Answer:
Estimate of difference is 4.
Step-by-step explanation:
10.04 = 10 to nearest whole number
6.3 = 6 to nearest whole number.
10 - 6 = 4.
If the ratio of the model car was 2:35 then what would be the actual length of the car??
PLEASE HELP AND SHOW WORK
The actual length of the car based on the model will be 350 inches.
How to calculate the ratio?Ratio in Mathematics simply demonstrates how many times one number can be able to fit into another number. It should be noted that ratios contrast two numbers by dividing them. In this case, A/B will be the formula if one is comparing one variable (A) to another variable (B).
Since the ratio of the model car was 2:35. If the model is 20 inches long, the actual length of the car will be:
= 20 ÷ 2/35
= 20 × 35/2
= 350 inches.
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Complete question
The ratio of the model car was 2:35. If the model is 20 inches long. then what would be the actual length of the car??
Determine which point from the specified that satisfies each system of equations
The required point that satisfies each system of equation is (12, -1). Option B is correct.
What are simultaneous linear equations?Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.
To equations are given as,
y = -1/3x + 3 - - - - (1)
y = -3/4x + 8 -- - - - (2)
From equation 1 and 2
-1/3x + 3 = -3/4x + 8
3/4x - 1/3x = 8 - 3
9x - 4x / 12 = 5
5x = 60
x = 12
Now, to evaluate the y ordinate put the above x in equation 1
y = -1/3(12) + 3
y = -4 + 3
y = -1
Thus, the required point that satisfies each system of equation is (12, -1). Option B is correct.
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The correlation coefficient r between the size of a lake in square miles, x, and the yearly number of registered boaters, y, is 0.743. What percent of the variation in the number of yearly boaters can be explained by differences in lake sizes
Answer:
55.2049%
Step-by-step explanation:
R², the coefficient of determination, is the proportion of the variation in the dependent variable that is predictable from the independent variable.
In this case, if we square our correlation coefficent r=0.743, we get R²=0.552049, which means that 55.2049% of the variation in the number of yearly boaters can be explained by differences in lake sizes
What is the coefficient of the variable in the expression 4 − 3x − 7 6 3 points?
The coefficient of the variable x in the expression 4a+3x +a is 3
Given that,
The expression is 4a+3x +a which can be simplified as 4a+3x +a
To find : The coefficient of the variable x in the expression
The coefficient of x in 4a+3x +a is 3
A number multiplied by a variable is referred to as the coefficient of a number. For instance, the x2 coefficient in the formula 6x2+99 is 6
Coefficient is the number that is beside the variable in the equation.
As we can see that the term that is in multiplication with x is 3
Hence, the coefficient of x in the equation is 3
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Devin drove her truck. The fuel remaining in the truck's tank (in liters) as a function of distance (in kilometers) is graphed.
What is the approximate average fuel consumption rate, between the 100th kilometer and 400th kilometer?
Between the 100th and 400th kilometers, the average fuel consumption rate is 5/6 liters per kilometer.
What is the rate of change?A consequence of dividing the variation in a variable's function by the variation in the variable. For example The rate at which a distance changes in relation to time is known as velocity.
Given, Devin drove her truck. The fuel remaining in the truck's tank (in liters) as a function of distance (in kilometers) is graphed.
Since in the domain [100, 400] graph shows somewhat linear properties.
from the graph Fuel consumption at 100 = 375 liters
Fuel consumption at 400 = 125 liters
rate in a linear function between two points = (y₂ - y₁)/ (x₂ - x₁)
thus rate between [100, 400] = (375-125)/400-100
rate between [100, 400] = 250/300
rate between [100, 400] = 5/6litere per kilometer
therefore, the average fuel consumption rate, between the 100th kilometer and 400th kilometer is 5/6 liters per kilometer.
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In the figure where 3x20 and 4x-36, which value of x will make aob, a traight line
The value of x that will make aob a straight line equation is x = -6.
