The mpg and hp is print(paste("The car with the best combination of mpg and hp is",rownames(myCars[myCars$combination==max(myCars$combination),])))
MPG is an abbreviation for 'miles-consistent with gallon and it is used to give you an indication of the fuel financial system of a vehicle or industrial vehicle. It helps you to work out how reasonably-priced or high-priced an automobile might be to run.
As for a new vehicle, an awesome MPG is around 50 to 60. Having this stepped forward gasoline economy facilitates minimizing going for walks fees and cheaper avenue tax due to decreased CO2 emissions. consistent with What car's featured blog, here are a number of the most gasoline-green automobiles in the marketplace nowadays.
The gas economic system of a car relates to the space traveled via an automobile and the quantity of fuel fed on. intake can be expressed in terms of the volume of gasoline to travel a distance, or the gap traveled according to the unit quantity of gasoline consumed.
The car has the “best” combination of mpg and hp, where mpg and hp must be given equal weight.
#sum the scaled values of mpg and hp and add a new column called combination
myCars$combination<-scale(myCars$hp)+scale(myCars$mpg)
#Which car has the best combination of mpg and hp?
print(paste("The car with the best combination of mpg and hp is",rownames(myCars[myCars$combination==max(myCars$combination),])))
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give the volume of the cylinder below
Answer:
B. 126 [tex]\pi[/tex]
Step-by-step explanation:
The volume for a cylinder is: Area of the base x Height
The base is a circle with a diameter of 6cm.
The formula to find the area of the base is: radius x radius x pi
In this case, the radius is 3 (6/2) so the area will be 9pi.
Now that we know that area of the base, all we need to do is multiply the height, which is 14cm.
9 pi x 14 = 126 pi
So B. 126 pi is the correct answer!
Answer:
126π
option B is the correct answer
determine the quotient of 2/3 divided by 4/5
The question is down below. Which option i correct?
Answer:
Step-by-step explanation:
overall the answer is wrong i say it is the first one
Answer:
The first one is theost plausible because root 16 is 4, 30 + 4 is 34, the actual answer is 25.38
Wayne made cookies. He used 4/5 of a cup of flour and 2/5 of a cup of sugar. How much more flour than sugar did Wayne use?
Write your answer as a fraction or as a whole or mixed number.
Answer:
Wayne used 2/5 more flour than sugar
Step-by-step explanation:
If you add 2/5 to 2/5 you get 4/5
peaches are 1.79 a pound at the grocery store.if kim bought 3.5 pounds of peaches how much did she spend?
please help me 15 points
Three students want to estimate the mean word length of the same book. To do this, each student randomly chose 4 words from the book and recorded their lengths. The samples are shown in the table. (a)Fill in the sample means in the table. Do not round your answers. Sample Word length (number of letters) Sample means 1 - 5, 6, 6, 2
2 - 4, 8, 4, 3
3 - 4, 2, 6, 5 (
b)Use the table to calculate the range of the sample means. Rangeofsamplemeans:
(c)The students are going to use the sample means to estimate the mean word length in the book. Select all the true statements below. The closer the range of the sample means is to 0, the more confident they can be in their estimate. The farther the range of the sample means is from 0, the more confident they can be in their estimate. The mean of the sample means will tend to be a better estimate than a single sample mean. A single sample mean will tend to be a better estimate than the mean of the sample means.
The range of the means is 0.5
How to find the meanMean of S1
= 5 + 6+ 6+ 2
= 19/4
= 4.75
Mean of S2
= 4 + 8 + 4 + 3
= 19/4
= 4.75
Mean of s3
= 4 + 2 + 6+ 5
= 17/4
= 4.25
Range of sample means = 4.75 - 4.25
= 0.5
c. the true statements here is that
The closer the range of the sample means is to 0, the more confident they can be in their estimate. The mean of the sample means will tend to be a better estimate than a single sample mean.Read more on mean here:
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Determine the factors of a2b 2a2 3b 6. (b 3)(a2 2) (b 2)(a2 3) (ab 2)(a 3) (ab 3)(a 2)
The factor of [tex]$a^{2} b+2 a^{2}+3 b+6$[/tex] exists [tex]$\left(a^{2}+3\right)(b+2)$$[/tex].
