Find a power series representations of the following
functions.
(a) f(x) = tan-1(3x)
(b) f(x) = x^3 / (1+x)^2
(c) f(x) = ln(1 + x)
(d) f(x) = e^(2(x-1)^2)
(e) f(x) = sin (3x^2) / x^3
(f) f(x) = Z e^
Answers
a)power series representation of
\(\[f(x) = \tan^{-1}(3x) = (3x) - \frac{(3x)^3}{3} + \frac{(3x)^5}{5} - \frac{(3x)^7}{7} + \ldots\]\)
b)power series representation of
\(\[f(x) = x^3 - 2x^4 + 3x^5 - 4x^6 + \ldots\]\)
c)power series representation of
\(\[f(x) = \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\]\)
d)power series representation of
\(\[f(x) = e^{2(x-1)^2} = 1 + 2(x-1)^2 + \frac{4(x-1)^4}{2!} + \frac{8(x-1)^6}{3!} + \ldots\]\)
e)power series representation of
\(\[f(x) = \frac{\sin(3x^2)}{x^3} = 3 - \frac{9x^2}{2!} + \frac{27x^4}{4!} - \frac{81x^6}{6!} + \ldots\]\)
f)power series representation of
\(\[f(x) = Z e^x = Z + Zx + \frac{Zx^2}{2!} + \frac{Zx^3}{3!} + \ldots\]\)
What is power series representation?
A power series representation is a way of expressing a function as an infinite sum of powers of a variable. It is a mathematical technique used to approximate functions by breaking them down into simpler components. In a power series representation, the function is expressed as a sum of terms, where each term consists of a coefficient multiplied by a power of the variable.
\((a) $f(x) = \tan^{-1}(3x)$:\)
The power series representation of the arctangent function is given by:
\(\[\tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots\]\)
To obtain the power series representation of \(f(x) = \tan^{-1}(3x)$,\) we substitute \($3x$\) for \($x$\) in the series:
\(\[f(x) = \tan^{-1}(3x) = (3x) - \frac{(3x)^3}{3} + \frac{(3x)^5}{5} - \frac{(3x)^7}{7} + \ldots\]\)
(b)\($f(x) = \frac{x^3}{(1+x)^2}$:\)
To find the power series representation of\($f(x)$\), we expand \($\frac{x^3}{(1+x)^2}$\)using the geometric series expansion:
\(\[\frac{x^3}{(1+x)^2} = x^3 \sum_{n=0}^{\infty} (-1)^n x^n\]\)
Simplifying the expression, we get:
\(\[f(x) = x^3 - 2x^4 + 3x^5 - 4x^6 + \ldots\]\)
(c)\($f(x) = \ln(1+x)$:\)
The power series representation of the natural logarithm function is given by:
\(\[\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\]\)
Thus, for \(f(x) = \ln(1+x)$,\) we have:
\(\[f(x) = \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\]\)
(d)\($f(x) = e^{2(x-1)^2}$:\)
To find the power series representation of \($f(x)$\), we expand \($e^{2(x-1)^2}$\) using the Taylor series expansion:
\(\[e^{2(x-1)^2} = 1 + 2(x-1)^2 + \frac{4(x-1)^4}{2!} + \frac{8(x-1)^6}{3!} + \ldots\]\)
Simplifying the expression, we get:
\(\[f(x) = e^{2(x-1)^2} = 1 + 2(x-1)^2 + \frac{4(x-1)^4}{2!} + \frac{8(x-1)^6}{3!} + \ldots\]\)
(e) \(f(x) = \frac{\sin(3x^2)}{x^3}$:\)
To find the power series representation of \($f(x)$\), we expand \($\frac{\sin(3x^2)}{x^3}$\)using the Taylor series expansion of the sine function:
\(\[\frac{\sin(3x^2)}{x^3} = 3 - \frac{9x^2}{2!} + \frac{27x^4}{4!} - \frac{81x^6}{6!} + \ldots\]\)
Simplifying the expression, we get:
\(\[f(x) = \frac{\sin(3x^2)}{x^3} = 3 - \frac{9x^2}{2!} + \frac{27x^4}{4!} - \frac{81x^6}{6!} + \ldots\]\)
(f)\($f(x) = Z e^x$:\)
The power series representation of the exponential function is given by:
\(\[Z e^x = Z + Zx + \frac{Zx^2}{2!} + \frac{Zx^3}{3!} + \ldots\]\)
Thus, for \($f(x) = Z e^x$\), we have:
\(\[f(x) = Z e^x = Z + Zx + \frac{Zx^2}{2!} + \frac{Zx^3}{3!} + \ldots\]\)
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Related Questions
How many counterexamples does it take to show that a conditional statement is false
Answers
Answer:
A counterexample is sufficient, as long as it is well built and complete.
