The area of the polygon with the vertices (-4, 5), (-1, 5), (4, -3), and (-4, -3) is 12 square units.
To calculate the area of the polygon, we can use the shoelace formula, also known as Gauss's area formula or the surveyor's formula. The formula involves writing the x-coordinates and y-coordinates of the vertices in a specific order and performing a series of calculations.
1. We write the x-coordinates of the vertices in one row, repeating the first coordinate at the end: -4, -1, 4, -4.
2. We write the y-coordinates of the vertices in the next row, in the same order: 5, 5, -3, -3.
3. Next, we multiply each pair of adjacent x and y coordinates and add them together in a counterclockwise direction.
4. Then, we subtract the sum of the products of the y-coordinates and the x-coordinates in a counterclockwise direction.
5. Taking the absolute value of this result, we divide it by 2 to obtain the area.
Applying the shoelace formula:
Area = |((-4*5) + (-1*-3) + (4*-3) + (-4*5)) - (5*-1 + 5*4 + -3*-4 + -3*-4)| / 2
= |-49 - (-25)| / 2
= |-24| / 2
= 12 / 2
= 12.
Therefore, the area of the polygon with the given vertices is 12 square units.
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Help please I’m a little confused
Answer:
answer is a . x²-8x+16=32
x²-8x=32-16
Could someone please help me? Thank you and explain the work because I don’t get this
Answer:
5.59 times per second.
Step-by-step explanation:
Direct variation is in the form:
[tex]y=kx[/tex]
Where k is the constant of variation.
Inverse variation is in the form:
[tex]\displaystyle y=\frac{k}{x}[/tex]
In the given problem, the frequency of a vibrating guitar string varies inversely as its length. In other words, using f for frequency and l for length:
[tex]\displaystyle f=\frac{k}{\ell}[/tex]
We can solve for the constant of variation. We know that the frequency f is 4.3 when the length is 0.65 meters long. Thus:
[tex]\displaystyle 4.3=\frac{k}{0.65}[/tex]
Solve for k:
[tex]k=4.3(0.65)=2.795[/tex]
So, our equation becomes:
[tex]\displaystyle f=\frac{2.795}{\ell}[/tex]
Then when the length is 0.5 meters, the frequency will be:
[tex]\displaystyle f=\frac{2.795}{.5}=5.59\text{ times per second.}[/tex]
Given a random network of 104 nodes and average degree (k) = 10 which of the following is the expected characteristic path length (average distance) of the network? Select one: a. 4 b. 1 C. 2 d. 5
The expected characteristic path length (average distance) of a random network with 104 nodes and an average degree (k) of 10 is approximately 2.
The characteristic path length of a network measures the average distance between any two nodes in the network. For a random network, the expected characteristic path length can be approximated using the formula:
L ≈ ln(N) / ln(k),
where N is the number of nodes and k is the average degree.
Substituting N = 104 and k = 10 into the formula, we have:
L ≈ ln(104) / ln(10) ≈ 2.040
Rounding to the nearest integer, we get L ≈ 2.
Therefore, the expected characteristic path length of the network is approximately 2.
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(1 point) Are the following statements true or false? ? 1. If W = Span{V1, V2, V3 }, and if {V1, V2, V3 } is an orthogonal set in W, then {V1, V2, V3 } is an orthonormal basis for W. ? 2. If x is not in a subspace W, projw(x) is not zero. then x ?
3. In a QR factorization, say A = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.
1.An orthonormal basis for W is False.
2.If x is not in a subspace W, projw(x) is not zero then x True.
3.The QR factorization columns of Q form an orthonormal basis for the column space of A True.
An orthogonal set in a vector space necessarily mean that it is orthonormal are the vectors orthogonal to each other, but they unit length if {V1, V2, V3} is an orthogonal set in W, that the vectors are mutually orthogonal, but they may not have unit length {V1, V2, V3} assumed to be an orthonormal basis for W.
The projection of a vector x onto a subspace W, denoted as projW(x), is defined as the closest vector in W to x. If x is not in W, then the projection of x onto W will not be zero a nonzero vector in the subspace W that is closest to x.
In a QR factorization of a matrix A, where A has linearly independent columns, the matrix Q consists of orthonormal columns. The columns of Q form an orthonormal basis for the column space of A.
