Solve the differential equation by variation of parameters.
y'' + y = sin2 x
The solution to the differential equation y'' + y = sin2x is y = -2sin2x + (Acos2x + Bsin2x).
This differential equation can be solved by the method of variation of parameters. This method involves finding two arbitrary functions, U and V, such that U' = V and V' = -U + sin2x, so that the general solution of the differential equation is y = U + V. Solving the two equations yields U = Acos2x + Bsin2x and V = -2sin2x, where A and B are arbitrary constants. Substituting U and V into the general solution of the differential equation yields the final solution y = -2sin2x + (Acos2x + Bsin2x).
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A whale is swimming due north at a speed of 40 kilometers per hour. Just 6 kilometers away,
a whale-watching tour boat is traveling south, directly toward the whale, at a speed of 83
kilometers per hour. How long will it be before they meet?
Answer:
O hrs and 30 minutes is the answer
Step-by-step explanation:
want proof?
Answer:
about 0.04878 hours ≈ 2 minutes 56 seconds
Step-by-step explanation:
You want the time until a whale-watching boat and a whale meet if the boat is traveling south at 83 km/h and the whale is traveling north at 40 km/h when they start 6 km apart.
Closing speedThe speed at which the gap between the boat and the whale is closed is the sum of their respective speeds:
closing speed = 83 km/h + 40 km/h = 123 km/h
TimeThe time it takes to close the 6 km gap is ...
time = distance/speed
time = (6 km)/(123 km/h) ≈ 0.04878 h
Converted to minutes and seconds, that is about ...
0.04878 h ≈ 2 minutes 55.6 seconds
It will be about 2 minutes 56 seconds before the boat and whale meet.
__
Additional comment
NOAA recommends a distance of at least 100 yards be maintained between a boat and a whale. In some areas, the recommended distance is 400 yards when approaching head-on. The recommended boat speed is 15 knots or less, about 28 km/h.
Yoshi is standing on top of a building, looking at a park in the distance. The angle of depression is
53°. If the building he's standing on is 100 feet tall, how far away is the park to the nearest
hundredth of a foot?
105
53°
Park
100
Step-by-step explanation:
because after 105 it's 100
1. Write 7 example of mathematical expreion. 2. Larry had 5 3/6 yard of fabric. He ued 2 2/3 to make a bag for hi computer. Write an expreion that will how how much of the fabric Larry will have left. PLEASE HURRY UPPPPP!!! I HAVE 1 HOUR LEFT UNTIL SUBMISSION!!!
a) 7 examples are:
[tex]2x + 3y\\(4a - 5b) / 2c\\(5 + 8) * 3\\8^2 - 6\\√(64)\\(1/3) * π * r^2\\2(3x + 4y) - 5z[/tex]
b) Fabric Larry will have left after using [tex]2\frac{2}{3}[/tex] yard out of [tex]5\frac{5}{6}[/tex] yard to make a bag for his computer is: [tex]3\frac{3}{6} yard[/tex]
What kinds of expressions are examples?Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
7 examples of mathematical expression are:
[tex]2x + 3y\\(4a - 5b) / 2c\\(5 + 8) * 3\\8^2 - 6\\√(64)\\(1/3) * π * r^2\\2(3x + 4y) - 5z[/tex]
For the second part of the question:
Larry had 5 3/6 yards of fabric. He used 2 2/3 yards to make a bag for his computer. Write an expression that will show how much of the fabric Larry will have left.
The mathematical expression that represents the amount of fabric Larry has left is:
5 3/6 - 2 2/3 = 2 7/18
You can simplify this further:
2 7/18 = 2 1/6
The amount of fabric Larry will have left is 2 1/6 yards.
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How do you write no solution?
No solution is when a system of equations has no solution. This means that the equations are inconsistent, and there is no set of values that can make the equations true.
For example, if the system of equations is 2x + 3y = 4 and 2x + 3y = 5, then there is no solution. This is because no matter what values are chosen for x and y, the equations cannot both be true. This means that the system of equations is inconsistent, and there is no solution.
To symbolically represent no solution, you can use the following notation:
2x + 3y = 4, 2x + 3y = 5 No solution
This notation is used to indicate that the equations are inconsistent and that there is no solution.
Complete question: What is the solution to the equation 2x + 3y = 4 and 2x + 3y = 5?
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Reduce the fraction and find the domain
The simplification of all the given algebra fractions are;
1) a/c
2) (a - b)/(a - 1)
3) (x + 1)/(x - 4)
How to solve algebraic fractions?An algebraic fraction is defined as a fraction whose numerator and denominator are algebraic expressions.