Step-by-step explanation:
A counterexample is a statement used to show that a previously spoken statement is false.
The counterexample must be direct, clear, simple, well constructed and concise to be effective. If the counterexample has all of these characteristics, only one counterexample is sufficient to show that a conditional statement is flawed.
As an example of this, we can use a square as a counterexample of a statement that states that every rectangle has 4 equal sides.
Given the scatter plot, choose the function that best fits the data.
(See photo attached)
A. f(x) = 2^x
B. f(x) = 2x
C. f(x) = -2x
D. f(x) = 2x^2
Answers
Answer:
A is the answer to your question because it's the only answer in exponential form.
B and C are in slope forms.
D is in quadratic form.
in exercise 4 and 5 find the unkown or marked angles
Answers
Answer:
Ф = 110°, β = 110°
Step-by-step explanation:
∠Ф and the 110° angle are corresponding angles since the vertical lines are parallel and a horizontal line cut across both lines. Thus Ф = 110°.
∠β and ∠Ф are alternate angles since the horizontal lines are parallel and a vertical line cut across both lines. Thus Ф = 110° and ∠β = 110°.
What is the reflexive property of congruence?
Answers
The reflexive property of congruence states that every figure is congruent to itself.
In other words, if a figure is reflected over its center, the resulting image is congruent to the original figure.
For example, consider a triangle. If we reflect the triangle over its center, we get an image that is exactly the same as the original triangle. Therefore, the triangle is congruent to itself.
This property holds for all figures, regardless of their size or shape. Every figure is congruent to itself because it can be reflected over its center to produce an image that is exactly the same as the original.
The reflexive property of congruence states that every figure is congruent to itself.
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Given the function below, what is the value of g(4)?
g(x)=3x2−3x−9
Select one:
A. 15
B. 27
C. 38
D. 56
Answers
Answer:
g(x) = 3x^2-3x-9
g(4) =3(4)^2-3(4)-9
g(4)=3(16)-12-9
g(4)=48-12-9
g(4)=27
So B is the correct answer.
Hope this helps!
Is this function odd, even, or neither
Answers
The function in this problem is neither odd nor even.
What are even and odd functions?In even functions, we have that the statement f(x) = f(-x) is true for all values of x.In odd functions, we have that the statement f(-x) = -f(x) is true for all values of x.If none of the above statements are true for all values of x, the function is neither even nor odd.
For this problem, we have that f(-0.5) = 0 is different of f(0.5) and of -f(0.5), hence we can prove by contradiction that the function is neither odd nor even.
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A while back, either James borrowed $12 from his friend Rita or she borrowed $12 from him, but he can’t quite remember which. Either way, he is planning to pay her back or ask that she pay him back this afternoon. If he has $42.80 in his wallet, which equation represents the amount of money he may have after he sees Rita?
Answers
Answer:
42.8 + or - 12
Step-by-step explanation:
He's either gaining or losing $12 so you can write 42.8 plus or minus 12 as the equation
The follow through is where you "pause" and hold the bow still for a few seconds after releasing the arrow.
