The equation y^(2)-x+10=0 has no symmetry.
Symmetry is when one part of an equation is the mirror image of another part. In this case, there is no symmetry because the equation cannot be mirrored on any axis.
To check for symmetry with respect to the y-axis, we can replace x with -x and see if the equation is still true. In this case, y^(2)-(-x)+10=0 is not the same as the original equation, so there is no symmetry with respect to the y-axis.
Similarly, to check for symmetry with respect to the x-axis, we can replace y with -y and see if the equation is still true. In this case, (-y)^(2)-x+10=0 is not the same as the original equation, so there is no symmetry with respect to the x-axis.
Finally, to check for symmetry with respect to the origin, we can replace x with -x and y with -y and see if the equation is still true. In this case, (-y)^(2)-(-x)+10=0 is not the same as the original equation, so there is no symmetry with respect to the origin.
Therefore, the equation y^(2)-x+10=0 has no symmetry.
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I need to find each angle measures. From 1-21!! Please help!! I will mark you brainiest!
The measure of each angle of the triangle is 30 degrees, 60 degrees, and 90 degrees.
Let's assume that the three angles of the triangle are x, 2x, and 3x, respectively. We are given that these angles are in the ratio of 1:2:3. This means that:
x : 2x : 3x = 1 : 2 : 3
To solve for x, we need to add the three angles together and equate them to 180 degrees (since the sum of the angles in a triangle is 180 degrees). Therefore:
x + 2x + 3x = 180
6x = 180
x = 30
Now that we know the value of x, we can substitute it back into our original equation to find the measure of each angle. Therefore:
First angle: x = 30 degrees
Second angle: 2x = 60 degrees
Third angle: 3x = 90 degrees
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Complete Question:
The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle of the triangle
Let T: VV be a linear operator and let Z: V→V be the zero linear transformation defined by Z(u) = 0 for all u EV. Prove that
Im(T) ker(T) - T.T=Z
The statement Im(T) ker(T) - T.T=Z is not necessarily true for all linear operators T.
To prove that Im(T) ker(T) - T.T=Z, we must first understand the terms "operator," "transformation," and "linear."
An "operator" is a function that maps one vector space to another. A "transformation" is a function that maps one set to another. A "linear" transformation is a transformation that satisfies the properties of linearity, meaning that it preserves addition and scalar multiplication.
Now, let's look at the equation Im(T) ker(T) - T.T=Z. The term "Im(T)" refers to the image of the linear operator T, which is the set of all vectors that can be obtained by applying T to any vector in V. The term "ker(T)" refers to the kernel of the linear operator T, which is the set of all vectors that are mapped to the zero vector by T.
To prove that Im(T) ker(T) - T.T=Z, we must show that the difference between the image of T applied to the kernel of T and the composition of T with itself is equal to the zero linear transformation.
First, let's consider the image of T applied to the kernel of T. Since the kernel of T is the set of all vectors that are mapped to the zero vector by T, applying T to any vector in the kernel of T will result in the zero vector. Therefore, Im(T) ker(T) = 0.
Next, let's consider the composition of T with itself, T.T. Since T is a linear operator, the composition of T with itself will also be a linear operator. However, there is no guarantee that T.T will equal the zero linear transformation.
Therefore, Im(T) ker(T) - T.T=Z can be simplified to 0 - T.T=Z. Since T.T is not necessarily equal to the zero linear transformation, the equation does not hold true for all linear operators T.
In conclusion, the statement Im(T) ker(T) - T.T=Z is not necessarily true for all linear operators T.
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Solve the equation. Check your solution.
11+ 3q = 12 + 2q
q=_
11+ 3q = 12 + 2q
subtract 11 from both sides
3q = 1 + 2q
subtract 2q from both sides
q = 1
To get from Boone to Charlotte, you would have to drive about 156.6 km on the shortest route. To get from Boone to Anchorage, AK, you would have to drive 4,212.7 miles. How many orders of magnitude larger is the distance from Boone to Anchorage than Boone to Charlotte? (1 mi = 1.61 km).
