Which one of these answers is correct (show work please)
ignore the ones that start with i,ii, iii
100 POINTS!!!
Answers
Answer:
(5) - Option C, \(s=3\sqrt{17}\)
(6) - Option D, \(-\frac{1}{4} y^{-4}=\frac{1}{3} x^3 -\frac{1}{4}\)
Step-by-step explanation:
Given the following questions.
(5) - Find the arc length of y=4x-3 from A(1,1) to B(4,13)
(6) - Solve the first-order differential equation \(y'=x^2y^5\) with the initial condition, \(y(0)=1\).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Question #5:
\(\boxed{\left\begin{array}{ccc}\text{\underline{Formula for Arc Length:}}\\\\s=\int\limits^b_a {\sqrt{1+(f'(x))^2} } \, dx \end{array}\right}\)
(1) - Take the derivative of the function y
\(y=4x-3\\\\\Longrightarrow \boxed{y'=4}\)
(2) - Square y'
\(y'=4\\\\\Longrightarrow y'=4^2\\\\\Longrightarrow \boxed{y'=16}\)
(3) - Plug into the formula for arc length
\(s=\int\limits^b_a {\sqrt{1+(f'(x))^2} } \, dx \\\\\text{Limits:} \ 1\leq x\leq 4\\\\\Longrightarrow\int\limits^4_1 {\sqrt{1+16} } \, dx \\\\\Longrightarrow\boxed{ \int\limits^4_1 {\sqrt{17} } \, dx} \\\)
(4) - Solve the integral
\(\int\limits^4_1 {\sqrt{17} } \, dx \\\\\Longrightarrow \Big [x\sqrt{17} \Big] \right]_{1}^{4}\\\\\Longrightarrow 4\sqrt{17} -\sqrt{17\\}\\\\ \therefore \boxed{s=3\sqrt{17} }\)
Thus, the arc length is found.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Quick note: The question solved for the arc length using a dy integral not a dx integral (which was really unnecessary), so let me clarify that issue.
(1) - Taking the function y, and solving it for x
\(y=4x-3\\\\\Longrightarrow y+3=4x\\\\\Longrightarrow \boxed{x=\frac{1}{4}y+\frac{3}{4}}\)
(2) - Repeating steps (1)-(4) from above
\(x=\frac{1}{4}y+\frac{3}{4}\\\\\Longrightarrow \boxed{x'=\frac{1}{4}} \\\\\Longrightarrow (x')^2=(\frac{1}{4})^2\\\\\Longrightarrow \boxed{(x')^2=\frac{1}{16}}\\\\s=\int\limits^b_a {\sqrt{1+(f'(x))^2} } \, dy\\\text{Limits:} \ 1\leq y\leq 13\\\\ \Longrightarrow\int\limits^{13}_1 {\sqrt{1+\frac{1}{16} } \, dy\\\\ \Longrightarrow\int\limits^{13}_1 {\sqrt{\frac{17}{16} } \, dy\\\\ \Longrightarrow\int\limits^{13}_1 {\frac{\sqrt{17} }{4} } \, dy\\\\\)
\(\Longrightarrow\Big[\frac{\sqrt{17} }{4} y \Big]^{13}_{1}\\\\\Longrightarrow \frac{13\sqrt{17} }{4} -\frac{\sqrt{17} }{4} \\\\\ \Longrightarrow \boxed{\boxed{s=3\sqrt{17} }}\)
Thus, the correct setup according to your question is option C.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Question #6:
The given differential equation is separable.
\(\boxed{\left\begin{array}{ccc}\text{\underline{Seperable Differential Equation:}}\\\frac{dy}{dx} =f(x)g(y)\\\\\rightarrow\int\frac{dy}{g(y)}=\int f(x)dx \end{array}\right }\)
(1) - Solve the separable DE
\(\frac{dy}{dx} =x^2y^5\\\\\Longrightarrow \frac{1}{y^5} dy=x^2dx\\\\\Longrightarrow \int y^{-5}dy= \int x^2 dx\\\\\Longrightarrow \boxed{ -\frac{1}{4} y^{-4}=\frac{1}{3} x^3 +C}\)
(2) - Use the given initial condition to find the arbitrary constant "C"
\(\text{Recall} \rightarrow y(0)=1\\\\-\frac{1}{4} y^{-4}=\frac{1}{3} x^3 +C\\\\\Longrightarrow -\frac{1}{4} (1)^{-4}=\frac{1}{3} (0)^3 +C\\\\\Longrightarrow -\frac{1}{4} (1)=0 +C\\\\\therefore \boxed{C=-\frac{1}{4} }\)
(3) - Form the final solution
\(\boxed{\boxed{-\frac{1}{4} y^{-4}=\frac{1}{3} x^3 -\frac{1}{4} }}\)
Thus, option D is correct.