1) (abc + ab²)/(b²c + bc²)
Factorizing numerator and denominator gives;
ab(c + b)/bc(b + c)
b(b + c) will cancel out to get; a/c
2) (a² - b²)/(a² - a + ab - b)
Factorizing numerator and denominator gives;
[(a + b)(a - b)]/[a(a - 1) + b(a - 1)]
= [(a + b)(a - b)]/(a + b)(a - 1)
(a + b) will cancel out to get;
(a - b)/(a - 1)
3) (x² - 2x - 3)/(x² - 7x + 12)
Expanding the numerator and denominator gives;
(x² - 3x + x - 3)/(x² - 4x - 3x + 12)
= ((x + 1)(x - 3))/((x - 3)(x - 4)
(x - 3) will cancel out to get;
(x + 1)/(x - 4)
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Find the length of the third side. If necessary, round to the nearest tenth
Answer:
6.3
Step-by-step explanation:
This is a right triangle, so in order to solve you can use the Pythagorean theorem; which is a² + b² = c² (The hypotenuse, or the longest side, equals C. A and B equal the legs, or the shorter sides.) So once you plug in the values, your equation should look like this:
3² + b² = 7²
Square each side, or in other words multiply it by itself.
9 + b² = 49
Subtract 9 from both sides.
b² = 40
Take the square root of 40 and your done!
b = 6.3
I hope this helped you! :)
How many solutions does the system y 3x 4 and 6x 2y 8 have?
The system y = 3x + 4 and 6x - 2y = 8 has no solutions.
A system of equations is a set of equations with multiple variables. To determine the number of solutions for a system of equations, one can use the following methods: graphing, substitution, or elimination. In this case, we can use the elimination method by adding or subtracting the equations to eliminate one of the variables.
If we add the two equations, we get y = 3x + 4 + 6x - 2y = 8, which results in 3x + 4 = 8. This is an inconsistent equation and does not have any solutions. Therefore, the system y = 3x + 4 and 6x - 2y = 8 has no solutions.
This shows that the system is inconsistent, which means that there's no solution that satisfies both equations.
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If star a is closer to us than star b, then star a's stellar parallax is __________.
O Larger than that of star B
O Larger than that of star A
O Smaller than that of star B
O Smaller than that of star A
If star A is closer to us than star B, then Star A's parallax angle is: Larger than that of Star B.
Hence, option (A) is correct choice.
The parallax of a star at a distance of 10 parsecs would be 0.1′′ since parallax is inversely proportional to distance. The parallax of Proxima Centauri, a star in the triple system of Alpha Centauri, is 0.76813′′, indicating that it is 4.24 light-years away from Earth at a distance of 1/0.76813, or 1.302 parsecs.
The apparent shift in an object's location caused by a change in the observer position is known as parallax. This apparent shift is measured (in arc seconds) is reversed to provide the distance from the background stars (in parsecs).
The definition of an object's absolute magnitude is the apparent magnitude the object would have if it were observed at a distance of precisely 10 parsecs (32.6 light-years), without light extinction.
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Each side of a square is increasing at a rate of 7 cm/s. At what rate (in cm2/s) is the area of the square increasing when the area of the square is 81 cm2
At the rate of 126cm/s is the area of the square increasing when the area of the square is 81cm2.
If each side of a square is increasing at a rate of 7 cm/s.
The area of the square increasing rate,
Area, A = s2
each side of the square is increasing at the rate of 7cm / s (= dx / dt).
s is the side length
=> dA/dt = 2s. ds/ dt
ds/dt = 7cm/s
Area of the square = 81 cm2
81 cm2 = s2
s = 9 cm
so, dA /dt = 2. 9. 7
= 126 cm2/s
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Just do the second one…
The L.C.M. defines the least number that is exactly divisible by two or more numbers, whereas the G.C.F. defines the greatest factor present between any given pair of two or more numbers.
What is the difference between LCM and GCF?LCM is also known as the Least Common Multiple (LCM), and HCF is also known as the Greatest Common Factor (GCF).
The highest common factor (HCF) between two or more numbers is two or more. The LCM, on the other hand, is the lowest common multiple that exactly divides these two figures. The greatest common factor between any two numbers is also known as HCF. Between two or more numbers, LCM is also known as Least Common Divisor.
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The sum of -3 times x +6.3 what are all the possible values of x
So x = 2.1 is the only possible value of x that makes the equation true.
