Yes , the arithmetic sequence ( written as sequences of the form (a , a+k , a+2k , a+3k for some constants a and k ) is a subspace of V .
To determine whether subsets of V in the arithmetic sequences are subspaces of V, we check if they satisfy the three conditions for a subset to be a subspace:
(i) The subset must contain the zero vector.
(ii) The subset must be closed under vector addition.
(iii) The subset must be closed under scalar multiplication.
Let S be a subset of V consisting of arithmetic sequences.
Now we check the three conditions :
(i) We find an arithmetic sequence in S that has all its terms equal to zero. The only arithmetic sequence that satisfies this is the sequence (0, 0, 0, 0, ...), which is in S.
So , the subset "S" contains zero vector.
(ii) Next we need to show that if u and v are two arithmetic sequences in S, then their sum "u + v" is also an arithmetic sequence in S.
Let U = (a, a+k, a+2k, a+3k, ...) and V = (b, b+l, b+2l, b+3l, ...) be two arithmetic sequences in S.
Then, their sum is ⇒ u+v = (a+b, a+k+b+l, a+2k+b+2l, a+3k+b+3l, ...).
This is also an arithmetic sequence,
So , subset "S" is closed under vector addition.
(iii) Next , we show that if U is an arithmetic sequence in S and C is a scalar, then CU is also an arithmetic sequence in S.
Let U = (a, a+k, a+2k, a+3k, ...) be an arithmetic sequence in S, and let c be a scalar.
we get , CU = (ca, ca+ck, ca+2ck, ca+3ck, ...) is also an arithmetic sequence,
So , subset "S" is closed under scalar multiplication.
we see that , S satisfies all three conditions, it is a subspace of V.
Therefore, any subset of V consisting of arithmetic sequences is a subspace of V.
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The given question is incomplete , the complete question is
Let V be the space of all infinite sequences of real numbers , Which of the subsets of V in The arithmetic sequences [i.e. sequences of the form (a , a+k , a+2k , a+3k for some constants a and k] are subspaces of V ?
Match each equation from the first set with an equivalent equation from the second set. Some of the answer choices are not used.
3x+6=4x+7
3(x+6)=4x+7
4x+3x=7-6
9x=4x+7
3x+18=4x+7
3x=4x+7
3x-1=4x
7x=1
Simplifying the equations will give 3x -1 = 4x, 3x + 18 = 4x + 7 and 7x = 1
Match each equation from the first set with an equivalent equation from the second setLets simplify each of the equations in order to match with the equivalent equation from the second set
3x+6=4x+7
Collecting like terms
3x +6 - 7 = 4x
3x -1 = 4x
The second equation
3(x+6)=4x+7
expanding the bracket
3x + 18 = 4x + 7
The third equation
4x+3x=7-6
simplifying the equation
7x = 1
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Help with this question
According to the figure of the right triangle
a = b = 10How to find angle a and b in the right triangleWhen a right triangle has one of the angles as 45 degrees the angle must be 45 degrees two. Also, the sides are equal, hence we have that
a = b
Using trigonometry we solve for b as follows
cos 45 = b / 10√2
multiplying both sides by 10√2
10√2 * cos 45 = 10√2 * b / 10√2
10√2 * cos 45 = b
rearranging
b = 10√2 * cos 45
where cos 45 = 1/√2 = √2/2
b = 10√2 * 1/√2
b = 10
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Each of the following statements is true. In each case write the converse of the statement, and give a counterexample showing that the converse is false. a. If n is any prime number that is greater than 2, then n+1 is even. b. If m is any odd integer, then 2m is even. c. If two circles intersect in exactly two points, then they do not have a common center.
a. Original statement: If n is any prime number that is greater than 2, then n+1 is even.
Converse: If n+1 is even, then n is prime and greater than 2.
Counterexample: n = 4, which is not a prime number but 4+1=5 is a prime number greater than 2.
b. Original statement: If m is any odd integer, then 2m is even.
Converse: If 2m is even, then m is odd.
Counterexample: m = 2, which is not an odd integer but 2×2=4 is an even integer.
c. Original statement: If two circles intersect in exactly two points, then they do not have a common center.
