Answer:
Tengo que convertir los siguientes números decimales a fracciones, y también tengo que simplificarlos a su mínima expresión. Aquí un ejemplo ;)
(no pido que me digan las respuestas, si no que me ayuden a entenderla)
0.44 = 44/100 = 22/50 = 11/25
0.62=
0.29=
2.05=
1.28=
0.08=
0.50=
0.375=
0.60=
0.7=
Step-by-step explanation:
Find the circumference of the circle below. (Use 3.14 for Pi)
The answer is 75.36 m
Please see the attached picture for full solution
Hope it helps
Answer:
\(75.36m\)
Step-by-step explanation:
\(c = 2\pi \: r \\ = 2 \times 3.14 \times 12 \\ = 75.36m\)
hope this helps
brainliest appreciated
good luck! have a nice day!
Explain why it is possible that one graph could represent two different equations.
Different equations can result in identical graphs. For example, −=0
y−x=0
and (−)(2+2)=0
(y−x)(x2+y2)=0
both have the same graph on the real xy-plane, a diagonal line through the origin. The first equation is a linear equation while the second is a cubic equation.
Step-by-step explanation:
To compare two equations or to find the solution of two equation we represent two different equations on the same graph.
We need to explain why it is possible that one graph could represent two different equations.
What is the graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them. The length of the lines and position of the points do not matter.
On the same graph we can represent two different equations, to compare two equations or to find the solution of two equation.
Therefore, to compare two equations or to find the solution of two equation we represent two different equations on the same graph.
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*5. give the percent of the area under the normal curve represented by the shaded region. a normal distribution has a mean of 28 and a standard deviation of 3. find the probability that a randomly selected x-value from the distribution is in the given interval.between 25 and 31
The probability that a randomly selected x-value from the distribution is in the given interval is 68.27%.
The area under the normal curve is calculated using the cumulative normal distribution.
Step 1: Calculate z-score for lower boundary (25):
z = (25-28)/3
= -1
Step 2: Calculate z-score for upper boundary (31):
z = (31-28)/3
= 1
Step 3: Calculate the area under the normal curve between the two z-scores using the cumulative normal distribution:
Area = 0.6827
Step 4: Calculate the probability that a randomly selected x-value from the distribution is in the given interval:
Probability = Area
= 0.6827
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To calculate depletion expense, first determine the depletion per unit. depletion per unit can be calculated by taking (cost ______)/total units of capacity.
To calculate depletion expense, you need to determine the depletion per unit. The depletion per unit can be calculated by taking (cost of the resource) / total units of capacity.
Depletion is an accounting method used to allocate the cost of natural resources over their useful life. It is commonly used in industries such as mining, oil and gas extraction, and timber harvesting, where companies extract and deplete natural resources as part of their operations.
The first step in calculating depletion expense is to determine the cost of the resource. This includes all costs incurred to acquire and develop the resource, such as exploration costs, development costs, and acquisition costs. These costs are typically capitalized and then allocated over the estimated recoverable units of the resource.
Once you have determined the cost of the resource, you need to calculate the total units of capacity. This refers to the estimated amount of resource that can be extracted or harvested from the reserve. For example, in mining, it could be measured in tons of ore, while in oil and gas extraction, it could be measured in barrels or cubic feet.
To calculate the depletion per unit, divide the cost of the resource by the total units of capacity. This will give you the amount of depletion expense that should be recognized for each unit extracted or harvested.
It's important to note that there are different methods for calculating depletion expense, such as units-of-production method and percentage depletion method. The units-of-production method calculates depletion based on actual production or extraction during a given period. On the other hand, the percentage depletion method allows for a fixed percentage deduction from gross income based on statutory rates.
In conclusion, to calculate depletion expense, you need to determine the depletion per unit by dividing the cost of the resource by the total units of capacity. This allows for an appropriate allocation of costs over the useful life of natural resources.
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welppppppppppppppppppppppppppppppppppppppp
Answer:
They are all acute triangles or iscoles triangles
Step-by-step explanation:
Tom is 4 inches taller than Sally together their heights equal 124 inches
a.create an equation to represent the situation.
b.Solve your equation
c. How tall is Sally?
d. How tall is Tom?