How to determine the factor of [tex]$a^{2} b+2 a^{2}+3 b+6$[/tex]?
Let the given factor be [tex]$a^{2} b+2 a^{2}+3 b+6$[/tex]
To factor this we use the grouping method
We group the first two terms and last two terms then we factor out the greatest common factor (GCF) from each group.
[tex]$\left(a^{2} b+2 a^{2}\right)+(3 b+6)$$[/tex]
Take out GCF from each group
[tex]$a^{2}(b+2)+3(b+2)$$[/tex]
Now factor out b+2, we get
[tex]$\left(a^{2}+3\right)(b+2)$$[/tex]
The factor of [tex]$a^{2} b+2 a^{2}+3 b+6$[/tex] exists [tex]$\left(a^{2}+3\right)(b+2)$$[/tex].
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Find the term that must be added to the equation x2 6x=1 to make it into a perfect square.
The value added to the equation [tex]$x^2-6x=1[/tex] exists [tex]$x^2-6x+9=10[/tex].
What is a perfect square?
A perfect square exists as a number that can be described as the product of an integer by itself or as the second exponent of an integer.
The perfect square trinomial exists
[tex](a[/tex] ± [tex]b)^2[/tex] = [tex]a ^2[/tex] ± 2ab + [tex]b ^2[/tex]
[tex]$x^2-6x=1[/tex]
[tex]x^2-2*3x=1[/tex]
then [tex]$2ab = 2 * 3*x = 2 * x *3[/tex]
The value of a = x and b = 3
[tex]$b^2=3^2=9[/tex]
[tex]$x^2-6x+9=1+9[/tex]
[tex]$x^2-6x+9=10[/tex]
The value added to the equation [tex]$x^2-6x=1[/tex] exists [tex]$x^2-6x+9=10[/tex].
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When conducting a hypothesis test for a given sample size, if the probability of a type i error decreases, then the __________.
Based on the type I error, if a hypothesis test leads to the probability of a type I error decreasing, then the a. probability of incorrectly accepting the null hypothesis decreases.
What is a type I error?A type I error refers to when we are are engaged in a hypothesis test and end up rejecting the Null hypothesis even though it is true.
If we are in hypothesis tests and the probability of a type 1 error decreases, then it means that the probability of us correctly rejecting the null hypothesis when it is false increases.
It also means that the probability of us correctly accepting the null hypothesis when it is true increases as well.
This can further be said as the probability of us accepting the null hypothsis when it is false, decreases. There are therefore less chances of us incorrectly accepting the null hypothesis.
Options for this question include:
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Given w = 10, what is the answer to w + (w – 12 ÷ 22 • 3) • 7? (1 point) 73 66 23 17
Answer:
68.54
Step-by-step explanation:
w + (w – 12 ÷ 22 • 3) • 7
10 + (10 – 12 ÷ 22 • 3) • 7
Parentheses first
(10 – 12 ÷ 22 • 3)
(10 – 36 ÷ 22)
(10 – 1.636)
(8.364)
-------
10 + (8.364) • 7
10 + 58.54
68.54
The value of the expression will be equal to 66.54. The correct option is B.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the expression is w + (w – 12 ÷ 2^ 2• 3) • 7 and the value of w is 10.
By substituting the value of w equal to 10 the value of the expression is,
E = w + (w – 12 ÷ 22 • 3) • 7
E = 10 + (10 – 12 ÷ 22 • 3) • 7
Solve the Parentheses first.
P = (10 – 12 ÷ 22 • 3)
P = (10 – 36 ÷ 22)
P = (10 – 1.636)
P = (8.364)
Solve further to get the final value.
E = 10 + (8.364) • 7
E = 10 + 56.54
E = 66.54.
Therefore, the value of the expression will be equal to 66.54.
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Find the surface area of the composite figure
Step-by-step explanation:
I answered this already yesterday.
the composite figure is actually the combination of 2 figures :
1. a 7cm × 6cm × 2cm box (purple)
2. a 8cm × 7cm × 6cm triangular shaped half-box (pink) with 10cm length of the rectangular "roof".
