The differential equation [tex]\frac{dy}{dx} = xy^{4}[/tex] is considered and the solutions for the following questions are found.
What is a differential equation?
A differential equation is an equation that links the derivatives of one or more unknown functions. Differential equations contain derivatives, which might be either partial derivatives or ordinary derivatives.
The differential equation describes a relationship between the variable that is constantly varying with respect to the change in another quantity, and the derivative represents a rate of change.
Given,
[tex]\frac{dy}{dx} = xy^{4}[/tex]
a) Now finding [tex]\frac{d^{2} y}{dx^{2} }[/tex] = [tex]y^{4} + 4xy^{3} \frac{dy}{dx}[/tex] = [tex]y^{4} + 4xy^3 (xy^4) = y^4 + 4x^2y^7[/tex]
In quadrant II, the values of x < 0 and the values of y >0.
So [tex]\frac{d^{2} y}{dx^{2} }[/tex] = [tex]y^4 + 4x^2y^7[/tex] > 0 for all values of x and y in quadrant II.
Therefore the concavity is up for of all solution curves of the given differential equation.
b)
[tex]\frac{dy}{dx} = xy^{4}\\\\\frac{dy}{y^4} =x dx\\\\\int {\frac{dy}{y^4} } \, = \int {x} \, dx \\\\\frac{y^{-4+1} }{-4+1} = \frac{x^{2} }{2} + c\\\\\frac{y^{-3} }{-3} = \frac{x^{2} }{2} +c[/tex]
Now it is given that initial condition is f(4) = -1
So substituting x =4 and y = -1 is the above equation to find c, we get
[tex]\frac{-1^{-3} }{-3} = \frac{4^2}{2} +c \\\\\frac{1}{3} = 8 + c\\\\c = \frac{1}{3} - 8 = \frac{-23}{8}[/tex]
Therefore the solution y = f(x) is to the given differential equation with initial condition f(4)=−1.
[tex]\frac{y^{-3} }{-3} = \frac{x^{2} }{2} -\frac{23}{8}\\ \\ \frac{1}{3y^3} = \frac{46 - 8x^2 }{16} = \frac{23 - 4x^2 }{8}\\\\y = \sqrt[3]{\frac{8}{3(23-4x^2)} }[/tex]
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Identify or mark the missing side or angle that would make triangle ABC congruent to triangle PDF by AAS
Answer:
F and C
Step-by-step explanation:
CUZ AAS means angle angle side
Answer:
see explanation
Step-by-step explanation:
for the triangles to be congruent by the angle- side- angle (ASA postulate )
if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle ( included side is the side between the vertices of the two angles ) , then the triangles are congruent.
given
∠ B ≡ ∠ D and AB ≡ PD
then the included sides are AB and PD
the other 2 angles for congruency are ∠ A ≡ ∠ P
-----------------------------------------------------
By the AAS postulate
If two angles and a non included side of one triangle are congruent to the corresponding parts of the other triangle, then the triangles are congruent
for AAS then ∠ C ≡ ∠ F
write a paragraph defending, refuting, or qualifying William Wordsworth's claim about the value of nature. Use a quote from one of his poems to support your claim OR present one of the quotes as a counterargument, if you are refuting his claim.
Answer: William Wordsworth, a prominent figure in the Romantic movement, believed that nature had an immense value and could have a profound impact on the human spirit. In his poem "I Wandered Lonely as a Cloud," he wrote, "Nature never did betray the heart that loved her," which is a claim that nature is a constant source of comfort and inspiration. In this poem he praises nature and its ability to be a source of beauty and joy.
One could argue that Wordsworth's claim is true based on the evidence that nature can have a positive impact on mental and physical health. Studies have shown that spending time in nature can reduce stress, improve mood, and boost overall well-being. Additionally, nature can be a source of inspiration for many artists and poets, including Wordsworth himself, who drew inspiration from the natural world for his poetry.
On the other hand, one could argue that Wordsworth's claim is idealistic and may not apply to everyone or every situation. Some people may not find nature as comforting or inspiring, especially those who live in urban areas with limited access to natural spaces. Additionally, there are negative impacts of human activities on nature, such as pollution and deforestation, which can make it harder for some people to find value in nature.
Overall, while Wordsworth's claim that "Nature never did betray the heart that loved her" is a strong statement, and it holds true for many people, but it's a personal perspective that may not apply to everyone. Each individual has their own relationship with nature and it's not always a constant source of comfort, however, it's a treasure that we should take care of, respect and appreciate in order to achieve a better quality of life.
Step-by-step explanation:
TRUE OR FALSE???!!!
The Axis of Symmetry is a VERTICAL line that runs through the VERTEX?