Answer:
Tom's height = x = 64 inches
Sally's height = y = 60 inches
Step-by-step explanation:
Let us represent
Tom's height = x
Sally's height = y
Tom is 4 inches taller than Sally
x = y + 4
Together their heights equal 124 inches
x + y = 124
We substitute y + 4 for x
y + 4 + y = 124
2y + 4 = 124
Collect like terms
2y = 124 - 4
2y = 120
y = 120/2
y = 60
Solving for x
x = y + 4
x = 60 + 4
x = 64
Therefore,
Tom's height = x = 64 inches
Sally's height = y = 60 inches
Ket
Question 1
Points A, O, and B, are collinear. Find the measure of ZCOD.
A
O x = 90°
Ox= 103°
0x=77°
Ox=23°
C
50°
O
D
B
1 pts
The measure of angle ZCOD is 27°.
Since A, O, and B are collinear, angles AOC and COB form a linear pair, which means their sum is 180°. Therefore:
x + 50° = 180°
Solving for x, we get:
x = 130°
Next, we can use the fact that angles AOC and BOA are vertical angles, which means they are congruent. Therefore:
x = 103°
Substituting x with 130°, we get:
130° = 103° + COD
Solving for COD, we get:
COD = 27°
Finally, since angles AOD and COB are alternate interior angles, they are congruent. Therefore:
77° = 50° + ZCOD
Solving for ZCOD, we get:
ZCOD = 27°
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--The complete question is, Given that points A, O, and B are collinear, find the measure of angle ZCOD, where:
Angle AOC is denoted by x and measures 90°
Angle BOA measures 103°
Angle AOD measures 77°
Angle COB measures 50°.
Please note that point O is located between points A and B.--
Macy made a 220 grooming dogs one day in her mobile grooming business she charges 60 per appointment and 40 earned in tips write an equation to represent the situation and solve the equation to determine how many appointments Messi had part B Logan made a profit of 300 as a mobile groomer he charge $70 per appointment and received $50 in tips but he had to pay a rental fee for the truck of $20 per appointment write an equation to represent the situation and solve the situation to determine how many appointments Logan had
Macy had 3 appointments.
Logan had 5 appointments.
What is the quadratic equation?
A quadratic equation is a type of polynomial equation of degree 2, which is written in the form of "ax^2 + bx + c = 0", where x is the variable and a, b, and c are constants. The solutions to a quadratic equation can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac))/2a.
Part A:
Let x be the number of appointments Macy had.
We know that the total income (60x + 40) must equal 220.
Therefore, the equation representing the situation is:
60x + 40 = 220
To solve for x, we can subtract 40 from both sides:
60x = 180
Finally, we divide both sides by 60 to get:
x = 3
Macy had 3 appointments.
Part B:
Let y be the number of appointments Logan had.
We know that the total profit (70y + 50 - 20y) must equal 300.
Therefore, the equation representing the situation is:
50y + 50 = 300
To solve for y, we can subtract 50 from both sides:
50y = 250
Finally, we divide both sides by 50 to get:
y = 5
Logan had 5 appointments.
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The length of a rectangle is 3 more than twice the width. the perimeter is 36 inches. what is the area?
If the length of a rectangle is 3 more than twice the width and the perimeter is 36 inches, then the area of the rectangle is equal to 65 in².
The length and width of the rectangle can be determined by using the formula for the perimeter of the rectangle,
P = ( L + W ) × 2
P = 2L + 2W
Here L is equal to the length of the rectangle and W is the width of the rectangle.
As the length is 3 more than twice the width, we can write the length of the rectangle as,
L = 3 + 2W
Putting L = 3 + 2W in the equation of perimeter,
P = 2L + 2W
As P = 36
36 = 2( 3 + 2W) + 2W
36 = 6 + 4W + 2W
6W = 30
W = 30 ÷ 6
W = 5
The length of the rectangle can be found by putting W = 5 in the equation,
L = 3 + 2W
L = 3 + 2(5)
L= 3 + 10
L = 13
Therefore, the length and width of the rectangle are equal to 13 and 5 respectively
As, area = width × length
area = 5 × 13
area = 65
Hence, the area of the rectangle is equal to 65 square inches.
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determine which function has the greater rate of change in problems 1−3
1.
x y
-------
-1 0
0 1
1 2
2 3
(1 point)
The rates of change are equal.
The graph has a greater rate of change
The table has a greater rate of change.
none of the above
2. y = 2x + 7
The slopes are equal.
The graph has a greater slope.
The equation has a greater slope.
none of the abov
3. As x increases by 1, y increases by 3
The slopes are equal.
