The locus of the midpoints of all chords that can be drawn from a given fixed point [tex](a,b)[/tex] on a circle with a radius of 6 units, is a circle of radius 3 units with center at a point whose x & y coordinates are shifted from the center of the given circle by [tex]\frac{a}{2}[/tex] and [tex]\frac{b}{2}[/tex] respectively.
Given: A circle of radius 6 units
To find: The locus of the midpoint of all chords that can be drawn from a given point on the circle.
To find the required locus, we need to know the following:
Locus of a moving point is the trajectory of that point. It is the geometrical figure represented by the equation which is satisfied by the coordinates of the moving point.A chord of a circle is a line segment joining any points of a circle.Equation of a circle with center at origin and radius of [tex]r[/tex] units is [tex]x^{2} +y^{2} =r^{2}[/tex] According to the midpoint formula, the coordinates of the midpoint of the line segment joining the points [tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] is [tex](\frac{x_{1}+x_{2} }{2} ,\frac{y_{1}+y_{2} }{2} )[/tex]Let, without loss of generality, the given circle be centered at the origin. Even if it is not, we can shift the origin to the center of the given circle with coordinate transformation.
Then, the equation of the given circle is [tex]x^{2}+y^{2} =6^{2}[/tex], that is, [tex]x^{2}+y^{2} = 36[/tex]
Let the coordinates of the given fixed point be [tex](a,b)[/tex]
Let the coordinates of any point on the circle be [tex](p,q)[/tex] and let the coordinates of the midpoint of the chord joining the points [tex](a,b)[/tex] and [tex](p,q)[/tex] be [tex](h,k)[/tex]
We have to find the locus of [tex](h,k)[/tex]
Then, using the midpoint formula,
[tex](h,k)=(\frac{a+p}{2} ,\frac{b+q}{2})[/tex]
On solving, we get,
[tex]p=2h-a,q=2k-b[/tex]
Since [tex](a,b)[/tex] and [tex](p,q)[/tex] are both points on the given circle, they satisfy the equation of the circle, [tex]x^{2}+y^{2} = 36[/tex]
Then,
[tex]a^{2} +b^{2} =36[/tex]
[tex]p^{2} +q^{2} =36[/tex]
Substituting [tex]p=2h-a,q=2k-b[/tex] in [tex]p^{2} +q^{2} =36[/tex], we have,
[tex](2h-a)^{2} +(2k-b)^{2} =36[/tex]
[tex](2(h-\frac{a}{2}) )^{2} +(2(k-\frac{b}{2}))^{2} =36[/tex]
[tex]4(h-\frac{a}{2})^{2} +4(k-\frac{b}{2})^{2} =36[/tex]
[tex](h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =\frac{36}{4}[/tex]
[tex](h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =9[/tex]
[tex](h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =3^{2}[/tex]
This is the locus of the point [tex](h,k)[/tex]
Replace [tex](h,k)=(x,y)[/tex] to get,
[tex](x-\frac{a}{2})^{2} +(y-\frac{b}{2})^{2} =3^{2}[/tex]
This is the equation of a circle with center at [tex](\frac{a}{2} ,\frac{b}{2} )[/tex] and radius 3 units.
Thus, we can conclude that the locus of the midpoints of all chords that can be drawn from a given fixed point [tex](a,b)[/tex] on a circle with a radius of 6 units, is a circle of radius 3 units with center at a point whose x & y coordinates are shifted from the center of the given circle by [tex]\frac{a}{2}[/tex] and [tex]\frac{b}{2}[/tex] respectively.
Learn more about locus here:
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Can you please solve these equations by elimination? -10x -10y=20 , -7x -7y=14.
Answer:
-10x -10y=20
-10x - 20=10y
-10x - 20/10=y
-10x - 2=y
y=-10x-2
-7x -7y=14
-7x - 14 = 7y
-7x - 14/7=y
-7x - 2=y
y=-7x-2
Find the missing side. Round your answer to the nearest tenth
Answer:
x = 24.8
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
Sin theta = opp / hypotenuse
sin 75 = 24 /x
x sin 75 = 24
x = 24/ sin 75
x=24.84662
Rounding to the nearest tenth
x = 24.8
Find the area of a rectangle whose length is 14cm and breadth is 6cm
Answer:
Ellos dan las pistas de algunos problemas se pueden resolver de forma automática, los valores numéricos tienen ninguna importancia en los distintos ejemplos.