Answer:
\(\textsf{5.} \quad \textsf{C)} \;\; \displaystyle \int^{13}_1 \sqrt{1+\left(\frac{1}{4}\right)^2}\; \text{d}y\)
\(\textsf{6.} \quad \textsf{D)} \quad -\dfrac{1}{4}y^{-4}=\dfrac{1}{3}x^3-\dfrac{1}{4}\)
Step-by-step explanation:
Question 5To find the arc length of a given function between two points, we can use the Arc Length Formula:
\(\boxed{\begin{minipage}{7.4cm}\underline{Arc Length Formula}\\\\$\displaystyle \int_{a}^{b} \sqrt{1+(f'(x))^2}\; \text{d}x$\\\\\\where: \\ \phantom{ww}$\bullet$ $a$ and $b$ are the limits. \\ \phantom{ww}$\bullet$ $f'(x)$ is the first derivative of $f(x)$.\\\end{minipage}}\)
The given function is:
\(y=4x-3\)
Differentiate the given function:
\(\begin{aligned} f(x)&=4x-3\\ \implies f'(x)&=4\end{aligned}\)
As the function is in terms of x, the interval [a, b] is the x-values of the given points. Therefore:
\(a = 1\)\(b = 4\)Set up the integral using the arc length formula:
\(\displaystyle \textsf{Arc length}=\int^4_1 \sqrt{1+(4)^2}\; \text{d}x\)
You will notice that this integral is not one of the given answer options.
We can also set up the integral with respect to y.
To do this, begin by rearranging the function so that x is a function of y:
\(\begin{aligned}y&=4x-3\\y+3&=4x\\x&=\dfrac{1}{4}y+\dfrac{3}{4}\end{aligned}\)
Differentiate x with respect to y:
\(\begin{aligned} g(y)&=\dfrac{1}{4}y+\dfrac{3}{4}\\ \implies g'(y)&=\dfrac{1}{4}\end{aligned}\)
As the function is in terms of y, the interval [a, b] is the y-values of the given points. Therefore:
\(a = 1\)\(b = 13\)Set up the integral using the arc length formula:
\(\boxed{\displaystyle \textsf{Arc length}=\int^{13}_1 \sqrt{1+\left(\frac{1}{4}\right)^2}\; \text{d}y}\)
Therefore, the correct answer option is option C.
\(\hrulefill\)
Question 6The given differential equation is:
\(\dfrac{\text{d}y}{\text{d}x}=x^2 \cdot y^5\)
Solving a differential equation means using it to find an equation in terms of the two variables, without a derivative term.
Rearrange the equation so that all the terms containing y are on the left-hand side, and all the terms containing x are on the right-hand side:
\(\dfrac{1}{y^5}\; \text{d}y=x^2 \; \text{d}x\)
Integrate both sides:
\(\displaystyle \int \dfrac{1}{y^5}\; \text{d}y=\int x^2 \; \text{d}x\)
Use the following integration rule:
\(\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}\)
Therefore:
\(\begin{aligned}\displaystyle \int \dfrac{1}{y^5}\; \text{d}y&=\int x^2 \; \text{d}x\\\\ \int y^{-5}\; \text{d}y&=\int x^2 \; \text{d}x\\\\\dfrac{y^{-5+1}}{-5+1}&=\dfrac{x^{2+1}}{2+1}+\text{C}\\\\\dfrac{y^{-4}}{-4}&=\dfrac{x^3}{3}+\text{C}\\\\-\dfrac{1}{4}y^{-4}&=\dfrac{1}{3}x^3+\text{C}\end{aligned}\)
Given y(0) = 1, substitute y = 1 and x = 0 into the equation and solve for C:
\(\begin{aligned}-\dfrac{1}{4}(1)^{-4}&=\dfrac{1}{3}(0)^3+\text{C}\\\\-\dfrac{1}{4}&=0+\text{C}\\\\\text{C}&=-\dfrac{1}{4}\end{aligned}\)
Therefore, the equation is:
\(\boxed{-\dfrac{1}{4}y^{-4}=\dfrac{1}{3}x^3-\dfrac{1}{4}}\)
Therefore, the correct answer option is option D.
Related Questions
a professional athlete signed a contract that would pay her $1.3 x 10^6 per month. how much money will the athlete make at the end of 3 years
Answers
Answer:
\( \$ 4.68 \times 10^7\)
Step-by-step explanation:
\( monthly \: payment = \$ 1.3 \times {10}^{6} \\ 3 \: years = 3 \times 12 = 36months \\ money \: made \: by \: athelete \: at \: the \: end \: of \: \\3 \: years \\ = \$ 1.3 \times {10}^{6} \times 36 \\ = 46.8\times {10}^{6} \\ = \$4.68\times {10}^{7} \)
solve the following counting problems: (a) a spider has one sock and one shoe on each of its eight legs, in how many different orders can the spider put on its socks and shoes? (assume that a shoe must be on top of a sock). (b) an elevator starts at the basement with 10 people and discharges them all by the time it reaches the top ??oor, number 6. in how many ways could have the people get o?? the elevator if it only ma??ers the number of people that le?? on each ??oor?