Converse: If two circles do not have a common center, then they intersect in exactly two points.
Counterexample: Two identical circles with centers coinciding have a common center and intersect at all points on the circle, which is more than two points.
What is converse statement?In mathematics, the converse of a conditional statement is formed by interchanging the hypothesis and conclusion.
For example, if the original statement is "If p then q",
the converse statement is "If q then p".
The converse statement may or may not be true, and it needs to be proven separately.
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One and Seventy One Hundredths
wright the number for the decimal names..
When one and seventy one hundredth is written in decimal form, it would be 1. 71
How to write as decimal ?A single part of something that has been evenly divided into one hundred parts is referred to as a hundredth in mathematics. The tenths place is represented by the first digit following the decimal. The hundredths place is represented by the digit that follows the decimal.
This means that when given the number one and seventy One Hundredths in words, then the number, " one " would be written before the decimal and the 71 would be 0. 71 as a hundredth figure.
When put together, this becomes:
= 1 + 0. 71
= 1. 71
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To save for retirement, a student invests $70 each month in an ordinary annuity with 3% interest compounded monthly. Determine the accumulated amount in the student's annuity after 30 years.
The accumulated amount will be $_______?
(Round to the nearest cent as needed.)
The accumulated amount will be $133.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
A student invests $70 each month in an ordinary annuity with 3% interest compounded monthly.
We know that;
⇒ Simple interest = P × r × t
Where, P is principle amount , 'r' is rate and t is time.
Here, P = $70
r = 3%
n = 30 years
Substitute all the values, we get;
Hence, We get;
⇒ Interest = P × r × t
⇒ Interest = 70 × 0.03 × 30
⇒ Interest = $63
Hence, The accumulated amount will be ;
⇒ P + Interest
⇒ $70 + $63
⇒ $133
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Find the quotient of these complex numbers (6-7i)-(4-5i)=
2 - 2i is the quotient of these complex numbers (6-7i)-(4-5i).
What is Division?A division is a process of splitting a specific amount into equal parts.
To find the quotient of complex numbers, we need to use the formula:
(a + bi) / (c + di) = [(a + bi) x (c - di)] / (c^2 + d^2)
where a, b, c, and d are real numbers and i is the imaginary unit.
In this case, we are subtracting two complex numbers:
(6-7i) - (4-5i)
= 6 - 7i - 4 + 5i (distribute the negative sign)
= 2 - 2i
Hence, 2 - 2i is the quotient of these complex numbers (6-7i)-(4-5i).
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Using the same table as before, which was the worst month for the team in terms of results?
The worst month for the team in terms of results was October, with a total of three losses in five matches.
What is number?Number is a mathematical entity used to represent a computer magnitude it can be symbol or a combination of simple you should in order to take quantity such as the two, five, seven, three or eight number are used to verify the context including counting measuring and computing.
The team won only one match and drew one, resulting in a total of five points. This was the lowest points total of any month during the season, a stark contrast to the other months where the team achieved much higher totals. The team's form during October was a far cry from their form during the rest of the season, with only one win in five matches. This poor form contributed to the team's overall lack of success during the season and led to their eventual relegation from the top division.
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The following table gives the frequencies of the nest size (number of eggs) of 153 Great Blue Heron nests
recorded on a recent survey in Indian River County. Use the data to construct a discrete probability
distribution.
The discrete probability distribution is given as follows:
P(X = 0) = 0.3268.P(X = 1) = 0.2353.P(X = 2) = 0.0392.P(X = 3) = 0.0065.P(X = 4) = 0.1438.P(X = 5) = 0.2484.How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.
There are 153 nests, hence the distribution in this problem is constructed dividing each of the amounts by 153, as follows:
P(X = 0) = 50/153 = 0.3268.P(X = 1) = 36/153 = 0.2353.P(X = 2) = 6/153 = 0.0392.P(X = 3) = 1/153 = 0.0065.P(X = 4) = 22/153 = 0.1438.P(X = 5) = 38/153 = 0.2484.More can be learned about probability at https://brainly.com/question/24756209
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Find the volume of a pyramid whose height is 15.7 inches and whose base is a rectangle with
dimensions of 7.6 inches and 12.4 inches.