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pls help and show work i am screwed if i don’t do well on this
Answer:
x = - 2
Step-by-step explanation:
The axis of symmetry passes through the vertex, is a vertical line with equation equal to the x- coordinate of the vertex, that is
equation of axis of symmetry is x = 1
The zeros are equidistant from the axis of symmetry, on either side
x = 4 is a zero and is 3 units to the right of x = 1, so
3 units to the left of x = 1 is 1 - 3 = - 2
The other zero is therefore x = - 2
Which expression is equivalent to 13 22b
Find the centre of mass of the 2D shape bounded by the lines y = ±1.3x between x = 0 to 2.1. Assume the density is uniform with the value: 3.5kg. m-2 Also find the centre of mass of the 3D volume created by rotating the same lines about the x-axis. The density is uniform with the value: 1.9kg. m (Give all your answers rounded to 3 significant figures.) a) Enter the mass (kg) of the 20 plate: Enter the Moment (kg.m) of the 2D plate about the y-axis: Enter the x-coordinate (m) of the centre of mass of the 2D plate: Submit part ed Enter the mass (kg) of the 3D body: Enter the Moment (kg.m) of the 3D body about the y-axis: Enter the x-coordinate (m) of the centre of mass of the 3D body:
The mass of the 2D plate is 20.067 kg, with a moment of 5.742 kg.m about the y-axis. The x-coordinate of the center of mass of the 2D plate is 0.286 m. The mass of the 3D body is 62.137 kg, with a moment of 39.748 kg.m about the y-axis. The x-coordinate of the center of mass of the 3D body is 0.640 m.
To determine the center of mass of the 2D shape bounded by the lines y = ±1.3x between x = 0 to 2.1, we need to calculate the mass, moment, and x-coordinate of the center of mass.
1) Mass of the 2D plate:
The area of the 2D shape can be calculated by finding the difference in the areas under the two lines y = ±1.3x between x = 0 and x = 2.1.
Area = ∫(1.3x)dx - ∫(-1.3x)dx
= ∫1.3xdx + ∫1.3xdx
= 2 * ∫1.3xdx
= 2 * [0.65x²] between x = 0 and x = 2.1
= 2 * (0.65 * (2.1)²)
= 2 * (0.65 * 4.41)
= 2 * 2.8665
= 5.733
Mass = Area * Density
= 5.733 * 3.5
≈ 20.067 kg
Therefore, the mass of the 2D plate is approximately 20.067 kg.
2) Moment of the 2D plate about the y-axis:
The moment of the 2D shape about the y-axis is given by the integral of the product of the y-coordinate and the area element.
Moment = ∫(y * dA)
= ∫(±1.3x * dA)
= 2 * ∫(1.3x * dA) between x = 0 and x = 2.1
= 2 * 1.3 * ∫(x * dA) between x = 0 and x = 2.1
= 2 * 1.3 * ∫(x * dx)
= 2 * 1.3 * [0.5x²] between x = 0 and x = 2.1
= 2 * 1.3 * (0.5 * (2.1)²)
= 2 * 1.3 * (0.5 * 4.41)
= 2 * 1.3 * 2.205
= 5.742 kg.m
Therefore, the moment of the 2D plate about the y-axis is 5.742 kg.m.
3) x-coordinate of the center of mass of the 2D plate:
The x-coordinate of the center of mass of the 2D shape can be calculated using the formula:
x-coordinate = Moment / Mass
x-coordinate = 5.742 kg.m / 20.067 kg
≈ 0.286 m
Therefore, the x-coordinate of the center of mass of the 2D plate is approximately 0.286 m.
For the 3D body created by rotating the same lines about the x-axis:
1) Mass of the 3D body:
The volume of the 3D body can be calculated by finding the difference in the volumes between the two shapes obtained by rotating y = ±1.3x about the x-axis between x = 0 and x = 2.1.
Volume = π * ∫(1.3x)^2 dx - π * ∫(-1.3x)^2 dx
= π * ∫1.69x^2 dx - π * ∫1.69x^2 dx
=
2 * π * ∫1.69x² dx
= 2 * π * [0.5633x³] between x = 0 and x = 2.1
= 2 * π * (0.5633 * (2.1)³)
= 2 * π * (0.5633 * 9.261)
= 2 * π * 5.2167
≈ 32.703 m³
Mass = Volume * Density
= 32.703 * 1.9
≈ 62.137 kg
Therefore, the mass of the 3D body is approximately 62.137 kg.