To make aob a straight line, we need to make the gradients of both lines equal.
The gradient of the first line is 3x20 = 60.
The gradient of the second line is 4x-36 = 4x-36.
To make the gradients equal, we need to solve the equation 4x - 36 = 60.
Subtracting 36 from both sides gives us: 4x = 96.
Dividing both sides by 4 gives us x = -6.
Therefore, the value of x that will make aob a straight line is x = -6
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I need help please! In each case below, list the lettered measures for angles or sides in descending order (FROM GREATEST TO LEAST).
Answer:
β, α, γ
Step-by-step explanation:
The figure only shows us the side lengths of the triangle, but is asking for the angles.
Thus, we have to use the angle relationships of a triangle to answer the question.
⭐What are the angle relationships of a triangle?
the angle opposite of the largest side is the largest anglethe angle opposite of the smallest side is the smallest angleα° is opposite of the side length with 9 cm.
β° is opposite of the side length with 12 cm.
γ° is opposite of the side length with 5 cm.
∴ The angles from highest to the least are β, α, γ
IF THIS RESPONSE HELPED YOU, PLEASE MARK IT THE BRAINLIEST :-)
Answer:
Step-by-step explanation:
The answer should be BAY but if i check it don't work
Bart and April are purchasing a house with a 15-year, 2/1 ARM for $315,000 at 5. 85% with a 2/8 cap structure. What will the difference in
payments be from year 2 to year 3?
$309. 69
$595. 18
$333. 68
$86. 56
The monthly payment for this difference would be $1,361.21. An Adjustable Rate Mortgage (ARM) is a type of mortgage in which the interest rate on the loan changes over time.
An Adjustable Rate Mortgage (ARM) payment is based on various factors such as the loan amount, interest rate, and the terms of the ARM. The interest rate is usually tied to a specific financial index, such as the U.S.
Here is the calculation format for the difference in payments between year 2 and year 3 for Bart and April's mortgage:
Total amount to be paid for the mortgage: $315,000
Rate of interest: 5.85% (Depreciating rate)
Time = 2 years:
a. Remaining amount to be paid after 2 years:
Principal × (1 - Rate/100)^time
= $315,000 × (1 - 5.85/100)²
= $315,000 × (1 - 0.0585)²
= $279,223.01
Time = 3 years:
a. Remaining amount to be paid after 3 years:
Principal × (1 - Rate/100)^time
= $315,000 × (1 - 5.85/100)³
= $315,000 × (1 - 0.0585)³
= $262,888.46
Difference in payments from year 2 to year 3:
= Remaining amount after 2 years - Remaining amount after 3 years
= $279,223.01 - $262,888.46
= $16,334.55
The monthly payment for this difference would be $1,361.21, which is calculated by dividing the difference in payments by 12:
$16334.546/12 = $1361.21
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Philip is having a party and wants to spend less than $572. He has already spent $500. The only item left on his list is sodas, which cost $8 per case. How many cases of soda, x, can Philip purchase and stay under his budget? Select the number line that includes the largest number of cases of soda that Philip can purchase and still stay under his budget.
Answer:
x = 8
Step-by-step explanation:
572 - 500 = 72
72 / 8 = 9.
But in order to spend less than 572, he has to buy 8 cases. If he buys 9 cases he will be exactly 572.
Answer:
9 or fewer cases
Step-by-step explanation:
Solving a two step equation:
First, determine how much he has left to spend
572 - 500 = 72
He has 72 dollars left to spend on soda
Each case is 8 dollars.
Divide how much he has to spend by the cost of each case
72/8 = 9
He can buy up to 9 cases and stay under budget.
In the figure, mzEDF = 42°. Which other angle must have the same measure?
OA. ZGDF
B. ZEGF
OC. ZDEG
D.