Define Equation.The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
Define Algebraic expression.An algebraic expression in mathematics is an expression created using variables, constant algebraic numbers, and algebraic operations. A good example of an algebraic expression is 3x2 2xy + c.
The equation is given as follows:
-3x + 6.3 = 0
To find the value of x, we need to isolate x on one side of the equation. We can do this by adding 3x to both sides:
-3x + 3x + 6.3 = 0 + 3x
6.3 = 3x
Now we can divide both sides by 3:
x = 6.3/3
x = 2.1
So x = 2.1 is the only possible value of x that makes the equation true.
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Amelie is making a recipe that uses 134 cups flour. She is making 212 times the original recipe. Amelie draws this model to represent the number of cups of flour she needs. How many cups of flour does Amelie need
Amelie needed 335 cups of flour for the recipe.
How is quantity calculated?The whole centre line length, as well as the construction's breadth and depth, must be multiplied in order to determine the required amount of quantity. Every time the main wall is attached to a cross wall, a partition, or a verandah, the centre line length will be halved at that intersection.Any number, variable, or algebraic combination of other numbers can be a quantity in a mathematical equation. Seven, ten, x, and the product of x and seven, or x + seven, are the four numbers in the equation x + 7 = 10.The unit price is the cost per unit of the product you are purchasing. The amount might be given per thing or per measurement type, such ounces, grammes, gallons, or litres.Given data :
She is making 2 1/2 or 2.5 times the original recipe.
Cups in original recipe= 134 cups
cups Amelie will need= 2.5 × 134
= 335 cups
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Clare made an error when solving -4x+3<23−4x+3<23.
The solution of the inequality is,
⇒ x > - 5
What is Inequality?A relation by which we can compare two or more mathematical expression is called an inequality.
Given that;
The inequality is,
⇒ - 4x + 3 < 23
Now,
Since, The inequality is,
⇒ - 4x + 3 < 23
Solve as;
⇒ - 4x < 23 - 3
⇒ - 4x < 20
⇒ - x < 5
⇒ x > - 5
Thus, The solution is,
⇒ x > - 5
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Find the general solution y''-y'-2y=-2t+4t^2
The general solution to the given differential equation is y = c₁e⁻t + c₂te⁻t + t².
The given differential equation can be written as y'' - y' - 2y = -2t + 4t², which is a second-order linear differential equation with constant coefficients. To solve this equation, the first step is to solve the associated homogeneous equation, which is y'' - y' - 2y = 0. This homogeneous equation can be solved by using the characteristic equation, r² - r - 2 = 0, which has two roots, r = 2 and r = -1.
The general solution of the homogeneous equation is then y = c₁e²t + c₂e⁻t. To find the particular solution of the non-homogeneous equation, the method of undetermined coefficients is used. Since the non-homogeneous term is 4t², the particular solution of the non-homogeneous equation is yp = t².
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Write an equation of the line passing through the point (1,8) and parallel to y=−3x+10
The equation of the line is 3x + y - 11 = 0.
The slope-intercept formula y = mx + b is used when you know the slope of the line to be examined and the point given is also the y intercept (0, b). In the formula, b represents the y value of the y intercept point.
First we find the slope of the line y=-3x+10 by placing it into slope-intercept form:
y = -3 x + 10
Therefore, the slope of the line is m= -3
Now since the equation of the line with slope m passing through a point
(x₁ ,y₁) is y-y₁ = m(x-x₁)
Here the point is (1,8) and slope is m= -3, therefore, the equation of the line is:
y - 8 = -3(x - 1)
y - 8 = -3x +3
3x + y - 11 = 0
Hence, the equation of the line is 3x + y - 11 = 0.
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If Jenny earns $375 when she works 5 hours, how much will she earn if she
works 12 hours?
Answer:
She will earn $900.00 in 12 hours.
Step-by-step explanation:
Her unit rate is 75 dollars per hour.
375/5 = $75
75 x 12 = 900
Answer: $900
Step-by-step explanation: 375/5x12.
tripple of a number added to its half part its equal to the difference between four times the number and 12. What number is it
If triple of a number added to its half part its equal to the difference between four times the number and 12 then the number is 24.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let us consider the number be x
tripple of a number added to its half part
3x+x/2
Difference between four times the number and 12
4x-12
So tripple of a number added to its half part its equal to the difference between four times the number and 12
3x+x/2=4x-12
6x+x=2(4x-12)
Apply distributive property on RHS
7x=8x-24
Subtract 7x on both sides and add 24
24=x
Hence, if triple of a number added to its half part its equal to the difference between four times the number and 12 then the number is 24.