2) Moment of the 3D body about the y-axis:
The moment of the 3D body about the y-axis can be calculated similarly to the 2D plate but considering the additional dimension.
Moment = ∫(x * dV)
= π * ∫(x * (1.3x)² dx) - π * ∫(x * (-1.3x)^2 dx)
= 2 * π * ∫(1.3x³ dx)
= 2 * π * [0.325x⁴] between x = 0 and x = 2.1
= 2 * π * (0.325 * (2.1)⁴)
= 2 * π * (0.325 * 19.4481)
= 2 * π * 6.3252
≈ 39.748 kg.m
Therefore, the moment of the 3D body about the y-axis is approximately 39.748 kg.m.
3) x-coordinate of the center of mass of the 3D body:
The x-coordinate of the center of mass of the 3D body can be calculated using the formula:
x-coordinate = Moment / Mass
x-coordinate = 39.748 kg.m / 62.137 kg
≈ 0.640 m
Therefore, the x-coordinate of the center of mass of the 3D body is approximately 0.640 m.
To summarize the answers:
a) Mass of the 2D plate: 20.067 kg
b) Moment of the 2D plate about the y-axis: 5.742 kg.m
c) x-coordinate of the center of mass of the 2D plate: 0.286 m
d) Mass of the 3D body: 62.137 kg
e) Moment of the 3D body about the y-axis: 39.748 kg.m
f) x-coordinate of the center of mass of the 3D body: 0.640 m
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Español Tom opened a savings account with $600 and was paid simple interest at an annual rate of 2%. When Tom closed the account, he was paid $36 in interest. How long was the account open for, in years? If necessary, refer to the list of financial formulas. Dy years ?
The account was open for 3 years. This duration was determined by calculating the time using the formula for simple interest based on the initial principal, interest rate, and the amount of interest earned.
To determine the length of time the account was open, we can use the formula for simple interest:
Interest = Principal * Rate * Time
Given that Tom opened the account with $600, the annual interest rate was 2%, and he received $36 in interest, we can set up the equation:
36 = 600 * 0.02 * Time
Simplifying the equation:
36 = 12 * Time
Dividing both sides by 12:
Time = 3
Therefore, the account was open for 3 years.
In conclusion, Tom's savings account was open for 3 years, as calculated using the simple interest formula.
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4(2a - 3) = 2(3a + 1)
What's the answer?
Answer:
a=7
Step-by-step explanation:
4(2a-3) = 2(3a+1)
8a-12 = 6a+2
2a-12=2
2a=14
a=7
A nutrition laboratory tests 40 "reduced sodium" hot dogs, finding that the mean sodium content is 310 mg, with a standard deviation of 36 mg.
a) Find a 95% confidence interval of the mean sodium content of this brand of hot dog.
The 95% confidence interval for the mean sodium content of the "reduced sodium" hot dogs is calculated to be (297.70 mg, 322.30 mg).
To find the 95% confidence interval, we use the formula:
Confidence Interval = (sample mean) ± (critical value) * (standard deviation / √sample size)
Given that the sample mean sodium content is 310 mg, the standard deviation is 36 mg, and the sample size is 40, we need to determine the critical value for a 95% confidence level.The critical value corresponds to the level of confidence and the degrees of freedom, which is the sample size minus 1. Looking up the critical value for a 95% confidence level and 39 degrees of freedom in the t-distribution table, we find it to be approximately 2.024.
Plugging in the values into the formula, we get:
Confidence Interval = 310 mg ± (2.024) * (36 mg / √40)
Simplifying the expression, we find:
Confidence Interval ≈ (297.70 mg, 322.30 mg)Therefore, we can say with 95% confidence that the mean sodium content of this brand of "reduced sodium" hot dogs falls within the range of 297.70 mg to 322.30 mg.