2 sides are completely blocking each other, so they are not part of the combined surface area.
let's start with the purple box. its contribution to the surface area is :
top and bottom 7×2 rectangles
front and back 6×2 rectangles
no left (blocked by the half-box)
right 6×7 rectangle
so, we get
2 × 7×2 = 2×14 = 28 cm²
2 × 6×2 = 2×12 = 24 cm²
6×7 = 42 cm²
in total that is : 94 cm²
the half-box contributes to the surface area :
top 10×7 rectangle
bottom 8×7 rectangle
front and back 8×6/2 triangles
no left (due to the triangular shape)
no right (blocked by the box)
so, we get
10×7 = 70 cm²
8×7 = 56 cm²
2 × 8×6/2 = 2×24 = 48 cm²
in total that is : 174 cm²
and so, the total surface area of the composite figure is
174 + 94 = 268 cm²
You deal 7 cards off of a 52-card deck and line them up in a row. how many possible lineups are there in which no card is a club?
Answer:
39C7 or 15380937
Step-by-step explanation:
There are 13 clubs in a 52 card deck. We can remove these from the possibilities. 52-13=39. The rest is simple, choose 7 from 39 using the choose function is 39!/(32!*7!)=15380937. Or just say 39C7 as your answer.
What is the slope of the line through (1,-1)(1,−1)left parenthesis, 1, comma, minus, 1, right parenthesis and (5,-7)(5,−7)left parenthesis, 5, comma, minus, 7, right parenthesis? Choose 1 answer: Choose 1 answer: (Choice A) A -\dfrac32− 2 3 minus, start fraction, 3, divided by, 2, end fraction (Choice B) B \dfrac23 3 2 start fraction, 2, divided by, 3, end fraction (Choice C) C \dfrac32 2 3 start fraction, 3, divided by, 2, end fraction (Choice D) D -\dfrac23− 3 2
The answer choices which represents the slope of the line described by the points, (1,-1) and (5,-7) is; Choice A; -3/2.
What is the slope of the line passing through the points; (1,-1) and (5,-7)?The line in discuss as described in the task content is passing through the points; (1,-1) and (5,-7).
Hence, it follows conventionally from the slope formula that the slope of the line is;
slope, m = (-7-(-1))/(5-1)
m = -6/4
m = -3/2.
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Here are yesterday's high temperatures (in Fahrenheit) in 12 U.S. cities. 48, 50, 54, 56, 63, 63, 64, 68, 74, 74, 79, 80 Notice that the temperatures are ordered from least to greatest. Give the five-number summary and the interquartile range for the data set. Five-number summary
Minimum:
Lower quartile:
Median:
Upper quartile:
Maximum:
Interquartile range:
The five number summary and interquartile range for the data-set is given by:
Minimum: 48Lower quartile: 54.Median: 63.5.Upper quartile: 74Maximum: 80.Interquartile range: 20How to find the five number summary and interquartile range of the data-set?The five number summary is composed by the minimum and maximum value, and the first quartile, median and third quartile. As for each of these data, they are explained below.
The minimum value is the smallest value from the data-set, as the maximum value is the greatest value of the data-set.The median of the data-set separates the bottom half from the upper half, that is, it is the 50th percentile.The first quartile is the median of the first half of the data-set.The third quartile is the median of the second half of the data-set.The interquartile range is the difference of the third quartile and the first quartile.In this problem, we have that:
The minimum value is the smallest value, of 48.The maximum value is the smallest value, of 80.The data-set has even cardinality, hence the median is the mean of the middle elements, which are 63 and 64, hence the median is of 63.5.The first quartile is the median of the five elements of the first half, hence it is of 54.The third quartile is the median of the five elements of the second half, hence it is of 74.The IQR is the difference between the quartiles, hence 74 - 54 = 20.More can be learned about five number summaries at brainly.com/question/17110151
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Nereo and Azeneth start recording video at the same time. Nereo's phone starts with 1,0001{,}0001,0001, comma, 000 megabytes (MB\text{MB}MBstart text, M, B, end text) of free space and uses 7 MB7\,\text{MB}7MB7, start text, M, B, end text of space per second of video. Azeneth's phone has 1,255 MB1{,}255\,\text{MB}1,255MB1, comma, 255, start text, M, B, end text of free space and uses 10 MB10\,\text{MB}10MB10, start text, M, B, end text of space per second of video. How much free space will they have when they first have the same amount of space left?