True or false
Answers
True-... i think because like i am not sure-
i need hlep on this pls
Answers
It's not a typo that both answers are 24
======================================================
Explanation:
Each time the weight goes up by 1 ounce, the cost goes up by 24 cents. So the unit cost is 24 cents per ounce. This is exactly the slope of the line because the slope is the rate of change. It tells us how fast something is changing.
You can use the slope formula to answer this question, but I think it's easier to simply locate the point (1,24) and then note that slope = rise/run = y/x = 24/1 = 24. This shortcut only works because the line goes through the origin.
The equation of this line is y = 24x
x = number of ounces
y = cost in cents
Step-by-step explanation:both are 24
The sum of the first four terms of an arithmetic progression is 14. If the sum of the first eight terms is 108, find the sixth term of this progression.
Answers
The sixth term of the arithmetic progression is 94.
An arithmetic progression is a sequence of numbers such that the difference between any two successive members of the sequence is a constant.
Let's call the first term of the arithmetic progression "a" and the common difference "d". Then the nth term of the progression can be found using the formula:
an = a + (n-1)d
We know that the sum of the first four terms is 14, so we can set up the following equation:
a + (a+d) + (a+2d) + (a+3d) = 14
We also know that the sum of the first eight terms is 108, so we can set up another equation:
a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) + (a+6d) + (a+7d) = 108
Now we can use the first equation to substitute for the first four terms of the second equation:
14 + (a+4d) + (a+5d) + (a+6d) + (a+7d) = 108
This gives us:
a + 4d = 94
So the sixth term can be found by substituting n = 6 into the first formula:
a + (6-1)d = a + 5d = 94
Therefore, the sixth term of the arithmetic progression is 94.
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Calculate the following probability: Given that a couple has an Education Level = 4, what is the probability that it has SC Index = 10?
o 0.94
o 17
o 0.06
o 0
Answers
The correct option is option C: 0.06.
Given that a couple has an Education Level = 4, the probability that it has SC Index = 10 is 0.06.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number that ranges from 0 to 1.
When an event is certain to occur, its probability is 1, while when an event is impossible to occur, its probability is 0.
The probability of an event A is denoted by P(A). It can be calculated using the following formula: P(A) = (number of favorable outcomes)/(total number of outcomes)
In this question, we need to calculate the probability of the event that a couple has an Education Level = 4 and SC Index = 10.
Given that a couple has an Education Level = 4, there are a total of 50 such couples. Of these, 3 have SC Index = 10.
Therefore, the probability that a couple has an Education Level = 4 and SC Index = 10 is: P(Education Level = 4 and SC Index = 10) = 3/50 = 0.06Hence, the correct option is option C: 0.06.
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algernon (customer) [00:00:00] a school system is reducing the amount of dumpster loads of trash removed each week. in week 5, there were 60 dumpster loads of waste removed. in week 10, there were 40 dumpster loads removed. assume that the reduction in the amount of waste each week is linear. write an equation in function form to show the amount of trash removed each week.
Answers
The equation in function form to show the amount of trash removed each week is:
y = -4x + 100, where y represents the number of dumpster loads and x represents the week number.
To find the equation in function form, we need to determine the slope and y-intercept of the linear relationship between the week number and the amount of trash removed.
Given that the reduction in the amount of waste each week is linear, we can use the two data points provided: (5, 60) and (10, 40).
Using the slope-intercept form of a linear equation (y = mx + b), we can calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values from the data points:
m = (40 - 60) / (10 - 5)
m = -20 / 5
m = -4
Now, we can determine the y-intercept (b) by substituting the slope (m) and the coordinates of one of the data points into the slope-intercept form:
60 = -4(5) + b
60 = -20 + b
b = 60 + 20
b = 80
Therefore, the equation in function form is:
y = -4x + 100
The equation y = -4x + 100 represents the amount of trash removed each week, where y is the number of dumpster loads and x is the week number. As the weeks progress, the amount of trash removed decreases by 4 dumpster loads per week.