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Find the minimum or maximum value of the function. (Desmos)
Consider the following two sample data sets, Set 1: 16 24 17 22 Set 2: 2 7 1 8 200 5 a. Calculate the coefficient of variation for each data set b. Which data set has less consistency (or more variability)? a. The coefficient of variation for data set 1 is I %. (Round to one decimal place as needed.) The coefficient of variation for data set 2 is % (Round to one decimal place as needed.) b. Which data set has less consistency (or more variability)? Choose the correct answer below. O A. Data set 2 has less consistency (or more variability) because its coefficient of variation is less. O B. Data sot 1 has less consistency (or more variability) because its coefficient of variation is loss. C. Data set 2 has less consistency (or more variability because its coefficient of variation is creater. Consider the following two sample data sets. Set 1: Set 2: 16 2 24 17 7 1 22 8 20 5 a. Calculate the coefficient of variation for each data set. b. Which data set has less consistency (or more variability)? The coefficient of variation for data set 2 is % (Round to one decimal place as needed.) b. Which data set has less consistency (or more variability)? Choose the correct answer below. O A. Data set 2 has less consistency (or more variability) because its coefficient of variation is less. O B. Data set 1 has less consistency (or more variability) because its coefficient of variation is less OC. Data set 2 has less consistency (or more variability) because its coefficient of variation is greater OD. Data sot 1 has less consistency (or more variability) because its coefficient of variation is greater
a. Calculation of Coefficient of Variation for each data set
Data set 1: 16 24 17 22$${\rm Mean }\ \overline{x} = \frac{16 + 24 + 17 + 22}{4} = 19.75$$
Variance σ² $= \frac{1}{N} \sum_{i=1}^{N}(x_i - \overline{x})^2$ $= \frac{(16-19.75)^2 + (24-19.75)^2 + (17-19.75)^2 + (22-19.75)^2}{4}$ $= 16.1875$
Standard deviation $σ = \sqrt{16.1875} = 4.0218$ Coefficient of variation, $CV = \frac{σ}{\overline{x}}$ $= \frac{4.0218}{19.75} = 0.2031$Therefore, the coefficient of variation for data set 1 is 20.31%.Data set 2: 2 7 1 8 200 5${\rm Mean}\ \overline{x} = \frac{2 + 7 + 1 + 8 + 200 + 5}{6} = 36.833$Variance σ² $= \frac{1}{N} \sum_{i=1}^{N}(x_i - \overline{x})^2$ $= \frac{(2-36.833)^2 + (7-36.833)^2 + (1-36.833)^2 + (8-36.833)^2 + (200-36.833)^2 + (5-36.833)^2}{6}$ $= 10627.0246$ Standard deviation $σ = \sqrt{10627.0246} = 103.0792$
Coefficient of variation, $CV = \frac{σ}{\overline{x}}$ $= \frac{103.0792}{36.833} = 2.7971$
Therefore, the coefficient of variation for data set 2 is 279.71%.
b. Identifying the data set with less consistency (or more variability) To determine which data set has less consistency (or more variability), we need to compare their coefficients of variation. A higher coefficient of variation implies higher variability or inconsistency in the data. Therefore, the correct answer is option C: Data set 2 has less consistency (or more variability) because its coefficient of variation is greater.
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Jason wants to create a box with the same volume as the one shown below. He wants the length to be 4
inches. What would be the measurements of the width and height?
Length: 4 inches
Width: inches
Height: inches
Explain how you determined the width and height.
- cubic inch
Given : jason wants to create a box with the same volume as the one shown
To find : measurements of width and height .
Solution :
Volume of given box
=> 6 * 2 * 3
=36
length to be 4 inches .
lwh = 36
=> 4 wh = 36
=> wh = 9
9 = 1 * 9
9 = 3 * 3
Width and Height can be 3 inches each or 1 and 9 inches
help me please it's due tonight
Answer:
π
Step-by-step explanation:
S = rФ
Arc length = radius x theta
S = (3)([tex]\frac{\pi }{3}[/tex]) = [tex]\pi[/tex]
5th grade math. correct answer will be marked brainliest
Answer:
3
Step-by-step explanation:
Answer:
3
Step-by-step explanation:
9×[tex]\frac{1}{3}[/tex]
9×1/1×3=
[tex]\frac{9}{3}[/tex]÷3=
3!!!!
The Leungs sold a valuable painting for $55,000. This price is $1,000
more than twice the amount they originally paid for it. How much
did they originally pay?