Traza 1
Uno de los lados de un rectángulo es 20 cm de largo; un segundo lado del rectángulo es de 0,85 m de largo. Calcular el perímetro y el área del rectángulo.
Traza 2
Calcular el área de un rectángulo cuyas dimensiones son 85 cm de largo y 20 cm respectivamente.
Traza 3
La base de un rectángulo es 20 cm de largo; la área es de 300 cm². Calcular la altura del rectángulo.
Traza 4
La altura de un rectángulo es 15 cm de largo; la área es de 300 cm². Calcula la base del rectángulo.
Traza 5
Un rectángulo tiene la altura que es de 3/8 de la base; la suma de las longitudes de los dos segmentos es 44 cm. Determinar el área del rectángulo y el perímetro.
Traza 6
La base de un rectángulo es de 0,40 m de largo; La altura del rectángulo es 30 cm. Calcular la diagonal.
Traza 7
Un tamaño de un rectángulo es un medio del lado de un cuadrado que tiene el perímetro de 20 cm. Sabiendo que los dos polígonos tienen el mismo perímetro, calcula la medida del tamaño del rectángulo.
Traza 8
La diagonal de un rectángulo es de 50 cm; la base es de 3/4 de la altura. Calcular el perímetro y el área del rectángulo.
Traza 9
La diagonal de un rectángulo mide 50 cm; ella es 5/3 de altura. Calcular el perímetro y el área del rectángulo.
Traza 10
Una mesa rectangular tiene lados de 180 cm y 90 cm respectivamente. Cuál es el perímetro y el área de un mantel que cuelga de 20 cm alrededor de la mesa?
Traza 11
Calcular el área de un rectángulo que tiene la altura 10 cm de largo, sabiendo que la medida de la base es el doble de la altura.
Traza 12
La diferencia entre el tamaño de un rectángulo es 12 cm y una es el triple de la otra. Calcular el área del rectángulo
Traza 13
La suma entre el tamaño de un rectángulo es 12 cm y una es el triple de la otra. Calcular el área del rectángulo
Traza 14
La suma de la base y la altura de un rectángulo es 50 cm; la base es superior a la altura de 4 cm. Calcular el área del rectángulo.
Traza 15
El semi-perímetro de un rectángulo es 32 cm y una dimensión es de 3/5 de la otra. Calcular el área del rectángulo.
Traza 16
El semi-perímetro de un rectángulo es 30 cm y una dimensión es igual a los sus 2/5. Calcular el área del rectángulo.
Traza 17
Un rectángulo tiene una base de 20 cm y una altura igual a 2/5 de la base. Calcular el perímetro y el área del rectángulo.
Traza 18
Un rectángulo tiene el área de 600 cm² y la base es 20 cm de largo. Cuál es su perímetro ?
Traza 19
Un rectángulo tiene un perímetro de 100 cm y la base es 30 cm de largo. Calcula su área.
Traza 20
Un rectángulo tiene un perímetro de 120 cm. Sabiendo que un tamaño es tres veces la otra, calcula el área del rectángulo.
Traza 21
La diferencia entre el tamaño de un rectángulo es 10 dm. Sabiendo que el perímetro es 100 dm, calcula el área del rectángulo.
Traza 22
Un rectángulo tiene un perímetro de 100 cm. Calcula su área sabiendo que la medida de la base es superior a la de la altura de 10 cm.
Traza 23
En el perímetro de un rectángulo es de 100 cm y la altura es de 20 cm de largo. Calcular el perímetro de un rectángulo equivalente a el mismo y que tiene su base de 40 cm de largo.
Traza 24
Un rectángulo es formado por dos cuadrados congruentes que tienen cada uno el perímetro de 24 cm. Calcular el perímetro y el área del rectángulo.
Traza 25
Un rectángulo es formado por tres cuadrados congruentes con cada lado 20 cm de largo. Calcular el perímetro y el área del rectángulo.
Traza 26
Un rectángulo es formado por dos cuadrados congruentes. Sabiendo que el perímetro del rectángulo es de 180 cm, calcular su área.
