Answers
[tex] \frac{dy}{dx} = \frac{x}{y} [/tex]
Let y² = x - v
Then
[tex]2y \frac{dy}{dx} =1 - \frac{dv}{dx} \\\\ \frac{dy}{dx} = \frac{1}{2y} (1- \frac{dv}{dx} )[/tex]
Substitute in the original ODE.
[tex] \frac{1}{2y} (1- \frac{dv}{dx}) = \frac{x}{y} \\\\ 1- \frac{dv}{dx} =2x \\\\ \frac{dv}{dx} =1-2x [/tex]
Integrate with respect to x to obtain
v = x - x² + b
Hence obtain
y² = x² + b
When y(x) passes through (0,0), obtain
b = 0
y² = x²
When y(x) passes through (1,0), obtain
0 = 1 + b => b = -1
y² = x² - 1
When y(x) passes through (0,1), obtain
1 = 0 + b => b = 1
y² = x² + 1
A graph of the three solutions is shown below.
Related Questions
The current (I) in an electrical conductor varies inversely as the resistance (r) of the conductor. The current is 2 amperes when the resistance is 960 ohms. What is the current when the resistance is 480 ohms?
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[tex]\frac{2}{1} = \frac{k}{960} \\k=2\cdot960=1920[/tex].
Since we know the data to find the current when the resistance is 480 Ohms [tex]\Rightarrow[/tex] [tex]I=\frac{k}{r}=\frac{1920}{480}=4 \ amperes[/tex].
Which equation represents a direct variation?
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Hope this helps!
Answer:
a y=.5x
Step-by-step explanation:
please dont look at my name
convert y=2x-7 to slope intercept form
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So y = 2x - 7
A northbound train and a southbound train meet each other on parallel tracks heading in opposite directions. The northbound train travels 4 miles per hour faster than the southbound train. After 2.5 hours, they are 330 miles apart. At what speeds are the two trains traveling?
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if, the southbound train is going at a speed of say " r " mph, since we know that the northbound one is going faster by 4 miles, than is really going at " r + 4 " mph then.
now, we know in 2.5 hours, they're 330 miles from each other, thus the southbound train has been rolling for 2.5 hours, and the northbound train has also been rolling for 2.5 hours.
if the northbound train has covered on those 2.5 hours say " d " miles, then the southbound train has covered the slack, or " 330 - d ".
[tex]\bf \begin{array}{lccclll} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{t}\\ &------&------&------\\ Northbound&d&r+4&2.5\\ Southbound&330-d&r&2.5 \end{array} \\\\\\ \begin{cases} \boxed{d}=2.5(r+4)\\ 330-d=2.5r\\ ----------\\ 330-\boxed{2.5(r+4)}=2.5r \end{cases} \\\\\\ 330-2.5r-10=2.5r\implies 320=5r\implies \cfrac{320}{5}=r\implies 64=r[/tex]
how fast is the northbound going? well, r + 4.
Which triangle has 0 reflections of symmetry
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Answer:
The answer would be option A the scalene triangle
Answer:
The first triangle, or, A !!
Step-by-step explanation:
a diamond's density is 3.2 g/cm3
determine the volume of a diamond that has a mass of 1 g
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________________________________________________________
1 g * (1 cm³ / 3.2g) = __?___ cm³ ;
The units of "g" cancel ;
and we have: (1/3.2) cm³ ;
= (1 ÷ 3.2) cm³ ;
= 0.3125 cm³ ; or; write as: 13/40 cm³ .
_____________________________________
Suppose you are taking the bus home, and can take either the 51 or the 82. you know that the 82 will arrive in exactly 10 minutes, but because the 51 is unreliable, you suspect that the amount of time until it arrives is uniformly distributed between 0 and 30 minutes. you take the first bus that arrives. let t be the number of minutes you wait until you board a bus. find e[t].
Answers
Final answer:
The expected waiting time for a bus, E[t], is 8.33 minutes, which is calculated considering the possibility of either bus 51, which has a uniform arrival time between 0 to 30 minutes, or bus 82, which arrives in exactly 10 minutes.