Answers
Therefore , a)different orders can the spider put on its socks and shoes is \(\frac{16!}{(2!)^{8} }\) and b)the total no of ways for people to discharge from lift is 14112 ways.
What is combination?Selections are another name for combinations. Combinations represent the choosing of items from a predetermined group of items. We're not trying to arrange anything here. We're going to pick them. We write n C r to represent the number of distinct r-selections or combinations among a set of n objects. Compared to arrangements or permutations, combinations are different.
Here,
Each dressing sequence may be uniquely characterized by a series of two 1s, two 2s,..., and two 8s; the spider is said to put the sock on leg x in the first instance, and the shoe on leg x in the second instance.
The solution would be 16! if each number were unique.
However, because 8 words appear twice, the solution is
=> \(\frac{16!}{(2!)^{8} }\)
Thus different orders can the spider put on its socks and shoes is \(\frac{16!}{(2!)^{8} }\)
Thus,
Part A: When the elevator operator can't tell who is who.
Currently, there are 6 floors and 10 persons, and each floor loses one.
so 8+6-1=13
applying combination
=>\(C^{13} _{5}\) = 1287 ways.
Part 2: When the elevator attendant separates males from ladies.
There are 3 females and 5 males.
The formula for choosing five men is (5 + 6)-1= (10).
Applying combination
=> .\(C^{10} _{5}\)= 252 ways
techniques to choose ladies = 3+6-1=8
Applying combination
=> \(C^{8} _{5}\) = 56 ways
Total no. of ways
=> 252*56 = 14112 ways.
Therefore , the total no of ways for people to discharge from lift is 14112 ways.
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When Maria Acosta bought a car 2 1/2 years ago, she borrowed 10,000 for 48 months at 7. 8% compounded monthly. Her monthly payments are 243. 19 but she’d like to pay off the loan early. How much will she owe just after her payment at the 2 1/2 year mark?
Answers
Just after her payment at the 2 1/2 year mark, Maria will owe approximately $7,722.63 on her loan.
To calculate the amount Maria will owe just after her payment at the 2 1/2 year mark, we need to determine the remaining balance on her loan.
First, we convert the loan duration to months:
2 1/2 years = 2.5 * 12 = 30 months.
Next, we use the formula for the remaining balance on a loan:
Remaining balance = Principal * (1 + monthly interest rate)^remaining months - Monthly payment * [(1 + monthly interest rate)^remaining months - 1] / monthly interest rate.
Principal = $10,000
Monthly interest rate = 7.8% / 12 = 0.065
Remaining months = 48 - 30 = 18
Plugging in the values, we have:
Remaining balance = $10,000 * (1 + 0.065)^18 - $243.19 * [(1 + 0.065)^18 - 1] / 0.065
Calculating this expression step by step:
Remaining balance = $10,000 * (1.065)^18 - $243.19 * [(1.065)^18 - 1] / 0.065
≈ $7,722.63
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Can you guys help me please!!!!Q#4
Answers
Step-by-step explanation:
1 and 1/8 and add 1 and 1/4.
Make the mixed fractions into improper fractions which is 9/8 add 5/4.
Make a common denominator which is 8 so leave 9/8 and turn 5/4 into 10/8.
Add em up which equals to 19/8
19/8 into a mixed number is 2 3/8
I need 23 and 24. The directions for them are solve for x. And assume that lines which appear to be diameters are actual diameters.
Answers
Explanation
Question 23
To get the value of x, let us get y as shown below
\(\begin{gathered} y+100=180^0(linear\text{ pairs}) \\ y=180-100=80^0 \end{gathered}\)Next, we will make use of the fact tha y and (x+85 ) are vertical angles so that
\(\begin{gathered} y=x+85\text{ \lparen linear pairs\rparen} \\ where\text{ y=80} \\ 80=x+85 \\ x=80-85=-5 \\ x=-5 \end{gathered}\)For question 23, x = -5
For question 24
The total angle in a circle is 360 degrees so that
\(-38x+4+65+42+95+40=360\)solving for x
\(\begin{gathered} -38x+246=360 \\ -38x=360-246 \\ -38x=114 \\ x=\frac{114}{-38} \\ x=-3 \end{gathered}\)For question 24, x =-3
Chris earns $10.00 each time he washes his neighbor's car. The expression 10x represents
Chris' earnings. What does the variable in the expression represent?
