Verify that both y₁(t) = 1 - t and y₂(t) = -t²/4 are solutions of the initial value problem Where are these solutions valid? Explain why the existence of two solutions of the given problem does not contradict the uniqueness part of Show that y = ct + c², where c is an arbitrary constant, satisfies the differential equation in part the initial condition is also satisfied, and the solution y = y₁(t) is obtained. Show that there is no choice of c that gives the second solution y = y₂(t).
Answers
Replacing y(t) into the initial value, we reach an identity, meaning that it can be verified that both y(t) = 1 - t and y(t) = -t²/4 are solutions to the initial value problem.
How to verify the solutions to the initial problem?The initial value problem is defined as follows:
[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]
The first solution is given as follows:
y(t) = 1 - t.
Hence the derivative is given as follows:
y'(t) = -1.
Replacing into the initial value problem, we have that:
[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]
[tex]-1 = \frac{-t + \sqrt{t^2 + 4(1 - t)}}{2}[/tex]
[tex]-1 = \frac{-t + \sqrt{t^2 - 4t + 4}}{2}[/tex]
[tex]-1 = \frac{-t + \sqrt{(t - 2)^2}}{2}[/tex]
[tex]-1 = \frac{-t + t - 2}}{2}[/tex]
-1 = -1.
Identity, verifying the solution.
The second solution is given as follows:
y(t) = -t²/4.
Hence the derivative is of:
y' = -0.5t.
Replacing into the initial value problem, we have that:
[tex]y^{\prime} = \frac{-t + \sqrt{t^2 + 4y}}{2}[/tex]
[tex]-0.5t = \frac{-t + \sqrt{t^2 - 4\frac{t^2}{4}}}{2}[/tex]
[tex]-0.5t = \frac{-t + \sqrt{0}}{2}[/tex]
-0.5t = -0.5t.
Identity, verifying the solution.
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Related Questions
is math invented or created
Answers
the answer to your question is mathematics was invented not created.
There are 11 girls and 5 boys taking taekwondo lessons. Write the ratio that compares the number of girls taking
taekwondo lessons to the total number of students taking taekwondo lessons
Answers
Answer:
11 : 16
Step-by-step explanation:
There are 11 girls and 5 boys taking taekwondo lessons. This means that in total there are 16 people taking lessons, we can turn this into a ratio that equals 11 : 16
A grilled chicken salad at a popular fast food restaurant contains 640 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount, in milligrams, of sodium? (Round your answer to one decimal place.)
mg
Answers
By percentage method the total recommended daily amount, in milligrams, of sodium is calculated to be 2370 mg.
What is percentage?
A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100.
Amount of sodium in chicken salad - 640 mg
Percentage 640 mg is of recommended amount - 27%
Let x be the total recommended daily amount (in mg) of the sodium.
According to the question -
27% of x = 640
x = (640 × 100)/27
x = 2370.37 ≈ 2370
Therefore, the recommended amount is 2370 mg.
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In order for line p to be parallel to line q, what must be the value of x?
Answers
Answer:
107
Step-by-step explanation:
By the converse of the corresponding angles theorem, [tex]x=107[/tex].
Suppose y varies directly with x. Write a direct variation equation that relates x and y. Then find the value of y when x = 14.
y=3 when x=2
Write a direct variation equation that relates x and y.
y = _______(Simplify your answer.)
Answers
George purchased a new car with the special 2.9% financing if the cars price was 19999 and he financed it for five years find his monthly payment
Answers
The monthly payment for car at rate of 2.9% is $358.57.
What is compound interest?The interest that is calculated using both the principal and the interest that has accrued during the previous period is called compound interest. It differs from simple interest in that the principal is not taken into account when determining the interest for the subsequent period with simple interest. Compound interest is commonly abbreviated C.I. in mathematics.
Given the cost of car = $19,999
rate per annum = 2.9% = 0.029
interest compounded monthly,
n = 12
time 5 years
compound interest formula for amount P =[tex]\frac{Cr/12}{[(1+r/12){12t} - 1]}}[/tex]
P = 19999*{(0.029/12)/[(1+0.029/12)^12*5 - 1}
p = $358.47
Hence the monthly payment is $358.47.
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Help please!!!! It’s on illustrative mathematics and I need help!!