Traza 27
Un rectángulo y un cuadrado tienen el mismo perímetro. El lado de un cuadrado de 45 cm y las dimensiones del rectángulo son una 1/2 de la otra. Calcular el área del rectángulo.
Traza 28
Dos rectángulos son equivalentes. Sabiendo que las dimensiones de el primero miden respectivamente 30 cm y 20 cm, y que la base del segundo rectángulo es 40 cm de largo, calcula la diferencia entre los dos perímetros.
Traza 29
Calcular el perímetro de la figura y el área de la parte interior con la obtención de las medidas a partir del dibujo:
Traza 30
Calcular el perímetro de la figura y el área de la parte interior con la obtención de las medidas a partir del dibujo:
Traza 31
Un constructor ha comprado un terreno que tiene la planta mostrada en el dibujo y las dimensiones en metros se indican en la figura. Calcula el área y el perímetro de la tierra.
Traza 32
Una parcela de tierra tiene una forma rectangular con unas dimensiones de 50 m y de 30 m de largo. En el interior se ha construido una casa que ocupa una superficie rectangular de longitud 20 m y de 8 m de ancho. Calcular el área de la tierra permanecida libre.
Traza 33
Step-by-step explanation:
Answer:
A= 84cm
Step-by-step explanation:
length x width= area
plug in the given information.
14cm x 6cm = A
A=84
with a length of 14cm and a width of 6cm multiply them for an area of 84cm.
Does this graph represent a function?
Answer:
I think it's a function
Step-by-step explanation:
as you can see in the picture curve drawn in a graph represents a function, if every vertical line intersects the curve in at most one point. So I think its a function.
Answer:
yes
Step-by-step explanation:
it's a cubic function having maximum and minimum turning points
it has a point of inflation, y - intercept and x-intercept
Two buses leave towns 576 kilometers apart at the same time and travel toward each other. One bus travels 12
h
slower than the other. If they meet in 3 hours, what is the rate of each bus?
km
Rate of the slower bus:
Rate of the faster bus:
Answer:
102 km/hr
90 km/hr
Step-by-step explanation:
Given that :
Total distance = 576
Time taken = 3 hours
Bus A :
Speed = x
Bus B :
Speed = x - 12
Recall :
Distance = speed * time
Hence,
Distance covered by A + Distance covered by B = total distance covered
(x * 3) + ((x - 12) * 3) = 576
3x + 3x - 36 = 576
6x = 576 + 36
6x = 612
x = 612 / 6
x = 102
Speed of faster bus, x = 102 km/h
Speed of slower bus, x - 12 = (102 - 12) = 90 km/hr
I really need help please
Answer:
Step-by-step explanation:
We have two sides; the Adjacent and the Hypotnuse
Meaning we will use Cos
Cos = A/H
Cos X = 16/19
Use the inverse of Cos to find the angle
X = cos-1 (16/19)
X = 0.45499141546
X = 0.45
Rewrite the equation below so that it does not have fractions.
2/3x -2=3/4
4
Do not use decimals in your answer.
9514 1404 393
Answer:
8x -24 = 9y
Step-by-step explanation:
We assume your equation is ...
2/3x -2 = 3/4y
__
Fractions can be eliminated by multiplying the equation by the least common denominator of the fractions.
LCM(3, 4) = 12
Multiplying the equation by 12 gives you ...
12(2/3x -2) = 12(3/4y)
8x -24 = 9y
Identify if the line has a positive or negative slope. Calculate the slope of the line 2y = x - 8
Answer: m = 1/2
Step-by-step explanation: To find the slope of the line, we would first put the equation in y = mx + b form to get y by itself on the left side.
So divide both sides by 2 to get y = 1/2x - 4.
Now the slope is the coefficient of the x term which is 1/2.
This is a positive slope.
A metal rod will be cut into pieces that are each 1/20 meters long. The rod is 4/5 meters long. How many pieces will be made from the rod?
Write your answer in simplest form.
Solve for X Solve for X Solve for X Solve for X Solve for X
Answer:
13.5
Step-by-step explanation:
We can use a ratio to solve
2 3
---- = ------
11 3+x
Using cross products
2(3+x) = 3*11
Distribute
6+2x = 33
Subtract 6
6+2x-6 = 33-6
2x = 27
Divide by 2
2x/2 = 27/2
x = 13.5
Now we have to,
→ solve for x
Then use a ratio to solve,
→ 2/11 = 3/3+x
Now see the further steps,
→ 2(3+x) = 3 × 11
→ 6+2x = 33
→ 2x = 33-6
→ 2x = 27
→ x = 27/2
→ x = 13.5
Hence, 13.5 is value of x.