Using a linear function, it is found that they will have 405 MB of free space when they first have the same amount of space left.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.The free space for each of Nereo and Azeneth can be modeled by a linear function, in which the slope is negative with the amount that each second of video takes and the y-intercept is the initial free space. Her the functions for the amount of free space after x seconds are given by:
N(x) = 1000 - 7x.A(x) = 1255 - 10x.They will have the same amount of free space when:
N(x) = A(x)
1000 - 7x = 1255 - 10x
3x = 255
x = 255/3
x = 85.
The space will be of:
N(85) = 1000 - 7 x 85 = 405 MB.
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Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.
What is the distance between points F and G?
A number line ranges from minus 5 to 5 with increment of 1 unit. Left, Point F is plotted on the number line at minus 3. Right, point G is plotted on the number line at 1.50.
The distance between F and G is
.
A number line ranges from minus 5 to 5 with increment of 1 unit. Left, Point F is plotted on the number line at minus 3. Right, point G is plotted on the number line at 1.50, then the distance between F and G is 4.50.
Calculating the Distance between F and G:
It is given that, on a number line,
Point F is located at -3
Point G is at 1.50
Since, 1.5 > (-3) and each number is at a 1 unit difference form the adjacent number on the number line,
The difference between F and G = 1.50 - ( -3 )
This implies that the distance between F and G = 1.50 + 3
= 4.50 units
The numbers lying between the points F and G on the number line are: -2, -1, 0, and 1
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Answer: 4 1/2
Step-by-step explanation:
Got it right
a sphere is inscribed in a cylinder use complete sentences and geometric formulas to compare the surface area of the sphere and the lateral area of the cylinder
Answer:
They are the same
Step-by-step explanation:
LA cyl =
[tex]2\pi \: rh[/tex]
SA sphere =
[tex]4\pi {r}^{2} [/tex]
If the sphere is inscribed in the cylinder, they have the same radius. The height of the cylinder is the diameter (Or 2× radius) of the sphere.
If you substitute 2r for h, then both formulas are
[tex] 4\pi{r}^{2} [/tex]
Measures of central tendency are called such because __________. a. they all find a different "center" of a data set b. they all find approximately the same "center" of a data set c. they are all different ways to find the exact same "center" of a data set d. they all find a different type of "center" of a data set, which may or may not be the same value
Measures of central tendency are called such because they all find approximately the same "center" of a data set (option B).
What are measures of central tendency?A measure of central tendency attempts to describe a data set by determining the central value of the data set. Measures of central tendency are mean, median and mode
Mean is the average of a set of numbers. It is determined by adding the numbers together and dividing it by the total number.
Median can be described as the number that occurs in the middle of a set of numbers that are arranged either in ascending or descending order.
Mode refers to a value that appears most frequently in a data set.
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What are the solutions to this quadratic equation x2+6x-5=0
Answer:
See Below.
Explanation:
Quadratic Form:
[tex]a {x}^{2} + bx + c[/tex]
Quadratic Formula:
[tex]x = \frac{ - b + - \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]
Based on the information given from the question, we can deduce:
a = 1
b = 6
c = -5
Now we can substitute all these values into the formula to find x.
[tex]x = \frac{ - 6 + - \sqrt{ {6}^{2} - 4(1)( - 5)} }{2(1)} \\ = - 3 + \sqrt{14} \: or \: - 3 - \sqrt{14} [/tex]
Find all solutions of the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.A) 3 sec2(θ) − 4 = 0B) cot(θ) + 1 = 0
The solution of the assumed equation is
θ = 135 + 360k
and
θ = -45 + 360k (or 315 + 360k)
Assuming the equation is
csc^2(θ) = 2cot(θ) + 4
and not
Assuming the equation to be:
csc^2(θ) = cot^2(θ) + 1
Solving these equations usually begins with algebra and/or trigonometry. ID for transforming equations to have one or more equations of the form:
trigfunction(expression) = number
It is not always easy to understand how to perform the desired transformation. If it's not obvious, first use the trigonometric identity to reduce the number of different arguments or functions in the equation. In this expression, all arguments are θ. Therefore, there is no need to reduce the number of arguments. But he has two different functions, csc and cot.
csc^2(θ) = cot^2(θ) + 1
Substituting the right side of this equation into the left side of the equation, we get: :
cot^2 (θ) + 1 = 2 cot(θ) + 4
Now that we have only one function cot and one argument θ, we are ready to find the form we need. Subtracting the entire right-hand side from both sides gives:
cot^2(θ) - 2cot(θ) - 3 = 0
The left-hand side factorizes:
(cot(θ)-3)(cot (θ) ) + 1 ) = 0
Using the properties of the zero product,
cot(θ) = 3 or cot(θ) = -1
These two equations are now in the desired form.