LEGD
D
42°
C
E
IL
In the quadrilateral, ∠GEF will be same as angle ∠EDF=42°°
A quadrilateral shape is what?A polygon with four sides, four angles, and four vertices is called a quadrilateral. The Latin words quadri, which means four, and latus, which means side, were combined to create the English word quadrilateral. A quadrilateral is shown in the above image as an example.
A parallelogram is a trapezoid, right?A parallelogram has two pairs of parallel sides, whereas a trapezoid only has one pair. Consequently, a parallelogram is a trapezoid.
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Answer:
EGF
Step-by-step explanation:
Got it right on plato.
A passenger elevator in the Sears Tower in Chicago can lift people 960 feet in 1 minute and 30 seconds. How fast must the freight elevator be traveling if it lifts cargo the same height but in 2 minutes and 30 seconds? [
D=rt
D=rt
]
Pilihan jawaban
6.4 feet per second
10.7 feet per second
19.2 feet per second
38.4 feet per second
If the freight elevator lifts goods the same height in 2 minutes and 30 seconds, it will travel at a pace of 6.4 feet per second.
What is speed?Speed is the pace at which an item moves along a route in time, whereas velocity is the rate and direction of movement. In other words, velocity is a vector, whereas speed is a scalar number. The sign for speed is v (italic), whereas the symbol for velocity is v. (boldface). A bar is placed over the symbol to represent average values. Speed is defined as the rate at which distance changes over time. It has a distance by time dimension. Thus, the SI unit of speed is defined as a combination of the fundamental units of distance and time. As a result, the SI unit of speed is the metre per second.
Here,
Given that a passenger elevator in the sears tower in Chicago can lift people 960 feet in 1 minute and 30 seconds (1.5 minutes). This means that the lift is travelling with a speed of:
=960/1.5
=640 ft/min
If the elevator lifts cargo through the same height in 2 minutes and 30 seconds (2.5 minutes), then the speed is given by:
=640/2.5
=384 ft/min
=384/60
=6.4 ft/second
The freight elevator be traveling at the speed of 6.4 ft/second if it lifts cargo the same height but in 2 minutes and 30 seconds.
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F(x)=5-3x find f(x+2)
Answer:
f(x + 2) = - 3x - 1
Step-by-step explanation:
substitute x = x + 2 into f(x) , that is
f(x + 2)
= 5 - 3(x + 2)
= 5 - 3x - 6
= - 3x - 1
the swim team trains 4 days every week. Mon, Tues, and Thurs, the swim tea spends 1/2h in the gym followed by 1 1/4 in the pool. On Saturday there is no gym work but double the swim team time. how many hours per week does the team train?
The number of hours that the team trains per week is 17 hours.
How to illustrate the fraction?A fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.
Since the swim tea spends 1/2h in the gym followed by 1 1/4 in the pool. On Saturday there is no gym work but double the swim team time. The number of hours will be:
( 1 1/4 + 1/2 + 2 1/2) × 4
= 4.25 × 4
= 17 hours.
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PLEASE HELP THIS IS URGENT!!!!!! Decide which of the following points are on the graph of the function y=2x+1
(0.1), (2,5), (1/2,2), (-1,-1) are the points on graph.
What is a graph, exactly?In mathematics, a graph is a visual representation or diagram that shows data or values in an ordered way. The relationships between two or more things are frequently represented by the points on a graph.
what a graph example?A graph is a type of non-linear data structure composed of nodes, also known as vertices and edges. Any two nodes, also referred to as vertices, in a graph are connected by edges. The vertex numbers in this graph are 1, 2, 3, and 5, and the edge numbers are 1, 2, 1, 3, 2, 4, and 5, respectively.
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What is Destiny's hourly wage?
the slope goes by several names
• average rate of change
• rate of change
• deltaY over deltaX
• Δy over Δx
• rise over run
• gradient
• constant of proportionality
however, is the same cat wearing different costumes.
or in this case, hourly wage :)
to get the slope of any straight line, we simply need two points off of it, let's use those in the picture below.
