(a) The particular solution, y_p is 7
(b) y_p is -4x
(c) y_p is -4x + 7
(d) y_p is 8x + (7/2)
To find a particular solution to a differential equation by inspection - is to assume a trial function that looks like the non-homogeneous part of the differential equation.
(a) Given y'' + 2y = 14.
Because the no non-homogeneous part of the differential equation, 14 is a constant, our trial function will be a constant too.
Let A be our trial function:
We need our trial differential equation y''_p + 2y_p = 14
Now, we differentiate y_p = A twice, to obtain y'_p and y''_p that will be substituted into the differential equation.
y'_p = 0
y''_p = 0
Substitution into the trial differential equation, we have.
0 + 2A = 14
A = 6/2 = 7
Therefore, the particular solution, y_p = A is 7
(b) y'' + 2y = −8x
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x
2Ax + 2B = -8x
By inspection,
2B = 0 => B = 0
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x
(c) y'' + 2y = −8x + 14
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x + 14
2Ax + 2B = -8x + 14
By inspection,
2B = 14 => B = 14/2 = 7
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x + 7
(d) Find a particular solution of y'' + 2y = 16x + 7
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = 16x + 7
2Ax + 2B = 16x + 7
By inspection,
2B = 7 => B = 7/2
2A = 16 => A = 16/2 = 8
The particular solution y_p = Ax + B
is 8x + (7/2)
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Identify the solution of -4/3x+4/9=x^2
7 - Which of the following tables represents a linear function?
x −2 −1 0 1 2
y 4 1 −2 −5 −8
x −2 −1 0 1 2
y 4 1 0 1 4
x 2 2 0 2 2
y −2 −1 0 1 2
x 0 1 2 3 4
y −2 1 0 1 −2
A linear function has a constant rate of change, which means that the change in the value of y is proportional to the change in the value of x. This means that if we plot the points (x,y) for a linear function, they will fall on a straight line.
To determine which of the tables represents a linear function, we can calculate the rate of change between the points. If the rate of change is the same for all pairs of points, then the table represents a linear function.
For the first table:
Rate of change between (−2,4) and (−1,1) = (1−4)/(−1−(−2)) = −3
Rate of change between (−1,1) and (0,−2) = (−2−1)/(0−(−1)) = −3
Rate of change between (0,−2) and (1,−5) = (−5−(−2))/(1−0) = −3
Rate of change between (1,−5) and (2,−8) = (−8−(−5))/(2−1) = −3
Since the rate of change is constant (equal to −3) for all pairs of points, the first table represents a linear function.
For the second table:
Rate of change between (−2,4) and (−1,1) = (1−4)/(−1−(−2)) = −3
Rate of change between (−1,1) and (0,0) = (0−1)/(0−(−1)) = 1
Rate of change between (0,0) and (1,1) = (1−0)/(1−0) = 1
Rate of change between (1,1) and (2,4) = (4−1)/(2−1) = 3
Since the rate of change is not constant (it changes from −3 to 1 to 3), the second table does not represent a linear function.
For the third table:
Rate of change between (2,−2) and (2,−1) is undefined since x does not change.
Rate of change between (2,−1) and (0,0) = (0−(−1))/(0−2) = 0.5
Rate of change between (0,0) and (2,2) = (2−0)/(2−0) = 1
Rate of change between (2,2) and (2,−2) is undefined since x does not change.
Since the rate of change is not constant (it changes from undefined to 0.5 to 1 to undefined), the third table does not represent a linear function.
For the fourth table:
Rate of change between (0,−2) and (1,1) = (1−(−2))/(1−0) = 3
Rate of change between (1,1) and (2,0) = (0−1)/(2−1) = −1
Rate of change between (2,0) and (3,1) = (1−0)/(3−2) = 1
Rate of change between (3,1) and (4,−2) = (−2−1)/(4−3) = −3
Since the rate of change is not constant (it changes from 3 to −1 to 1 to −3), the fourth table does not represent a linear function.
Therefore, the only table that represents a linear function is the first one.
Noah wrote that 6+6=12. Then he wrote that 6+6-n=12-n. Are his equations balanced?Explain
Answer: Yes
Step-by-step explanation:
We don't know the value of n and you are subtracting it from both sides.