Step-by-step explanation:
The axis of symmetry is the vertical line that goes through the vertex, dividing the parabola into two equal parts. If h is the x-coordinate of the vertex, then the equation for the axis of symmetry is x=h
somebody pls help me
Answer:
y=-52
Step-by-step explanation:
-5*9=-45
-5*9=-45 +7=
y=-52
which of the following phrases translate to the expression y+7 ?
The required phrase to translate the expression x + 7 is, A number increases by 7 or an Addition of number and 7.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
Here,
Given expression x + 7,
The expression can be translated as a number increase by 7 or the Addition of the number and 7.
Thus. the required phrase to translate the expression x + 7 is, A number increases by 7 or an Addition of number and 7.
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Which is the definition of a ray?
a part of a line that has one endpoint and extends indefinitely in one direction
the set of all points in a plane that are a given distance away from a given point
lines that lie in the same plane and do not intersect
a part of a line that has two endpoints
Definition of Ray:
'A part of a line that has one endpoint and extends indefinitely in one direction'.
The correct statement is A.
What is a ray?A ray is a part of the line having one fixed point and the other point does not have an end. It means that a ray has one terminating end and the other end is extending infinitely.
Given definitions of ray:
A. a part of a line that has one endpoint and extends indefinitely in one direction.
B. the set of all points in a plane that are a given distance away from a given point.
C. lines that lie in the same plane and do not intersect.
D. a part of a line that has two endpoints.
Only statement A satisfies the definition of ray.
Hence, the definition of ray is described by statement A.
A part of a line that has one endpoint and extends indefinitely in one direction.
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Find the value of xx in the equation below.
11=
11=
\,\,x -6
x−6
The required value of x is 17 for the given equation 11 = x - 6.
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.
The equation is given in the question, as follows:
11 = x - 6
We can determine the value of x by adding 6 to both sides of the equation.
As per the question, we have
⇒ 11 = x - 6
⇒ 11 + 6 = x - 6 + 6
⇒ x = 11 + 6
Apply the addition operation, and we get
⇒ x = 17
Therefore, the required value of x is 17.
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Is (- 2 4 a solution to the system?
yes, (-2,4) is the solution of the system of equation 4x² - 4y = 0.
To check whether the given points are the solution of the given system of equation 4x² - 4y = 0. We have to put the values of x = -2 and y = 4 in the given equation. If we get the result equals zero then (-2, 4) is the solution of 4x² - 4y = 0 and if the result is a non-zero number then (-2, 4) is not the solution of 4x² - 4y = 0.
⇒4x² - 4y = 0
⇒4(-2)² - 4 × 4 = 0
⇒4 × 4 - 4 × 4 = 0
⇒0 = 0
Here we get the result equal to zero, hence (-2,4) is the solution of the system of equation 4x² - 4y = 0.
--This question is incomplete, the complete question is
''Is (-2,4) a solution of the system of equation 4x² - 4y = 0?''--
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if u answer correct u r a math god its hard trust me
Answer:
27 square units
Step-by-step explanation:
Area of triangle = 1/2(b)(h)
Base = 6
Height = 9
1/2(6)(9)
1/2(54)
Area = 27 units
Help now please help me with this assignment
Answer:
Since 3⁵= 243 you can just multiply by 3 to find 3⁶
Answer:
You can multiply 243 by 3 to get the answer to 3^6
Step-by-step explanation:
Write and solve the inequality that represents negative one third is greater than or equal to the product of negative two fifths and a number.
negative one third is greater than or equal to negative two fifths y where y is greater than or equal to negative five sixths
negative one third is greater than or equal to negative two fifths y where y is greater than or equal to five sixths
negative two fifths is greater than or equal to negative one third y where y is less than or equal to negative two fifteenths
negative two fifths is less than or equal to negative one third y where y is greater than or equal to two fifteenths
The inequality that represents negative one third is greater than or equal to the product of negative two fifths and a number is: B. negative one third is greater than or equal to negative two fifths y where y is greater than or equal to five sixths.
What is an inequality?In Mathematics, an inequality simply refers to a mathematical relation that can be used to compare two (2) or more numerical values, numbers, and variables in an algebraic equation, especially based on any of the following inequality symbols:
Greater than (>).Less than (<).Greater than or equal to (≥).Less than or equal to (≤).Note: Let the variable y represent the unknown number.
Next, we would translate the word problem "negative one third is greater than or equal to the product of negative two fifths and a number" into an algebraic inequality as follows;
-1/3 ≥ -2/5 × y
-1/3 ≥ -2y/5
Multiplying both sides of the algebraic inequality by -5/2, we have:
-1/3 × -5/2 ≥ -2y/5 × -5/2
5/6 ≤ y
y ≥ 5/6.
