Answers
Answer:
1.
centre(h,k)=(-13,9)
radius (r)=6
we have
equation of the circle is
(x-h)²+(y-k)²=r²
(x+13)²+(y-9)²=6²
x²+26x+169+y²-18y+81=36
x²+y²+26x-18y+169+81-36=0
x²+y²+26x-18y +214=0
is a required equation of the circle.
2.
centre(h,k)=(1,-1)
radius (r)=11
we have
equation of the circle is
(x-h)²+(y-k)²=r²
(x-1)²+(y+1)²=11²
x²-2x+1+y²+2y+1=121
x²+y²-2x+2y=121-2
x²+y²-2x+2y=119
is a required equation of the circle.
Related Questions
The scale on a map is 1 in: 55 miles. What is the distance on the map between two cities that are 99 miles apart?
Answers
Answer:
1.8 inches
Step-by-step explanation:
Create a proportion where x is the distance on the map
[tex]\frac{1}{55}[/tex] = [tex]\frac{x}{99\\}[/tex]
Cross multiply and solve for x
55x = 99
x = 1.8
So, the distance on the map is 1.8 inches
Answer:
1.8 inches
Step-by-step explanation:
Scales on a map always represent the direct proportion.
Compare the distances as fractions:
Inches on the map
________________ = [tex]\frac{1}{55} = \frac{x}{99}[/tex]
Miles on the ground
[tex]x = \frac{1x99}{55}[/tex]
[tex]x = \frac{99}{55}[/tex]
[tex]x = \frac{9}{5} = 1\frac{4}{5}[/tex]
x = 1.8 inches
Is Y +7 = 5X a linear function
Answers
3 apples cost $1.00. How many apples for $2.00
Answers
Answer:
6 apples
Step-by-step explanation:
3 apples cost $1.00
so for $2.00, 2x3÷1=6
hope it helps. plz mark me as brainliest.
Answer:
6 apples
Step-by-step explanation:
if three apples are 1 dollar then every time you add a dollar you would get three more apples
so it would look somthing like
1$ = 3 apples
2$= 6 apples
3$= 9 apples
4$= 12 apples
so on
HELP MEEEEEE i dont get it my teaher didnt teach me dis??
Answers
Answer:
6
Step-by-step explanation:
because i had the same question and the departments where 6
Answer:
Hey! I can teach you this.
The range is the distance between the highest and lowest numbers.
In this case, the lowest number is 3, and the highest is 54, making the range 54 - 3, which is 51.
Because 3 is 51 away from 54. I hope this helps you!
The relation R is defined on set A = {23, 51, 36, 75, 35, 11,
102, 9, 10, 29}, and aRb means a ≡ b (mod 3)
Explain and Draw R in Digraph Notation
Answers
relation R on set A = {23, 51, 36, 75, 35, 11, 102, 9, 10, 29}, aRb means a ≡ b (mod 3), which indicates that a and b have the same remainder when divided by 3.
In the given relation R on set A = {23, 51, 36, 75, 35, 11, 102, 9, 10, 29}, aRb means a ≡ b (mod 3), which indicates that a and b have the same remainder when divided by 3.
To represent this relation R in digraph notation, we can draw a directed graph where each element of set A is represented as a node, and there is a directed edge from node a to node b if aRb holds true.
Let's go through each element of set A and determine the directed edges based on the given relation R:
1. For 23, its remainder when divided by 3 is 2. Therefore, there will be an edge from 23 to itself.
2. For 51, its remainder when divided by 3 is 0. There will be an edge from 51 to itself.
3. For 36, its remainder when divided by 3 is 0. There will be an edge from 36 to itself.
4. For 75, its remainder when divided by 3 is 0. There will be an edge from 75 to itself.
5. For 35, its remainder when divided by 3 is 2. There will be an edge from 35 to itself.
6. For 11, its remainder when divided by 3 is 2. There will be an edge from 11 to itself.
7. For 102, its remainder when divided by 3 is 0. There will be an edge from 102 to itself.
8. For 9, its remainder when divided by 3 is 0. There will be an edge from 9 to itself.
9. For 10, its remainder when divided by 3 is 1. There will be an edge from 10 to itself.
10. For 29, its remainder when divided by 3 is 2. There will be an edge from 29 to itself.
In this digraph, each node represents an element from set A, and the directed edges indicate the relation R (a ≡ b mod 3).