Need help with math. Need now
Answer:
Step-by-step explanation:
the volume of top triangular prism :
(16x21)/2x15=2520 yd^3
the volume of bottom:
22x16x15=5280 yd^3
so the volume of the figure:
2520+5280=7800 yd^3
Are there any online Line Integral Calculators?
Calculating antiderivatives, definite line integrals, double and triple integrals, and improper integrals is made easy using Wolfram | Alpha.
Wolfram| | Alpha does integral computations differently than people. It uses the Integrate function in Mathematica, which is the result of much mathematical and computational study. Integrate does integrals differently than individuals do. Instead, it makes use of robust, broad algorithms, which frequently need quite complex arithmetic. One includes determining an integral's general form, differentiating this form, and then resolving equations to match ambiguous symbolic parameters. The equations produced in this technique can be quite complicated, even for relatively basic integrands, and are best solved using Mathematica's robust algebraic computation capabilities. Converting integrals to generalized hypergeometric functions and then using collections of relations about these extremely complex mathematical functions is another method Mathematica utilizes to calculate integrals.
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Question 3(Multiple Choice Worth 2 points)
(Linear Functions LC)
Which of the following graphs represents a linear function?
coordinate plane with a graph that passes through the points negative 3 comma 0 and negative 1 comma negative 4 and 1 comma 0
coordinate plane with a graph drawn that passes through the points negative 4 comma 0 and 0 comma negative 4 and 4 comma 0 and 0 comma 4
coordinate plane with a graph drawn that passes through the points negative 4 comma negative 2 and negative 4 comma negative 1 and negative 4 comma 0
coordinate plane with a graph that passes through the points negative 4 comma 0 and negative 2 comma negative 1 and 0 comma negative 2
The second and fourth graphs represent linear functions.
What is a linear function?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
To determine which of the given graphs represents a linear function, we need to check whether the graph is a straight line.
The first graph passes through the points (-3, 0), (-1, -4), and (1, 0). We can plot these points and see that they do not lie in a straight line. Therefore, the first graph does not represent a linear function.
The second graph passes through the points (-4, 0), (0, -4), (4, 0), and (0, 4). We can plot these points and see that they do lie on a straight line. Therefore, the second graph represents a linear function.
The third graph passes through the points (-4, -2), (-4, -1), and (-4, 0). We can plot these points and see that they lie on a vertical line. Therefore, the third graph does not represent a linear function.
The fourth graph passes through the points (-4, 0), (-2, -1), and (0, -2). We can plot these points and see that they do lie on a straight line. Therefore, the fourth graph represents a linear function.
Therefore, the second and fourth graphs represent linear functions.
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Answer: It is the second one
Step-by-step explanation:
Because It makes the most sense
2. The mean age of a family of five is 22 years. The median is 13, the mode is 42, and the
range is 39 years. Find the ages of the five family members.
The ages of the five family members as required are; 3, 10, 13, 42 and 42.
What are the ages of the five family members?As evident, in the task content, let a, b , c , d , e represents the ages of the family members in ascending order.
Since the median of the ages is; 13, it follows that the the third term, c = 13.
Also, since the mode is 42, it follows that; d = e = 42.
Also, since the range of the ages is 39, it follows that; 42 - a = 39 and consequently a = 3.
Finally, since the mean age is 22; it follows from the mean formula that;
22 = ( 3 + b + 13 + 42 + 42 ) / 5
110 = b + 100
b = 10.
Ultimately, the ages of the five family members as required are; 3, 10, 13, 42 and 42.
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A bike shop in Belleville bought a mountain bike for $723 and marked it up 50% from the original cost. Later on, Andy purchased the mountain bike and paid Belleville sales tax of 14%. How much, including tax, did he pay for the mountain bike? You must include a $ sign and you must round to the nearest cent
The tax money given by Andy is equivalent to $151.83.
What is tax?Tax is a type of financial charge imposed by governmental organization on the individual to fund government spending and various public expenditures.
Given is that a bike shop in Belleville bought a mountain bike for $723 and marked it up 50% from the original cost.
Total cost of bike after marking it up -
{x} = $723 + ${50% of 723}
{x} = $723 + ${50/100 x 723)
{x} = $723 + $361.5
{x} = $1084.5
Money given by Andy as tax -
{t} = 14% of 1084.5
{t} = 14/100 x 1084.5
{t} = $151.83
Therefore, the tax money given by Andy is equivalent to $151.83.
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What is 20oz to lbs ?