The circumference of a
square orchard is 1600
meters. How many square
meters does the orchard
cover? How many hectares?
Answer:
A square is 4 even sides.
the circumference around the square area is 1600 meters. This means that each side is 400 meters.
Square meters is the area of the square.
400 x 400 = 160000 m^2
To get to Hectares, you divide the squared measurement by 10,000.
Answer:
160000m^2 = 16ha
Step-by-step explanation:
Bit of a nit pick first the word is perimeter not circumference circumference only applies to circles. 1600/4=400 (divide by 4 because a square has 4 sides) 400^2=160000 (A=L*H the length and height are the same so you square it) 160000/10000=16 (1 hectare = 10000m^2), Hope this helps. :)
what is the greatest common factor for 9x^2 6x and show work
Answer:
3x
Step-by-step explanation:
9x^2 ?6x
9x? 6x
÷3x. ÷3x
=3x and 2x
highest number is 3x
100 mice eat 100 cakes. If each big mouse eats 3 cakes, and 3 baby mice eat 1 cake, how many big mice and baby mice are there?
Edgar accumulated $5,000 in credit card debt. If the interest rate is 10% per year and he does not make any payments for 3 years, how much will he owe on this debt in 3 years by compounding continuously?
the discrete compounding formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is then number of time periods
in your problem, you are given:
f = what you want to find
p = 5000
r = 30% per year / 100 = .3 per year (percent / 100 = rate).
n = 3 years
if you compound annually, the formula becomes:
f = 5000 * (1 + .3) ^ 3 = 10985
if you compound quarterly, the formula becomes:
f = 5000 * (1 + .3 / 4) ^ (3 * 4) = 11908.898
if you compound monthly, the formula becomes:
f = 5000 * (1 + .3 / 12) ^ (3 * 12) = 12162.67658
if you compound continuously, a different formula is used.
that formula is f = p * e ^ (r * n)
f is the future value
p is the present value
e is the scientific constant of 2.718281828.......
r is the interest rate per time period
n is the number of time periods.
with this formula, you leave the time periods in terms of years.
it will make no difference what time periods and compounding periods you use, the answer will be the same.
most of the time you will just give it the interest rate per year and the number of years.
the reason is as follows:
r * n = .3 * 3 = .9 when giving it rate and time in terms of years.
r * n = .3 / 4 * 3 * 4 = .9 when giving it rate and time in terms of quarters.
r * n = .3 / 12 * 3 * 12 = .9 when giving it rate and time in terms of months.
in your problem, the formula becomes f = 5000 * e ^ (.3 * 3) = 12298.01556.
the more compounding periods per year, the higher the future value.
the highest is when you compound continuously.
this is apparent from the data.
A , B, C , D probability
Answer:
your laptop is nice really
I need to know what goes in those two boxes and why I would really appreciate it if someone help me with this question
Answer:
below
Step-by-step explanation:
the is the procedure above
4x - 2 - 1 = 2 hep plz
what is the image of ( 4, -8 ) after a dilation by a scale factor of 1/4 centered at the origin ?
what we know?:
* scale factor of 1/4
* the point (4, -8)
all we have to do is put 4/4 (because we are dilating by 1/4)
4/4= 1
same for the other one: -8/4= -2
FINAL ANSWER: (1, -2)
Tolong bantuin pakai cara
Answer:
1364
Step-by-step explanation:
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364
a1+a2 = a3, a2+a3=a4 etcetera..
1+3 =4
3+4 =7
4+7=11
.
.
a13+a14 = a15
521+843 = 1364
so, 1364 is the answer
Explain why a + b = d.
B
lbº
aº
dº
A
С C
Assume that blood pressure readings are normally distributed with a mean of 115 and a standard deviation of 8. If 100 people are randomly selected, find the probability that their mean blood pressure will be less than 117.