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Gerald deposits $50 every week into a savings account. At week 16, the account balance is $1650. Write an equation that models the balance, y, and the number of weeks, x.
Answers
Becauseee y is the balance and x is the weeks so if we plug in the information we’re given about the balance being 1650 at 16 weeks, it’d be
1650=50(16)+850
Which if you do the math it’s 1650=1650
What is the value for y in the following expression y=-2x+8 when x=20
Answers
Answer:
y = -32Step-by-step explanation:
\(y=-2x+8\\\\x=20\)
Substitute ; x =20 in given equation
\(y =-2(20 )+8\\\\y = -40+8\\\\y = -32\)
what is the value of x in the equation 2x + 5 = 21
Answers
Answer:
x=8
Step-by-step explanation:
2x + 5 = 21
Subtract 5 from each side
2x+5-5 = 21-5
2x= 16
Divide each side by 2
2x/2 = 16/2
x = 8
Answer:8
Step-by-step explanation:
2 times eight plus five is 21
A shipping container will be used to transport several 120-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 27500 kilograms. Other shipments weighing 12400 kilograms have already been loaded into the container. Which inequality can be used to determine x, the greatest number of
120-kilogram crates that can be loaded onto the shipping container?
Answers
Inequality can be used to determine x, the greatest number of 120-kilogram crates that can be loaded onto the shipping container is 8000 + 120C <= 27500
What is inequality?An inequality is a relationship that allows us to contrast two or more mathematical expressions.
Knowing that;
Each carton weighs 120 kg.
The container's maximum weight when loaded is 27500 kg.
Weight of the container as it is currently loaded: 8000 kg
Now that we have merged, we have;
Solution
because 8000 grams have already been loaded kilograms
into the container
each crate weighs 120 kilograms.
so 8000 + 120C <= 27500
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Seven less than 15 times a number
Answers
Answer: 15x-7
Step-by-step explanation:
15x-7
Consider the following generic C comparison function and its assembly language representation C code: byte compbyte a,byte b)/a in rdi,b in rsi Assembly code cmpb %rsi,%rdi set_inst %a1 ret Your jobs(fill-in blank):now sh given values of a and b g SET instruction and the A.5 points set CI SF OF %al setg 47 23 B.5 points set h SF OF %a setl 23 47 C.5 points ZA SF OF %al set sete 23 23 D.5 points CF ZF SF OF 00%1 set b setne 23 47
Answers
The correct answer is D. setne 23 47. Based on the provided information, I understand that you have a comparison function in C code and its corresponding assembly code. You are asked to fill in the blanks by selecting the appropriate instructions based on the given values of a and b and the status flags SF, OF, ZF, and CF. Let's go through the options:
A. setg 47 23: This option is incorrect because setg is used to set a byte to 1 if the Greater flag (ZF=0 and SF=OF) is set, but the given values of a and b are 47 and 23, respectively, so it does not satisfy the condition for setg to be set.
B. setl 23 47: This option is incorrect because setl is used to set a byte to 1 if the Less flag (SF≠OF) is set, but the given values of a and b are 23 and 47, respectively, so it does not satisfy the condition for setl to be set.
C. sete 23 23: This option is incorrect because sete is used to set a byte to 1 if the Zero flag (ZF=1) is set, but the given values of a and b are 23 and 23, respectively, so it does not satisfy the condition for sete to be set.
D. setne 23 47: This option is correct. setne is used to set a byte to 1 if the Zero flag (ZF=0) is not set, which means the values of a and b are not equal. In this case, the given values of a and b are 23 and 47, respectively, so they are not equal, and setne should be used.