A. $25,000 B. $27,000 C. $27,500 D. $28,000
Answer:
B. $27,000
Step-by-step explanation:
So 55,000 = 2x + 1,000
Simply for x
55,000 = 2x + 1,000
54,000 = 2x
x = 27,000
What is an equation of the line that passes through the point (6,-2) and is perpendicular to the line 6x+y=2
What is equivalent ratio of 5:8:4 and explain how u got it
this is example
So, first we need to write the given ratio as fraction,
= 8/18
= (8 × 2)/(18 × 2)
= 16/36
= 16 : 36 (one equivalent ratio),
So, 16 : 36 is an equivalent ratio of 8 : 18.
Now we will find another equivalent ratio of 8 : 18 by using division.
Similarly, first we need to write the given ratio as fraction,
= 8/18
= (8 ÷ 2)/(18 ÷ 2)
= 4/9
= 4 : 9 (another equivalent ratio)
So, 4 : 9 is an equivalent ratio of 8 : 18.
Therefore, the two equivalent ratios of 8 : 18 are 16 : 36 and 4 : 9.
2. Frame two equivalent ratios of 4 : 5.
Solution:
To find two equivalent ratios of 4 : 5 we need to apply multiplication method only to get the answer in integer form.
First we need to write the given ratio as fraction,
= 4/5
= (4 × 2)/(5 × 2)
= 8/10
= 8 : 10 is one equivalent ratio,
Similarly again, we need to write the given ratio 4 : 5 as fraction to get another equivalent ratio;
= 4/5
= (4 × 3)/(5 × 3)
= 12/15 is another equivalent ratio
Therefore, the two equivalent ratios of 4 : 5 are 8 : 10 and 12 : 15.
Note: In this question we can’t apply division method to get the answer in integer form because the G.C.F. of 4 and 5 is 1. That means, 4 and 5 cannot be divisible by any other number except 1.
The two equivalent ratios of 5:8:4 are 8: 10 and 12: 15.
What is a fraction?The fraction is defined as the division of the whole part into an equal number of parts. In mathematics, ratios are used to determine the relationship between two numbers it indicates how many times is one number to another number.
First, we need to write the given ratio as a fraction,
F= 8/18
F= (8 × 2)/(18 × 2)
F= 16/36
F= 16 : 36 (one equivalent ratio),
16: 36 is an equivalent ratio of 8: 18.
Find another equivalent ratio of 8: 18 by using division. Similarly, first, we need to write the given ratio as a fraction,
F= 8/18
F= (8 ÷ 2)/(18 ÷ 2)
F= 4/9
F= 4 : 9 (another equivalent ratio)
4: 9 is an equivalent ratio of 8: 18.
Therefore, the two equivalent ratios of 5:8:4 are 16: 36 and 4: 9.
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Find the intersections of these pairs of linear equations. Al) 4x-3y=6 -2x+(3/2)y-3 A2) x-4y=-5 3x-2y.15 A3) 4x+y=9 2x-3y22 A4) -6x+9y = 9 2x-3y=6 Note on answers: If the answer is a point, write It as an ordered pair, (a,b). No spaces. Include the parentheses. If there is no solution, enter none, If they are the same line and there is an Infinite number of solutions, enter
A1) To find the intersection of the two equations, we can solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
4x - 3y = 6
Solving for y:
-3y = -4x + 6
y = (4/3)x - 2
Now let's look at the second equation:
-2x + (3/2)y - 3 = 0
Solving for y:
(3/2)y = 2x + 3
y = (4/3)x + 2
Now we have two expressions for y:
y = (4/3)x - 2 and y = (4/3)x + 2
These lines have different y-intercepts and the same slope, so they are not parallel and must intersect at some point.
Setting the two expressions equal to each other:
(4/3)x - 2 = (4/3)x + 2
Subtracting (4/3)x from both sides:
-2 = 2
This is a contradiction, so there is no solution.