Answers
Answer:
See attached
Step-by-step explanation:
You want the slopes of the various graphs, and a description of how you got it.
SlopeAs you know from your real-world experience a steep slope has more vertical change in the same horizontal distance than one that is less steep. When the height increases over a horizontal distance, we say the slope is positive. When the height decreases, the slope is negative.
In the real world, a downhill (negative) slope turns into a positive (uphill) slope if you just turn around and go the other direction.
On an x-y (Cartesian) coordinate plane, we define the direction as positive to the right. So, a line or curve with positive slope goes up to the right. A line or curve with negative slope goes down to the right. (The slope of a horizontal line is zero. The slope of a vertical line is "undefined.")
Measure of slopeSlope can be described different ways. The natural slope of a pile of loose material (sand, gravel, rock, salt, coal, and the like) can be described by its angle of repose. Dry sand may have an angle of repose of as much as 34°. When it is filled with water, it may slump to 15° or so.
The slope, or grade, of a roadway or other path is often described as a percentage (feet of rise or drop per 100 horizontal feet). Many places a 10% grade (about 5.7°) would be considered a pretty steep slope. The steepest hills in San Francisco have a 31.5% grade. A "black diamond" ski run is one with a slope of 40% (about 22°) or more.
In Algebra, we usually give a slope a numerical value. Its units will be the ratio of the units of the vertical measure on a graph to the units of the horizontal measure on the graph. For a graph with no particular units, or where the horizontal and vertical units are the same, the slope is simply a "pure number" that has no units.
Calculating slopeSlope is defined as the ratio of vertical change to the corresponding horizontal change. On a graph, the vertical change between two points is the difference of their vertical (y) coordinates. The horizontal change is the difference of the horizontal (x) coordinates taken in the same order.
The formula for the slope between two points (x₁, y₁) and (x₂, y₂) is ...
[tex]\text{slope}=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
The points can be designated as "point 1" and "point 2" in any order, but the chosen designation needs to be used consistently in the calculation.
If the denominator is zero, the slope is "undefined."
1. Given graphsThe graphs show the coordinates of the points you are to use to calculate the slope. In the attachment, we use the point on the right as "point 2" and the point on the left as "point 1". That makes the difference of x-coordinates positive, so the sign of the slope will be the sign of the difference of corresponding y-coordinates. (The slope is positive in all cases here.)
2. ProcedureThe "procedure" is simply a description of what you did to find the slope. It is described above, and also in the attachment. You know what you did, so all you need to do is write that down.
3. ExpressionThe answer to question 3 is the same as the answer to any of the parts of question 1, except you're using different symbols for the coordinates of the point. Considering graph A, you're using "u" and "v" instead of "4" and "9". The letters go in the same places the numbers do in the equation for slope.
The attachment has the answers to the questions.
Which question can be represented by the equation ? * 1/8 = 3?
1. How many 1/8's are in 3?
2. How many 3s are in 1/8?
3. What is 1/8 of 3?
4. What is 3 groups of 1/8?
Answers
The question that can be represented by the equation ? * 1/8 = 3 is (1). How many 1/8's are in 3?
How to determine the question of the equation
From the question, we have the following parameters that can be used in our computation:
? * 1/8 = 3
Represent the expression ? with x
So, we have
x * 1/8 = 3
Divide both sides of the equation by 1/8
This gives
x = 3/(1/8)
This means that how much 1/8 can be gotten from 3
Hence, the question is (1)
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A fair die with its faces numbered from 1 to 6 will be rolled. Which of the following is the best interpretation of the probability that the number landing face up will be less than 3?
For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 1/3
For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 1/2
For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 2/3
For three rolls of the die, a number less than 3 will land face up one time.
It will take three rolls of the die before a number less than 3 lands face up for the first time.
Answers
The best interpretation of the probability that the number landing face up will be less than 3 is [tex]\frac{1}{3}[/tex].
For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is [tex]\frac{1}{3}[/tex].
As per the given question a fair die with its faces numbered from 1 to 6 will be rolled.
Here we have to calculate the probability that the number landing face up will be less than 3.
The event of the number landing face up will be less than 3 is exhaustive, mutually exclusive, and independent face of a die = 6
(1, 2, 3, 4, 5, 6)
Favorable number of faces of a die = 2
(1, 2)
Since the number landing faces up will be less than 3.