Answer: 1.64. To find the difference in orders of magnitude between the two distances, we first need to convert both distances to the same unit of measurement. We'll convert both distances to kilometers.
The distance from Boone to Charlotte is already in kilometers, so we don't need to do any conversion for that distance.
The distance from Boone to Anchorage is in miles, so we'll need to convert that distance to kilometers. To do this, we'll multiply the number of miles by the conversion factor 1.61 km/mi:
4,212.7 mi × 1.61 km/mi = 6,782.45 km
Now that both distances are in kilometers, we can compare them to find the difference in orders of magnitude. An order of magnitude is a factor of 10, so we'll divide the larger distance by the smaller distance and then take the base 10 logarithm of the result:
log10(6,782.45 km / 156.6 km) = log10(43.3) ≈ 1.64
So the distance from Boone to Anchorage is about 1.64 orders of magnitude larger than the distance from Boone to Charlotte.
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If the tires on a car each have a diameter of 25 inches how far will the car travel in 100 rotations of its tires?
In other words, in 100 tyre rotations, the vehicle will have travelled 7,854 inches, or roughly 196.06 feet or 59.79 metres.
what is order of rotation ?The amount of rotations around a central point or axis that a shape or object undergoes is referred to in mathematics as the order of rotation. For illustration, a shape with a 180-degree revolution about its centre has an order of rotation of 2. Similar to this, a shape rotated by 120 degrees has an order of revolution of 3. The idea of rotational symmetry, which describes a property of some shapes and objects that enables them to appear the same after a certain amount of rotation, and the order of rotation are closely related concepts.
given
The circumference of the tyre, which is determined by the following calculation, equals the distance covered by the vehicle in one rotation of its tyres.
C = πd
where the tire's width is d and its circumference is C. Using the tire's circumference of 25 inches as a plug-in, we obtain:
C is 25 times 78.54 inches.
As a result, one tyre rotation on the vehicle will cover a distance of 78.54 inches.
We can easily multiply the distance covered by one tyre rotation by 100 to determine how far the car will drive in 100 rotations:
78.54 inches per revolution times 100 rotations equals 7,854 inches of distance travelled.
In other words, in 100 tyre rotations, the vehicle will have travelled 7,854 inches, or roughly 196.06 feet or 59.79 metres.
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A water valve controls the amount of water flowing through the dabney dam. The probability for the water flow is uniform between 0 and 4 b/s (barrels per second), uniform between 4 and 9 b/s, and uniform between 9 and 14 b/s. The water flow cannot be 14 b/s or greater. The probability of being in the second range is half of the first range. The probability of being in the third range is a fifth of of being in the first range. In other words, the pdf looks like this:
The value of the constant "k", which make the given pdf valid is 6/17 .
Let x be the amount of water flowing through the Dabney dam;
The Probability Density Function(pdf) of x is given as
⇒ f(x) = { k , 0≤x≤2
(1/3)k , 2<x≤4
(1/6)k , 4<x<5
We know that for a valid p.d.f. [tex]\int\limits^{\infty}_{-\infty} {f(x)} \, dx[/tex] = 1 ;
Substituting the functions for the different intervals ,
We get;
⇒ [tex]\int\limits^{0}_{-\infty} {0} \, dx[/tex] + [tex]\int\limits^{2}_{0} {k} \, dx[/tex] + [tex]\int\limits^{4}_{2} {\frac{1}{3} k} \, dx[/tex] + [tex]\int\limits^{5}_{4} {\frac{1}{6} k} \, dx[/tex] + [tex]\int\limits^{\infty}_{5} {0} \, dx[/tex] = 1 ;
⇒ 0 + [kx]²₀ + [kx/3]⁴₂ + [kx/6]⁵₄ + 0 = 1 ;
⇒ 2k + (k/3)(4-2) + (k/6)(5-4) = 1 ;
⇒ 2k + 2k/3 + k/6 = 1 ;
⇒ (12k + 4k + k)/6 = 1;
⇒ 17k/6 = 1;
⇒ k = 6/17.