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Sample size = 100, sample mean = 39, sample standard deviations 13. Find the 95% confidence interval for the population mean.
Answers
Given that the sample mean is 100, the sample mean is 39, and the sample standard deviation is 13.
To find the 95% confidence interval for the population mean, we use the formula as follows:
Confidence Interval formula: CI = X ± Z* σ/√nWhere CI = Confidence IntervalX = Sample Mean
Z* = Z-Scoreσ = Standard Deviationn = Sample SizeHere, the sample size(n) is 100, the sample mean(X) is 39, and the sample standard deviation (σ) is 13.The formula for finding the Z-Score is:Z = 1 - α/2,
where α is the level of significance. α is the probability of the event not occurring, so we subtract it from one to get the probability of the event occurring.
Here, the level of significance is 0.05 since we need to find the 95% confidence interval.
Z = 1 - α/2 = 1 - 0.05/2 = 0.975Then we find the Z-Score from the Z-Score table, which is 1.96.
Therefore, the 95% confidence interval is:CI = X ± Z* σ/√n= 39 ± 1.96 (13/√100)= 39 ± 2.548Thus, the 95% confidence interval for the population mean is (36.452, 41.548).
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The 95% confidence interval for the population mean is
[tex]\[\large \left( 36.452,41.548 \right)\][/tex].
Sample size = 100
Sample mean = 39
Sample standard deviation = 13
Confidence level = 95%
To find the confidence interval, we use the formula given below:
Confidence interval formula is as follows:
[tex]\[\large \left( \overline{X}-z\frac{\sigma }{\sqrt{n}},\overline{X}+z\frac{\sigma }{\sqrt{n}} \right)\][/tex]
We are given, sample mean is 39
[tex]\(\overline{X}=39\)[/tex],
sample standard deviation is 13
[tex]\(\sigma=13\)[/tex],
sample size is 100
i.e. n=100, and confidence level is 95%
z=1.96 (From Z table)
By substituting all the given values in the formula, we get the confidence interval as,
[tex]\[\large \left( 39-1.96\frac{13}{\sqrt{100}},39+1.96\frac{13}{\sqrt{100}} \right)\][/tex]
Simplifying the above expression, we get,
[tex]\[\large \left( 39-2.548,39+2.548 \right)\][/tex]
Therefore, the 95% confidence interval for the population mean is
[tex]\[\large \left( 36.452,41.548 \right)\][/tex].
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Can someone please help me on this
Answers
Answer:
the 1 one
Step-by-step explanation:
Answer:
113.4=x(18)
x=6.3
Step-by-step explanation:
It says the product of a number, x, and 18 is 113.4. Since it says product there will be multiplication. The word "is" tells us that x and 18 equals 113.4. Is x= 6.3, 6.3 times 18 equals 113.4. This is a true equation.
Let 21, a2, a3 be a sequence defined by a1 = 1 and ak = 2ak-1 . Find a formula for an and prove it is correct using induction.
Answers
The formula for the sequence is an = [tex]2^n[/tex], where n is a positive integer. This formula is proven correct using mathematical induction.
To find a formula for the sequence defined by a1 = 1 and ak = 2ak-1, we can use mathematical induction to establish a pattern and then derive the formula. Here's how we can solve it step by step:
Step 1: Base case:
For k = 1, we have a1 = 1.
Step 2: Assume the formula holds for some positive integer n, where n ≥ 1.
Assume that an = [tex]2^{n-1[/tex] for some positive integer n.
Step 3: Use the assumption to prove the formula for the next term.
Now, let's prove that an+1 = [tex]2^n[/tex] holds.
Using the recursive formula ak = 2ak-1, we have:
an+1 = 2an
Substituting the assumed formula an = [tex]2^{n-1[/tex], we get:
an+1 = 2([tex]2^{n-1[/tex])
To simplify, we have:
an+1 = [tex]2^n[/tex]
Step 4: Conclusion:
Based on the assumption and the proof for the next term, we can conclude that the formula an = [tex]2^n[/tex] holds for all positive integers n ≥ 1.
Therefore, the formula for the sequence defined by a1 = 1 and ak = 2ak-1 is an = [tex]2^n[/tex].
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Which of the following functions are solutions of the differential equation y'' + y = 3 sin(x)? (Select all that apply.)
a. y = 3 sin(x)
b. y = 3/2x sin(x)
c. y = 3x sin(x)-4x cos(x)
d. y = 3 cos(x) e. y = -3/2x cos(x)
Answers
To determine which functions are solutions of the given differential equation y'' + y = 3 sin(x), we need to check if plugging each function into the differential equation satisfies the equation. We will examine each option and identify the functions that satisfy the equation.