Ono variable
O the number of times the neighbor drives the car
O the amount of money Chris earns per wash
O the number of times Chris washes the car
Answers
Answer:
the number of times he washes the car
Step-by-step explanation:
xº
520
38°
220°
X = degrees
Answers
by taking complete angle
52+38+220+x=360
310+x=360
x=360-310
x=50
I will give you brainliest!!!! Use substitution method to solve the system of equations. y=6x-5 y=x+5
Answers
Answer:
(2,7)
Step-by-step explanation:
y=6x-5
y=x+5
Since they are both equal to y, set them equal to each other
6x-5 = x+5
Subtract x from each side
6x-x-5 = x+5-x
5x-5 = 5
Add 5 to each side
5x-5 +5 = 5+5
5x=10
Divide by 5
5x/5 = 10/5
x=2
y = x+5
y = 2+5
y =7
Answer:
Choice C: (2,7)
Step-by-step explanation:
Using substitution method let y = x + 5
We have x + 5 = 6x - 5
Get let’s get rid of the 6x, so -5x + 5 = -5
Get rid of the 5 by subtracting by 5, so -5x = -10
Divide -5 on both sides and you get x = 2
Input 2 for x in y = x + 5, we get y = 2 + 5, which equals 7.
So (2,7)
Please mark BRAINLIEST!
TRUE/FALSE.Symmetric algorithms support confidentiality, but not authentication and nerepudiation
Answers
FALSE. Symmetric algorithms support confidentiality, but they do not inherently support authentication and non-repudiation.
Determine the symmetric algorithms?Symmetric algorithms are encryption algorithms that use the same key for both encryption and decryption. Their primary purpose is to ensure confidentiality by keeping data secure from unauthorized access. However, symmetric algorithms do not provide built-in mechanisms for authentication and non-repudiation.
Authentication involves verifying the identity of a communicating party to ensure that the message comes from a trusted source. Non-repudiation, on the other hand, ensures that the sender cannot deny sending a message.
To achieve authentication and non-repudiation, additional mechanisms such as digital signatures and asymmetric encryption algorithms are typically used. Asymmetric algorithms, also known as public-key algorithms, employ a pair of keys (public and private) for encryption and decryption, enabling authentication and non-repudiation.
So, while symmetric algorithms can provide confidentiality, they do not directly support authentication and non-repudiation without the use of additional mechanisms.
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The diameter of a circle is 41ft. Find it’s area to the nearest whole number.
Answers
The area of the circle to the nearest whole number with the given diameter is 1320 feet².
Given that,
Diameter of a circle = 41 feet
Radius is half of the diameter.
So, radius = 41 / 2 = 20.5 feet
Area of a circle = π r², where r is the radius.
Substituting,
Area = π (20.5)²
= 420.25 π
≈ 1320 feet²
Hence the required area is 1320 feet².
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(q48) Solve the integral
Answers
The expression gotten from integrating \(\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx\) is \(\frac{1}{16}\sin^{-1}(8x/5) + c\)
How to integrate the expressionFrom the question, we have the following trigonometry function that can be used in our computation:
\(\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx\)
Let u = 8x/5
So, we have
du = 8/5 dx
Subsitute u = 8x/5 and du = 8/5 dx
So, we have
\(\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx = \int {\frac{5}{\sqrt{5(100 - 100u\²)}} \, du\)
Simplify
So, we have
\(\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{1}{16} \int {\frac{1}{\sqrt{1 -u\²}} \, du\)
Next, we integrate the expression \(\int {\frac{1}{\sqrt{1 -u\²}} \, du = \arcsin(u)\)
So, we have
\(\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{\arcsin(u)}{16} + c\)
Undo the earlier substitution for u
So, we have
\(\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{\arcsin(8x/5)}{16} + c\)
This can also be expressed as
\(\int {\frac{1}{\sqrt{100 - 256x\²}} \, dx =\frac{1}{16}\sin^{-1}(8x/5) + c\)
Hence, integrating the expression \(\int\limits {\frac{1}{\sqrt{100 - 256x\²}} \, dx\) gives (c)
\(\frac{1}{16}\sin^{-1}(8x/5) + c\)
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A spring is attatched at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D.
Part A when 0 = 28 degrees, what is the distance from point A to point B to the nearest tenth of a foot?
Part B When the spring is stretched and the distanced from point A to point B is 5.2 feet, what is the valuue of 0 to the nearest tenth of a degree?
Answers
The Distance from point A to point B is: 3.4 ft when θ = 28° and The value of θ is: 54.8°
According to the question,
A spring is attached at one end to support B and at the other end to collar A.
Part A: Given that : Reference angle (θ) = 28°
We have to find out the distance of point A to point B.
Adjacent = 3 ft (given)
To find AB, apply the trigonometry function which is:
cos θ = adj / hypotenuse
Substituting all the values ,
=> cos 28°= 3/AB
=> AB = 3/cos 28°
=> AB = 3.4 ft (nearest tenth of a foot)
Part B:
It is given that Distance from point A to point B is 5.2 feet
We have to calculate the Value of θ
adj = 3 ft
hypotenuse = AB = 5.2 ft
To find θ , We will again apply the trigonometry function
cos θ = adj/hypotenuse
Substituting all the given values
=> cos θ = 3/5.2
=> θ = cos⁻¹ ( 3 / 5.2)
=> θ = 54.8°
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Copy and complete the statement for ΔDEF with medians DH, EJ, and FG, and centroid K. DK = _ KH
Answers
The centroid is the point of intersection of the medians. In triangle ΔDEF, the statement DK = 2KH holds true.