Answer: V = 493.19
Step-by-step explanation:
V=1/3bh
7.6*12.4=94
94*15.7=1479.568
1479.568*1/3=493.19
Evaluate 5r + p if p = 3 and r = 4
Work Shown:
5r + p
5*4 + 3
20 + 3
23
Arianys has $0.45 worth of pennies and nickels. She has a total of 21 pennies and
nickels altogether. Graphically solve a system of equations in order to determine the
number of pennies, x, and the number of nickels, y, that Arianys has.
Taking into account the definition of a system of linear equations, the number of pennies and nickles that Arianys has is 15 and 6 respectively.
Definition of system of linear equationsThe degrees of systems of linear equations are groupings of linear equations with the same unknowns, of which it is necessary to find a common solution.
Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied.
Number of pennies and nickelsIn this case, a system of linear equations must be proposed taking into account that:
"x" is the number of pennies that Arianys has."y" is the number of nickels that Arianys has.You know:
Arianys has $0.45 worth of pennies and nickels. She has a total of 21 pennies and nickels altogether.The system of equations to be solved is
x + y = 21
0.01x + 0.05y = 0.45
There are several methods to solve a system of equations, it is decided to solve it using the graphical method, which consists of representing the graphs associated with the equations of the system to deduce its solution. The solution of the system is the point of intersection between the graphs, since the coordinates of said point satisfy both equations.
In this case, graphing the system of equations (image attached) it is obtained that the system of intersection between both equations is (x,y)=(15,6). This means that the number of pennies that Arianys has is 15 and the number of nickels that Arianys has is 6.
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SOMEONE HELPPPPPLLLOLLL
A bag contains 7 green marbles, 2 white marbles, 5 orange marbles, 9 green marbles, and 7 red marbles. A marble will be drawn from the bag and replaced 180 times. What is a reasonable prediction for the number of times a white or red marble will be drawn? Answer A 15 B 54 C 57 D 63
A Reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Thus, option b is correct
If a marble is drawn and not replaced, what is the probability of drawing an orange marble on the second draw?If a marble is drawn and not replaced, the probability of drawing an orange marble on the second draw depends on the color of the marble drawn on the first draw. If an orange marble was drawn on the first draw, then there will be one less orange marble in the bag, and the probability of drawing another orange marble on the second draw will be 4 out of the remaining 29 marbles. If a non-orange marble was drawn on the first draw, then there will be 5 orange marbles in the bag, and the probability of drawing an orange marble on the second draw will be 5 out of 29.
There are a total of 30 marbles in the bag.
The probability of drawing a white or red marble is
2/30 + 7/30
= 9/30
= 3/10.
Therefore, a reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Answer B.
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Question 3 Charlie has a total of $7 in quarters and nickels. He has twice as many nickels as quarters. In which system of equations does q represent the number of quarters he has and n represent the number of nickels?
Answers are linked below. I need help like fast
A system of equations where n is the number of nickels and q is the number of quarters he has is represented by (C) 25q + 5n = 7, n = 2q.
What are equations?The equals sign is a symbol used in mathematical formulas to denote the equality of two expressions.
An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values. as in 3x + 5 = 15, for example.
There are many different types of equations, including linear, quadratic, cubic, and others.
Let's learn more about math equations in this tutorial.
So, we know that nickels are double quarters.
And nickels are represented by n and quarters are represented by q.
Then, we can write as:
n = 2q
Then, the equation that represents the situation will be:
25q + 5n = 7
Therefore, a system of equations where n is the number of nickels and q is the number of quarters he has is represented by (C) 25q + 5n = 7, n = 2q.
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what's the formula to find the area of a pentagon
Answer:
The formula to find the area of a regular pentagon (a polygon with five sides of equal length and equal interior angles) is:
Area = (1/4) * sqrt(5(5+2sqrt(5))) * s^2
Where s is the length of one side of the pentagon.
Step-by-step explanation:
simplify 4(x+5)-x+3n
3x+3n+20 is the answer
order the given functions in increasing order of their growth rate. note: all logarithms are base-10.
When comparing exponential functions, the one with the bigger base expands more quickly. A logarithmic function grows more quickly if its argument is bigger.