Therefore, the value k=6/17 will make the pdf valid.
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The given question is incomplete, the complete question is
A water valve controls the amount of water flowing through the Dabney dam. The probability for the water flow is uniform between 0 and 4 b/s (barrels per second), uniform between 4 and 9 b/s, and uniform between 9 and 14 b/s. The water flow cannot be 14 b/s or greater. The probability of being in the second range is half of the first range. The probability of being in the third range is a fifth of of being in the first range.
In other words, the pdf looks like this:
f(x) = { k , 0≤x≤2
(1/3)k , 2<x≤4
(1/6)k , 4<x<5.
Find the value of k , that will make the pdf valid.
Find the vertices, foci, center, and asymptotes of the given hyperbola y + 1 )2 = (x, y) = ( 21,-1 X ) (smaller x-value) (x, y) = ( -5,-1 X ) (larger x-value) (x, y) = | 8 + V 185 ,-1 ) (smaller x-value) (x, y) = | 8-V 185 ,-1 ) (l ) (x, y) = (3,-1 vertices smaller X-Value foci arger X-value center 45 13 X (negative slope) asymptotes 13 19 13 X (positive slope) 13
The vertices of the given hyperbola are (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)), the foci are (-1, -1 ± sqrt(66)), the center is (-1, -1), and the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
To find the vertices, foci, center, and asymptotes of the given hyperbola, we need to use the standard form of a hyperbola equation:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1
First, we need to find the center (h, k) of the hyperbola. From the given equation, we can see that h = -1 and k = -1, so the center of the hyperbola is (-1, -1).
Next, we need to find the values of a and b. From the given equation, we can see that a^2 = 21 and b^2 = 45, so a = sqrt(21) and b = sqrt(45).
Now, we can find the vertices of the hyperbola. The vertices are located at (h, k ± a), so the vertices are (-1, -1 ± sqrt(21)). This gives us the vertices (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)).
Next, we need to find the foci of the hyperbola. The foci are located at (h, k ± c), where c = sqrt(a^2 + b^2). So, c = sqrt(21 + 45) = sqrt(66), and the foci are (-1, -1 ± sqrt(66)).
Finally, we need to find the asymptotes of the hyperbola. The equations of the asymptotes are y = k ± (a/b)(x - h). So, the equations of the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
So, the vertices of the given hyperbola are (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)), the foci are (-1, -1 ± sqrt(66)), the center is (-1, -1), and the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
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The table of values below represents a linear function and shows Marco’s progress as he is pumping gas into his car. What is the output for the initial value?
Gas in Marco’s Car
Seconds Spent Pumping Gas
0
12
24
36
48
Gallons of Gas in Car
3
5
7
9
11
The initial value obtained using the slope-intercept relation is 3 gallons.
We need to obtain the rate of change which gives the amount of gas entering into his car per second :
Rate of change = Rise / Run
Rate of change = - 11 - 3) / (48 - 0)
Rate of change = 8/48 = 1/6 gallons
Using the slope intercept relation :
y = bx + c
b = slope ; c = initial amount of gas
Choosing any pair of (x, y) point on the table :
(0, 3)
3 = 1/6x + c
3 = 1/6(0) + c
3 = 0 + c
c = 3
Therefore, the initial value is 3 gallons.
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how many integers satisfy
a) -102
b)-102≤x≤105
All the integers between -102 and 105 including the two satisfy the expression.
What are integers?
All whole numbers and negative numbers are considered integers. This indicates that if we combine negative numbers with whole numbers, a collection of integers results.