Therefore the probability that the number landing face up will be less than 3
[tex]=\frac{\text { Favourable Number }}{\text { Total Number }}$[/tex]
[tex]& =\frac{2}{6} \\[/tex]
[tex]& =\frac{1}{3}[/tex]
Therefore the best interpretation of the probability that the number landing face up will be less than 3 is [tex]\frac{1}{3}[/tex]
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The long-run relative frequency of a number less than 3 landing face up is 2/3.
The best interpretation of the probability that the number landing face up will be less than 3 is that for many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 2/3. This can be expressed mathematically as P(x<3) = 2/3, where P(x) is the probability of the number landing face up being less than 3.
For a single roll of the die, the probability of a number less than 3 landing face up is 1/2. This is because there are three numbers (1,2,3) which are less than 3, and 6 possible outcomes. Therefore, the probability of any one of those three numbers landing face up is 3/6, or 1/2.
For multiple rolls of the die, the probability of a number less than 3 landing face up is the sum of the probabilities of each of those three numbers landing face up. This can be expressed mathematically as P(x<3) = (1/2) + (1/2) + (1/2), or 2/3.
Therefore, for many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 2/3.
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Can someone please help me?
Answers
Answer:
Step-by-step explanation:
no
For the solution: (1,-5)
y=4x-9 => -5=4(1)-9 => -5=4-9 => -5=-5
This equation is TRUE, making the ordered pair, (1,-5) a solution to the equation.
Now for the solution: (-2,-10)
y=4x-9 => -10=4(-2)-9 => -10=-8-9 => -10=-19 => -10≠-19
This equation is FALSE, thus the ordered pair, (-2,-10) is NOT a solution to the given equation.
Hope this helps!
In a regular polyhedron, the number of vertices is 4 and the number of edges is 6. Find the number of faces by using Euler's formula.
Answers
In this case, the number of vertices is 4 and the number of edges is 6, so we can substitute these values into the equation to find the number of faces:
4 - 6 + F = 2
F = 2 - 4 + 6
F = 4
Therefore, the number of faces in the regular polyhedron is 4.
May someone please help me my work is do tomorrow 1/4x - 3 = 1/2x + 12
Answers
X/4-3=1/2x+12
X/4+4(-3)/4=1/2x+12
X+4(-3)/4=1/2x+12
X-12/4=1/2x+12
the following data points represent the yearly salaries of high school cheerleading coaches in dakota county (in thousands of dollars). 41,38,36,57,43, 41,find the mean absolute deviation (mad) of the data set. thousand dollars
Answers
The data set's mean absolute deviation (MAD) is 5.6.
What is mean absolute deviation (MAD)?The average of the absolute deviations from a central point makes up the average absolute deviation of a data collection.
It is a statistical summary of statistical variability or dispersion.
We know that:
Dakota County high school cheerleading coaches' yearly salaries.
Data:
41, 38, 36, 57, 43
Mean = (41 + 38 + 36 + 57 + 43)/5
= 215/5
= 43
mean absolute deviation (MAD) = (1/5) ∑ | Mean - xi | i = 1 to 5
| 43 - 41| = 2
| 43 - 38 | = 5
| 43 - 36| = 7
| 43 - 57} = 14
| 43 - 43 | =0
mean absolute deviation (MAD) = (1/5) (2 + 5 + 7 + 14 + 0)
=28/5
= 5.6
Therefore, the data set's mean absolute deviation (MAD) is 5.6.
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Name
Level of Education
Weekly Income
Miley
High School Dropout
$440.50
Niko
High School Graduate
$650.35
Taylor
2-Year College Graduate
$771.25
Pinky
4-Year College Graduate
$1,099.20
Estimate about how much money Pinky will earn in a month.
About how long will Miley have to work to earn the same amount?
Answers
Answer:
274.8 dollars
Step-by-step explanation:
Mr. SANTOS owns a piece of land that measures 950 square meters. He planted 42% of it with fruit-bearing trees. How many square meters of land were palnted with fruit-bearing trees?
Answers
The square meters of land that were palnted with fruit-bearing trees is 399 m².
What is percentage in math?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion therefore refers to a component per hundred.
The measurement of the piece of land is 950 square meters
Then we were told to find the 42% of it with fruit-bearing trees which is the 42% of the land.