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Which of the following is equivalent to (X - 2) (x + 9) =0
The equivalent quadratic equation to (x - 2)(x + 9) = 0 is x² + 7x - 18 = 0.
What is a quadratic equaton?A quadratic equation is an algebraic expression in the form of variables and constants.
A quadratic equation has two roots as its degree is two.
We know if α and β are the roots of a quadratic equation ax² + bx + c = 0,
Then we can construct the equation by the factors (x - α)(x - β).
Given, We have to construct a quadratic equation with (x - 2)(x + 9) = 0.
Now, Multiply the terms we have,
x² + 9x - 2x - 18 = 0.
x² + 7x - 18 = 0.
So, Our required quadratic equation is x² + 7x - 18 = 0.
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(5 points) 2. The coordinates of the vertices of ARST are R(1, 6), S(5, 3), and T(3,-2). Triangle R'S'T' is the image of ARST after a reflection in x = -1. Graph and state the coordinates of the vertices of AR'S'T'.
The coordinates of the reflected points are R' (1, -6), S' (5, -3), T' (3, 2) and the graph is attached.
What is reflection of graph?Reflections of graphs involve reflecting a graph over a specific line.Reflecting functions are functions whose graphs are reflections of each otherGiven are the coordinates of the vertices of ΔRST are R(1, 6), S(5, 3), and T(3,-2).
The rule for a reflection over the x -axis is (x, y) →(x, −y). Now we can write the coordinates of the reflection as -
R' (1, -6)
S' (5, -3)
T' (3, 2)
The graph is attached.
Therefore, the coordinates of the reflected points are R' (1, -6), S' (5, -3), T' (3, 2) and the graph is attached.
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Be so kind and help me please!
Answer:
2A/h - b2= b1
Step-by-step explanation:
cos(-0)=3, sin 0 <0
What is the value of sin ?
2√3/ 3
√6/3
-2 √3/3
-√6/3
On solving the provided question we can say that - here in trigonometry [tex]sinx(2cosx - 1) = 0\\[/tex] => sinx = 0 and 2 cosx -1 =0
what is trigonometry?The area of mathematics known as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, around the third century BC. from the use of geometry in astronomical study. The area of mathematics known as exact methods deals with specific trigonometric functions and how they might be used in calculations. There are six popular trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics of triangles, particularly right triangles, is called trigonometry. The study of geometry, however, is the characteristics of all geometric figures.
[tex]sin 2x - sinx = 0\\-sinx + 2cosx sinx=0\\sinx(2cosx - 1) = 0\\[/tex]
sinx = 0 and 2 cosx -1 =0
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Graph the Following parent function
Answer:
see attached
Step-by-step explanation:
You want the graph of g(x) = -f(x-3), given the graph of the parent function f(x) = log₂(x).
TransformationThe replacement of x with x-3 as the function argument causes the graph to be translated 3 units to the right.
The replacement of f(x) with -f(x) causes the graph to be reflected over the x-axis.
The translated, reflected graph is shown in blue in the attachment.
a+ 1 1/6 = 11 7/9
...........................
Answer:
a = 191/81
Step-by-step explanation:
a+ 1 1/6 = 11 7/9
1 1/6 = 7/6
11 7/9 = 106/9
So, our equation is
a + 7/6 = 106/9
Subtract 7/6 from both sides
a = 191/81
So, the answer is
a = 191/81
PLEASE HELP
Using the ratio 26 black marbles to 2 white marbles, match the following to create the equivalent ratios.
Step-by-step explanation:
1:13
2:26
3:39
4:52
5:65
6:78
Rule - ×13
Can u please help? .
Answer:
y = -3x -27
Step-by-step explanation:
Parallel lines have the same slope. So the slope will be -3
We need to solve for the y-intercept. Us the x and y values from the point (-6,-9) to find the y-intercept
m = -3 given
y = -9 from the point( -6,-9)
x = -6 from the point (-6,-9)
y = mx + b substitute in what you know and solve for b
-9 = (-3) (-6) + b
-9 = 18 + b Subtract 18 from both sides
-9 -18 = 18 - 18 +b
-27 = b
The equation of the line is y = -3x -27
The probability that a student studied is 0.85. The probability that a student will not pass, given that the student studied is 0.10.
a) Find the probability that a student will study and will pass.
b) Suppose that the probability that a student will either study or pass is 0.8. Find the probability that a student will pass.
a) The probability that a student will study and will pass is 0.765.
b) The probability that a student will pass is 0.715.
What is probability?
Simply put, probability is the likelihood that something will occur. When how an event will turn out is not known, one can discuss the likelihood of several outcomes.