From the given information, by doing unit conversion 20 ounces is equal to approximately 1.25 pounds.
Unit conversion is the process of converting a given quantity from one unit of measurement to another. It involves using conversion factors or conversion equations to change the numerical value of the quantity while keeping the physical quantity constant. Common units of measurement include length, mass, volume, time, temperature, and many others.
To convert ounces to pounds, you can use the following conversion formula:
1 lb = 16 oz
So, to convert 20 ounces to pounds, you can divide 20 by 16:
20 oz ÷ 16 oz/lb = 1.25 lb (rounded to two decimal places)
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Determine which integer in the solution set will make the equation true. 2p + 11 = 19 S: {−1, 1, 3, 4}
Answer:
4
Step-by-step explanation:
2x4=8, 8+11=19
Suppose a population of cells starts at 200 and triples every day. Let p equal the cells and d equal the number of days since the 200 cells were observed. True or False: You could use the equation the equation below to model this situation.
p = 200 · 3d
True
False
The model [tex]p = 200.3^d[/tex] satisfies the statement, A population of cells starts at 200 and triples every day.
What is the formula for exponential growth and exponential decaying function?The formula for exponential growth is [tex]y = y_0.e^{(kt)}.[/tex]
The formula for exponential decay is [tex]y = y_0.e^{(-kt)}.[/tex]
Given, A population of cells starts at 200 and triples every day.
Where 'p' is the number of cells and 'd' is the number of days.
The situation is represented by [tex]p = 200.3^d[/tex].
Now, Three times 200 is 600 and Nine times 200 is 1800.
At d = 1.
[tex]p = 200.3^1[/tex].
[tex]p = 600[/tex].
And,
At, d = 2.
[tex]p = 200.3^2[/tex].
[tex]p = 1800[/tex].
So, The model satisfies the statement and it is true.
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Is 0 0 is infinity or indeterminate form?
In calculus, 0^0 is an indeterminate form. It is not equal to infinity. But 0x0 is equal to zero.
Some of the indeterminate forms 0/0, 0×∞,∞/∞, ∞ −∞, ∞/0, [tex]0^{0}[/tex].
An unknown value is referred to as being "indeterminate." Even after inserting the limits, we are still unable to determine the original value, according to the indeterminate form of mathematics. When two functions are considered as a ratio, the indeterminate form frequently appears when both functions go closer to 0 in the limit. The term "indeterminate form 0/0" refers to these circumstances. The indeterminate form can also be created, much as addition, subtraction, multiplication, and exponential operations. If the limiting behavioral of separate components of the provided expression cannot identify the overall limit, a limit is said to be indeterminate.
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Given m lil n, find the value of x and y
The numerical values of x and y are 14 and 54 respectively.
What is the numerical value of x and y?The sum of angles on a straight line is equal to 180 degrees.
From the diagram, angle (9x + 2)° and angle (5x - 18)° are on a straight line,
Since the sum of angles on a straight line is equal to 180 degrees.
(9x + 2)° + (5x - 18)° = 180°
Solve for x
9x + 2 + 5x - 18 = 180
14x - 16 = 180
14x = 180 + 16
14x = 196
x = 196/14
x = 14
Now, angle (9x + 2)° and angle (2y + 20)° are corresponding angles.
Since corresponding angles are congruent.
(9x + 2)° = (2y + 20)°
Plug in x = 14 and solve for y
9( 14 ) + 2 = 2y + 20
126 + 2 = 2y + 20
128 = 2y + 20
128 - 20 = 2y
108 = 2y
2y = 108
y = 108/2
y = 54
Therefore, the value of x is 14 and y is 54.
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In 6 and 7, construct a line plot for each data set.
133
2'4' 41,1,0,
113
2' 2'4
6.
According to the information, the line plots of points 6 and 7 have several values that are repeated more than once as shown in the graphs.
How to make a line plot?To make a line plot we must identify the values that we must locate on the number line. Once we know the values we must place a point or symbol above the corresponding numerical value. For each number we must put a point, so we can count the frequency of different values.
Based on the above, we can infer that the line plots of points 6 and 7 would look like this:
attached images
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CAMBRILEARN
-0-7
b=-11
CAMBRILEARN
•-6√301
CAMBRILEARN
Бу+8
866.150
21:22
A
CAMBRILEARN
1.316
236
11346
·cm
CAMBRILEARN
and
10. Determine the sizes of angles a, b and c in the diagram, with reasons.
Remember to give your answer in degrees.
a =
32°
850
(Reason)
b
NOT TO
SCALE
E
[3]
What is 0900 military time?