Explanation:
The problem you are dealing with involves finding the expected waiting time, E[t], for either bus 51 or bus 82 to arrive. The expected value of a uniformly distributed random variable on an interval [a, b] is given by the formula (a + b)/2. However, we need to account for the fact that if bus 82 arrives in exactly 10 minutes, then if the 51 bus hasn't arrived by then, you will take the 82 bus.
To compute E[t], if the 51 bus arrives within the first 10 minutes, the expected waiting time is the average of 0 and 10 minutes, which is 5 minutes. If the 51 bus does not arrive within 10 minutes, you will take the 82 bus at exactly 10 minutes. Therefore, the expectation must only account for the time if the 51 bus arrives before the 82. This gives us:
The probability that the 51 bus arrives in the first 10 minutes is 10/30 = 1/3.
The expected waiting time if 51 arrives in the first 10 minutes is 5 minutes.
If the 51 bus does not arrive within 10 minutes, you'll wait exactly 10 minutes for the 82 bus.
So, E[t] = (1/3) * 5 minutes + (2/3) * 10 minutes = (5 + 20)/3 = 25/3 minutes.
Therefore, the expected waiting time E[t] is 8.33 minutes (rounded to two decimal places).
which function is an even function?
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All the functions in the problem are constant functions, because no x is involved.
For example, consider the first function: [tex]f(x)=sin(-3 \pi)[/tex],
[tex]sin(-3 \pi )[/tex] is just a constant value.( it is actually equal to 0)
Thus for any x, f(x) = 0 and f(-x) =0
This can be said for all constant functions: all of them are even functions.
Answer: all
baby sister took nap for 2 hours and 22 minutes and 1 hour and 35 minutes yesterday. how many more minutes did she sleep yesterday than today
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Fifteen more than twice the hours Carla worked last week is the same as three times the hours she worked this week decreased by 15. She worked the same number of hours each week. How many hours did she work?
Answers
2x-15=3x-45
2x+30=3x
30=x
Let's check this work!
2(30)-15=3(30-15)
60-15=3(15)
45=45
She worked 45 hours each week
*the reason it is 2x-15 and not 15+2x is because "more than" is considered a "turn around word" meaning it is the opposite
To find the number of hours Carla worked, we'll set up and solve equations based on the information given.
Explanation:To solve this problem, we'll set up an equation based on the information given.
Let x be the number of hours Carla worked last week and y be the number of hours she worked this week.
According to the problem, 15 more than twice the hours Carla worked last week is the same as three times the hours she worked this week decreased by 15:
2x + 15 = 3y - 15
Next, we'll set up another equation based on the fact that Carla worked the same number of hours each week:
x = y
We can now solve this system of equations to find the value of y, which represents the number of hours Carla worked this week:
Substituting y for x in the first equation: 2y + 15 = 3y - 15
Combining like terms: 15 = y - 15
Adding 15 to both sides: 30 = y
So, Carla worked 30 hours this week.
How to solve -5 3/14 to the thousandths place?? Please help
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[tex]\bf \stackrel{mixed}{-5\frac{3}{14}}\implies -\cfrac{5\cdot 14+3}{14}\implies \stackrel{improper}{-\cfrac{73}{14}}\implies -73\div 14\approx -5.21428571 \\\\\\ -5.\stackrel{tenth}{2}~\stackrel{hundreth}{1}~\stackrel{thousandth}{4}[/tex]
from january to june a company spent 60.00 per month on office supplies. in july the price increased by 15% and remained the same for the rest of the year. how much did the company spend on supplies for the year?
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There are 45 new house bulky in the neighborhood. Last month 1/3 of the were sold . This month 1/5 of the remaining houses were sold . How many houses are left to be sold
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Anyway, after last month, 2/3 of the houses are left, or 45*2/3=30 houses.
This month, 1/5 of those were sold leaving 4/5*30=24 houses.
24 houses are left to be sold.
Hope this helps!
Your fortune is "Ask your mom."
Consider the exponential function f(x) = 3 and its graph. Which statements are true for this function and graph? Check all that apply. The initial value of the function is . The growth value of the function is . The function shows exponential decay. The function is a stretch of the function f(x) = . The function is a shrink of the function f(x) = 3x. One point on the graph is (3, 0).