Therefore, the correct answer is D. setne 23 47
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the united nations stores data about energy production from animal waste and the use around the world. the following data set gives information for converting animal waste into electricity. country or area commodity - transaction year quantity unit united states animal waste - transformation in electricity 2014 171907 terajoules united states animal waste - transformation in electricity 2013 149233 terajoules united states animal waste - transformation in electricity 2012 104847 terajoules united states animal waste - transformation in electricity 2011 141347 terajoules united states animal waste - transformation in electricity 2003 185912 terajoules united states animal waste - transformation in electricity 2002 197120 terajoules united states animal waste - transformation in electricity 2001 104725 terajoules united states animal waste - transformation in electricity 2000 126745 terajoules united states animal waste - transformation in electricity 1999 133206 terajoules united states animal waste - transformation in electricity 1998 224425 terajoules united states animal waste - transformation in electricity 1997 129113 terajoules united states animal waste - transformation in electricity 1996 5274 terajoules united states animal waste - transformation in electricity 1993 21748 terajoules united states animal waste - transformation in electricity 1992 242924 terajoules united states animal waste - transformation in electricity 1991 210448 terajoules united states animal waste - transformation in electricity 1990 215437 terajoules what is the slope of the linear equation of the entire data set? round to nearest hundredth. provide your answer below:
Answers
The slope of the linear equation of the entire data set is -720.98
1. Bring data in to excel sheet
2.Select both x and y variable and insert scatter plot chart. Select any data point in the chart and add trend line. In the format trend line option choose "Linear" and also select "Display an equation on chart" & "Display R-squared value on chart"
To find: What is the slope of the linear equation of the entire data set? Round to nearest hundredth.
Regression equation : y = -720.98x + 15,90,277.32
Slope = -720.98 (Round to nearest hundredth.)
Intercept = 15,90,277.32
So, the slope of the linear equation of the entire data set is -720.98
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Show why PX=2) = P(X= 3) in a binomial distribution where n = 5 and p=0.5. [3]
Answers
P(X = 2) is not equal to P(X = 3)
How to show that P(X = 2) = P(X = 3) in a binomial distribution with n = 5 and p = 0.5?To show that P(X = 2) = P(X = 3) in a binomial distribution with n = 5 and p = 0.5, we need to use the formula for the probability mass function (PMF) of a binomial distribution.
The PMF of a binomial distribution is given by the formula:
P(X = k) = C(n, k) *\(p^k * (1-p)^{(n-k)}\)
where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) represents the binomial coefficient.
In our case, n = 5 and p = 0.5. Let's calculate P(X = 2) and P(X = 3) using the formula:
P(X = 2) = \(C(5, 2) * (0.5)^2 * (1-0.5)^{(5-2)}\)
= 10 * 0.25 * 0.125
= 0.3125
P(X = 3) = C(5, 3) * \((0.5)^3 * (1-0.5)^{(5-3)}\)
= 10 * 0.125 * 0.125
= 0.125
As we can see, P(X = 2) = 0.3125 and P(X = 3) = 0.125.
Therefore, P(X = 2) is not equal to P(X = 3) in this specific case of a binomial distribution with n = 5 and p = 0.5.
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A line that includes the points (-10, 6) and (-9, h) has a slope of -8. What is the value of h?
Answers
Hope this helps you !!!!
there are 6 studenjts in the class. one of them is to be selcted as the best studen and another student is to be selceted as a runner-up. how mant different ways can this be done?
Answers
As per the combination method, there are 15 different ways are possible for this situation.
Combination method:
Combination method also known as NCR is a formula of combination and arrangement, where the object's order does not matter so much.
Given,
Here we have to find the number of ways to be selected as the best student and another student is to be selected as a runner-up among the 6 students.
Here we know that that the value of
Total number of students = 6
Best student = 1
Runner - up = 1
So, as per the NCR formula, the value of n = 6 and the value of r = 2
Then the possible combinations are calculated as,
=> ⁶C₂ = 6! / 2!(6 - 2)!
=> ⁶C₂ = 6!/2!4!
=>⁶C₂ = 15
Therefore, the resulting number of ways is 15.