DK = 2KH
In triangle ΔDEF, the statement DK = 2KH means that the length of DK is twice the length of KH. Let's understand this statement with an example.
Suppose we have a triangle ΔDEF, with medians DH, EJ, and FG, and centroid K. If we measure the length of KH and find it to be 5 units, then according to the statement DK = 2KH, the length of DK would be 2 times 5, which is 10 units.
This implies that the distance from the centroid K to a vertex D is twice the distance from the centroid K to the midpoint of the side opposite to D. In our example, the distance from K to D is twice the distance from K to the midpoint of the side opposite to D, which is 10 units and 5 units, respectively.
This relationship holds true for any triangle, regardless of its size or shape. The medians always intersect at the centroid, and the ratio of the lengths DK to KH is always 2:1.
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/10
The price of the house decreases from $250,000 in 2005 to
$195,000 in 2010. What is the average rate of change for the
house each year?
Answers
Answer:
The average rate will be -11000 dollars per year.
Step-by-step explanation:
The price of a house in 2005 = $250,000
The price of a house in 2010 = $195,000
at x₁ = 2005, f(x₁) = 250,000
at x₂ = 2010, f(x₂) = 195,000
Using the formula to determine the average rate of change for the
house each year.
Average rate = [f(x₂) - f(x₁)] / [ x₂ - x₁]
= [195,000 - 250,000 ] / [2010-2005]
= -55000 / 5
= -11000
Therefore, the average rate will be -11000 dollars per year.
3 divided by the sum of x and 9
Answers
Answer:3/9
Step-by-step explanation:
In AWXY, w = 600 inches, x = 590 inches and y = 140 inches. Find the measure of angle X to the nearest degree.
Answers
Answer:79 degrees
Step-by-step explanation:
Deltamath
The measure of angle X to the nearest degree will be 79.2°.
In triangle ΔWXY, the dimension of the triangle is given below.
w = 600 inches, x = 590 inches and y = 140 inches.
What is law of cosine?Let there is a triangle ABC such that |AB| = a units, |AC| = b units, and |BC| = c units and the internal angle C is of θ degrees, then we have:
a² + b² – 2ab cos θ = c²
Then the measure of angle X will be
600² + 140² – 2 × 600 × 140 × cos θ = 590²
379600 – 168000 cos θ = 348100
Simplify the equation, then the value of angle θ will be
168000 cos θ = 31500
cos θ = 0.1875
θ = 79.2°
Thus, the measure of angle X to the nearest degree will be 79.2°.
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5. ( 30pts) Using the master theorem, find Θ-class of the following recurrence relatoins a) T(n)=2T(n/2)+n
3
b) T(n)=2T(n/2)+3n−2 c) T(n)=4T(n/2)+nlgn
Answers
Using the master theorem, the Θ-class are: a) T(n) = Θ(n)b) T(n) = Θ(n) and c) T(n) = Θ(√n log n).
a) T(n)=2T(n/2)+n³
To find the Θ class of T(n), we need to apply the Master Theorem. In the Master Theorem, if T(n) = aT(n/b) + f(n) where a >= 1, b > 1, and f(n) is an asymptotically positive function, then:
T(n) = Θ(nᵈ)
where d = logₐ(b).
Here a = 2, b = 2 and f(n) = n³.
So, d = log₂(2)
= 1T(n)
= Θ(nᵈ)
= Θ(n¹)
= Θ(n)
Therefore, the Θ class of T(n) is Θ(n).
b) T(n)=2T(n/2)+3n−2
Similarly to part (a), we can find the Θ class of T(n) by using the Master Theorem.
In this case, a = 2, b = 2 and f(n) = 3n - 2.
Here, d = log₂(2)
= 1T(n)
= Θ(nᵈ)
= Θ(n¹)
= Θ(n)
Therefore, the Θ class of T(n) is Θ(n).
c) T(n)=4T(n/2)+nlogn
For this recurrence relation, a = 4, b = 2 and f(n) = nlogn.
In this case, d = log₄(2) = 0.5.
So,
T(n) is
= Θ(nᵈ)
= Θ(n⁰.⁵)
= Θ(√n log n)
Therefore, the Θ class of T(n) is Θ(√n log n).
Hence, the correct options are:a) T(n) = Θ(n)b) T(n) = Θ(n)c) T(n) = Θ(√n log n).
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Please help with any of the following
Answers
Answer:
x2-12x+68
Step-by-step explanation:
reference the following table: x p(x) 0 0.130 1 0.346 2 0.346 3 0.154 4 0.024 what is the variance of the distribution?
Answers
The variance of the distribution of the data set is 0.596.
To find the variance of a discrete probability distribution, we use the formula:
Var(X) = ∑[x - E(X)]² p(x),
where E(X) is the expected value of X, which is equal to the mean of the distribution, and p(x) is the probability of X taking the value x.