Compared to polynomial functions, which in turn expand more quickly than logarithmic functions, exponential functions grow more quickly.
When comparing exponential functions, the one with the bigger base expands more quickly.
A logarithmic function grows more quickly if its argument is bigger.
We can arrange the provided functions in this manner by applying these rules:
log n, n, n log n, n, n2, n!, nn
Taking the function with the fastest rate of growth first: log n n n log n n 2
Each of these variables has a polynomial or logarithmic growth rate. The exponential functions will now be discussed:
2^n > n^2
Since polynomial functions increase more slowly than exponential functions, n2 grows more slowly than 2n.
The two categories with the fastest growth are: n! > n^2
n! grows more quickly than 2n and n2, as factorial functions expand even more quickly than exponential functions do. Finally:
n^n > n!
Therefore, nn grows more quickly than n! because exponential functions with bigger bases scale up more quickly than those with smaller bases.
Consequently, the following functions are listed in ascending order of growth rate:
log n n n log n n n log n n n! n n.
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There are 3 second grade classrooms that are 10 packs of paper. how much should each classroom get?
By solving an expression, we can find that the number of packs of paper that each classroom should get is 10/3.
Define Expression.Expressions are mathematical statements that comprise either numbers, variables, or both and at least two terms associated by an operator. Addition, subtraction, multiplication, and division are examples of mathematical operations. An example is the expression x + y, which combines the terms x and y with an addition operator. In mathematics, there are two different types of expressions: algebraic expressions, which also include variables, and numerical expressions, which solely comprise numbers.
Given,
Number of classrooms= 3
Number of packs of paper= 10
Now let the number of packs of paper per class be x.
x= Number of packs of paper/Number of classrooms
= 10/3.
Therefore, each classroom will get 10/3 packs of paper.
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Which of the following could be the areas of
the three squares shown?
A. 40 ft², 55 ft², 95 ft²
B. 35 ft²2, 25 ft², 45 ft²
C. 10 ft2, 10 ft², 100 ft²
D. 40 ft2, 30 ft², 1200 ft²
The areas of the squares are calculated from the Pythagoras relation of right angled triangle and it is 40 ft², 55 ft², 95 ft²
What is a Triangle?A triangle is a plane figure or polygon with three sides and three angles.
A Triangle has three vertices and the sum of the interior angles add up to 180°
Let the Triangle be ΔABC , such that
∠A + ∠B + ∠C = 180°
The area of the triangle = ( 1/2 ) x Length x Base
For a right angle triangle
From the Pythagoras Theorem , The hypotenuse² = base² + height²
if a² + b² = c² , it is a right triangle
if a² + b² < c² , it is an obtuse triangle
if a² + b² > c² , it is an acute triangle
Given data ,
Let the right angle triangle be represented as ΔABC
Now , the measure of side AB = √40 feet
The measure of side BC = √55 feet
The measure of side AC = √95 feet
For a right angle triangle
From the Pythagoras Theorem , The hypotenuse² = base² + height²
Substituting the values in the equation , we get
AC² = AB² + BC²
( √95 )² = ( √40 )² + ( √55 )²
On simplifying the equation , we get
95 = 40 + 55
95 = 95
Therefore , the area of squares formed is 40 ft², 55 ft², 95 ft²
Hence , the triangle is having sides √40 feet , √55 feet and √95 feet
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Mr. Sanchez made a apple pie. The circumstances of his pie was 28.26 inches.
? What was the diameter of the pie?
The diameter of the pie is approximately 9 inches
What is a diameter ?
Diameter can be defined as the double of the radius
To determine the diameter of the pie, we need to use the formula for the circumference of a circle:
C = 2πr
where C is the circumference, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
In this case, we are given that the circumference (circumstances) of the pie is 28.26 inches, so we can set up the equation:
28.26 = 2πr
To solve for the radius, we can divide both sides of the equation by 2π:
r = 28.26 / (2π)
r ≈ 4.5
So the radius of the pie is approximately 4.5 inches. To find the diameter, we just need to double the radius:
d = 2r
d ≈ 9
Therefore, the diameter of the pie is approximately 9 inches.
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4 m
/60°
5m
4m
5m
Find the area of the figure. Round your answer to the nearest tenth.