[tex](\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{10}~,~\stackrel{y_2}{200}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{200}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{10}-\underset{x_1}{0}}} \implies \cfrac{ 200 }{ 10 } \implies \cfrac{20}{1}\cfrac{dollars}{hour}\qquad \textit{\$20 per hour}[/tex]
find the area of the region enclosed by one loop of the curve. r = sin(6θ)
Area of the region enclosed by one loop of the curve is pi/288.
To find the area of the region enclosed by one loop of the curve, we can use the formula for the area of a polar region:
A = 1/2 * ∫(r^2)dθ
We can use this formula with the equation for r given:
A = 1/2 * ∫(sin^2(6θ))dθ
This integral can be simplified by noting that sin^2(x) = (1/2)(1 - cos(2x))
Thus,
A = 1/2 * ∫(1/2)(1 - cos(12θ))dθ
This integral can be solved by noting that the antiderivative of (1/2)(1 - cos(12θ)) is (1/24)(sin(12θ) + θ).
Therefore, the area enclosed by one loop of the curve is
A = 1/2 * ((1/24)(sin(12θ) + θ)) evaluated from 0 to pi/6
A = (1/48)(sin(2pi) + pi/6)
= (1/48)(0 + pi/6)
= pi/288
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You have $15 in savings. Your friend has $25 in savings. You both start saving $5 per week. Write a system of linear equations that represents this situation. Will you ever have the same amount of savings as your friend? Explain.
Zachary plans to collect $0. 30 from each person that comes to the party to cover the cost of the candy.
How many people will need to come to the party to cover the entire cost of the candy
In order to pay for the sweets, Zachary intends to take $0.30 from each guest that attends the celebration. To pay for the total cost of the sweets, 4 guests are required to attend the party.
How much candy per person?We suggest 8 ounces of sweets per visitor when utilizing ounces. A 5 pound bag of candy will thus be enough to satisfy 10 people. The candy bar may have been distributed evenly in the following manner. Since there are four individuals, split each according to who will receive. About half a pound of sweets is given to each visitor. Therefore, you should buy 50 pounds of sweets at the very least if 100 people are expected.
Setting up the confectionery should minimize guest contact with it for hygienic reasons. Tongs and candy scoops in your theme-coordinating color should be provided for bulk candies. Since you don't want your guests to consume too much sugar, smaller serving utensils are preferred.
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Select the values that make the inequality 5u> -80 true. Then write an equivalent
inequality, in terms of u.
(Numbers written in order from least to greatest going across.)
-26
0 -17
0-13
0 -21
☐ -16
Submit Answer
0 -11
Equivalent Inequality: u
□ -19
0-15
-6
The value u > -16 that makes the inequality 5u > -80 is true.
What is an inequality?In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to each other. To solve the inequality, you may multiply or divide each side by the same positive number, add the same amount to each side, take the same amount away from each side, and more. You must flip the inequality sign if you multiply or divide either side by a negative number.
Given:
We have the inequality,
5u > -80
To find the solution set of the inequality;
Divide by 5 to both sides,
5u/5 > -80/5
u > -16.
That means, the inequality is true for u > -16.
Therefore, the inequality is true for u > -16.
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what is -5.6 opposite ?
Answer: 5.6
Step-by-step explanation:
the opposite of an negative number...
So your answer would be....
=5.6
Answer:
[tex]5.6[/tex] or [tex]\dfrac{1}{-5.6}[/tex]
Step-by-step explanation:
The additive inverse of [tex]-5.6[/tex] is [tex]-1 \cdot -5.6 = \boxed{5.6}[/tex]
or...
If you mean the multiplicative inverse of [tex]-5.6[/tex], then the answer would be [tex]\dfrac{1}{-5.6}[/tex]
Suppose a historian wants to estimate the true mean number of children per household in New York City in 1900. The historian randomly selects 200 household census records from New York City. She records the number of children residing in each household. The historian then computes the mean number of children per household among these 200 households.