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what is the equation?
Answers
===================================================
First find the slope
m = (y2-y1)/(x2-x1)
m = (-7-1)/(7-3)
m = -8/4
m = -2
The slope is -2.
Now pick either point for the (x,y) coordinate values. I'll pick the first point. So (x,y) = (3,1)
We'll plug m = -2, x = 3, y = 1 into the equation below to solve for b
y = mx+b
1 = -2*3+b
1 = -6+b
1+6 = b ... add 6 to both sides
7 = b
b = 7
The y intercept is 7.
Since m = -2 and b = 7, we go from y = mx+b to y = -2x+7
--------------------
As a check:
Plug in (x,y) = (3,1)
y = -2x+7
1 = -2(3)+7
1 = -6+7
1 = 1
So (3,1) is confirmed.
Let's check the other point
y = -2x+7
-7 = -2(7)+7
-7 = -14+7
-7 = -7
The other point (7,-7) is confirmed as well. Both points are on the line. This confirms the answer.
Answer:
Y=2x+1
Step-by-step explanation:
Slope= -7-1/7-3 =2
Point slope form:
y-y₁=m(x-x₁)
y-3=2(x-1)
+3 +3
y=2x+1
A problem states: "There are 9 more children than parents in a room. There are 25 people in the room in all. How many children are there in the room?"
What are the unknowns in this problem?
Responses
the total number of people in the room
the total number of people in the room
the total number of parents in the room
the total number of parents in the room
the total number of children in the room
the total number of children in the room
the total number of children and the total number of parents in the room
Answers
By solving a system of equations, we can see that there are 17 children and 8 parents in the room.
How to get the total number of children in the room?Let's define the variables:
x = number of children.
y = number of parents.
We know that there are 9 more children than parents, then:
x = y + 9
And there is a total of 25 people, so:
x + y = 25
So we have a system of equations:
x = y + 9
x + y = 25
Replacing the first equation into the second one, we get:
(y +9) + y = 25
2y + 9 = 25
2y = 25 - 9 = 16
y = 16/2 = 8
then the value of x is:
x = y + 9 = 8 + 9 = 17
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Write the decimal as a percent.
20= .......?%
Answers
Answer:
20%
Step-by-step explanation:
Rachel is a stunt driver, and she's escaping from a building that is about to explode!
The variable d models Rachel's distance from her exit (In meters) t seconds after the cameras began recording the stunt
What is Rachel's speed?
Answers
Answer:
The variable d models Rachel's distance from her exit (in meters) t seconds after the cameras began recording the stunt. d = - 38t
Step-by-step explanation:
From given, we have, The expression that represents the Rachel's distance is given by, d = - 38t + 220
where, d represents the distance in meters
t represents the time in seconds
We know that, speed = distance by time. Therefore, we need to differentiate the given equation w.r.t to time to find the speed.
we have, d/dt (d) = d/dt [- 38t + 220]
| d/dt (d) | = | - 38 + 0 |
s = d/dt (d) = 38 m/s
Therefore, the speed of Rachel is 38 m/s.
Answer:
38 meters per second
Step-by-step explanation:
Hope this helps!
I need to find the foci of this equation to answer my problem!
Answers
Correct option is A, The foci of the hyperbola is (0,±√17).
A hyperbola is a plane curve that is produced by a point that is moving so that the difference between the distances from two fixed points is constant. It is created when a double right circular cone intersects with a plane that cuts through both ends of the cone.
The difference in distances between a group of points that are situated in a plane and two fixed points—which is a positive constant—is what is referred to as the hyperbola.