The next step is to write the general solution for each equation. The general solution represents all solutions of the equation.
cot(θ) = 3
Note that 3 is not a specific angle value for cot. That's why you need a calculator. Your computer probably doesn't have the crib button, so you'll need to switch it to tan
Since tan is the reciprocal of cot, if cot = 3...
tan(θ) = 1/3
Inverse tan tan^-1(1/3) can be used to find the reference angle. You should get a reference angle of 18.43494882 degrees. Using this reference angle and cot (and tan) being positive in the 1st and 3rd quadrants, we get the general solutions
θ = 18.43494882 + 360k
and
θ = 180 + 18.43494882 + 360k
.
θ = 198.43494882 + 360k
where
cot(θ) = -1
-1 should be recognized as a special angle value for cot. So you don't need a calculator. This reference angle is 45 degrees. Using this reference angle, cot is negative in the 2nd and 4th quadrants, so
θ = 180 - 45 + 360k
and
θ = -45 + 360k (or 360 - 45 + 360k) the general solution for
must get. to:
θ = 135 + 360k
and
θ = -45 + 360k (or 315 + 360k)
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According to the Coase theorem, negative externalities can be internalized if Select one: a. the government takes action to solve the problem. b. property rights are assigned to the party who is being damaged. c. property rights are assigned to either party. d. property rights are assigned to the party who is doing the damage.
Answer:
B. Property right are assigned to the party who is being
A quadrilateral must be a parallelogram if one pair of opposite sides is: a. congruent and the other pair is parallel b. congruent and parallel c. parallel d. congruent
Answer:
b
Step-by-step explanation:
the opposite sides of a parallelogram are parallel and congruent
thus option b is the correct one
!!!!!50 POINTS IF AND BRAINLEST IF ANSWERED CORRECTLY !!!!!!
By the end of the summer, Tammy and Keith had mowed the same number of lawns and raked the same
number of yards. Keith had met his goal of earning more than $180 but Tammy did not meet her goal of
earning more than $200.
B.) What is a possible combination of the number of lawns they could have each mowed and the
number of yards they could have raked? Explain how you chose the combination and why you
know that combination is correct.
1) The Inequalities for Tammy and Keith are respectively; For Tammy: 10x + 5y > 200 and for Keith: 12x + 3y > 180
2) The possible combination of the number of lawns they could have each mowed and the number of yards they could have raked are;
(11, 17), (14, 5), (16, 0), (18, 2)
How to create Inequalities?Tammy earns $10 for each lawn mowed.
Tammy earns $5 for each yard raked.
We are told that She wants to earn more than $200 from her part-time jobs. Thus, the inequality for Tammys would be;
10x + 5y > 200
Keith earns $12 for each lawn mowed
Keith earns $3 for each yard raked.
We are told that Keith wants to earn more than $180 for each part time job. Thus, the inequality is;
12x + 3y > 180
B) To know the possible combination of the number of lawns they could have each mowed and the number of yards they could have raked, i have drawn a graph of both inequalities and the possible points from there are;
(11, 17), (14, 5), (16, 0), (18, 2)
The complete question is;
Tammy and Keith each work two part-time jobs in the summer mowing lawns andraking yards . Tammy earns $10 for each lawn she mows and $5 for each yard sherakes . She wants to earn more than $200 from her part-time jobs . Keith earns $12 for each lawn he mows and $3 for each yard he rakes. He wants to earn more than $180 for each part time job.
A) Write a system of Inequalities to model the number of lawns they each mow (x) and the number of yards they each rake.
B.) What is a possible combination of the number of lawns they could have each mowed and the number of yards they could have raked? Explain how you chose the combination and why you know that combination is correct.
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Solve. if there is more than one solution, separate them with comas. (y-4)(y+7)=0
Answer:
y= 4
Step-by-step explanation:
(y-4) (y+7)=0
(4-4) (4+7)=0
(0) (11)=0
11*0=0
Match the scenarios to their corresponding boundaries.