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The function shows exponential growth a stretch of the function f(x) = (1/3)^x .
What are exponential functions?When the expression of a function is such that it involves the input to be present as the exponent (power) of some constant, then such a function is called an exponential function. Their usual form is specified below. They are written in several equivalent forms.
Given that the exponential function f(x) = 3
The initial value of the function is 2.
The base of the function is 3.
When the intial value is where x=0
where x=0, the value of f(0)=3
The growth is 1/3 but it is decay, that might be correct, or not because it is decay.
It is a stretch of the function f(x)=(1/3) power of x
when x=3, then f(3)=1/9
The function shows exponential growth a stretch of the function f(x) = [tex](1/3)^x[/tex].
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what is the basic ratio for 30:42
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The lowest form for that is 5/7. Hope that helped. Good luck
The basic ratio of 30:42 is 5:7. This was found by simplifying the ratio to its lowest terms, which is done by dividing both sides of the ratio by the largest number that can evenly divide both numbers, the greatest common factor (GCF).
Explanation:The basic ratio of 30:42 is found by simplifying the ratio to its lowest terms. To do this, find the greatest common factor (GCF) of the two terms, which is the largest number that can evenly divide both numbers. In this case, the GCF of 30 and 42 is 6. Therefore, by dividing both sides of the ratio by 6, we get the simplified ratio: 5:7. So, the basic ratio for 30:42 is 5:7.
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If twice a number decreased by 5 and the result is multiplied by 4, the result is 36. find the number
Answers
(2x-5)4=36
2x-5=9 <----- divide by 4
2x=14 <----- add 5
x=7 <----- divide by 2
Someone please help me!
Answers
check the picture below.
describe how you would Use the distributive property to simplify 39 * 5
Answers
~PutarPotato
Is y=7x+2 x+7y=8 parallel, perpendicular, or neither?
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y = 7x + 2
y = -(1/7) x + 8
product of their slopes is 7 * -1/7 = -1 so they are perpendicular
What is the solution to the system of equations?
Answers
Answer:
C. [tex](0,-5)[/tex]
Step-by-step explanation:
We have been given a graph of system of equation on coordinate plane. We are asked to find the solution for the system of equations.
We know that we can find solution of a system of linear equations by graphing. The solution of the system is the point, where both lines intersect.
Upon looking at our given graph, we can see that both lines intersect at point [tex](0,-5)[/tex] that is at y-axis.
Therefore, the solution to the system of equations is [tex](0,-5)[/tex] and option C is the correct choice.
Holland added three decimals that made a sum of 2. One of the decimals was 0.34. What are two other decimals Holland could have used to make a sum of 2
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To solve, 1st subtract 0.34 on both sides to get 2x = 1.66. Now divide by 2 on both sides to get x = 0.83.
So your decimals are 0.34, 0.83, and 0.83
Answer:
In general the two other decimals are x and y that satisfy x + y = 1.66 where x ≤ .99 and y ≤ .99
One particular pair would be x = 0.99 and y = 0.67
Step-by-step explanation:
If Holland added three decimals that made a sum of 2 and one of them was 0.34, we are going to name x and y to the other 2 decimals.
We are going to write the equation:
[tex]0.34 + x +y= 2\\x+y=2-0.34\\x+y = 1.66[/tex]
So we have that both the other numbers have to sum 1.66, however, we know both numbers are decimals so neither one of them can be greater than 1.
So this problem has many answers and the answers would be all the numbers that satisfy x + y = 1.66 where x ≤ .99 and y ≤ .99
For example, if x = 0.99, y would be:
y = 1.66 -0.99 = 0.67.
And our pair of numbers would be 0.99 and 0.67.
If x = .80, y would bey = 1.66 - 0.80 = 0.86
And our pair of numbers would be 0.80 and 0.86
This way, we can find more than 1 answer for the other two decimals.