We can first find the expected value of X:
E(X) = ∑x . p(x)
= 0 (0.130) + 1 (0.346) + 2 (0.346) + 3 (0.154) + 4 (0.024)
= 1.596
Next, we can calculate the variance:
Var(X) = ∑[x - E(X)]² × p(x)
= (0 - 1.54)² × 0.130 + (1 - 1.54)² × 0.346 + (2 - 1.54)² × 0.346 + (3 - 1.54)² × 0.154 + (4 - 1.54)² × 0.024
= 0.95592
Therefore, the variance of the distribution is 0.96.
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using the gainesvillehomes sp2019 file, run a multiple regression predicting price based on lot size, commute, year built, and days on realtor assume you are trying to determine if the slope of commute is different than 0: what is the p value? group of answer choices 0.3189 <0.0001 0.1733 0.8312
Answers
The p-value for testing the hypothesis that the slope of commute is different from zero in the multiple regression predicting price based on lot size, commute, year built, and days on realtor.
In statistical analysis, the p-value is used to determine the significance of a predictor variable in a regression model. A p-value less than a chosen significance level (usually 0.05) indicates that there is strong evidence to reject the null hypothesis, suggesting that the predictor variable has a significant effect on the dependent variable.
In this case, the multiple regression model is used to predict the price of homes based on lot size, commute, year built, and days on realtor. The specific focus is on the slope of the commute variable, and we want to determine if it is significantly different from zero. The p-value obtained from the regression analysis is less than 0.0001, indicating strong evidence against the null hypothesis. Therefore, we can conclude that the slope of the commute variable is indeed different from zero, meaning that it has a significant impact on the price of homes in the Gainesvillehomes SP2019 dataset.
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Find the 9th term of the geometric sequence 5, -25, 125
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Answer:
The previous number keeps getting multiplied by - 5
Step-by-step explanation:
5 × -5 = -25
-25 × -5 = 125
125× -5 = -625
-625 × -5 = 3,125
3125 × -5 = -15,625
-15,625 × -5 = 78,125
78,125 × -5 = -390, 625
-390,625 × -5 = 1,953,125
1,953,125 × -5 = 9,765,625
:)
The 9th term of the geometric sequence is 5⁹.
What is a geometric sequence?A geometric sequence is a special type of sequence where the ratio of every two successive terms is a constant. This ratio is known as a common ratio of the geometric sequence.
The formula to find nth term of the geometric sequence is \(a_n=ar^{r-1}\). Where, a = first term of the sequence, r= common ratio and n = number of terms.
The given geometric sequence is 5, 25, 125,..
To find the common ratio, divide the second term by the first term:
r = 25/5
r = 5
The formula for the nth term of a geometric sequence is:
aₙ = a₁ rⁿ⁻¹
Substitute n = 9, r = 5, and a₁ = 5:
a₉ = 5 (5)⁹⁻¹
a₉ = 5 (5)⁸
a₉ = 5⁹
Hence, the 9th term of the geometric sequence is 5⁹.
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recall exercise 6.1.1 where x1, x2,...,xn is a random sample on x that has a γ(α = 4, β = θ) distribution, 0 <θ< [infinity].
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After solving the question, the Fisher information I(θ) is 4n/θ².
Let X₁, X₂, ... , \(X_{n}\) is a random sample on X that has a (α = 4, β = θ) distribution with PDf as
\(f(\frac{x}{\theta} )=\frac{x^3}{\theta^44!}e^{-\theta x}\); x ≥ 0, θ > 0.
As we know that
E(x) = 4θ and var(x) = 4θ²
The likelihood function for θ is
L(θ) = \(\prod_{i=1}^{n}f\left(\frac{x_{i}}{\theta}\right)\)
L(θ) = \(\prod_{i=1}^{n}\left(\frac{x_{i}^3}{\theta^4\cdot4!}e^{-\frac{x_{i}}{\theta}}\right)\)
L(θ) = \(\frac{\prod_{i=1}^{n}x_{i}^3}{\theta^{4n}\cdot4!^{n}}e^{-\frac{\prod_{i=1}^{n}x_{i}}{\theta}}\)
The likelihood function for θ
L(θ) = In(θ)
L(θ) = In(\(\frac{\prod_{i=1}^{n}x_{i}^3}{\theta^{4n}\cdot4!^{n}}e^{-\frac{\prod_{i=1}^{n}x_{i}}{\theta}}\))
L(θ) = \(3\prod_{i=1}^{n}x_{i}^3}- 4n\text{In}{\theta}-n\text{In}(4!)-{\frac{\Sigma x_{i}}{\theta}}\)
∂L(θ)/∂θ = -4n/θ + \({\frac{\Sigma x_{i}}{\theta^2}}\)
∂²L(θ)/∂θ² = 4n/θ² - \({\frac{2\Sigma x_{i}}{\theta^3}}\)
Then the fisher information as
I(θ) = E[-∂²L(θ)/∂θ²]
I(θ) = -4n/θ² + \({\frac{2\Sigma x_{i}}{\theta^3}}\)
I(θ) = -4n/θ² + 8nθ/θ³
I(θ) = -4n/θ² + 8n/θ²
I(θ) = 4n/θ²
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The complete question is:
Recall exercise 6.1.1 where X₁,X₂,...,\(X_{n}\) is a random sample on x that has a γ(α = 4, β = θ) distribution, 0 < θ < [infinity].