Define some events -
A: A student studies.
B: A student will pass.
Given that P(A)=0.85 and
P(B|A)=1-P([tex]\bar{B}[/tex]|A)
=1-0.10
=0.90
a) P(a student will study and will Pass)= P(A∩B)
= P(A)·P(B|A)
= 0.85×0.90
= 0.765
b) Given that P( student will either study or pass), the probability is -
=P(A∪B)= 0.8 then P(a student will pass)
=P(B)
It is known that P(A∪B)= P(A)+P(B)-P(A∩B)
Then P(B)=P(A∪B) -P(A) +P(A∩B)
=0.8-0.85 + 0.765
=0.715
Therefore, the probabilities are 0.765 and 0.715.
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Write an exponential function to represent the spread of Ben's social media post. Write an exponential function to represent the spread of Carter's social media post
So, y = ab^x and y = ac^x are the exponential functions to represent the spread of Ben's and Carter's social media posts respectively.
An exponential function has the form y = ab^x, where a and b are constants and x is the independent variable (in this case, time).
For Ben's social media posts:
y = ab^x
For Carter's social media post:
y = ac^x
Please note that the value of a, b, and c will be different for Ben and Carter's post, as it will depend on the number of shares, likes, views, etc., and the rate of spread of the post.
This is an exponential function that represents the spread of Ben's social media posts and Carter's social media post.
Complete Question:
1. Write an exponential function to represent the spread of Ben's social media post 2. Write an exponential function to represent the spread of Carter's social media post. 3. Graph each function using at least three points for each curve. All graphs should be placed together on the same coordinate plane, so be sure to label each curve. You may graph your equation by hand on a piece of paper and scan your work, or you may use graphing technology. 4. Using the functions for each student, predict how many shares each student's post will be received on Day 3 and then on Day 10. Justify your answers 5. If Amber decides to mail copies of her photo to the 45 residents of her grandmother's assisted living facility, the new function representing her photo shares is . How does this graph compare with the original graph of Amber's photo share? 6. Based on your results, which students' post travels the fastest? How is this shown in the equation form of the functions? 7. If you had to choose, would you prefer a post with fewer friends initially but more shares, like Amber, or more friends initially but fewer shares? Justify your answer with your calculations from previous questions
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Courtney is in the process of buying a minivan. The car she wants to buy is $29,999. The required down payment is 20% . How much will she need to put down in order to purchase this car
Car insurance coverage covers damages caused by or other mishaps. The following damages are covered claims except :
A. A head on collisions .
B. A tree falls in your car during a windstorm
C. The sun weathers the paint on your car and it must be repainted
D. A thief breaks your back window and steals the guitar from your back seat
On observing the provided question we can say that - the function f(x) = a^x is Range : (−∞,∞) or all real numbers
What is Range?Range: the discrepancy between the top and bottom numbers. To get the range, locate the greatest observed value of the number and deduct the least observed value (the minimum). The data points between the two extremes of the distribution are not taken into consideration by the range; just these two values are considered. Between the lowest and greatest numbers, there is a range. Values at the extremes make up the range. The data set 4, 6, 10, 15, 18, for instance, has a range of 18-4 = 14, a maximum of 18, a minimum of 4, and a minimum of 4.
The answer to the question is that all real numbers or (−∞,∞)
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A ratio is a comparison of two quantities. For example, the ratio of cows to pigs on a farm could be 1 to 3. For every 1 cow, there are 3 pigs. The ratio of chocolate chips to raisins in a cookie could be 5 to 9. For every 5 chocolate chips in the cookie, there are 9 raisins. Do you notice a repeating phrase in these examples? “For every” is key ratio language. It tells you how much of one quantity exists in comparison to the other quantity. A ratio consists of just the numbers, not the units. If someone asked the ratio of chocolate chips to raisins in the cookie example, you’d say 5 to 9.
Which of these could you represent with a ratio?
The solution is given by the equation A = how many diet sodas there are in the fridge compared to how many regular sodas
What is Proportion?The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.
The proportional equation is given as y ∝ x
And , y = kx where k is the proportionality constant
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Given data ,
Let the proportion be represented as A
Now , the equation will be
The ratio of cows to pigs on a farm could be 1 to 3
So , the proportion is 1 : 3
Now , The ratio of chocolate chips to raisins in a cookie could be 5 to 9
So , the proportion is 5 : 9
Now , the number of diet sodas there are in the fridge compared to number of regular sodas is also a proportion
Hence , the proportion is number of diet sodas to number of regular sodas
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