The area is about square meters.
Both measurements are below the minimum area of 169 square feet required.
What is area?Area is a measurement of size of two dimensional surface such as Square rectangle or circle. It is calculated by multiplying the length of the surface by width area also can measured in terms of square units such as square feet or square meters area is an important concept in mathematics and is used to measure the size of safe and object it is also used to find the total area of a group of shape.
The measurements of the parking spaces shown do not meet the requirements of the town. The first parking space has an area of 144 square feet, while the second parking space has an area of 128 square feet. Both measurements are below the minimum area of 169 square feet required.
The area of the figure can be found by using the formula for the area of a regular polygon. Since the figure is composed of four 60-degree angles and four sides, the formula can be expressed as:
A = (1/2) * (a * s) * n
Where a is the length of one side of the polygon, s is the length of the apothem (a line perpendicular to the center of the polygon to a corner), and n is the number of sides of the polygon.
Since the figure has four sides of equal length (5 meters each), the length of the apothem can be calculated by using the Pythagorean theorem. The equation for the apothem is:
s = [tex](4^{2} - 5^{2} ) ^{0.5}[/tex]
Plugging these values into the area formula yields:
A = (1/2) * (5 * 4.472) * 4
A = (1/2) * (22.36) * 4
A = 44.72 square meters
Rounding to the nearest tenth yields an area of 44.7 square meters.
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A 1.08 km diameter asteroid has a density of 3000 kg/m3 and impact the surface at 23.0 km/sec. Assuming a spherical asteroid, what is the kinetic energy (in joule) of the asteroid? Convert the answer to megatons of TNT, where 1 megaton is about 4e15 joule. For comparison, the most energetic weapon in the human arsenal is about 100 megatons.
The kinetic energy (in joule) of the asteroid is, 33 million
What is kinetic energy ?The kinetic energy is the energy associated with the particle having mass m and moving with the velocity v.
To find the kinetic energy of the asteroid, we need to use the formula:
KE = (1/2)mv²
where KE is the kinetic energy, m is the mass of the asteroid, and v is its velocity.
The mass of the asteroid can be calculated using its density and volume:
V = (4/3)πr³
where V is the volume of the asteroid, r is its radius, and π is the mathematical constant pi.
Since the asteroid is spherical, its radius is half its diameter, which is 0.54 km.
So,
V = (4/3)π(0.54 km)³
= 0.65 km³
The mass of the asteroid is then:
m = density × volume
= 3000 kg/m³ × 10⁹ m³/km³ × 0.65 km³
= 1.95 × 10¹² kg
Now we can calculate the kinetic energy:
KE = (1/2)mv²
= (1/2) × 1.95 × 10¹² kg × (23.0 km/sec)²
= 1.32 × 10²³ joule
To convert this to megatons of TNT, we divide by the energy of one megaton:
= 1.32 × 10³ joule / (4 × 10¹⁵ joule/megaton)
= 3.3 × 10⁷ megatons
Hence, the KE of asteroid is 33 million
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This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Show that if n is an integer and n^3 + 5 is odd, then n is even usinga proof by contradiction. Rank the options below. Suppose that n^3 + 5 is odd and that n is odd. We know that the sum of two odd numbers is even. As n is odd. n^3 is odd. Therefore, our supposition was wrong; hence n is even. As n^3 and 5 are odd, their sum n^3 + 5 should be even, but it is given to be odd. This is a contradiction.
Assume n is odd. Since n^3 is odd, n^3 + 5 should be even, but it's odd. Therefore, n is even.
The correct ranking of the options is:
As n^3 and 5 are odd, their sum n^3 + 5 should be even, but it is given to be odd. This is a contradiction.
Suppose that n^3 + 5 is odd and that n is odd. We know that the sum of two odd numbers is even. As n is odd, n^3 is odd. Therefore, our supposition was wrong; hence n is even.
The proof by contradiction shows that the assumption that n is odd leads to a contradiction with the given fact that n^3 + 5 is odd. Therefore, the assumption that n is odd must be false, and hence n is even.