The differential equation y'' + y = 3 sin(x) represents a second-order linear homogeneous differential equation with a particular non-homogeneous term.
(a) Plugging y = 3 sin(x) into the differential equation gives 0 + 3 sin(x) ≠ 3 sin(x). Therefore, y = 3 sin(x) is not a solution.
(b) Plugging y = (3/2)x sin(x) into the differential equation gives (3/2) sin(x) + (3/2)x sin(x) = (3/2)(1 + x) sin(x), which is not equal to 3 sin(x). Therefore, y = (3/2)x sin(x) is not a solution.
c) Plugging y = 3x sin(x) - 4x cos(x) into the differential equation gives 6 cos(x) - 4 sin(x) + 3x sin(x) - 3x cos(x) = 3 sin(x), which satisfies the equation. Therefore, y = 3x sin(x) - 4x cos(x) is a solution.
(d) Plugging y = 3 cos(x) into the differential equation gives -3 sin(x) + 3 cos(x) = 3 sin(x), which is not equal to 3 sin(x). Therefore, y = 3 cos(x) is not a solution.
(e) Plugging y = (-3/2)x cos(x) into the differential equation gives (3/2) sin(x) - (3/2)x cos(x) = (-3/2)(x cos(x) - sin(x)), which is not equal to 3 sin(x). Therefore, y = (-3/2)x cos(x) is not a solution.
Based on the analysis, the only function that is a solution to the given differential equation is y = 3x sin(x) - 4x cos(x) (option c).
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The cutting department of Pharoah Manufacturing has the following production and cost data for July.
Production
Costs
1.
Completed and transferred out 12,500 units.
Beginning work in process
$-0-
2.
2,850 units in ending work in process inventory are 60% complete
Direct materials
42,980
in terms of conversion costs and 100% complete
Direct labour
15,700
in terms of materials at July 31.
Manufacturing overhead
18,404
Materials are entered at the beginning of the process. Conversion costs are incurred uniformly throughout the process.
(a)
Correct answer icon
Your answer is correct.
Determine the equivalent units of production for materials and conversion costs.
Direct Materials
Conversion Costs
Total equivalent units
enter a number of units enter a number of units
eTextbook and Media
Attempts: unlimited
(b)
Calculate unit costs and prepare a cost reconciliation schedule. (Round unit costs to 2 decimal places, e.g. 15.25.)
Unit costs
Direct materials
$enter a dollar amount rounded to 2 decimal places
Conversion costs
$enter a dollar amount rounded to 2 decimal places
Cost Reconciliation Schedule
Costs accounted for
Completed and transferred out
$enter a dollar amount
Work in process inventory, July 31
Direct materials
$enter a dollar amount
Conversion costs
enter a dollar amount enter a subtotal of the two previous amounts
Total costs
Answers
a) The determination of the equivalent units of production for materials and conversion costs is as follows:
Equivalent units of production:Units Materials Conversion
Completed and transferred out 12,500 12,500 12,500
Ending work in process 2,850 2,850 1,710
Total equivalent units 15,350 15,350 14,210
b) The calculation of the unit costs is as follows:
Unit costs:
Direct Materials Conversion Costs
Production costs $42,980 $34,104
Total equivalent units 15,350 14,210
Unit costs $2.80 $2.40
c) The preparation of the cost reconciliation schedule is as follows:
Cost Reconciliation Schedule:Direct Materials Conversion Costs Total Costs
Beginning work in process $0 $0 $0
Cost to be accounted for $42,980 $34,104 $77,084
Total production costs $42,980 $34,104 $77,084
Costs accounted for units:
Completed / transferred out $35,000 $30,000 $65,000
Ending work in process $7,980 $4,104 $12,084
Total costs accounted for $42,980 $34,104 $77,084
What are equivalent units?Equivalent units are the multiplication of the number of physical (or actual) units on hand by the percentage of completion of the units.
1. Completed and transferred out 12,500 units.
2. Beginning work in process = $0
Ending work in process = 2,850 units 60% complete
Production Costs
Direct materials costs = $42,980 100% complete in terms of materials at July 31.