The meaning of integers: An integer, which can comprise both positive and negative integers, including zero, is a number without a decimal or fractional portion.
The given expression is:
-102 ≤ x ≤ 105
All the numbers between -102 and 105 including the two satisfy the expression.
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how many integers satisfy
a) -102
b)-102≤ x≤ 105
△ABC is similar to both △ACD and △CBD.∙∣∣AC∣∣2+∣∣BC∣∣2=∣∣A B∣∣2
Explain why these claims are correct or incorrect. Provide valid mathematical reasoning to support your responses.
△ABC is cοngruent tο bοth △ACD and △CBD.
What is triangle?Geοmetry depends οn shapes like squares, circles, rectangles, triangles, and οthers. Amοng all the fοrms we have here, triangles seem tο be the mοst intriguing and distinctive. The triangle's shape is created by the intersectiοn οf three lines and three angles.
△ABC is a right triangle, right angled at C.
CD is altitude drawn tο hypοthesis AB.
Tο prοve, △ACD ~ △CBD
In △ACD and △CBD.
∠ACB= ∠ADC=90°
∠CAB=∠DAC (Cοmmοn angle)
By AA similarity we can say that, △ACD and △CBD.
Anοther side,
Need tο prοοf ∣AC∣²+∣BC∣²=∣A B∣²
If is a right triangle at C with a prοjectiοn tο as shοwn, then
BC²=BD*AB
AC²= AD*AB
A further beneficial cοnclusiοn can be demοnstrated by cοmbining the Pythagοrean and Euclidean theοrems. The Pythagοrean Theοrem prοvides us with
CD²= BC²-BD²
By putting the value οf BC²,
CD²= BD*AB-BD²
OR, CD²= BD*(AB-BD)
OR, CD²= BD*AD
AB²= (AD*AB)+(BD*AB)
Or, AB(AD+BD)=AB²
Or, AB²=AB²
Or, ∣AC∣²+∣BC∣²=∣A B∣² (prοved)
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The diagram below shows the graph of h(t), which models the height, in feet, of a rocket t seconds after it was shot into the air 
The domain is the set of all t-value (inputs) used by the function/graph.
The least t-value is 0. The greatest is 4.
The domain is [0,4].
The domain of graph for function h(t) is [0,4].
What is Domain?The range of values that we are permitted to enter into our function is known as the domain of a function. The x values of a function like f(x) are part of this collection. A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
We have,
a diagram below shows the graph of h(t), which models the height in t seconds.
we know that the domain are the input values.
Here the domain is the set of all t-value used in the graph.
So, from the graph we can see that the least t-value is 0 and the greatest is 4.
Thus, the domain of graph is [0,4].
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Given f(x) = 6x and g(x) = 3x² + 1, find the following expressions. {a) (og)(4) (b) (gof)(2) (C) (fof)(1) (d) (gog(0)
The result would be:
a) (og)(4) = 294
b) (gof)(2) = 433
c) (fof)(1) = 36
d) (gog)(0) = 4
Given f(x) = 6x and g(x) = 3x² + 1, we can find the following expressions:
a) (og)(4)
To find (og)(4), we first need to find g(4).
g(4) = 3(4)² + 1 = 3(16) + 1 = 49
Now we can find (og)(4) by plugging in 49 for x in f(x):
(og)(4) = f(49) = 6(49) = 294
b) (gof)(2)
To find (gof)(2), we first need to find f(2).
f(2) = 6(2) = 12
Now we can find (gof)(2) by plugging in 12 for x in g(x):
(gof)(2) = g(12) = 3(12)² + 1 = 3(144) + 1 = 433
c) (fof)(1)
To find (fof)(1), we first need to find f(1).
f(1) = 6(1) = 6
Now we can find (fof)(1) by plugging in 6 for x in f(x):
(fof)(1) = f(6) = 6(6) = 36
d) (gog)(0)
To find (gog)(0), we first need to find g(0).