Mohammad would like to graph the line that represents the total number of baseball cards he will have collected, T, after m months, given that he buys 5 cards each month and he started with no cards. He would like the graph to show the number of cards after 6 and 12 months. Select from the drop-down menus to correctly complete each statement. To best show this information, the scale for the T-axis of his graph should go from 0 to at least , and the m-axis of his graph should go from 0 to at least .
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Solution:
Number of Baseball cards bought by Mohammad each month = 5
As number of baseball cards is represented by T , which are
T = 0,5,10,15,20,25,30,35,40,45,50,55,60 →→ Y axis
Number of months(m) from which Mohammad started buying cards is given as
m = 0,1,2,3,4,5,6,7,8,9,10,11,12 →→ X axis
The given information represented in equation form
T = 5 m
This is a equation of line in two variable passing through origin.
Slope of line = 5 =Amount of baseball card bought each month
Number of cards bought after 6 months = 5 × 6=30
Number of cards bought after 12 months = 5 ×12 =60
Graph is depicted below.
The scale for the T-axis of his graph should go from 0 to at least 60 , and the m-axis of his graph should go from 0 to at least 12 .
Find the volume of the solid region. the solid between the planes z = 3x + 2y + 1 and z = x + y, and above the triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane.
Answers
The volume of the solid region. the solid between the planes z = 3x + 2y + 1 and z = x + y, and above the triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane is 6 cubic unit.
What is volume?It is defined as a three-dimensional space enclosed by an object or thing.
It is given that
The solid between the planes z = 3x + 2y + 1 and z = x + y, and above the
triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane.
r(u) = (1, 2)
r(v) = (-2, -1)
|r(u)xr(v)| = 3
The volume of the solid:
∬S(3x + 2y + 1 − x − y)ds = ∫₀¹ ∫(u, 0)(2x + y +1)3 dv du
After solving:
= 3(4/3 - 5 /6 + 3/2)
= 6
Thus, the volume of the solid region. the solid between the planes z = 3x + 2y + 1 and z = x + y, and above the triangle with vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0) in the xy-plane is 6 cubic unit.
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Final answer:
The volume of the solid region between the planes z = 3x + 2y + 1 and z = x + y, above the triangular region in the xy-plane, can be found using a double integral over the triangular area defined by vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0).
Explanation:
To find the volume of the solid region between the planes z = 3x + 2y + 1 and z = x + y, and above the triangular region in the xy-plane, we can use a double integral over the triangular region. The volume is given by the difference between the two planes integrated over the area of the triangle formed by the vertices (1, 0, 0), (2, 2, 0), and (0, 1, 0).
First, define the vertices of the triangle in the xy-plane: A(1, 0), B(2, 2), and C(0, 1).
Second, find the equations of the lines forming the sides of the triangle, which helps in defining the limits of the integral.
Lastly, set up the double integral:
∫∫ (3x + 2y + 1) - (x + y) dA,
after solving, we get volume = 6
where dA is the differential area element in the xy-plane, and the limits of integration are determined by the triangular region.
The integration can be simplified by determining a suitable ordering for dx and dy, often choosing to integrate in x first and then y, or vice versa, based on the geometry of the triangle. The exact bounds for x and y need to be worked out from the equations of the lines defining the triangle's edges.
The result of this integral will give the volume of the solid region bounded by the two planes and above the triangle.
Evaluate the series 50 + 10 + 2 + . . .
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[tex]r=\dfrac{a_2}{a_1}=\dfrac{10}{50}=\dfrac{1}{5}[/tex]
so:
[tex]S=50+10+2+\ldots=\dfrac{a_1}{1-r}=\dfrac{50}{1-\frac{1}{5}}=\dfrac{50}{\frac{4}{5}}=\dfrac{50\cdot5}{4}=\dfrac{250}{4}=\boxed{62.5}[/tex]
The table represents a linear function.
Answers
[tex] \frac{ - 6 - - 16}{ - 2 - - 4} [/tex]
y2 - y1 / x2 - x1 = slope
Answer:
The slope of the function is 5.