Then this can be written as :
42/100 * 950 = 399 m²
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39. Sophia has one of each of these cards. 1 3 5 71 9 How many three digit numbers divisible by three can she make with her cards? (Write only the number on your answer sheet)
Answers
Answer: Sophie can use 225
Step-by-step explanation:
The angles of a AABC are given in terms of a variable x as follows,
angle A = 3x + 28°
angle B = 5x + 52°
angle C = 2x - 10°
Answers
The values of angles are ∠A = 61°, ∠B = 107°, ∠C = 12°.
What is triangle?Simple polygons with three sides and three internal angles make up triangles. One of the fundamental geometric shapes, it is represented by the symbol △ and consists of three connected vertices. Triangles can be divided into several categories according to their sides and angles.
Given
∠A = 3x + 28°
∠B =5x + 52°
∠C = 2x - 10°
sum of angles of triangle is 180°
∠A + ∠B + ∠C = 180°
3x + 28° + 5x + 52° + 2x - 10° = 180
10x + 80 - 10 = 180
10x = 180 - 70
10x = 110
x = 11
∠A = 3x + 28° = 3(11) + 28
∠A = 61°
∠B =5x + 52° = 5(11) + 52
∠B = 107°
∠C = 2x - 10° = 2(11) -10
∠C = 12°
Hence measure of angles is ∠A = 61°, ∠B = 107°, ∠C = 12°.
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The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Answers
The equation that can be used to solve for the length, y , is y²-5y = 750
What is area of a rectangle?Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. Area is measured in squared unit.
The area of a rectangle is given by the expression:
A = l × w
where l is the length and w is the width of the rectangle.
The width of the rectangle is 5 feet less than that of the length. The length is represented by y. this means that w = y-5
Area = length × width
area = 750
therefore 750 = y ×( y-5)
opening the parentheses,
750 = y² - 5y
therefore the equation that can be used to solve for the length of the rectangle is
y²-5y = 750
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please help me on math
Answers
Answer:
the populate the same it increst then decrest the same amont
Step-by-step explanation:
Find the reciprocal of 123
Answers
Answer:
5/3
Step-by-step explanation:
The first step in finding the reciprocal of 1 2/3 is to make 1 2/3 into a fraction, which would be 5/3. The reciprocal of a given fraction is simply that fraction flipped over. The reciprocal of 5/3 would be 3/5.
Answer:
1/123
Step-by-step explanation:
reciprocal is just 1/ 123
Mariano tosses a quarter into a wishing well. The top of the well is represented by 0 feet. The location of the quarter after the fall is represented by -22 feet. How far did the quarter fall?
Answers
The quarter tossed by Moriano fall a distance of 22 feet
How to determine how far did the quarter fall?The given parameters in the question are
Location of the quarter after the fall = -22 feet.
Top of the well = 0 feet
The required distance can be calculated using the following equation
Distance = Top of the well - Location of the quarter after the fall
Next, we substitute the given values in the above equation
When this is done, the equation becomes
Distance = 0 + 22
Evaluate the sum
Distance = 22
Hence, the value of the distance is 22 feet
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Please solve everything here in this page
Answers
Answer:
Step-by-step explanation:
Write an equation of the line that passes through the given points.
(-2,6) and (4,0)
The equation is _____. (Type your answer in slope-intercept form.)
Answers
The equation of the line that passes through the points (-2,6) and (4,0) is
y = -x + 4
How to identify the equation of the lineThe slope, m of the linear function is calculated using the points (-2, 6) and (4, 0)
m = (y₀ - y₁) / (x₀ - x₁)
m = (6 - 0) / (-2 - 4)
m = (6) / (-6)
m = -1
equation passing through point (4,0) using point slope formula given by
(y - y₁) = m' (x - x₁)
y - 0= -1(x - 4)
y = -x + 4
the equation of the line is y = -x + 4
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Use the figure to solve the problem.
The sail on a model boat has the shape of a triangle. What is the area of the sail?
Formula for area of a triangle:
A=1/2(b × h)
A=1/2( . × 12)
A=1/2(72)
Solution: A= sq in.
Answers
The area of the model boat sail, given the shape of the sail being a triangle, is :
A = 1 / 2 ( 6 x 12 )A = 36 inch ²How to find the area ?The area of a triangle can be found by the formula :
= 1 / 2 x base x height
In the model boat given, the sail has the base of 6 inches and has a height of 12 inches.