The correct choices based on the integers are: 1a, 2b, 3d, and 4c.
In the question, we are asked to match the scenarios to their corresponding boundaries.
miles traveled by a car in one hour: this can be non-negative numbers as the distance traveled by a car cannot be negative, but it is not infinitely possible, making the right option a. numbers between 0 and 70.average Celsius temperature in Antarctica: is mostly negative, and even if positive, very low. making the right option b. numbers between -100 and 20.amount of money owed on a car: this can be any non negative number without limit, making the right option d. no negative numbers.age when a baby takes their first step: as its an age it wont be a negative number, and its a baby age so it will be very small, making the right option, c. no negative numbers and positive numbers less that 2.Thus, the correct choices based on the integers are: 1a, 2b, 3d, and 4c.
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Find the values of x and y.
Write answers in simplest radical form.
x=______ y=_____
Using the Geometric mean theorem and the Pythagorean theorem:
x = 3√3, and y = 6.
What is the Right Triangle Altitude Theorem/Geometric Mean Theorem?The Geometric mean theorem or the right triangle altitude theorem states that the geometric mean of the the two segments equals he length of the altitude of a right triangle.
The geometric mean theorem is expressed by the equation, h = √(ab), where:
a is the length of one segmentb is the length of the other segmenth is the length of the altitude of the right triangle.Given the following:
h = x
a = 9
b = 3
x = √(9 × 3)
x = 3√3
Find y using the Pythagorean theorem
Based on the Pythagorean theorem, we would have the following:
y = √(x² + 3²)
y = √((3√3)² + 3²)
y = √(27 + 9)
y = √(36)
y = 6
Thus, in the simplest radical form, the value of x = 3√3, and y = 6.
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Determine what type of model best fits the given situation: an internet phone company presently provides service to 5,000 customers at a monthly rate of $20 per month. after a market survey, it was determined that for each $1 decrease in the monthly rate an increase of 500 new customers would result.
The expression [tex]$-500 \times 18+15000=-9000+15000=6000[/tex] which best fit exists in the Linear model.
How to estimate the linear model?
Given: Monthly Rate = $20
Number of customers = 5000
If there exists a decrease of $1 in the monthly rate, the number of customers increases by 500.
Let us decrease the monthly rate by $1.
Monthly Rate = $20 - $1 = $19
Number of customers = 5000 + 500 = 5500
Let us decrease the monthly rate by $1 more.
Monthly Rate = $19 - $1 = $18
Number of customers = 5500 + 500 = 6000
Linear change in the number of customers whenever there exists a decrease in the monthly rate.
We have 2 pairs of values here,
x = 20, y = 5000
x = 19, y = 5500
The equation in slope-intercept form: y = mx + c
The slope of a function: [tex]$&m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\[/tex]
[tex]$&m=\frac{5500-5000}{19-20} \\[/tex]
[tex]$&\Rightarrow-500[/tex]
So, the equation is y = -500x + c
Putting x = 20, y = 5000:
[tex]$&5000=-500 \times 20+c \\[/tex]
[tex]$&\Rightarrow c=5000+10000=15000 \\[/tex]
[tex]$&\Rightarrow \mathbf{y}=-500 \mathbf{x}+15000[/tex]
Whether (18,6000) satisfies it.
Putting x = 18
[tex]$-500 \times 18+15000=-9000+15000=6000[/tex]
Therefore, the expression [tex]$-500 \times 18+15000=-9000+15000=6000[/tex] which best fits exist Linear model.
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Solve the following initial value problem.
d²s
dt²
= -36cos(6t+n), s'(0) = 100, s(0) = 0
S=
(Type an exact answer, using * as needed.)
For starters,
[tex]\cos(6t+\pi) = \cos(6t) \cos(\pi) - \sin(6t) \sin(\pi) = -\cos(6t)[/tex]
Now by the fundamental theorem of calculus, integrating both sides gives
[tex]\displaystyle \frac{ds}{dt} = s'(0) + \int_0^t 36 \cos(6u) \, du = 100 + 6 \sin(6t)[/tex]
Integrating again, we get
[tex]\displaystyle s(t) = s(0) + \int_0^t (100 + 6\sin(6u)) \, du = \boxed{100t - \cos(6t) + 1}[/tex]
Alternatively, you can work with antiderivatives, then find the particular constants of integration later using the initial values.