0900 military time is 9:00 AM in the 24-hour time format used by the military, also known as the 24-hour clock.
The 24-hour clock is a timekeeping system that uses a continuous count of 24 hours, instead of two sets of 12 hours for AM and PM. In this system, each hour of the day is numbered from 0 to 23, and is referred to as "military time" or "24-hour time". So, 0900 military time is equivalent to 9:00 AM, with the "0" indicating the first hour of the day. This system is commonly used in the military, as well as in other fields where precise and standardized timekeeping is important, such as aviation and public transportation.
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Mr. Phil had a boatload of money he decided to leave it to his family but wanted to make sure his goldfish was taken care of today he decided to give 1/2 of his estate to his wife, 30,000 to his daughter half of what was left over to his butler, half of what was left to his goldfish. he realized he had $8000 remaining so he donated to charity. what was the original value of his estate?
Answer:
208,000
Step-by-step explanation:
e by 1
3. Mr. Bauer placed the letters in the word
BASKETBALL into a bag. What is the
probability of choosing the letter B, not
replacing it, and then choosing a vowel?
Answer:
1/15 chance
Step-by-step explanation:
BASKETBALL = 10 letters
taking out a B = 2/10 or 1/5
Then picking a vowel = 3/9 or 1/3
1/5 x 1/3 = 1/15 chance
John is a quarterback. This year, he completed
350 passes, which is
70percent of all the passes he's attempted this year.
How many passes has John attempted this year?
The number of passes that John has attempted this year will be 500.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
John is a quarterback. This year, he completed 350 passes, which is 70% of all the passes he's attempted this year.
Let 'y' be the total passes that John attempted. Then the equation is given as,
0.70 = (350 / y)
y = 350 / 0.70
y = 500
The number of passes that John has attempted this year will be 500.
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Part C: Charlie designs a square garden. Each side length is 3x + 5.
The perimeter of Charlie's garden is how much larger than the perimeter of Shawn's
garden?
Answer:
To answer this question, we first need to determine the perimeter of Shawn's garden. If we let x represent the side length of Shawn's garden, then the perimeter of Shawn's garden would be 4x.
Now, the perimeter of Charlie's garden is 12x + 20, which is 8x + 20 larger than the perimeter of Shawn's garden (4x). Therefore, the perimeter of Charlie's garden is 8x + 20 larger than the perimeter of Shawn's garden.
Use an algebraic equation to find the measure of each angle that is represented in terms of x.
Answer:
Step-by-step explanation: 9.17
Does someone mind helping me with this? Thank you!
The equation of the line that has a slope of -4 and passes through the point (4, -1) is y = -4x + 15.
What is a slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction
If a line passes through two points (x₁ ,y₁) and (x₂, y₂) ,
then the equation of a line is
y - y₁ = (y₂- y₁) / (x₂ - x₁) x (x - x₁)
To find the slope;
m = (y₂- y₁) / (x₂ - x₁)
Given:
An equation,
y = x/4 - 2.
The slope of the line that is perpendicular to the line y = x/4 - 2 is,
= -(1/1/4)
= -4.
The equation of the line that has a slope of -4 and passes through the point (4, -1) is,
y + 1 = -4(x - 4)
y = -4x +16 -1
y = -4x + 15
Therefore, the equation is y = -4x + 15.
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Find the length of Segment AB and Segment CD. How did you determine the lengths?
[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ A(\stackrel{x_1}{1}~,~\stackrel{y_1}{5})\qquad B(\stackrel{x_2}{4}~,~\stackrel{y_2}{2})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ AB=\sqrt{(~~4 - 1~~)^2 + (~~2 - 5~~)^2} \implies \implies AB=\sqrt{( 3 )^2 + ( -3 )^2} \\\\\\ AB=\sqrt{ 9 + 9 } \implies AB=\sqrt{ 18 }\implies AB\approx 4.24[/tex]
now, we could do the same thing for CD, however, notice, the y-coordinates are the same, meaning is just a horizontal line, we can just count the units off the grid, from 2 to 6, 4 units.
I really need help with this question
Answer: Can't help I tried to figure it out, but it was wrong.
Step-by-step explanation: I'm sorry.