A. 0.0062.
B. 0.8615.
C. 0.8819.
D. 0.9938.
Answer:
D. 0.9938.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 115 and a standard deviation of 8.
This means that [tex]\mu = 115, \sigma = 8[/tex]
100 people are randomly selected
This means that [tex]n = 100, s = \frac{8}{\sqrt{100}} = 0.8[/tex]
Find the probability that their mean blood pressure will be less than 117.
This is the p-value of Z when X = 117, so:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{117 - 115}{0.8}[/tex]
[tex]Z = 2.5[/tex]
[tex]Z = 2.5[/tex] has a p-value of 0.9938, and thus, the correct answer is given by option D.
A freshly inoculated bacterial culture of Streptococcus contains 100 cells. When the culture is checked 60 minutes later, it is determined that there are 450 cells present. Assuming exponential growth, determine the number of cells present at any time t (measured in minutes) and the find the doubling time.
Answer:
[tex]P(t) = 100e^{0.0251t}[/tex]
The doubling time is of 27.65 minutes.
Step-by-step explanation:
Exponential equation of growth:
The exponential equation for population growth is given by:
[tex]P(t) = P(0)e^{kt}[/tex]
In which P(0) is the initial value and k is the growth rate.
A freshly inoculated bacterial culture of Streptococcus contains 100 cells.
This means that [tex]P(0) = 100[/tex]. So
[tex]P(t) = 100e^{kt}[/tex]
When the culture is checked 60 minutes later, it is determined that there are 450 cells present.
This means that [tex]P(60) = 450[/tex], and we use this to find k. So
[tex]450 = 100e^{60k}[/tex]
[tex]e^{60k} = 4.5[/tex]
[tex]\ln{e^{60k}} = \ln{4.5}[/tex]
[tex]60k = \ln{4.5}[/tex]
[tex]k = \frac{\ln{4.5}}{60}[/tex]
[tex]k = 0.0251[/tex]
So
[tex]P(t) = 100e^{0.0251t}[/tex]
Doubling time:
This is t for which P(t) = 2P(0) = 200. So
[tex]200 = 100e^{0.0251t}[/tex]
[tex]e^{0.0251t} = 2[/tex]
[tex]\ln{e^{0.0251t}} = \ln{2}[/tex]
[tex]0.0251t = \ln{2}[/tex]
[tex]t = \frac{\ln{2}}{0.0251}[/tex]
[tex]t = 27.65[/tex]
The doubling time is of 27.65 minutes.
• The difference between a polynomial or rational equation and polynomial or rational inequality
Answer:
An equation has an equal sign between two expressions, while an inequality has a ≤ or ≥ sign.
Given limit f(x) = 4 as x approaches 0. What is limit 1/4[f(x)]^4 as x approaches 0?
Answer:
[tex]\displaystyle 64[/tex]
General Formulas and Concepts:
Calculus
Limits
Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Limit Rule [Variable Direct Substitution Exponential]: [tex]\displaystyle \lim_{x \to c} x^n = c^n[/tex]
Limit Property [Multiplied Constant]: [tex]\displaystyle \lim_{x \to c} bf(x) = b \lim_{x \to c} f(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \lim_{x \to 0} f(x) = 4[/tex]
Step 2: Solve
Rewrite [Limit Property - Multiplied Constant]: [tex]\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = \frac{1}{4} \lim_{x \to 0} [f(x)]^4[/tex]Evaluate limit [Limit Rule - Variable Direct Substitution Exponential]: [tex]\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = \frac{1}{4}(4^4)[/tex]Simplify: [tex]\displaystyle \lim_{x \to 0} \frac{1}{4}[f(x)]^4 = 64[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
Book: College Calculus 10e
Answer: C. 64
Step-by-step explanation:
Edge 100%
Please helpppppp
ASAPpppppp
Does anyone know how to take the fuzzy stuff off
Answer:
???
Step-by-step explanation:
I need the answers to this
In the power function f(x) = -2x3, what is the end
behavior of f(x) as x goes to infinity ?
O f(x) → negative infinity
O f(x) → -2
O f(x) → + 0
O f(x) → 2
O f(x) → infinity
Answer:
first option f(x) -> negative infinity
Step-by-step explanation:
the other answer is sadly wrong.
there is a "-" in front of the expression.
-2x³ had two parts
-2, which is multiplied to x³
now x³ goes to (positive) infinity for x going to (positive) infinity.
but because this is multiplied by a negative value (-2), all the functional values for positive x are negative.
therefore, the function goes to negative infinity for x going to (positive) infinity.