The data which is given by the historian is used to find the population, variable, parameter, sample, data, and statistic. He estimated the mean number of children per household in New York City in 1900.
Population - all of the households in New York City in 1900.
Variable - a quantity equivalent to the number of children in a randomly selected household
Parameter - the mean number of children per household in New York City in 1900
Sample - the 200 households selected by the historian
Data - the 200 records collected by the historian
Statistic - the mean number of children per household in the historian's sample.
Complete question:
Suppose a historian wants to estimate the true mean number of children per household in New York City in 1900. The historian randomly selects 200 household census records from New York City. She records the number of children residing in each household. The historian then computes the mean number of children per household among these 200 households. Classify each description as one of a parameter, variable, population, sample, data, or statistic.
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Pls help me do this i need hwlp if any one can figure if out
Considering that the quadrilateral EFGH is similar to quadrilateral IJKL the side LI is calculated to be
30.6How to find the side LIThe side LI is calculated using the concept of similar polygons. This concept allows for the sides to be proportionally equal to each other.
This implies that the factor used for any side will produce same result
let the factor be r and using side HG and JK to solve for the factor r
HG * r = JK
where
HG = 16
JK = 51
substituting the values and solving for r
HG * r = JK
16 * r = 51
r = 51/16
and HE * r = LI
where HE = 9.6
LI = 9.6 * 51/16
LI = 30.6
hence side LI = 30.6
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what is the next term in this sequence of cube numbers?
1,8,27,64...
Answer: 125
Step-by-step explanation:
The first term is 1³. The second term is 2³. The third term is 3³. And the Fourth term is 4³.
So the pattern is n term = n³
plug in 5
= 5³
= 125
HELP PLEASE IVE BEEN STUCK ON THIS PROBLEM ALL DAY
The table that is a justification for the given solution to the system of equations is table 1
How to determine the table that bears the correct values?The two linear equations are
y=4x-9 and
y=2x-5
We that the following points for x : -1, 0, 1, 2, 3
When we substitute the values in the two equations we will have
x y₁ y₂
-1 -13 -9
0 -9 -5
1 -5 -3
2 -1 - 1
3 3 1
Conclusively the values of y₁, and y₂ are shown in the table above
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Blank times 9 minus blank times 4 equals blank minus blank equals blank. You have to use the numbers 9,7,4,3 only.
The value for the blanks after calculating the as per the given data is found to be as 4 and 9 respectively.
On solving the question we will get the answer of the final blank as 0.
The question can be rephrased as ,
__x 9 - __ x 4 = ___ - ___ = ___.
Now if we will place 4 in first blank and 9 in second blank we will get the result as follows,
4 x 9 - 9 x 4 = 36 - 36 = 0
Therefore , we will get our desired answer.
A relationship between two quantities or, more broadly, two mathematical expressions that asserts that the quantities have the same value or that the expressions reflect the same mathematical object is referred to as equality.
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A phone company offers two monthly charge plans. In Plan A, the customer pays a monthly fee of $20 and then an additional 7 cents per minute of use. In Plan B, the customer pays a monthly fee of $26.30 and then an additional 4 cents per minute of use. For what amounts of monthly phone use will Plan A cost no more than Plan B? Use m for the number of minutes of phone use, and solve your inequality for m.
34.7 is the monthly phone use will Plan A cost no more than Plan B and 2.1 is the value of m in the inequality.
What is Inequality?a relationship between two expressions or values that are not equal to each other is called 'inequality.
In Plan A, the customer pays a monthly fee of $20 and then an additional 7 cents per minute of use
20+7m
In Plan B, the customer pays a monthly fee of $26.30 and then an additional 4 cents per minute of use
26.3+4m
We need to find the amounts of monthly phone use will Plan A cost no more than Plan B
20+7m<26.3+4m
7m-4m<26.3-20
3m<6.3
m<2.1
Hence, 34.7 is the monthly phone use will Plan A cost no more than Plan B and 2.1 is the value of m in the inequality.