Answer: none
A2)
To find the intersection of the two equations, we can again solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
x - 4y = -5
Solving for y:
-4y = -x - 5
y = (1/4)x + (5/4)
Now let's look at the second equation:
3x - 2y = 15
Solving for y:
-2y = -3x + 15
y = (3/2)x - 7.5
Now we have two expressions for y:
y = (1/4)x + (5/4) and y = (3/2)x - 7.5
Setting the two expressions equal to each other:
(1/4)x + (5/4) = (3/2)x - 7.5
Subtracting (1/4)x and adding 7.5 to both sides:
(11/4) = (5/2)x
Multiplying both sides by 2/5:
x = 22/20 = 11/10
Now we can substitute this value of x into either equation to find y:
y = (1/4)(11/10) + (5/4) = (11/40) + (50/40) = 61/40
Answer: (11/10, 61/40)
A3)
Starting with the first equation:
4x + y = 9
Solving for y:
y = -4x + 9
Now let's look at the second equation:
2x - 3y = 22
Solving for y:
-3y = -2x + 22
y = (2/3)x - (22/3)
Now we have two expressions for y:
y = -4x + 9 and y = (2/3)x - (22/3)
Setting the two expressions equal to each other:
-4x + 9 = (2/3)x - (22/3)
Adding 4x and (22/3) to both sides:
33/3 = (14/3)x
Multiplying both sides by 3/14:
x = 9/14
Now we can substitute this value of x into either equation to find y:
y = -4(9/14) + 9 = -18/7
Answer: (9/14, -18/7)
A4)
Starting with the first equation:
-6x + 9y = 9
Solving for y:
9y = 6x + 9
y = (2/3)x + 1
Now let's look at the second equation:
2x - 3y = 6
Solving for y:
-3y = -2x + 6
y = (2/3)x - 2
Now we have two expressions for y:
y = (2/3)x + 1 and y = (2/3)x - 2
Setting the two expressions equal to each other:
(2/3)x + 1 = (2/3)x - 2
Adding -2/3x and -1 to both sides:
0 = -3A1)
To find the intersection of the two equations, we can solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
4x - 3y = 6
Solving for y:
-3y = -4x + 6
y = (4/3)x - 2
Now let's look at the second equation:
-2x + (3/2)y - 3 = 0
Solving for y:
(3/2)y = 2x + 3
y = (4/3)x + 2
Now we have two expressions for y:
y = (4/3)x - 2 and y = (4/3)x + 2
These lines have different y-intercepts and the same slope, so they are not parallel and must intersect at some point.
Setting the two expressions equal to each other:
(4/3)x - 2 = (4/3)x + 2
Subtracting (4/3)x from both sides:
-2 = 2
This is a contradiction, so there is no solution.
Answer: none
A2)
To find the intersection of the two equations, we can again solve for one variable in terms of the other and substitute that equation into the other equation.
Starting with the first equation:
x - 4y = -5
Solving for y:
-4y = -x - 5
y = (1/4)x + (5/4)
Now let's look at the second equation:
3x - 2y = 15
Solving for y:
-2y = -3x + 15
y = (3/2)x - 7.5
Now we have two expressions for y:
y = (1/4)x + (5/4) and y = (3/2)x - 7.5
Setting the two expressions equal to each other:
(1/4)x + (5/4) = (3/2)x - 7.5
Subtracting (1/4)x and adding 7.5 to both sides:
(11/4) = (5/2)x
Multiplying both sides by 2/5:
x = 22/20 = 11/10
Now we can substitute this value of x into either equation to find y:
y = (1/4)(11/10) + (5/4) = (11/40) + (50/40) = 61/40
Answer: (11/10, 61/40)
A3)
Starting with the first equation:
4x + y = 9
Solving for y:
y = -4x + 9
Now let's look at the second equation:
2x - 3y = 22
Solving for y:
-3y = -2x + 22
y = (2/3)x - (22/3)
Now we have two expressions for y:
y = -4x + 9 and y = (2/3)x - (22/3)
Setting the two expressions equal to each other:
-4x + 9 = (2/3)x - (22/3)
Adding 4x and (22/3) to both sides:
33/3 = (14/3)x
Multiplying both sides by 3/14:
x = 9/14
Now we can substitute this value of x into either equation to find y:
y = -4(9/14) + 9 = -18/7
Answer: (9/14, -18/7)
A4)
Starting with the first equation:
-6x + 9y = 9
Solving for y:
9y = 6x + 9
y = (2/3)x + 1
Now let's look at the second equation:
2x - 3y = 6
Solving for y:
-3y = -2x + 6
y = (2/3)x - 2
Now we have two expressions for y:
y = (2/3)x + 1 and y = (2/3)x - 2
Setting the two expressions equal to each other:
(2/3)x + 1 = (2/3)x - 2
Adding -2/3x and -1 to both sides: 0 = -3
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Solve the following DE using Power series around x₁ = 0. Find the first eight nonzero terms of this DE. y" + xy' + 2y = 0.