[tex]\displaystyle \int \frac{d^2s}{dt^2} \, dt = \int 36\cos(6t) \, dt \implies \frac{ds}{dt} = 6\sin(6t) + C_1[/tex]
[tex]\displaystyle \int \frac{ds}{dt} \, dt = \int (6\sin(6t) + C_1) \, dt \implies s(t) = -\cos(6t) + C_1t + C_2[/tex]
Now,
[tex]s(0) = 0 \implies 0 = -1 + C_2 \implies C_2 = 1[/tex]
and
[tex]s'(0) = 100 \implies 100 = 0 + C_1 \implies C_1 = 100[/tex]
Then the particular solution to the IVP is
[tex]s(t) = -\cos(6t) + 100t + 1[/tex]
just as before.
pls help with math!!!
Find two power series solutions of the given differential equation about the ordinary point x = 0. y'' + xy = 0
Answer:
First we write y and its derivatives as power series:
y=∑n=0∞anxn⟹y′=∑n=1∞nanxn−1⟹y′′=∑n=2∞n(n−1)anxn−2
Next, plug into differential equation:
(x+2)y′′+xy′−y=0
(x+2)∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0
x∑n=2∞n(n−1)anxn−2+2∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0
Move constants inside of summations:
∑n=2∞x⋅n(n−1)anxn−2+∑n=2∞2⋅n(n−1)anxn−2+∑n=1∞x⋅nanxn−1−∑n=0∞anxn=0
∑n=2∞n(n−1)anxn−1+∑n=2∞2n(n−1)anxn−2+∑n=1∞nanxn−∑n=0∞anxn=0
Change limits so that the exponents for x are the same in each summation:
∑n=1∞(n+1)nan+1xn+∑n=0∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−∑n=0∞anxn=0
Pull out any terms from sums, so that each sum starts at same lower limit (n=1)
∑n=1∞(n+1)nan+1xn+4a2+∑n=1∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−a0−∑n=1∞anxn=0
Combine all sums into a single sum:
4a2−a0+∑n=1∞(2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an)xn=0
Now we must set each coefficient, including constant term =0 :
4a2−a0=0⟹4a2=a0
2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an=0
We would usually let a0 and a1 be arbitrary constants. Then all other constants can be expressed in terms of these two constants, giving us two linearly independent solutions. However, since a0=4a2 , I’ll choose a1 and a2 as the two arbitrary constants. We can still express all other constants in terms of a1 and/or a2 .
an+2=−(n+1)nan+1+(n−1)an2(n+2)(n+1)
a3=−(2⋅1)a2+0a12(3⋅2)=−16a2=−13!a2
a4=−(3⋅2)a3+1a22(4⋅3)=0=04!a2
a5=−(4⋅3)a4+2a32(5⋅4)=15!a2
a6=−(5⋅4)a5+3a42(6⋅5)=−26!a2
We see a pattern emerging here:
an=(−1)(n+1)n−4n!a2
This can be proven by mathematical induction. In fact, this is true for all n≥0 , except for n=1 , since a1 is an arbitrary constant independent of a0 (and therefore independent of a2 ).
Plugging back into original power series for y , we get:
y=a0+a1x+a2x2+a3x3+a4x4+a5x5+⋯
y=4a2+a1x+a2x2−13!a2x3+04!a2x4+15!a2x5−⋯
y=a1x+a2(4+x2−13!x3+04!x4+15!x5−⋯)
Notice that the expression following constant a2 is =4+ a power series (starting at n=2 ). However, if we had the appropriate x -term, we would have a power series starting at n=0 . Since the other independent solution is simply y1=x, then we can let a1=c1−3c2, a2=c2 , and we get:
y=(c1−3c2)x+c2(4+x2−13!x3+04!x4+15!x5−⋯)
y=c1x+c2(4−3x+x2−13!x3+04!x4+15!x5−⋯)
y=c1x+c2(−0−40!+0−31!x−2−42!x2+3−43!x3−4−44!x4+5−45!x5−⋯)
y=c1x+c2∑n=0∞(−1)n+1n−4n!xn
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