This means that the area of the triangle is :
= 1 / 2 x ( 6 x 12 ).
= 1 / 2 x 72
= 36 inch ²
The reason the units are squared is that the units for area is squared.
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determine the value of mjk in numerical value only
Answers
The numeric value of ∠MJK is 135°
What is an angle measure?When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.
Given:
Some measurements of angles of JKLM and PQRS are given. m∠MJK = (12x+15)°, m∠SPQ = (14x-5)°, m∠JKL = 115°,
m∠JML = (8y + 12)°, m∠PSR = = (13y-18)°.
And we know the quadrilaterals are similar.
Two quadrilaterals are similar, so their corresponding angles are equal.
JKLM and PQRS are shown, where JKLM is similar to PQRS.
So,
m∠MJK= (12x+15)° = m∠SPQ = (14x-5)°
12x + 15 = 14x - 5
2x = 20
x = 10
So,
m∠MJK= (12(10)+15)°
m∠MJK = (135)°.
Therefore, 135° is the measure of ∠MJK
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Does two cups : 2/3 have a unit rateof three
Answers
It should be noted that two cups : 2/3 have a unit rate of three. This is true.
How to illustrate the ratio?Ratio in Mathematics simply demonstrates how many times one number can be able to fit into another number. It should be noted that ratios contrast two numbers by dividing them. In this case, A/B will be the formula if one is comparing one variable (A) to another variable (B). This simply indicates that you're dividing the variable A by B.
In this case, two cups : 2/3 have a unit rate of three. This can be Illustrated as:
= 2 ÷ 2/3
= 2 × 3/2
= 3
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6/7m - 3 = 5/7m - 5
Please Explain step by step
Answers
Answer:
m = - 14
Step-by-step explanation:
[tex]\frac{6}{7}[/tex] m - 3 = [tex]\frac{5}{7}[/tex] m - 5
multiply through by 7 to clear the fractions
6m - 21 = 5m - 35 ( subtract 5m from both sides )
m - 21 = - 35 ( add 21 to both sides )
m = - 14
Answer:
Step-by-step explanation:
6/7m - 3 = 5/7m - 5
(as you can see, there are simple numbers and some with m besides them, you have to group the "m" together and the simple ones on the other side of the "=" which means that you have to take away numbers that are "on the wrong side")
(first with -3, you add 3 on both sides)
6/7m - 3 + 3 = 5/7m - 5 + 3
6/7m + 0 = 5/7m - 2
(then with 5/7m)
6/7m - 5/7m = 5/7m - 5/7m - 2
1/7m = 0m - 2
(good job ! now, since we only want "m", we take away 1/7 by dividing both sides by 1/7 since it's a multiplication)
1/7m /1/7 = - 2 / 1/7
m = - 2 * 7 (because dividing a fraction is multiplying it by the inverse )
m = - 14
verify if it's correct , and you can finish :)
I DON'T UNDERSTAND THE ANSWER HELP!!!
The average high temperatures in degrees for a city are listed.
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57
If a value of 120.5° is added to the data, how does the mean change, and by how much?
The mean stays at 83.5°.
The mean increases by 3.1°.
The mean increases by 3.5°.
The means stays at 80.4°.
Answers
The mean stays at 83.5°,Every value increases in mean and median by the same constant if a constant is added to each value.
What changes in the median and mean?Every value increases in mean and median by the same constant if a constant is added to each value. Take a set of scores, for instance, that have a mean of 5 and a median of 6. Every score will increase by 10, changing the median and mean to 15 and 16, respectively.
When we multiply the values in the data set by 2, we get the mean, median, mode, range, and IQR multiplied by 2.
Explanation:
58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57 the set multiplide by 2
The mean stays at 83.5°.
The mean stays at 83.5°,Every value increases in mean and median by the same constant if a constant is added to each value.
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Answer: stays at 83.5
Step-by-step explanation:
The two shapes on the grid below are similar.
What is the ratio of side x to side y?
Should this not have measurements?
Answers
1
10
Minos mas simples).
B. How long is 10% of 60 minutes?
minutes
Answers
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{10\% of 60}}{\left( \cfrac{10}{100} \right)60}\implies 6~minutes[/tex]