The graph has a greater slope.
The function rule has a greater slope.
none of the above
The table has a greater rate of change.
The rates of change are equal.
In the given problem, we have a table showing the relationship between x and y values. By comparing the change in y with the change in x, we can determine the rate of change. Looking at the table, we observe that for every increase of 1 in x, there is a corresponding increase of 1 in y. Therefore, the rate of change for this table is 1.
The slopes are equal.
The equation has a greater slope.
In problem 2, we are given a linear equation in the form y = mx + b, where m represents the slope. The given equation is y = 2x + 7, which means the slope is 2. To compare the rates of change, we compare the slopes. If the slopes are equal, the rates of change are equal. In this case, the slopes are equal to 2, so the rates of change are the same.
The function rule has a greater slope.
The slopes are equal.
In problem 3, we are told that as x increases by 1, y increases by 3. This information gives us the rate of change between x and y. The slope of a function represents the rate of change, and in this case, the slope is 3. Comparing the slopes, we find that they are equal, as both have a value of 3. Therefore, the rates of change are the same.
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Are the answers 200 cm and 1052 cm? Please help!
Check the picture below.
so hmmm let's find the distance AC, which is the only one missing for the perimeter.
\(\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2 + b^2} \qquad \begin{cases} c=\stackrel{hypotenuse}{AC}\\ a=\stackrel{adjacent}{6}\\ b=\stackrel{opposite}{8}\\ \end{cases} \implies AC=\sqrt{6^2 + 8^2} \\\\\\ AC=\sqrt{100}\implies AC=10~\hfill \underset{Perimeter}{\stackrel{6~~ + ~~8~~ + ~~10}{\text{\LARGE 24}cm}} \\\\[-0.35em] ~\dotfill\)
\(\textit{area of a triangle}\\\\ A=\cfrac{1}{2}bh \begin{cases} b=base\\ h=height\\[-0.5em] \hrulefill\\ b=6\\ h=8 \end{cases}\implies A=\cfrac{1}{2}(6)(8)\implies A=\text{\LARGE 24}~cm^2\)
HELP
PLSSSSSSSSSSSSSSSSSSSSSS
OH SO LIKE CHESTER WAS A REALLY BAD PERSON.
(35) MOVIE RENTALS The number of copies of a movie rented at a video kiosk decreased at a constant rate of 5 copies per week. The 6th week after the movie was released 4 copies were rented. How many copies were rented during the second week?
Answer:
25 copies rented
Step-by-step explanation:
r = r1 - 5t, t is time in weeks, r is the number of copies rented, r1 how many people were renting at the start
we know on the 6 weeks passed and 4 copies were rented
4 = r1 - 5(6)
4 = r1 - 30; add 30 on both sides
34 = r1
r = 34 - 5t now plug t = 2 in for this function
r = 34 - 5(2)
r = 35 - 10
r = 25 copies rented
Let g(x) = 2x and h(x) = x2 + 4.
Evaluate (h.g)(1).
A. 2
B. 10
C. 4
D. 8
Answer:
10
Step-by-step explanation:
Hello,
(h.g)(1)=h(1)*g(1)
h(1)=1+4=5
g(1)=2
so h(1)*g(1)=5*2=10
the correct answer is 10
Answer:
10
Step-by-step explanation:
g(x) = 2x
h(x) = x^2 + 4
(h*g)(1)
h(1) = 1^2 +4 = 1+4 = 5
g(1) = 2*2 =2
(h*g)(1) = 5*2 = 10
Help me!! All questions on statistics!!!!!
Problem 2
According to the Empirical Rule, roughly 95% of the normal distribution is within 2 standard deviations of the mean.
So the value 108 is 2 standard deviations above the mean, while 76 is 2 standard deviations below the mean.
The distance between those values is 108-76 = 32 units. Divide by 4 to get 32/4 = 8 to indicate the standard deviation itself. I'm dividing by 4 because we add the "2 standard deviations" to itself twice which is a total of 4 standard deviations in distance.
You could also solve like this
z = (x-mu)/sigma
2 = (108-92)/sigma
2 = 16/sigma
2sigma = 16
sigma = 16/2
sigma = 8
There's a similar set of steps if you were to use z = -2 and x = 76, which should lead to sigma = 8 as well.
Answer: 8===========================================================
Problem 3
The center of the distribution is at point c, and that's 170, since that's given to us. The center is always the mean.