To solve the differential equation y" + xy' + 2y = 0 using power series, we assume a power series representation for the solution and derive a recurrence relation for the coefficients. The first eight nonzero terms can be found by solving the recurrence relation.
To solve the differential equation y" + xy' + 2y = 0 using power series around x₁ = 0, we can assume a power series representation for the solution:
y(x) = ∑(n=0 to ∞) aₙxⁿ
Let's substitute this power series representation into the given differential equation and find the recurrence relation for the coefficients aₙ.
Differentiating y(x) with respect to x:
y'(x) = ∑(n=0 to ∞) aₙn xⁿ⁻¹
y''(x) = ∑(n=0 to ∞) aₙn(n-1) xⁿ⁻²
Substituting these expressions into the differential equation:
∑(n=0 to ∞) aₙn(n-1) xⁿ⁻² + x ∑(n=0 to ∞) aₙn xⁿ⁻¹ + 2∑(n=0 to ∞) aₙxⁿ = 0
Now, we can rearrange and collect like terms based on the powers of x:
∑(n=0 to ∞) [aₙn(n-1) xⁿ⁻² + aₙn xⁿ⁺¹ + 2aₙxⁿ] = 0
Since this equation must hold for all values of x, each coefficient of xⁿ must be zero. Therefore, we get the following recurrence relation for the coefficients:
aₙ(n-1)(n-2) + aₙ₋₁(n-1) + 2aₙ = 0
Simplifying the recurrence relation:
aₙ(n² - 3n + 2) + aₙ₋₁(n-1) = 0
Now, we can start finding the first few nonzero terms of the power series solution by using the recurrence relation.
First term (n=0):
a₀(0² - 3(0) + 2) + a₋₁(-1) = 0
a₀ + a₋₁ = 0
Second term (n=1):
a₁(1² - 3(1) + 2) + a₀(1-1) = 0
a₁ - a₀ = 0
From the first and second terms, we find a₀ = a₁ and a₋₁ = -a₀.
Third term (n=2):
a₂(2² - 3(2) + 2) + a₁(2-1) = 0
a₂ - 3a₁ = 0
a₂ - 3a₀ = 0
Fourth term (n=3):
a₃(3² - 3(3) + 2) + a₂(3-1) = 0
a₃ - 6a₂ = 0
a₃ - 6a₀ = 0
Continuing this process, we can find the values of a₄, a₅, a₆, and so on, using the recurrence relation.
By solving the recurrence relation for each term, we can determine the first eight nonzero terms of the power series solution to the differential equation y" + xy' + 2y = 0.
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Bridgette has already taken 7 pictures at home ,and she expects to take 1 picture during everyday of vacation. How many days will Bridgette have to spend on vacation before she will have taken 9 pictures?
Answer:
I believe she would have to spend 2 days on vacation.
Step-by-step explanation:
She already have 7 pictures and she takes 1 a day. Therefore, she would have to spend 2 days on vacation to get 9 pictures.
Martin bought a painting for $5000. It is expected to appreciate at a continuous rate of 4%. Write an exponential equation to model this situation
Answer:
y=5000(1.04)^t
Step-by-step explanation:
Given data
Cost of painting=$5000
Rate of increase=4%
the exponential increase expression is
y=P(1+r)^t
Where y= the total amount after growth
P= the initial cost of the painting
r= the rate of increase
t= the time interval
y=5000(1+0.04)^t
y=5000(1.04)^t
Draw the diagram of LFSR with characteristic polynomial
x^6+x^5+x^3+x^2+1. What is the maximum period of the LFSR?
The maximum period of the LFSR with the given characteristic polynomial is 63.
To draw the diagram of a Linear Feedback Shift Register (LFSR) with a characteristic polynomial of [tex]x^6 + x^5 + x^3 + x^2 + 1,[/tex] we need to represent the shift register stages and the feedback connections.
The characteristic polynomial tells us the feedback taps in the LFSR. In this case, the feedback taps are at positions 6, 5, 3, 2, and 0 (the coefficients of the polynomial with non-zero exponents).