Find the Fisher information I(θ).
10. (Modeling) Height of a Projectile A projectile is fired vertically upward, and its
height s(t) in feet after t seconds is given by the function defined by
s(t) = -16t² + 800t + 600.
(a) From what height was the projectile fired?
(b) After how many seconds will it reach its maximum height?
(c) What is the maximum height it will reach?
(d) Between what two times (in seconds, to the nearest tenth) will it be more than
5000 ft above the ground?
(e) How long to the nearest tenth of a second will the projectile be in the air?
fuel
Answers
The answer is a) 0 height, b) 25 seconds c) 10,600 feet d) 3 times e) (x+1)(x-3)(x+5).
A projectile is fired vertically upward, and its height s(t) in feet after t seconds is given by the function defined by
s(t) = -16t² + 800t + 600.
To find out
(a) From what height was the projectile fired
Given height functions: s(t) = -16t² + 800t + 600
Now,
\(s^{'}(t)\) - -32t+800
(b) Now, to find the time it will attain maximum height,
\(s^{'}(t)\) = 0
-32t = -800
t = 25 seconds
(c) The maximum height it will reach
= \(-16*(25)^{2} + 800*25+600\)
= -10000+20000+600
= 10,600 feet
(d) Between what two times (in seconds, to the nearest tenth) will it be more than 5000 ft above the ground
By putting x = 3
\(P(3) = 4*(3)^{3}-50 * (3)^{2}-8*3+6\)
= 108-90-24+6
= 114-114
= 0
Therefore x = 3 is a zero of P(x)
(e) How long to the nearest tenth of a second will the projectile be in the air
Now, factoring,
\(f(x) = x^{3}+3x^{2} -13x-15\\ \\= x^{3}+x^{2} +2x^{2}+2x-15x-15\\ \\ = (x+1)(x^{2} +5x-3x-15)\\\\= (x+1)(x(x+5)-3(x+5))\\\\=(x+1)(x-3)(x+5)\)
Hence the answer is a) 0 height, b) 25 seconds c) 10,600 feet d) 3 times e) (x+1)(x-3)(x+5)
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estimate the sum of 24,872 + 651 + 971 + 8,499 to the nearest hundred.
Answers
Answer: 34990
Step-by-step explanation:
When the number of trials, n, is large, binomial probability
tables may not be available. Furthermore, if a computer is not
available, hand calculations will be tedious. As an alternative,
the Poisson distribution can be used to approximate the binomial distribution when n is large and p is small. Here the mean of the Poisson distribution is taken to be μ = np. That is, when n is large and p is small, we can use the Poisson formula with μ = np to calculate binomial probabilities; we will obtain results close to those we would obtain by using the binomial formula. A common rule is to use this approximation when n / p ≥ 500.
To illustrate this approximation, in the movie Coma, a young female intern at a Boston hospital was very upset when her friend, a young nurse, went into a coma during routine anesthesia at the hospital. Upon investigation, she found that 11 of the last 30,000 healthy patients at the hospital had gone into comas during routine anesthesias. When she confronted the hospital administrator with this fact and the fact that the national average was 6 out of 80,000 healthy patients going into comas during routine anesthesias, the administrator replied that 11 out of 30,000 was still quite small and thus not that unusual.
Note: It turned out that the hospital administrator was part of a conspiracy to sell body parts and was purposely putting healthy adults into comas during routine anesthesias. If the intern had taken a statistics course, she could have avoided a great deal of danger.)
(a) Use the Poisson distribution to approximate the probability that 11 or more of 30,000 healthy patients would slip into comas during routine anesthesias, if in fact the true average at the hospital was 6 in 80,000. Hint: μ = np = 30,000 (6/80,000) = 2.2.
Probability =
b) Given the hospital's record and part a, what conclusion would you draw about the hospital's medical practices regarding anesthesia?
Hospitals rate of comas is =
Answers
a) The probability that 11 or more of 30,000 healthy patients would slip into comas during routine anesthesia's is 0.1%
b) The conclusion would you draw about the hospital's medical practices regarding anesthesia is the hospital's medical practices regarding anesthesia may be suboptimal or inadequate, and further investigation may be necessary to identify the cause and improve patient safety.
a) To apply the Poisson distribution, we need to calculate the expected number of events, which is μ = np, where n is the sample size and p is the probability of the event occurring in one trial. In this case, n = 30,000 and p = 6/80,000, so μ = 30,000 x (6/80,000) = 2.2.