The proof begins by assuming that n is odd and showing that this leads to a contradiction with the given fact that n^3 + 5 is odd. Specifically, since the sum of two odd numbers is even, n^3 + 5 should be even if n is odd. However, we know that n^3 + 5 is odd, so our assumption that n is odd must be false, meaning that n is even.
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11. A line through (3, 5) and (k, 12) is perpendicular to a line through (0, 7) and (2, 10). Find the value of k that makes the above statement true.
Answer:
k = 23/7
Step-by-step explanation:
To solve the problem, we can use the fact that the product of the slopes of two perpendicular lines is -1. We can start by finding the slope of the line passing through the points (3, 5) and (k, 12).
slope = (12 - 5) / (k - 3)
We can simplify this expression by multiplying both numerator and denominator by -1:
slope = (-7) / (3 - k)
Now, let's find the slope of the line passing through the points (0, 7) and (2, 10):
slope = (10 - 7) / (2 - 0) = 3 / 2
Since the two lines are perpendicular, we can set the product of their slopes equal to -1:
(-7) / (3 - k) * (3 / 2) = -1
Simplifying this equation, we get:
(7/2) * (3 - k) = 1
Multiplying both sides by 2/7, we get:
3 - k = 2/7
Subtracting 3 from both sides, we get:
-k = 2/7 - 3 = -21/7 - 2/7 = -23/7
Finally, dividing by -1, we get:
k = 23/7
Therefore, the value of k that makes the statement true is k = 23/7.
Answer:
Step-by-step explanation:
m1=y2-y1/x2-x1=12-5/k-3=7/k-3
m2=10-7/2-0=3/2
two lines with slopes m1 and m2 are perpendicular if m1×m2 = -1.
(7/k-3) *3/2=-1
3(7)=-2(k-3)
21=-2k+6
-2k=15
k=-15/2
HELP!!!!!!!!!!!!!!!!
The value of x and y are 13 and 3 respectively and the measure of angle mqr is 51 degree.
What are Exterior angle ?
It is an measure of rotation between one extended side(we extend it virtually) of the polygon with its adjacent side which is not extended.
WE are given that in the diagram, Ks, LT, MU are parallel to each other.
Angle knj = (2x-5)°, Angle lpn = (2y + 15), Angle pqu = (3x+4y)°.
Sine, the corresponding angles are equal
(2x-5) = (2y+15)
2x - 2y = 15 + 5
2x - 2y = 20
x - y = 10
x = 10 + y
Now,
(3x+4y) = (2y+15)
3x + 4y - 2y = 15
3x + 2y = 15
3(10 + y) + 2y = 15
30 + 5y = 15
y = 3
x = 13
Angle mqr = (3x+4y) = 51 degree.
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A payment on a $155,000 house is $850. If you pay this monthly for 30 years, how much interest will you have paid on this house?
To find out how much interest will be paid on a $155,000 house if a monthly payment of $850 is made for 30 years, we can use the formula for the total interest paid on a fixed-rate mortgage loan:
Total Interest = (Monthly Payment x Number of Payments) - Loan Amount
where Monthly Payment is $850, Number of Payments is 30 years x 12 months/year = 360 months, and Loan Amount is $155,000.
Plugging in these values, we get:
Total Interest = ($850 x 360) - $155,000
Total Interest = $306,000 - $155,000
Total Interest = $151,000
Therefore, the total interest paid on the $155,000 house over 30 years with a monthly payment of $850 is $151,000.
find the open intervals on which the function is increasing or decreasing. use a graphing utility to verify your results. (enter your answers using interval notation.) increasing incorrect: your answer is incorrect. decreasing
The function is decreasing on the interval (-∞, -2), has a local minimum at x = -2, is increasing on the interval (-2, 2), and is increasing on the interval (2, ∞). The function has a relative minimum at x = 3 (f(3) = -27) and no relative maximum.
Begin by finding the derivative of y.
y' = 3x^2 - 12
Find the critical numbers. (Enter your answers as a comma-separated list.)
To find the critical numbers, we need to solve y' = 0 for x:
3x^2 - 12 = 0
x^2 - 4 = 0
(x - 2)(x + 2) = 0
So the critical numbers are x = -2 and x = 2.
Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
To test the intervals, we can use a sign chart:
Interval (-∞, -2) (-2, 2) (2, ∞)
y' (-)(-) (-)(+) (+)(+)
y Decreasing Local Min Increasing
So the function is decreasing on the interval (-∞, -2), has a local minimum at x = -2, is increasing on the interval (-2, 2), and is increasing on the interval (2, ∞).
f(x) = x^4 - 4x^3
To find the relative extrema, we need to find the critical numbers and then evaluate the function at those points.
Find the derivative of f(x).
f'(x) = 4x^3 - 12x^2
Find the critical numbers by solving f'(x) = 0 for x.
4x^3 - 12x^2 = 0
4x^2(x - 3) = 0
x = 0 or x = 3
So the critical numbers are x = 0 and x = 3.
Evaluate the function at the critical numbers and the endpoints of the interval to find the relative extrema.
f(-∞) = ∞
f(0) = 0
f(3) = -27
f(∞) = ∞
So the function has a relative minimum at x = 3 (f(3) = -27) and no relative maximum.
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____The given question is incomplete, the complete question is given below:
Find the critical numbers and the open intervals on which the function is increasing or decreasing. Use a graphing utility to verity your results.
y = x^3 − 12x + 2
Step 1: Begin by finding the derivative of y.
y'=
Step 2: Find the critical numbers. (Enter your answers as a comma-separated list.)
x=
Step 3: Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
Increasing (___)
Decreasing (____)
2) Find all relative extrema of the function. (If an answer does not exist, enter DNE.)
f(x) = x4 − 4x3
relative max:(____)
relative min:(_____)
F and H are sets of real numbers defined as follows.
the rate of change in the number of americans over the age of 65 is n(t) = 0.00246t^2 + 0.118t + 0.183 million people per year
where is the number of years since 2000. in 2000, there were 34.42 million people over age 65 in the us. answer the following questions, rounding the answer to 2 decimal places: the average rate of change in the number of americans over the age of 65 from 2000 to 2015 is____
The average rate of change in the number of Americans over the age of 65 from 2000 to 2015 is approximately -2.09 million people per year.
To find the average rate of change in the number of Americans over the age of 65 from 2000 to 2015, we need to calculate the change in the number of people over 65 during that time period and divide it by the number of years:
Number of people over 65 in 2015:
n(15) = 0.00246(15)^2 + 0.118(15) + 0.183
n(15) = 2.781 million people
Change in the number of people over 65 from 2000 to 2015:
2.781 million - 34.42 million = -31.639 million people
Average rate of change:
-31.639 million / 15 years = -2.09 million people per year
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The population change for the ten most populous counties in the US from 2000 to 2010 are given in the following table:County & Percent Change: Los Angeles CA 3.1, Cook IL -3.4, Harris TX 20.3, Maricopa AZ 24.2, San Diego CA 10.0, Orange CA 5.8, Kings NY 1.6, Miami-Dade FL 10.8 Dallas TX 6.7, Queens NY 0.1a. What is the mean of the data? b. What is the median of the data? c. Which is the better description of the center of this data and why?
The mean and median of given data of population change for the ten most populous counties in the US from 2000 to 2010 is 7.2% and 6.25%.
To find the mean of the data, we need to add up all the percent changes and divide by the total number of counties:
Mean = (3.1 - 3.4 + 20.3 + 24.2 + 10.0 + 5.8 + 1.6 + 10.8 + 6.7 + 0.1) / 10 = 7.2%
Therefore, the mean percent change for the ten most populous counties in the US from 2000 to 2010 is 7.2%.
To find the median of the data, we first need to order the percent changes from smallest to largest:
-3.4, 0.1, 1.6, 3.1, 5.8, 6.7, 10.0, 10.8, 20.3, 24.2
Since there are an even number of values, the median is the average of the two middle values, which in this case are 5.8 and 6.7. Therefore, the median percent change for the ten most populous counties in the US from 2000 to 2010 is (5.8 + 6.7) / 2 = 6.25%.
The better description of the center of this data is the median, because it is less affected by outliers than the mean. In this case, the percent change for Maricopa County, AZ (24.2%) is much larger than the other values, and it pulls the mean upward. The median, on the other hand, is not affected by outliers, and it gives a better representation of the typical percent change for these counties.
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