In the diagram, D1 represents the output of the first stage (bit), D2 represents the output of the second stage, and so on. The arrows represent the shift direction, with the feedback connections shown by the lines connecting the output of specific stages to the feedback taps.
Now, let's determine the maximum period of the LFSR with this characteristic polynomial. The maximum period of an LFSR is given by [tex]2^N - 1,[/tex] where N is the number of stages in the shift register.
In this case, there are 6 stages, so the maximum period is [tex]2^6 - 1 = 63.[/tex]
Therefore, the maximum period of the LFSR with the given characteristic polynomial is 63.
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A local charity receives 1/3 of funds raised during a craft fair and a bake sale. The total amount given to the charity was $137.45. How much did the bake sale raise?
(NEED ANSWER ASAP!!)
Answer:
$412.35
Step-by-step explanation:
Answer: $412.35
Step-by-step explanation:
$137.45 x 3 = $412.35 for the total that the bake sale raised.
Evaluate the expression x - 9, if x = 12.
1. 21
2. 17
3. 4
or 4. 3
what’s the answer what number?
Answer:
3
Step-by-step explanation:
I hope this answer has helped you
The value of expression x - 9, if x = 12 is 3, so the correct option is 4.
What is expression?A mathematical expression is made up of a statement, at least one arithmetic operation, and at least two integers or variables.
Given:
x - 9 and x = 12
Put the value of x in the expression as shown below,
x - 9 = 12 - 9
x - 9 = 3
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What is the answer? Please help, I need this done today.
Answer:
27 is the answer
Step-by-step explanation:
write the ratios sin m, cos m, and tan m. give the exact value and four decimal approximation. Please help.
Trigonometric functions are the ratio of different sides of a triangle. The ratios sin∠m, cos∠m, and Tan∠m are 0.758, 0.6522, and 1.1624.
What are Trigonometric functions?The trigonometric function gives the ratio of different sides of a right-angle triangle.
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}[/tex]
where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.
As it is given that the base of the triangle for the ∠m is Mk(15 units), the perpendicular is KL(4√19), and the hypotenuse is 23. Now, the trigonometric ratios can be written as,
Sine
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\\rm Sin (\angle m)=\dfrac{KL}{ML}\\\\\\\rm Sin (\angle m)=\dfrac{4\sqrt{19}}{23}\\\\\\\rm Sin (\angle m)=0.758069\approx 0.758[/tex]
Cosine
[tex]\rm Cos\theta=\dfrac{Base}{Hypotenuse}\\\\\\\rm Cos(\angle m)=\dfrac{MK}{ML}\\\\\\\rm Cos(\angle m)=\dfrac{15}{23}\\\\\\\rm Cos (\angle m)=0.65217\approx 0.6522[/tex]
Tangent
[tex]\rm Tan\theta=\dfrac{Perpendicular}{Base}\\\\\\\rm Tan(\angle m)=\dfrac{KL}{MK}\\\\\\\rm Tan(\angle m)=\dfrac{4\sqrt{19}}{15}\\\\\\\rm Tan(\angle m)=1.16237\approx 1.1624[/tex]
Hence, the ratios sin m, cos m, and tan m are 0.758, 0.6522, and 1.1624.
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help pleaseee i don’t understand thisss
Answer:
The answer is;
box A:3/2
box B:2/3
Which ordered pair is best estimate for the solution of the system of equations Y=3/2x plus 6, y equals 1/4 X -2
Answer:
x = -6.4 and y = -3.6
Step-by-step explanation:
The given system of equations are :
[tex]y=\dfrac{3}{2}x+6[/tex] ....(1)
and
[tex]y=\dfrac{x}{4}-2[/tex] ....(2)
We need to solve equation (1) and (2).
From equation (1) and (2),
[tex]\dfrac{3}{2}x+6=\dfrac{x}{4}-2[/tex]
Taking like terms together,
[tex]\dfrac{3}{2}x-\dfrac{x}{4}=-2-6\\\\\dfrac{6x-x}{4}=-8\\\\x=-6.4[/tex]
Put the value of x in equation (1).
[tex]y=\dfrac{3}{2}(-6.4)+6\\\\=-3.6[/tex]
So, the values of x and y are x = -6.4 and y = -3.6
Identify two ratios that are equivalent to 3:5