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what is the approximate volume?
Answer:
A. 524m^3
Step-by-step explanation:
Formula for volume of sphere:
[tex]V = \frac{4}{3} \pi r^3[/tex]
If we use 3.14 for pi, we get:
V = 4/3(3.14)(5)^3
V = 523.33
. An interior designer plans to use triangular tiles that are 8 cm × 9 cm × 13 cm for a backsplash. In order to fit them together properly, she needs to know the size of the angles on the tile. Calculate the three angles on the tile, accurate to one decimal place. Include a diagram.
The measures of three angles are 40.5°, 43.5° and 95.8°.
What is sine rule?Law of Sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles.
The formula for sine rule is sinA/a=sinB/b=sinC/c
Given that, an interior designer plans to use triangular tiles that are 8 cm×9 cm×13 cm for a backsplash.
Using cosine rule, we get
9²= 8²+ 13²- 2(8)(13) cos θ (angle opposite 9 cm side.)
81=64+169-208 cos θ
81-233=-208 cos θ
-208 cos θ=-152
cos θ=-152/(-208)
cos θ=0.7307
θ=43.7°
Using sine rule,
The formula for sine rule is sinA/a=sinB/b=sinC/c
sinA/8 =sin43°/9=sinC/13
sinA/8 =0.7307/9
sinA/8=0.0811
sinA=0.6495
A=40.5°
So, ∠A+∠B+∠C=180°
40.5°+43.7°+∠C=180°
∠C=95.8°
Therefore, the measures of three angles are 40.5°, 43.5° and 95.8°.
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What is a 1 million sided shape called?
A one million sided shape called as Megagon.
We know that a polygon is noting but a closed two-dimensional figure which is enclosed by line segments.
These line segments are known as sides.
A polygon has at least three straight sides and angles.
Polygons are classified mainly into four categories. Regular polygon, Irregular polygon, Convex polygon and Concave polygon
We know that a polygon with n sides is called n-gon.
For n = 1000000, i.e., a polygon with one million sides is called as Megagon.
A number of edges and vertices = 1000000
And the internal angle of megagon is about 179.99 degrees.
Therefore, a shape with 1 million sides is known as a Megagon
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can someone help me do my homework
Answer:
(0, -8)
Step-by-step explanation:
Because the question asks for the y-axis intercept, the x coordinate will automatically be 0. Then we just look at where parabola intersects the y axis (-8), and that's the y coordinate. Finally, we put those two numbers together:
(0, -8)
Answer:
(0, -8)
Step-by-step explanation:
That is where the y-intercept happens.
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If a business discovered that changing the price of its muffins created an absolute value of .35 < 1, what did its price change mean
If a company found that raising the price of its muffins resulted in an absolute value of.35 < 1, then the demand for muffins is inelastic.
What is meant by mean?The mean is the set of two or more numbers' mathematical average. It is possible to calculate two different sorts of means: the arithmetic mean and the geometric mean. Add the numbers in a set together and divide by the total number of numbers in the set to find the arithmetic mean. To determine the mean, only numerical variables whether continuous or discrete can be employed.It is calculated by merely dividing the total number of values in a data collection by the sum of all the values in the data set. The mean, or average of the supplied numbers, is computed by dividing the total number of numbers by the sum of the given numbers.To learn more about mean, refer to:
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Please solve with explanation
Answer:
a. 49/2
b. -2/9
If I make any mistakes, I'm sorry.
Find whether the following expressions are functions in the ranges they are defined.
The options that shows expressions which are functions in the ranges they are defined are;
Option A: f(x) = 2x + 5
Option B: f(x) = x² + 1
How to identify real numbers and Integers?Real numbers are defined as the combination of rational and irrational numbers, in the number system. This means that, all the arithmetic operations can be performed on real numbers and they can be represented in the number line.
Now, integers are defined as whole numbers that can be positive, negative or zero.
a) We are told that the input of real numbers must yield only real numbers. Yes this is true because any real number we put in for x in f(x) = 2x + 5 will yield an output that is also a real number.
b) We are told that the input of real numbers must yield only real numbers. Yes this is true because any real number we put in for x in f(x) = x² + 1 will yield an output that is also a real number.
c) We are told that the input of real numbers must yield only real numbers. No this is not true because if we put -1 which is a real number for x in the denominator, we will get an undefined function.
d) We are told that the input of integers must yield only integers. No this is not true because if we put 1 for x which is an integer, we will get a fraction which is not an integer.
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I need help with area of a rectangle! Plese help me out this is a starr review!
