
quarts of juice.
The two-way frequency table represents data from a survey asking a random sampling of people whether they can see the sunrise or sunset from the front of their home.
Which is the joint relative frequency for the people who can only see the sunset?
A) 5/38
B) 7/38
C) 12/38
D) 14/38
The joint relative frequency for people who can only see the sunset is 7/38.
To find the joint relative frequency for people who can only see the sunset, we need to look at the corresponding cell in the two-way frequency table. Let's assume the cell value is x. The total number of observations in the table is the sum of all the cell values, which is 38 in this case.
The joint relative frequency is the ratio of the cell value to the total number of observations. Therefore, the joint relative frequency for people who can only see the sunset is x/38.
Out of the given options, the value of x/38 that equals 7/38. Therefore, 7/38 represents the joint relative frequency for people who can only see the sunset based on the provided two-way frequency table.
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On May 1, you sign a $1000 note with simple interest of 8.5% and a maturity date of December 19. You make partial
payments of $475 on June 2 and $200 on November 4. How much will you owe on the date of maturity?
A) $355.79
B) $354.39
C) $359.53
D) $358.96
Answer:
The amount to be repaid is $379.26.
Step-by-step explanation:
Period of note from May 1 to December 19 = 233 days
Amount of note or principal = $1,000
Simple interest rate = 8.5%
Maturity date = December 19
Repayments:
June 2 = $475
Nov. 4 = $200
Total paid $675
Simple interest = $54.26 ($1,000 * 8.5% * 233/365)
Total amount to be repaid = $1,054.26
Total amount repaid = 675.00
Balance to be paid on maturity $379.26
According to a study done by Pierce students, the height for Hawaiian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Hawaiian adult male is randomly chosen. Let X = height of the individual. What is the proper expression for this distribution? X-C. O X-N(66, 2.5) O X-U(2.5, 66) O X-N(2.5, 66) OX-E(66, 2.5)
The proper expression for the distribution of the height of a randomly chosen Hawaiian adult male is X ~ N(66, 2.5). This means that X follows a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches.
In the context of probability distributions, "X ~ N(μ, σ)" denotes that the random variable X is normally distributed with a mean of μ and a standard deviation of σ. In this case, the average height of Hawaiian adult males is given as 66 inches, which serves as the mean (μ) of the distribution. The standard deviation (σ) is specified as 2.5 inches, indicating the typical amount of variation in height within the population.
By using the notation X ~ N(66, 2.5), we explicitly state that X follows a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches, as determined by the study conducted by Pierce students. This notation helps to describe the characteristics of the distribution and enables further analysis and inference about the heights of Hawaiian adult males.
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given:quadrillateral ABCD inscribed in a circle
prove angel A and angel C are supplementary angel B and D are supplementary
Answer:
Step-by-step explanation:
Given: quadrilateral ABCD inscribed in a circle
To Prove:
1. ∠A and ∠C are supplementary.
2. ∠B and ∠D are supplementary.
Construction : Join AC and BD.
Proof: As, angle in same segment of circle are equal.Considering AB, BC, CD and DA as Segments, which are inside the circle,
∠1=∠2-----(1)
∠3=∠4-----(2)
∠5=∠6-------(3)
∠7=∠8------(4)
Also, sum of angles of quadrilateral is 360°.
⇒∠A+∠B+∠C+∠D=360°
→→∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8=360°→→→using 1,2,3,and 4
→→→2∠1+2∠4+2∠6+2∠8=360°
→→→→2( ∠1 +∠6) +2(∠4+∠8)=360°⇒Dividing both sides by 2,
→→→∠B + ∠D=180°as, ∠1 +∠6=∠B , ∠4+∠8=∠B------(A)
As, ∠A+∠B+∠C+∠D=360°
∠A+∠C+180°=360°
∠A+∠C=360°-180°------Using A
∠A+∠C=180°
Hence proved.
credit: someone else
Eliza's backpack weighs pounds with her math book in it. Without her math book, her backpack weighs pounds. How much does Eliza's math book weigh?
a. 2 pounds
b. 3 pounds
c. 4 pounds
d. 5 pounds
Eliza's math book weighs 3⁶⁵/₇₂ pounds, based on fractional subtractions.
What is fractional subtraction?Fractional subtraction involves the subtraction of a number with fractions from another.
Subtraction is one of the four basic mathematical operations, including addition, multiplication, and division.
Fractions are portions or parts of a whole value and may be classified as proper, improper, or complex.
The weight of the backpack with Eliza's math book = 18⁷/₉ pounds
The weight of the backpack without Eliza's math book = 14⁷/₈ pounds
The weight of the math book = 3⁶⁵/₇₂ (18⁷/₉ - 14⁷/₈) pounds
Thus, using fractional subtractions, we can conclude that Eliza's math book weights 3⁶⁵/₇₂ pounds.