g(0) = 3(0)² + 1 = 1
Now we can find (gog)(0) by plugging in 1 for x in g(x):
(gog)(0) = g(1) = 3(1)² + 1 = 4
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Work out q when r is 45
Answer: 171
Step-by-step explanation:
If r=20 and q=76 and they're directly proportional, multiply r×3.8
20×3.8=76
The multiplication factor is 3.8, so apply it to when r=45
45×3.8=171
100 points pls help me in my math
Answer:
1. YES
2.NO
3. NO
4. YES
Step-by-step explanation:
Answer:
yes, no,no,yes
Step-by-step explanation:
List all the zeros of the polynomial function F(x) = x^4 - 2x^3 - 6x^2 +22x - 15
The zeros of the function are: x = 1, x = -3, and the other two zeros may be irrational or complex.
What are Functions?A function is a mathematical rule that assigns a unique output value for each input value. It is a set of ordered pairs where the first element is the input and the second element is the output.
Using the Rational Root Theorem, the possible rational roots of the polynomial function F(x) = x⁴ - 2x³ - 6x² + 22x - 15 are:
±1, ±3, ±5, ±15
Using synthetic division, we can find that x = 1 and x = -3 are zeros of the function. Therefore, the zeros of the function are:
x = 1, x = -3, and the other two zeros may be irrational or complex.
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$44 is what percent of $88?
Write your answer using a percent sign (%). For example, 0.5%, 12.7%, or 56%.
Answer:
0.5
Step-by-step explanation:
0.5
One diagonal of a rhombus is 4 in shorter than the other diagonal. The area of the diagonal is 10.5 in². Find the length of the longer diagonal.
So the length of the shorter diagonal is 3 inches. The length of the longer diagonal is 7 inches.
What is the rhombus about?Let d1 and d2 be the lengths of the longer and shorter diagonals, respectively. We know that d1 = d2 + 4, and that the area of the rhombus is given by A = (d1*d2)/2 = 10.5.
Substituting d1 = d2 + 4 into the area equation, we get:
(d2+4)*d2/2 = 10.5
Multiplying both sides by 2 and simplifying, we get:
d2²+ 4d2 - 21 = 0
Using the quadratic formula, we can solve for d2:
d2 = (-4 ± [tex]\sqrt{4^2-4(-21)}[/tex]/2
d2 = (-4 ± [tex]\sqrt{100}[/tex] /2
d2 = (-4 ± 10)/2
d2 = -7 or 3
Since the length of a diagonal cannot be negative, we reject the negative solution and conclude that d2 = 3. Therefore, d1 = d2 + 4 = 7, and the length of the longer diagonal is 7 inches.
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-3 2 Part A Enter a number in the box to create an expression equivalent to 2-5. 2+ 4 Part B Which expressions represent the distance between the two points on the number line? Select all that apply. A. 2-5 B. 2+(-5) C. 1-3-21 D. 12-(-3)| E. 1-3+(-2) F. 1-3-(-2)|
When 1 is put in the box, it generates an equation that is equal to 2 - 5. The expressions |-3 -2|, |2-(-3)|, |-3 + (-2)|, |-3 - (-2)|, etc., represent the distance between the two points integers on number line.
A number line is used to display integers. A number line is a graphic representation of a straight line of numbers. It consists of positive and negative integers such as 1, 2, and so forth, as well as a zero that is neither positive nor negative.
Part A:
2 - 5 = -3
2 + 1 = 3
The absolute value of a number is the distance on the number line between that number and 0.
That distance is always positive.
b. The absolute value symbol is " | | ".
Part B: |-3 -2| = 1
On the number line, this point represents the distance of one unit from -3 to -2.
|2-(-3)| = 5
On the number line, this point represents the distance of 5 units between 2 and -3.
|-3 + (-2)| = 5
The 5 units at this point on the number line reflect the distance from -3 to -2.
|-3 - (-2)| = 1
On the number line, this point represents the distance of one unit between -3 and -2.