The probability of having 11 or more comas can then be approximated using the Poisson distribution formula:
P(X ≥ 11) = 1 - P(X ≤ 10) ≈ 1 - ∑(k=0 to 10) \([e^{-\mu} \times \mu^k / k!)]\)
where X is the number of comas, and the symbol ≈ means "approximately equal to."
Using a calculator or statistical software, we can compute this probability to be approximately 0.001, or 0.1%.
b) With a probability of 0.1%, it is highly unlikely that 11 or more healthy patients out of 30,000 would slip into comas during routine anesthesia if the true average rate is 6 in 80,000.
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The average mass of 5 boys is 50 kilograms. Another boy’s mass is added and the average goes to 40 kilograms. What was the mass of the sixth boy?
Answers
The mass of the sixth boy is 80 kilograms.
What was the mass of the sixth boy?It's important to note that a mean simply means the average of a set of numbers that are given. In this situation, the average mass of 5 boys is 50 kilograms. The total mass will be:
= 50 × 5
= 250 kilograms
In this case, when another boy’s mass is added and the average goes to 55 kilograms. The total mass will be:
= 55 × 6
= 330 kilograms.
The average mass of the last boy will be:
= 330 - 250
= 80 kilograms
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Complete questions
The average mass of 5 boys is 50 kilograms. Another boy’s mass is added and the average goes to 55 kilograms. What was the mass of the sixth boy?
Write a function P(x) that can be used to determine the profit for sales of x fixtures.The profit is the difference between the revenue function, R(x), and the cost function, C(x), shownR(x) = 252C(x) = 1500 + 1035x + 1500O 15x + 1500O 15x - 15000 -15x - 1500
Answers
P(X)= R(X) - C(X)
Substituing the functions on the P(X) functions:
P(X)=25x-(1500+10x)
P(X)=25x-1500-10x
Solving:
P(x)=15x-1500, The answer would be the third option!
data collected at toronto pearson international airport suggests that an exponential distribution with mean value 2.725 hours is a good model for rainfall duration (urban stormwater management planning with analytical probabilistic models, 2000, p. 69). a. what is the probability that the duration of a particular rainfall event at this location is at least 2 hours? at most 3 hours? between 2 and 3 hours?
Answers
The probability that the duration of rainfall at least 2 hours is 0.473, at most 3 hours is 0.683, and between 2 and 3 hours is 0.156.
How to find probability?1. Find the rate parameter (λ):
λ = 1/2.725 ≈ 0.367
2. To find P(X ≥ 2):
P(X ≥ 2) = 1 - P(X < 2).
The (pdf) is:
P(X < x) = 1 - e^(-λx)
P(X < 2) = 1 - e^(-0.367 * 2) ≈ 0.527
P(X ≥ 2) = 1 - 0.527 ≈ 0.473
3. To find P(X ≤ 3):
P(X ≤ 3) = 1 - e^(-0.367 * 3) ≈ 0.683
4. To find P(2 ≤ X ≤ 3):
P(2 ≤ X ≤ 3) = P(X ≤ 3) - P(X ≤ 2) = 0.683 - 0.527 ≈ 0.156
The probability that duration at least 2 hours is 0.473, at most 3 hours is 0.683, and between 2 and 3 hours is 0.156.
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Anna runs 3/4 miles in 6 minutes, or 1/10 hour. Assuming she runs at a constant rate, what is her speed, in miles per hour?
Answers
Speed of Ana = 7.5miles/hour
Step-by-step explanation:
Given,
Ana runs 3/4 mile in 6 minutes.
To find,
The speed of Ana in miles per hour
Recall the formula,
Speed = distance travelled/time taken
60 minutes = 1 hour
Solution:
Distance traveled = 3/4 mile
Time taken = 6 minutes
Since 60 minutes = 1 hour, we have
1 minute =1/60 hour
6 minutes = 6/60 hour
=1/10 hour
Hence speed = distance travelled/time taken
= 3/4 / 1/10 miles/hour
= 15/2 miles/hour
= 7.5miles/hour
∴ Speed of Ana = 7.5miles/hour
Alana has 2 1/4 bags of chocolate chips that she wants to use in 3 batches of chocolate chip cookies. How much of the chocolate chips will she use in each batch of cookies?
Answers
Answer:
1 1/3 bags of chocolate chips
Step-by-step explanation:
If Alana has 2 1/4 bags of chocolate chips that she wants to use in 3 batches of chocolate chip cookies, then we can say;
2 1/4 bags of cookies = 3 batches of chips
To determine how much of the chocolate chips will she use in each batch of cookies, we can say;
1 bag of cookies = x batches of chips
Equating both expressions
2 1/4 bags of cookies = 3 batches of chips
1 bag of cookies = x batches of chips
Cross multiply
2 1/4 x = 3
9/4 x = 3
9x/4 = 3
9x = 12
x = 12/9
x = 4/3
x = 1 1/3
Hence Alana will need 1 1/3 bags of chocolate chips