The end behavior of f(x) as x goes to infinity is (a) f(x) → negative infinity
How to determine the end behavior of the function?The function is given as:
f(x) = -2x^3
Next, we plot the graph of the function (see attachment)
From the graph, we can see that as the x values increase to infinity, the function values approaches negative infinity
Hence, the end behavior of f(x) as x goes to infinity is (a) f(x) → negative infinity
Read more about end behavior at:
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Test for exactness. If exact, solve it directly. Otherwise, use integrating factors to solve it. Solve the IVP (if given). 2xy + (x^2) y' = 0
sin(x) cos(y) + cos(x) sin(y) y' = 0
(x^2) + (y^2) - 2xyy' = 0
e^(2x).(2 cos(y) - sin(y) y') = 0, where y(0) = 0
• 2xy + x ² y' = 0
This DE is exact, since
∂(2xy)/∂y = 2x
∂(x ²)/∂x = 2x
are the same. Then there is a solution of the form f(x, y) = C such that
∂f/∂x = 2xy ==> f(x, y) = x ² y + g(y)
∂f/∂y = x ² = x ² + dg/dy ==> dg/dy = 0 ==> g(y) = C
==> f(x, y) = x ² y = C
• sin(x) cos(y) + cos(x) sin(y) y' = 0
is also exact because
∂(sin(x) cos(y))/∂y = -sin(x) sin(y)
∂(cos(x) sin(y))/∂x = -sin(x) sin(y)
Then
∂f/∂x = sin(x) cos(y) ==> f(x, y) = -cos(x) cos(y) + g(y)
∂f/∂y = cos(x) sin(y) = cos(x) sin(y) + dg/dy ==> dg/dy = 0 ==> g(y) = C
==> f(x, y) = -cos(x) cos(y) = C
• x ² + y ² - 2xyy' = 0
is not exact:
∂(x ² + y ²)/∂x = 2x
∂(-2xy)/∂y = -2x
So we look for an integrating factor µ(x, y) such that
µ (x ² + y ²) - 2µxyy' = 0
becomes exact, which would require that these be equal:
∂(µ (x ² + y ²))/∂y = (x ² + y ²) ∂µ/∂y + 2µy
∂(-2µxy)/∂x = -2xy ∂µ/∂x - 2µy
Observe that if µ(x, y) = µ(x), then ∂µ/∂y = 0 and ∂µ/∂x = dµ/dx, so we would have
2µy = -2xy dµ/dx - 2µy
==> -2xy dµ/dx = 4µy
==> dµ/µ = -2/x dx
Integrating both sides gives
∫ dµ/µ = ∫ -2/x dx ==> ln|µ| = -2 ln|x| ==> µ = 1/x ²
So in the modified DE, we have
(1 + y ²/x ²) - 2y/x y' = 0
which is now exact and ready to solve, since
∂(1 + y ²/x ²)/∂y = 2y/x ²
∂(-2y/x)/∂x = 2y/x ²
We get
∂f/∂x = 1 + y ²/x ² ==> f(x, y) = x - y ²/x + g(y)
∂f/∂y = -2y/x = -2y/x + dg/dy ==> dg/dy = 0 ==> g(y) = C
==> f(x, y) = x - y ²/x = C
• exp(2x) (2 cos(y) - sin(y) y' ) = 0
is exact, since
∂(2 exp(2x) cos(y))/∂y = -2 exp(2x) sin(y)
∂(-exp(2x) sin(y))/∂x = -2 exp(2x) sin(y)
Then
∂f/∂x = 2 exp(2x) cos(y) ==> f(x, y) = exp(2x) cos(y) + g(y)
∂f/∂y = -exp(2x) sin(y) = -exp(2x) sin(y) + dg/dy ==> dg/dy = 0 ==> g(y) = C
==> f(x, y) = exp(2x) cos(y) = C
Given that y = 0 when x = 0, we find that
C = exp(0) cos(0) = 1
so that the particular solution is
exp(2x) cos(y) = 1
state the hundred thousands place for 7,832,906,215
Answer:
Step-by-step explanation:
6 is the thousands place
0 (right next to it) is the 10 thousands place
9 is the hundred thousands place. There is only 1 nine present so the answer is unique.