The standard deviation is 7. Each tickmark on that horizontal number line shown represents a full standard deviation (aka 7 cm). This means going from c to d will lead to
d = c + sigma = 170+7 = 177
and
e = d + sigma = 177+7 = 184
or you could say
e = c+2*sigma = 170+2*7 = 184
To determine the other values, we go backwards by subtracting off multiples of sigma.
a = c - 3*sigma = 170 - 3*7 = 170 - 21 = 149
b = c - 1*sigma = 170 - 1*7 = 170 - 7 = 163
Answers:a = 149b = 163c = 170d = 177e = 184what is the most probable number of students born on january 1
Assuming a roughly even distribution of birthdays throughout the year, we can estimate that approximately 1/365th of students (or 0.27%) were born on January 1st.
Determining the most probable number of students born on January 1st requires some statistical analysis. To start, we would need data on the number of students enrolled in the relevant grade level, as well as data on the distribution of birthdays throughout the year. Assuming a roughly even distribution of birthdays throughout the year, we can estimate that approximately 1/365th of students (or 0.27%) were born on January 1st. However, this estimate may not hold true for all populations. For example, some cultures may place a greater emphasis on giving birth on auspicious dates, such as New Year's Day.To get a more accurate estimate, we could look at past enrollment data for the school or district and see how many students in that age range were born on January 1st. We could also look at national birth statistics to see if there are any trends in the number of babies born on this date.Ultimately, the most probable number of students born on January 1st will depend on a variety of factors, including the size of the student population and the specific demographics of the school or district. However, with the right data and analysis, we can arrive at a reasonably accurate estimate.For more such question on distribution
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The Datafile MedianHousehold contains the median household income for a family with two earners for each of the fifty states (American Community survey, 2013). Construct a frequency and percent frequency distribution of median household income. Begin the first class at 65.0 and use a class width of 5. Construct a histogram using Minitab. Comment on the shape of the distribution. Which state has the highest median income for two-earner households? Which state has the lowest median income for two-earner households?
State Median Income (000's)
Alabama 76.2
Alaska 98.4
Arizona 79.7
Arkansas 70.9
California 91.2
Colorado 89.3
Connecticut 107.5
Delaware 90.2
Florida 75.5
Georgia 79.7
Hawaii 89.7
Idaho 67.1
Illinois 89.7
Indiana 76.7
Iowa 81.3
Kansas 79.9
Kentucky 76.4
Louisiana 82.6
Maine 77.8
Maryland 108.5
Massachusetts 106.8
Michigan 81.0
Minnesota 90.1
Mississippi 70.4
Missouri 77.0
Montana 73.6
Nebraska 78.3
Nevada 75.1
New Hampshire 93.9
New Jersey 110.7
New Mexico 77.6
New York 95.2
North Carolina 76.5
North Dakota 87.0
Ohio 80.9
Oklahoma 74.5
Oregon 78.7
Pennsylvania 86.8
Rhode Island 95.1
South Carolina 77.1
South Dakota 72.0
Tennessee 73.4
Texas 82.0
Utah 75.0
Vermont 85.1
Virginia 97.2
Washington 91.6
West Virginia 76.8
Wisconsin 82.3
Wyoming 87.9
The median household income for two-earner households in each state was analyzed using the provided data. A frequency and percent frequency distribution were constructed with a class width of 5, starting from 65.0. A histogram was generated in Minitab to visualize the distribution. The shape of the distribution indicates that the majority of states have median incomes clustered around the middle range. The state with the highest median income for two-earner households is New Jersey (110.7), while Mississippi (70.4) has the lowest median income.
To construct a frequency distribution, we group the data into intervals or classes. Given a class width of 5 and starting at 65.0, we can determine the class boundaries and count the number of values falling within each class. The resulting frequency distribution would show the number of states within each income range. Additionally, the percent frequency distribution would represent the proportion of states in each income range.
Based on the histogram generated in Minitab using the frequency distribution, we can observe the shape of the distribution. With the class intervals and frequencies plotted, the histogram helps visualize the distribution pattern. In this case, since the provided data represents the median household income for two-earner households in each state, the shape of the distribution can provide insights into the overall income distribution among states.
From the provided data, we can determine that New Jersey has the highest median income for two-earner households with a value of 110.7 (in thousands), indicating a relatively higher income level. On the other hand, Mississippi has the lowest median income at 70.4 (in thousands), reflecting a lower income level. This information highlights the variation in median household incomes for two-earner households across different states.