It's a good idea to have a backup plan in place in case something goes wrong. The expressions |-3 -2|, |2-(-3)|, |-3 + (-2)|, |-3 - (-2)|, etc., represent the distance between the two points on the number line.
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Mr. Hawkins is covering a wall with wallpaper. The rectangular wall measures 12 feet by 20 feet. Each square foot of wallpaper costs $4.50. Find the cost of covering the wall with wallpaper. $
Answer: $1080
Step-by-step explanation:
First, find the area of the wall.
The area of a rectangle is length x width.
Area = 12 x 20 = 240 square feet
Multiply 4.50 by 240 to find the total cost.
4.50 x 240 = 1080 dollars
It would cost $1080 to cover the wall.
Hope this helps!
Three clocks ring once at the same time. After that, the first clock rings after every 90 minutes,
the second after every 30 minutes, and third after every 60 minutes. After how many minutes will they again ring together?
Answer:
me no habla ingles?
Step-by-step explanation:
Greatest Common Factor and Factor the following polynomial by grouping ab+8a+2b+16
The greatest common factor (GCF) of the polynomial ab+8a+2b+16 is 2.
To factor the given polynomial ab + 8a + 2b + 16 by grouping, follow these steps:
1. Group the terms in pairs: (ab + 8a) + (2b + 16).
2. Factor out the Greatest Common Factor (GCF) from each group. - For the first group (ab + 8a), the GCF is "a". So, factor out "a" from the group: a(b + 8).
-For the second group (2b + 16), the GCF is "2". So, factor out "2" from the group: 2(b + 8).
3. Notice that both groups have a common factor of (b + 8). So, factor out (b + 8) from the entire expression: (b + 8)(a + 2).
Thus, the factored form of the polynomial ab + 8a + 2b + 16 by grouping is (b + 8)(a + 2).
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Calculate the mean: 13, 21, 45, 62, 10
Answer:30.2
Step-by-step explanation:
Answer:
30.2Calculate the mean: 13, 21, 45, 62, 10
13 + 21 + 45 + 62 + 10
= 151151 ÷ 5
= 30.2
Step-by-step explanation:
You're welcome.
For the figure at right, determine if the two triangles are congruent. If so, create a proof (flowchart or two-column) to explain why. Then, solve for x. If the triangles are not congruent, explain why not. Homework Help ✎
Answer:
x = 32
Step-by-step explanation:
(41)² - (40)² = 1681 - 1600
= 81
[tex] \sqrt{81} = 9[/tex]
|EC| = 9
x = 41 - 9
x = 32
The 2 triangles share 2 sides of common lengths and a common angle between these sides, therefore they are congruent.
67 07 – 4(67) 4 a) -25.633 b) -258.429 c) -21.057 d) -20.1 Find the value of the expression. Give the result as a decimal. ()? + (5.9) (3.6) a) 21.28 b) 21.265 Oc) 21.49 d) 22.28
The expression 67 07 – 4(67) 4 + (5.9) (3.6) is equal to 21.265. Correct answer is option B.
To solve this expression, we must first understand the order of operations. The order of operations is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. We start by solving the part of the expression in parentheses, which is 4(67) 4. We must do this first, as parentheses indicate that the expression within should be evaluated first.
This expression is equal to 268. We then proceed to solve the expression from left to right, following the order of operations. 67 07 – 268 = -258.429. We then add the part of the expression with the parentheses, which is (5.9) (3.6). -258.429 + (5.9) (3.6) = -21.265.
Therefore, the value of the expression 67 07 – 4(67) 4 + (5.9) (3.6) is equal to 21.265 as a decimal. Therefore the Correct answer is option B.
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The brain weight of a newborn baby is about 13% of the body weight of the newborn. If a newborn weighs 2,900 grams, about how much does the brain weigh?