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Why do we use similarity transformation?
The goal of using similarity transformations is to make the evaluation of a matrix's eigenvalues less complicated. In fact, the computation of the eigenvalues would be instantaneous if a given matrix could be translated into a similar matrix in diagonal or triangular form.
A geometric similarity or a matrix transformation that produces a likeness are both covered by the term "similarity transformation."
A conformal mapping with a similarity transformation is one whose transformation matrix A' has the formula A'=BAB⁻¹, where A and A' are referred to as similar matrices.
Space-based items are converted to equivalent objects by similarity transformations. Fractals and iterated function systems have a lot to do with similarity transformations and the idea of self-similarity.
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What is the equation of a line with a slope of 3 and y-intercept equal to 5?
The equation of a line is given by y = mx + b, where m is the slope and b is the y-intercept. Given that the slope is 3 and the y-intercept is 5, the equation of the line is y = 3x + 5.
The equation of a line is a mathematical expression of the form y = mx + b, where m is the slope and b is the y-intercept. This equation can be used to describe the relationship between the x-coordinate and y-coordinate of points on a line. In this specific equation, the slope is 3 and the y-intercept is 5, which means that the line passes through the point (0, 5) and has a slope of 3. Substituting 3 for m and 5 for b in the equation, we get y = 3x + 5, which is the equation of the line with a slope of 3 and a y-intercept of 5. We can use this equation to determine the y-coordinate of any given point on the line, provided we know the x-coordinate of the point. It is important to note that this equation is only applicable to straight lines, and not curves.
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GEOMETRY:
I might just be slow but I don't understand what this question is asking at all. Can someone help?
Answer:
[tex]b = 90[/tex]
[tex]d = 100[/tex]
[tex]c = 80[/tex]
Step-by-step explanation:
The problem is asking for the values of the variables [tex]b[/tex], [tex]c[/tex], and [tex]d[/tex] in the given parallelogram with angles [tex]d\textdegree[/tex], [tex]c\textdegree[/tex], [tex](b-10)\textdegree[/tex], and [tex](b+10)\textdegree[/tex].
To solve for these variables, we can construct three equations using the knowledge that any two adjacent angles of a parallelogram are supplementary (their measures add to 180º).
The first equation shows that the top two angles are supplementary:
[tex]d\textdegree + c\textdegree = 180\textdegree[/tex]
The second equation shows that the bottom two angles are supplementary:
[tex](b-10)\textdegree + (b+10)\textdegree = 180\textdegree[/tex]
The third equations shows that the left two angles are supplementary:
[tex]d\textdegree + (b-10)\textdegree = 180\textdegree[/tex]
First, we can solve for [tex]b[/tex] in the second equation using simple algebraic manipulation.
[tex](b-10)\textdegree + (b+10)\textdegree = 180\textdegree[/tex]
[tex](b+b)\textdegree + (10-10)\textdegree = 180\textdegree[/tex]
[tex]2b\textdegree = 180\textdegree[/tex]
[tex]b\textdegree = 90\textdegree[/tex]
[tex]b = 90[/tex]
Using this [tex]b[/tex]-value, we can solve for [tex]d[/tex] by rewriting the third equation, plugging [tex]90[/tex] in for [tex]b[/tex]:
[tex]d\textdegree + (b-10)\textdegree = 180\textdegree[/tex]
[tex]d\textdegree + (90-10)\textdegree = 180\textdegree[/tex]
[tex]d\textdegree + 80\textdegree = 180\textdegree[/tex]
[tex]d\textdegree = 100\textdegree[/tex]
[tex]d = 100[/tex]
Finally, we can solve for [tex]c[/tex] in the first equation by plugging [tex]100[/tex] in for [tex]d[/tex]:
[tex]d\textdegree + c\textdegree = 180\textdegree[/tex]
[tex]100\textdegree + c\textdegree = 180\textdegree[/tex]
[tex]c\textdegree = 80\textdegree[/tex]
[tex]c = 80[/tex]
Solve the differential equation by variation of parameters.