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Graph: y - 10 = -2(x – 10) Y χ y 30 20 10 X -10 10 20 -10 Draw Click or tap the graph to plot a point.
Ms. D's 4th period geometry class has both freshman and sophomores. There are 27 kids in the class and it has twice as many freshman as sophomores. How many freshman are in the class?
let x=
let y=
Help pls it for math I’m doing a test and didnt take all notes help I will give the brainliest
Answer:
A:1/18 B:1/6 for 4 and 1/3 for red
Step-by-step explanation:
A mouse has made holes in opposite corners of a rectangular kitchen. Starting from its hole
in the northwest corner, the mouse scurries 20 feet along the length of the kitchen to reach a
piece of cheese in the southwest corner. Then the mouse scurries 15 feet along the width of
the kitchen to its other hole in the southeast corner. Finally the mouse scurries back to the
first hole. What is the total distance the mouse scurries?
Answer:
60 ft
Step-by-step explanation:
a² + b² = c²
20² + 15² = c²
c = √625
c = 25
total distance = 20 ft + 15 ft + 25 ft = 60 ft
a water well has an eight-inch casing (diameter) and is 175 feet deep. the water is 25 feet from the top of the well. determine the amount of work done in pumping the well dry, assuming that no water enters it while it is being pumped.
The required work done in pumping the given well completely dry is 326,560 lb-feet
The work done is calculated as follows:
\(W=\int_a^b F(x) d x\)Here, F(x) is the external force and a to b is the displacement.
The pressure exerted by the liquid on the object in the depth is
P=whHere, w is the weight of the liquid per unit volume and h is the depth.
And, the force of liquid is given as follows
F=PA
Here, P is the pressure and A is the area of the object
Consider the well with 8-inch diameter, 175 feet deep and water is 25 feet below the top.
In a situation, the diameter of water well is 8 inch.
And the depth of the well is 175feet.
The water is 25 feet from the top of the well.
To evaluate the work done in pumping it out dry
The level of water from top is h=x
And, area of the well is A
\($$\begin{aligned}A & =\pi\left(\frac{4}{12}\right)^2 \\& =\frac{16 \pi}{144}\end{aligned}$$\)
Since, the constant weight of water is 62.4 pounds per unit cubic foot
Thus the work done is
\($$\begin{aligned}W & =\int_a^b F(x) d x \\& =62.4 \int_{25}^{175} \frac{16 \pi}{144} x d x \text { since } \int x^n d x=\frac{x^{n+1}}{n+1} \\& =21.77\left[\frac{x^2}{2}\right]_{25}^{175} \\& =326560\end{aligned}$$\)
Hence, the required work done in pumping the given well completely dry is 326,560 lb-feet.
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A store pays $72 for a crystal vase. The store marks up the price by 50%. What is the amount of the mark-up?
Answer:
$36.00 markup
Step-by-step explanation:
$72 x 50% (0.50) = $36.00
pls help help help help
Log4(2m^3-14m^2)-log4(2m)=log4^8
Solving logarithmic equations
The solutions of the logarithmic equation above are m = 0 and m = 7 + 2¹⁰/7.
To solve the logarithmic equation Log4(2m^3-14m^2)-log4(2m)=log4⁸, we can use the properties of logarithms.
First, we can combine the two logarithms on the left side using the quotient rule, which states that log(a/b) = log(a) - log(b).
This gives us log4((2m³-14m²)/2m) = log4⁸.
Simplifying the fraction inside the logarithm, we get log4(m²-7m) = 8.
Using the definition of logarithms, we can rewrite this equation as 4⁸ = m² - 7m.
Rearranging and factoring, we get m(m-7) = 4⁸.
Thus, the solutions are m = 0 and m = 7 + 2¹⁰/7.
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an unnormalized relation is a table that has more than one row.
An unnormalized relation refers to a table in a relational database that contains more than one row. This means that there are duplicate rows in the table, which violates the rules of normalization.
Normalization is a process in database design that aims to eliminate data redundancy and ensure data integrity. It involves breaking down a database into multiple tables and defining relationships between them. By doing so, we can efficiently store and retrieve data while minimizing inconsistencies.