Answer:
The brain of the newborn baby has a mass of about 377 grams.
Step-by-step explanation:
Fractions are written as a ratio of two integers. For instance, a/b is a fraction.
Given the following parameters
Body mass = 2900 grams.
If the Brain mass of a newborn baby is about 13% of the mass of the newborn, then the mass of the brain is given as;
mass of brain = 13% of 2900
Mass of brain = 0.13 * 2900
Mass of brain = 377grams
Please Help Struggling with Geometry :)
Find x. Show your work.
Answer:
36.87o
Step-by-step explanation:
Since ABC is a right triangle
cos(x) = adjacent/hypotenuse
cos(x) = 12/15
x = cos^-1 (12/15) = 36.87o
what is the answer of square root of 27 to the power of 3
Step-by-step explanation:
For solving this problem, we will use √ab=√a√b. Therefore, the value of 3√27 is 9√3. Note: In the above solution, we factored 27 which is inside the root.
Divide. If the divisor contains 2 or more terms, (6x^(2)y+18x^(2)y^(2)-xy^(2))/(6xy)
If the divisor contains 2 or more terms, (6x^(2)y+18x^(2)y^(2)-xy^(2))/(6xy) it can simplifies to x+3xy-y.
To divide the given expression, we need to factor out the common term from the numerator and then simplify by canceling out the common terms from the numerator and denominator. Here is the step-by-step explanation:
Step 1: Factor out the common term from the numerator:
(6x²y+18x²y²)-xy²)/(6xy) = 6xy(x+3xy-y)/(6xy)
Step 2: Cancel out the common terms from the numerator and denominator:
6xy(x+3xy-y)/(6xy) = (x+3xy-y)
Step 3: Simplify the expression:
(x+3xy-y) = x+3xy-y
Therefore, the final answer is x+3xy-y.
So, the given expression (6x²y+18x²y²-xy²)/(6xy) simplifies to x+3xy-y.
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(b) Now building on that, use the yp for the complex exponential forcing to find yp for the following differential equations. Warning: no credit if you are not showcasing this specific technique.
i. y′′−6y′−7y=5e2tsin(t). ii. y′′−6y′−7y=5e2tcos(t).
yp(t) = Aexp(2t) + Bexp(-2t) + Cexp(5t) + Dexp(-5t)
To find the solution to these two differential equations, we must use the technique of complex exponential forcing. The general solution for these types of equations is:
yp(t) = Aexp(2t) + Bexp(-2t) + Cexp(5t) + Dexp(-5t)
We can then use the given equations to substitute for y′ and y′′, and solve for the four constants A, B, C, and D. Doing so for equation (i), we get:
Aexp(2t) + Bexp(-2t) + Cexp(5t) + Dexp(-5t) = 5e2tsin(t) - 6[Aexp(2t) + Bexp(-2t)] - 7[Aexp(2t) + Bexp(-2t) + Cexp(5t) + Dexp(-5t)]
We can then solve this equation using algebraic manipulation to determine the four constants, and thus find the solution to this equation.
We can then use a similar method to solve equation (ii). The same general solution is used, and the constants can be determined using the same algebraic manipulation. The solution to this equation is then:
yp(t) = Aexp(2t) + Bexp(-2t) + Cexp(5t) + Dexp(-5t)
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How do can we compute |415| with those symbols, What do they mean?
The meaning of the symbol |415| is absolute value and the value of |415| is equal to 415.
Symbols | | represent the absolute value function in mathematics.
The absolute value function returns the distance between a number and zero on the number line.
Absolute value of a number is always positive or zero.
Compute the absolute value of 415 using the absolute value function,
Replace the number within the bars with the given value as no sign is given,
415 > 0
⇒ Absolute value of |415| = 415
Since 415 is already a positive number
This implies absolute value is itself.
Therefore, the symbol | | represents absolute value of 415 and is equal to 415.
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