At a high school, the probability that a student is a senior is 0.25. The
probability that a student plays a sport is 0.20. The probability that a student
is a senior and plays a sport is 0.08.
What is the probability A that a randomly selected student plays a sport, given
that the student is a senior?
• A. 0.32
• B. 0.08
• C. 0.25
• D. 0.17
The probability a that a randomly selected student plays a sport, given that the student is a senior is 0.32
How to determine the probabilityFrom the question, we have the following parameters that can be used in our computation:
The probability that a student is a senior is 0.25. The probability that a student plays a sport is 0.20. The probability that a student is a senior and plays a sport is 0.08.Using the above as a guide, we have the following:
P(Sport Given Senior) = 0.08/0.25
Evaluate
P(Sport Given Senior) = 0.32
Hence, the probability is 0.32
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What is the probability of obtaining 3 heads in four flips of a fair coin?
The probability of obtaining 3 heads in four flips of a fair coin is 1/4.
The probability of obtaining 3 heads in four flips of a fair coin can be calculated using the binomial probability formula. The formula is:
[tex]P(X = k) = (n choose k) * p^k * (1 - p)^{(n - k)}[/tex]
where P(X = k) is the probability of obtaining k successes (in this case, 3 heads), and n is the total number of trials (in this case, 4 flips).
p is the probability of success on a single trial (in this case, 0.5 since the coin is fair), and (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.
Plugging in the values, we get:
[tex]P(X = 3) = (4 choose 3) * 0.5^3 * (1 - 0.5)^{(4 - 3)}[/tex]
= 4 * 0.125 * 0.5
= 0.25
Therefore, the probability of obtaining 3 heads in four flips of a fair coin is 1/4 or 0.25
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A floor plan of Gail’s bedroom and closet is shown.
Which expression could be used to determine the total area of the bedroom and closet in square meters?
The bedroom has a surface space of 17.4225 m² and the closet has a surface area of 2.5875 m².
What is area?In mathematics, the area is a measure of the amount of two-dimensional space taken up by a shape. It is a scalar quantity, meaning it has only magnitude and no direction. The unit of measurement used for area depends on the units used for the length of the sides of the shape. Common units used to measure area include square centimeters, square meters, square inches, and square feet. The formula for finding the area of a shape depends on the specific type of shape. For example, the area of a rectangle is given by the formula A = lw, where l is the length and w is the width of the rectangle. The area of a triangle is given by the formula A = (1/2)*bh, where b is the base and h is the height of the triangle.
Here,
area of bedroom=l*w
l=5.05 m
w=3.45 m
area of bedroom=5.05*3.45
=17.4225 m²
area of closet=l*w
l=0.75 m
w=3.45 m
area=0.75*3.45
=2.5875 m²
The area of the bedroom is 17.4225 m² and that of closet is 2.5875 m².
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passes through the point (4, 5);
slope = -1/4
The equation of line is y= -1/4x+ 6.
What is Slope?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. y and x are the net changes in the y and x coordinates, respectively. Therefore, it is possible to write the change in y coordinate with respect to the change in x coordinate as,
m = Δy/Δx where, m is the slope
Given:
passes through the point (4, 5);
slope = -1/4
Using the slope intercept form
y = mx + b
y= -1/4x + b
As the point passes through (4, 5) then
5 = -1/4 (4) + b
5 + 1 = b
b= 6
So, the equation of line is y= -1/4x+ 6.
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Help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
to tiny submit closer picture
A bee population is measured each week and the results are plotted on the graph.
Fill in the blanks to the answers below:
What is the bee population when it is first measured?
The bee population when it is first measured is ___ bees.
Is the bee population growing by the same factor each week? Yes or No
If you answered yes to question 2, then what is the growth factor?
The growth factor is ____
The population of the bees when it was first measured was 500 bees
The bees are growing by the same factor each week. The answer is yes
How to find the population of the beesAt the fist time the bees where measured, we have to measure from the origin of the graph.
The first point on the graph is at the population of 500
2. To get if the growth is by the same factor, we have to observe how the growth is taking place weekly
In the first week, it grew from 500 to 1000
1000 / 500 = 2
In the second week, the growth is from 2000 to 1000
2000 / 1000 = 2
In the third week. the growth is from 2000 to 4000
4000 / 2000 = 2
Hence there is a growth constant of 2 across the weeks
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