In an unnormalized relation, duplicate rows can lead to various problems. For example, it can result in data inconsistencies, as updating one instance of a row may not reflect changes in other duplicate rows. Additionally, it can cause unnecessary storage and maintenance overhead.
To normalize an unnormalized relation, we need to identify the functional dependencies in the table and create separate tables for related data. This helps organize the data and reduces redundancy.
In summary, an unnormalized relation is a table in a relational database that has duplicate rows. It is important to normalize such relations to ensure data integrity and efficiency.
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Kenji's electricity bill costs $21.56 per month plus $2.46 per kilowatt hour. How many kilowatt hours can he use and keep his bill to no more than $54?
Answer:
Kenjii can use 13.19 kilowatt-hour.
Step-by-step explanation:
Giving the following information:
Kenji's electricity bill costs $21.56 per month plus $2.46 per kilowatt-hour.
First, we will structure the total cost formula:
Total cost= 21.56 + 2.46*x
x= kilowatt-hour
54= 21.56 + 2.46x
32.44= 2.46x
13.19= x
Kenjii can use 13.19 kilowatt-hour.
x and y are declared as integer variables.
You are given the following expression:
x >= y
Which of these expressions is the opposite of the above expression and has valid syntax?
-x <= y
- x =< y
- There's not enough information to determine this.
- x == y
- x < y
- x =< y is the expression and acceptable syntax.
Explain about the expression?
A mathematical expression is made up of numbers, variables, and functions (such as addition, subtraction, multiplication or division etc.) It is possible to compare expressions and phrases.
A term may be a number, a variable, a product of two or more variables, or a combination of both. A single phrase or a set of terms can make up an algebraic expression. The two terms in the formula 4x + y, for instance, are 4x and y.
Algebra's associative, commutative, and distributive characteristics are the ones that are most frequently utilized to make algebraic expressions simpler.
The converse of the expression and having proper syntax is
= x >= y.
= -x <= y
=- x =< y
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The data set below provides the number of hours that a sample of 8 college students spent playingvideo games.7, 11, 19, 14, 16, 4, 8, 10Find the median of the data set.
1) Considering this data set, we need to rewrite them orderly:
4,7,8,10, 11, 14, 16, 19
2) Note that there are 8 data points. So the Median can be found this way:
\(Md=\frac{4th+5th}{2}=\frac{10+11}{2}=10.5\)You are given the polynomials 2−lx,1+fx
2
,lx+mx
3
, and 1−x+2x
2
. Write the polynomials with the values of f,m, and l filled in. Answer the two questions below based on these polynomials. (a) (5 pts) Check if the polynomials above form a basis for the vector space of polynomials having degree at most 3 . If they do not form a basis, change any one entry in place of f,m, or l, and rewrite the polynomials to prove that they form a basis. (b) (5 pts) With the original (or changed) polynomials as a basis, and in the order they are given, express the components of the polynomial 3−2x
2
+5x
3
. Note that the components of the given polynomial with respect to the standard basis are (3,0,−2,5).
a). If the only solution is A = B = C = D = 0, then the polynomials are linearly independent and form a basis.
b). We can solve this system of equations to find the values of A, B, C, and D that give the desired polynomial.
To check if the given polynomials form a basis for the vector space of polynomials with degree at most 3,
we need to determine if they are linearly independent and span the entire vector space.
(a) To check for linear independence, we set up the equation:
A(2−lx) + B(1+fx²) + C(lx+mx³) + D(1−x+2x²) = 0
where A, B, C, and D are constants.
Equating the coefficients of like powers of x, we get:
2A + D = 0 (for the constant term)
-Al + Cl + D = 0 (for x term)
B + 2D = 0 (for x² term)
Cm + 2B = 0 (for x³ term)
We can solve this system of equations to find the values of A, B, C, and D that make the equation true. If the only solution is A = B = C = D = 0, then the polynomials are linearly independent and form a basis.
(b) To express the components of the polynomial 3−2x²+5x³ with respect to the given basis, we need to find the constants A, B, C, and D such that:
A(2−lx) + B(1+fx²) + C(lx+mx³) + D(1−x+2x²) = 3−2x²+5x³
We can equate the coefficients of like powers of x to get a system of equations:
2A + D = 0 (for the constant term)
-Al + Cl + D = 0 (for x term)
B + 2D = -2 (for x² term)
Cm + 2B = 5 (for x³ term)
We can solve this system of equations to find the values of A, B, C, and D that give the desired polynomial.
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