y'' − y = cosh x
The general solution is [tex]$$y=c_3 e^x+c_4 e^{-x}+\frac{1}{2} x \sinh x \text {. }$$[/tex]
Consider the following differential equation: y'' − y = cosh x
The given differential equation in the operator form will be: (D^2-1)y=coshx
The auxiliary equation of the corresponding homogeneous equation will be:
m^2-1 = 0
=>m^2=1
=>m=+1,-1
So, the complementary function will be [tex]y_c=c_1 e^x+c_2 e^{-x}.\alpha[/tex]
Now, let [tex]y_1=e^x, y_2=e^{-x}.[/tex]
Compute the Wronskian determinant:
[tex]$$\begin{aligned}W\left(y_1, y_2\right) & =\left|\begin{array}{ll}y_1 & y_2 \\y_1^{\prime} & y_2^{\prime}\end{array}\right| \\& =\left|\begin{array}{ll}e^x & e^{-x} \\e^x & -e^{-x}\end{array}\right| \\& =-e^x e^{-x}-e^x e^{-x} \\& =-2\end{aligned}$$[/tex]
Since the given differential equation is already in the form [tex]$y^{\prime \prime}+P(x) y^{\prime}+Q(x) y=f(x)$[/tex] (that is, the coefficient of y'' is 1 ),
So identify:
f(x)=cosh x
Then, calculate the values of u_1' and u_2', using formulas as follows:
[tex]$$\begin{aligned}u_1^{\prime} & =-\frac{y_2 f(x)}{W} & u_2^{\prime} & =\frac{y_1 f(x)}{W} \\& =-\frac{e^{-x} \cdot \cosh x}{-2} & & =\frac{e^x \cdot \cosh x}{-2} \\& =\frac{e^{-x} \cdot\left(\frac{e^x+e^{-x}}{2}\right)}{2} & \text { and } & =-\frac{e^x \cdot\left(\frac{e^x+e^{-x}}{2}\right)}{2} \\& =\frac{1}{4}\left(1+e^{-2 x}\right) & & =-\frac{1}{4}\left(e^{2 x}+1\right)\end{aligned}$$[/tex]
Integrate on both sides of u_1' and u_2' with respect to ' x', to get:
[tex]$$u_1=\frac{x}{4}-\frac{e^{-2 x}}{8} \text { and } u_2=-\frac{x}{4}-\frac{e^{2 x}}{8} .$$[/tex]
So, the particular solution is,
[tex]$$\begin{aligned}y_p & =u_1 y_1+u_2 y_2 \\& =\left(\frac{x}{4}-\frac{e^{-2 x}}{8}\right) e^x+\left(-\frac{x}{4}-\frac{e^{2 x}}{8}\right) e^{-x} \\& =\frac{1}{4} x\left(e^x-e^{-x}\right)-\frac{1}{8}\left(e^{-x}+e^x\right) \\& =\frac{1}{2} x \sinh x-\frac{1}{8}\left(e^{-x}+e^x\right)\end{aligned}$$[/tex]
Thus, the general solution of the given differential equation is,
[tex]$$\begin{aligned}y & =y_c+y_p \\& =c_1 e^x+c_2 e^{-x}+\frac{1}{2} x \sinh x-\frac{1}{8} e^{-x}-\frac{1}{8} e^x \\& =\left(c_1-\frac{1}{8}\right) e^x+\left(c_2-\frac{1}{8}\right) e^{-x}+\frac{1}{2} x \sinh x \\& =c_3 e^x+c_4 e^{-x}+\frac{1}{2} x \sinh x\end{aligned}$$[/tex]
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