Given the equation of hyperbola is \(\frac{y^2}{12} - \frac{x^2}{5} = 1\)
In this equation,
a² = 12
b² = 5
Using the relation c² = a² + b²
c = √(12 + 5)
c = √ 17
As, the hyperbola is vertical,
The value of foci will be (0,±√17)
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savanna is sitting in a bucket of a farris wheen at the 3 oclock position. the ferris wheel has a radius 62 feet long. the ferris wheel begins moving clockwise. what is the measure of savanna's angle of roatation when she is 42 feet above the ferris wheel's horizontal diameter
Answers
Therefore, when Savannah is 42 feet above the ferris wheel's horizontal diameter, her angle of rotation is approximately 68.6 degrees.
Given by the question.
Let's first draw a diagram to visualize the situation.
*
* *
* * <-- Ferris wheel at 3 o'clock position
* *
* * <-- Highest point
*______________*______
62 ft
The Ferris wheel is a circle with radius 62 feet, and Savannah is sitting in a bucket that moves along the circle. When she is at the highest point, she is 62 feet away from the center of the circle, and when she is on the horizontal diameter, she is 62 feet - 42 feet = 20 feet away from the center of the circle.
To find Savannah's angle of rotation, we can use the cosine function.
cos(theta) = adjacent / hypotenuse
In this case, the adjacent side is 20 feet, and the hypotenuse is 62 feet.
cos(theta) = 20/62
To solve for theta, we need to take the inverse cosine of both sides.
theta = cos^-1(20/62)
Using a calculator, we get:
theta = 68.6 degrees
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What is the value of k for which the system of equation 2x +3y 7 and 8x +( K 4 y 0 has infinitely many solution?
Answers
The k value for the system of equation 2x + 3y = 7 and 8x + (K + 4)y - 28 = 0 that has infinitely many solution is 8.
How to calculate the k value?First for infinitely many solution the rule is all comparisons have the same value or \(\frac{a_1}{a_2} = \frac{b_1}{b_2} =\frac{c_1}{c_2} = k\) with the equation ax + by + c = 0.
So for the first equation we can written as 2x + 3y - 7 = 0.
Then for the second equation we can written as 8x + (K+4)y - 28 = 0.
The we can comparison two equation. So, for a value is 2/8 = 1/4 and for c value is -7/-28 = 1/4. So according to the rule, the comparisons value for b also 1/4. So,
\(\frac{b_1}{b_2}\) = 1/4
3/(K+4) = 1/4
we cross multiply
12 = K + 4
12 - 4 = K
8 = K
Thus, the k value for the equation 8x + (K + 4)y - 28 = 0 and 2x + 3y = 7 is 8.
You question is incomplete, but most probably your full question was
Find the value of k for which the system of equations 2x+3y=7 and 8x+(k+4)y−28=0 has infinitely many solution.
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Solve rational inequality give solution set incinterval
notation
Answers
The solution set, in interval notation, for the given rational inequality is (-∞, -3) U (2, ∞).
To solve a rational inequality, we follow a few steps. Let's consider the given rational inequality:
(3x - 2)/(x + 1) > 0
Find the critical points:Set the numerator and denominator equal to zero and solve for x. In this case, we have:
3x - 2 = 0 ---> 3x = 2 ---> x = 2/3
x + 1 = 0 ---> x = -1
So, the critical points are x = -1 and x = 2/3.
We need to determine the intervals where the rational function is positive or negative. To do this, we can use test points. Choose a test point from each interval:
Interval 1: (-∞, -1)
Test point: x = -2
Interval 2: (-1, 2/3)
Test point: x = 0
Interval 3: (2/3, ∞)
Test point: x = 1
Determine the solution:Plug in the test points into the original inequality and observe the signs:
For x = -2: (3(-2) - 2)/((-2) + 1) = -8 < 0
For x = 0: (3(0) - 2)/(0 + 1) = -2 < 0
For x = 1: (3(1) - 2)/(1 + 1) = 1 > 0
Based on the signs, we can conclude that the rational function is positive in the interval (2/3, ∞) and negative in the intervals (-∞, -1) and (-1, 2/3).
Finally, we express the solution set in interval notation:
(-∞, -1) U (2/3, ∞)
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