Answer:
B
Step-by-step explanation:
Nora needs at least $115 for a new cell phone. She has already saved $45. She earns $7 an hour at her job. Write and solve an inequality to find how many hours she will need to work to buy the phone.
Answer:
she only needs to work ten hours
Explanation: if you subtract 45 from 115 you well end up with 70 a and if you work for ten hours 7 ×10=70 so you would have enough.
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The circumference of a circle is 21π cm. What is the area, in square centimeters? Express your answer in terms of pi.
Answer:
110.25π sq cm
Step-by-step explanation:
circumference 2πr= 21π cm
so r = 21/2= 10.5 cm
area of circle = πr^2
=π (10.5)^2
=110.25 sq cm
Order the set of numbers from least to greatest: 5, -19, -25, 7, 22, -4, -15
Answer:
-25, -19, -15, -4, 5, 7, 22
Step-by-step explanation:
-25, -19, -15, -4, 5, 7, 22 is the order from least to greatest
The bigger the negative number, the smaller it is; that is why the order is
-25, -19, -15, -4, 5, 7, 22
Determine how many solutions exist for this system of equations:
y = -3x+6
y=-3x – 5
A= No solution
B= One solution
C= infinity many
D= Two solutions
I’ve been having a few mistakes with substitution so pls help
Answer:
No Solution
Step-by-step explanation:
-3x+6=-3x-5
+5 +5
-3x+11=-3x
+3 +3
11=0
No Solution
Simplify (4s^5t^-7/-2s^-2t^4). Write your answer using only positive exponents. Evaluate any numerical powers.
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In positive exponents, the simplified form of the above polynomial is s⁷/t¹¹.
What is polynomial?A polynomial is a mathematical statement made up of indeterminates and coefficients that solely includes the operations of addition, subtraction, multiplication, and positive-integer powers of variables. x2 4x + 7 is an example of a polynomial with a single indeterminate x. A polynomial is an expression in mathematics that consists of variables (also known as indeterminates) and coefficients and includes only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Here,
=4s⁵*t⁻⁷/-2s⁻²*-2t⁴
=s⁷/t¹¹
The simplified form of given polynomial is s⁷/t¹¹ in positive exponents.
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Samuel spends a winter day recording the temperature once every three hours for science class. At 9 am, the temperature was -14.7°F. Between 9am and noon, the temperature increased by 17.8°F. Between noon and 3pm, the temperature increased by 5.1°F. Between 3pm and 6pm, the temperature went down 14.5°F. What was the temperature at 6pm?
Answer:
The temperature at 9 am = -14.7°F
Between 9am and 12 pm = -14.7 + 17.8 = 3.1°F
Between 12pm and 3 pm = 3.1+5.1 = 8.2°
Between 3pm and 6 pm = 8.2-14.5 = -6.3°F
In conditional probability, the notation P(
AB) is read:
"The probability of event A occurring given that event B has
occurred."
For example: In the following two-way table
P(Walk to school Sophomore) = 37 (2 + 25 + 3) = 0.1
Grade Drive to school Take the bus
Walk
Sophomore
2
25
3
Junior
13
20
2
Senior
25
5
5
P(Take the bus Sophomore ) = [?]
Answer:
0.83
Step-by-step explanation:
It’s right use the formula they give you
The probability of a student taking the bus given that the student is sophomore is 0.83
Given that:
P(Walk to school | Sophomore) = 3 /(2 + 25 + 3) = 0.1
To cal culate the probability of a student taking the bus given that the student is sophomore, we make use of:
P(Take the bus | Sophomore ) = 25/(2 + 25 + 3)
Rewrite properly as:
[tex]Pr= \frac{25}{2 + 25 + 3}[/tex]
So, we have:
[tex]Pr= \frac{25}{30}[/tex]
Divide 25 by 30
[tex]Pr= 0.83[/tex]
Hence, the probability of a student taking the bus given that the student is sophomore is 0.83
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This probability distribution shows the
typical grade distribution for a Geometry
course with 35 students.
Grade
A
BCDF
Frequency
5
10
15
3
2
Find the probability that a student earns a
grade of A, B, or C.
p=[?]
Enter a decimal rounded to the nearest hundredth.
Answer:
85.71% probability
Step-by-step explanation:
30 of 35 of the students are in abc range so take 35/100% or 0.35 and go 30/0.35 and you get 85.71%
You deposit $111 and it increases by 2% per year. How much money do you have after 15 years?
Answer:
2/100=0.02
0.02*15=33.3
111+33.3
144.3
Sidney would like to buy a winter coat with an original price of $44.60. Which coupon should she use?