Step-by-step explanation:
Remember that to calculate the slope of any function you just need the equation of the function on slope form: y=mx+c or any two points in the graph that are result of the function, in this case we have a table, and we will use the first two points:
(-4,-16)
(-2.-6)
So the formula for slope is:
[tex]\frac{y2-y1}{x2-x1}[/tex]
We insert the values of our two points:
[tex]\frac{y2-y1}{x2-x1} \\\frac{-6-(-16)}{-2-(-4)}\\\frac{10}{2}\\ Slope=5[/tex]
So the slope of the function is 5.
A finite population consists of four elements: 6, 1, 5, 3. (a) how many different samples of size n = 2 can be selected from this population if you sample without replacement? (sampling is said to be without replacement if an element cannot be selected twice for the same sample.)
Answers
[tex]nCk= \frac{n!}{k!(n-k)!} [/tex]
So the number of ways to choose 2 elements from a set of 4 elements would be
[tex]4C2= \frac{4!}{2!(4-2)!}=6 [/tex]
Thus the answer is 6.
Consider the linear function that is represented by the equation y=-10x+6 and the linear function that is represented by the equation y-36=8(x-4). Which statement is correct regarding their slopes and y-intercepts?
a. The function that is represented by the equation y=-10x+6 has a steeper slope and a greater y-intercept.
b. The function that is represented by the equation y=-10x+6 has a steeper slope, and the function that is represented by the equation y-36=8(x-4) has a greater y-intercept.
c. The function that is represented by the equation y-36=8(x-4) has a steeper slope, and the function that is represented by the equation y=-10x+6 has a greater y-intercept.
d. The function that is represented by the equation y-36=8(x-4) has a steeper slope and a greater y-intercept.
Answers
This is about understanding slopes.
Option A is correct.
We are given two linear functions which are;y = -10x + 6 - - - (eq 1)
y - 36 = 8(x - 4) - - - (eq 2)
Formula for straight line equation is;y = mx + c
Where m is slope and c is the y intercept.
Let's expand eq 2 to be in the form of y = mx + c
Thus;
>>y - 36 = 8(x - 4)
>> y - 36 = 8x - 32
>> y = 8x - 32 + 36
>> y = 8x + 4 - - - (eq 3)
Thus, slope of eq 1 is -10 and y-intercept is 6.
Similarly, slope of eq 3 is 8 and y-intercept is 4.
This means y = -10x + 6 has a greater y-intercept than y - 36 = 8(x - 4).
Also, the slope in y - 36 = 8(x - 4) which is 8 will slope to the right while the slope of y = -10x + 6 which is -10 will slope to the left. This means that line represented by y = -10x + 6 has a steeper slope.
This is because the higher the slope, the steeper the line.
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After rewriting both equations in slope-intercept form, we find that the function y = -10x + 6 has a steeper slope and the function y - 36 = 8(x - 4) has a greater y-intercept, making option (b) the correct answer.
The question involves comparing the slopes and y-intercepts of two linear functions, represented by the equations y = -10x + 6 and y - 36 = 8(x - 4). To compare them, we need to write both equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
The first equation, y = -10x + 6, is already in slope-intercept form, with a slope of -10 and a y-intercept of 6. For the second equation, y - 36 = 8(x - 4), we first distribute the 8 to get y - 36 = 8x - 32 and then add 36 to both sides to get the slope-intercept form, y = 8x + 4, which has a slope of 8 and a y-intercept of 4.
Comparing the slopes, -10 for the first equation and 8 for the second, we see that the first equation has a steeper slope because -10 has a greater magnitude than 8. Comparing the y-intercepts, 6 for the first equation and 4 for the second, the first equation has a greater y-intercept. Therefore, the correct statement regarding their slopes and y-intercepts is (b) The function that is represented by the equation y = -10x + 6 has a steeper slope and the function that is represented by the equation y - 36 = 8(x - 4) has a greater y-intercept.
On a trip to the beach, you travel 200miles in 300 minutes. How fast did you travel?
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Why does multiply numbers by 10 move the decimal point to the right but multiplying by 0.10 move the decimal point to the left
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This graph shows a portion of an even function. Use the graph to complete the table of values.
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1
3
3
all you really have to look at is how the graph is set up and then you have the answers like up top
1, 1, 3, 3 - I just did it on e2020 and got it correct