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Question Completion
Eliza’s backpack weighs 18⁷/₉ pounds with her math book in it. Without her math book, her backpack weighs 14⁷/₈ pounds. How much does Eliza’s math book weigh?
Which scenarios below show ways governments or organizations work to solve large problems?
Answers
Red Cross volunteers travel to Haiti to help earthquake victims.
The United Nations passes a treaty to limit pollution into the Pacific Ocean.
A charity group from the United States travels to Africa to help victims of a virus.
Two ethnic groups begin fighting instead of negotiating.
A military dictator gives a threatening speech to other leaders.
A large company leaks pollutants into the Mississippi River
QUESTION 6 What is the main lesson that is demonstrated by the Saint Petersburg Paradox? Choose one 1 point
a. Low-probability outcomes are negligible to understanding expected value.
b. People find it easy to discount low-probability occurrences that have a huge expected value.
c. Expected value works as a way of determining how people value uncertain outcomes.
d. People overestimate easy to remember situations.
According to the question the correct option is c. Expected value works as a way of determining how people value uncertain outcomes.
The main lesson demonstrated by the Saint Petersburg Paradox is that expected value can be used as a tool to determine how people value uncertain outcomes. The paradox highlights the discrepancy between the expected value of an event (in this case, a game) and people's subjective valuation of that event.
Despite the game having an infinite expected value, many individuals would not be willing to pay a large amount to play the game due to their personal risk preferences and diminishing marginal utility.
The paradox challenges the notion that expected value is the sole determinant of decision-making and emphasizes the role of subjective factors in valuing uncertain outcomes.
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the object was turned 45 degrees around a fixed point that's called?
Answer:
A transformation that turns a figure around a fixed point, called the center of rotation. hope this helps
Step-by-step explanation:
What is the best definition for parabola?
Answer: curve shape of something
Step-by-step explanation:
Answer:
Hey mate......
Step-by-step explanation:
This is ur answer.......
A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and. a fixed straight line (the directrix)
Hope it helps!
mark me brainliest pls.......
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The line graph shows the balance for a business on each day. How much more of a loss occurred on Friday than on Thursday?
A. $0
B. $2.50
C. $5
D. $10
Answer: this link will help you out
Step-by-step explanation: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/constructing-linear-models-real-world/v/word-problem-solving-4
Let (V, f) an inner product space and let U be a subspace of V. Let w € V. Write w=u_w + v_w with u_w € U and v_w €U. Let u € U.
(a) Show that f(w-u, w-u) = ||u_w - u ||² + ||v||².
We have proved the given equation f(w - u, w - u) = ||u_w - u||² + ||v_w||².
The given inner product space is (V, f) and U is a subspace of V. It is given that w € V and it can be written as w = u_w + v_w with u_w € U and v_w €U.
Also, u € U. To show that f(w-u, w-u) = ||u_w - u ||² + ||v||², we have to prove it.
Let's consider the left-hand side of the equation. We can expand it as follows:
f(w - u, w - u) = f(w, w) - 2f(w, u) + f(u, u)
By the definition of w and the fact that u is in U, we know that w = u_w + v_w and u = u. So we can substitute these values:
f(w - u, w - u) = f(u_w + v_w - u, u_w + v_w - u) - 2f(u_w + v_w, u) + f(u, u)
Now, using the properties of an inner product, we can rewrite this as:
f(w - u, w - u) = f(u_w - u, u_w - u) + f(v_w, v_w) + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u)
The term f(v_w, v_w) is non-negative since f is an inner product. Similarly, the term f(u, u) is non-negative since u is in U. Hence we can write the above equation as:
f(w - u, w - u) = ||u_w - u||² + ||v_w||² + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u)
We can write f(u_w, v_w) as f(u_w - u + u, v_w) and then use the properties of an inner product to split it up:
f(u_w - u + u, v_w) = f(u_w - u, v_w) + f(u, v_w)
By definition, u is in U so f(u, v_w) = 0. Hence we can simplify:
f(u_w - u + u, v_w) = f(u_w - u, v_w) = f(u_w, v_w) - f(u, v_w)
Now we can substitute this back into the previous equation:
f(w - u, w - u) = ||u_w - u||² + ||v_w||² + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u) = ||u_w - u||² + ||v_w||² + 2f(u_w - u, v_w) + f(u, u)
Since U is a subspace, u_w - u is also in U. Hence, f(u_w - u, v_w) = 0.
Therefore,
f(w - u, w - u) = ||u_w - u||² + ||v_w||².
Therefore, we have proved the given equation.
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