Direct labour = $15,700
Manufacturing overhead = $18,404
Total conversion costs = $34,104 ($15,700 + $18,404)
Equivalent units of production:
Units Materials Conversion
Completed and transferred out 12,500 12,500 12,500
Ending work in process 2,850 2,850 (100%) 1,710 (60%)
Total equivalent units 15,350 15,350 14,210
Unit costs:
Direct Materials Conversion Costs
Production costs $42,980 $34,104
Total equivalent units 15,350 14,210
Unit costs $2.80 $2.40
($42,980 ÷ 15,350) ($34,104 ÷ 14,210)
Cost Reconciliation Schedule:
Direct Materials Conversion Costs Total Costs
Beginning work in process $0 $0 $0
Cost to be accounted for $42,980 $34,104 $77,084
Total production costs $42,980 $34,104 $77,084
Costs accounted for units:
Completed / transferred out $35,000 $30,000 $65,000
(12,500 x $2.80) (12,500 x $2.40)
Ending work in process $7,980 $4,104 $12,084
(2,850 x $2.80) (1,710 x $2.40)
Total costs accounted for $42,980 $34,104 $77,084
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What is the complement of an angle of 41°?
(A) 41°
(B) 49°
(C) 139°
(D) 90°
(E) 180°
Answers
2. find the surface area of 12 in 6 in 6 in Your answer
Answers
Answer:
276 in^2
Step-by-step explanation:
2((6*6)/2) + 2(6*12) + (8*12) = 276 in^2
Of 88 adults randomly selected from one town, 69 have health insurance.
(Q) Find 90% confidence interval for the true proportion.
Write the solution with two decimal places, for example: (X.XX, X.XX)
Answers
To find the 90% confidence interval for the true proportion of adults in the town with health insurance, we can use the formula:
[tex]\[\text{{Confidence Interval}} = \left( \hat{p} - Z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + Z \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right)\][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the sample proportion (69/88 in this case)
- [tex]\(Z\)[/tex] is the Z-score corresponding to the desired confidence level (90% corresponds to [tex]\(Z = 1.645\)[/tex] for a two-tailed test)
- \(n\) is the sample size (88 in this case)
Substituting the values into the formula, we have:
[tex]\[\text{{Confidence Interval}} = \left( \frac{69}{88} - 1.645 \cdot \sqrt{\frac{\frac{69}{88} \cdot \left(1-\frac{69}{88}\right)}{88}}, \frac{69}{88} + 1.645 \cdot \sqrt{\frac{\frac{69}{88} \cdot \left(1-\frac{69}{88}\right)}{88}} \right)\][/tex]
Evaluating the expression, we find the confidence interval to be approximately (0.742, 0.892).
The confidence interval is approximately (0.742, 0.892).
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Help please this math is hard
Answers
Sebastian is going to deposit $790 in an account that earns 6.8% interest compounded annually his wife Yolanda will deposit $815 in an account that earns 7.2% simple interest each year they deposit the money on the same day and no additional deposits or withdrawals for the accounts which statement is true concerning Sebastian's in Yolanda's account balances after 3 years
Answers
Answer:
Step-by-step explanation:
Complete question
A) Sebastian's account will have about $28.67 less than Yolanda's account. B) Sebastian's account will have about $9.78 less than Yolanda's account. C) Yolanda's account will have about $28.67 less than Sebastian's account. D) Yolanda's account will have about $9.78 less than Sebastian's account.
For Sebastian
Amount = [tex]P (1 + \frac{r}{n})^{nt}[/tex]
Substituting the given values we get
A =
[tex]790 (1 + \frac{6.8}{100*1})^{3*1} \\962.367[/tex]
For Yolanda
Amount [tex]= P(1+rt)[/tex]
[tex]A = 815 (1 + \frac{7.2}{100}*3)\\A = 991.04[/tex]
Yolanda's account will have about $28.67 less than Sebastian's account
Option C is correct
What happens to light when it strikes a smooth shiny surface?
Answers
Step-by-step explanation:
Light reflects from a smooth surface at the same angle as it hits the surface. For a smooth surface, reflected light rays travel in the same direction. This is called specular reflection. For a rough surface, reflected light rays scatter in all directions.Answer:
When light strikes an object, its rays can be either absorbed or reflected. A solid black object absorbs almost all light, while a shiny smooth surface, such as a mirror, reflects almost all light back. When reflected off a flat mirror, light bounces off at an angle equal to the angle it struck the object.
Step-by-step explanation:
I think it’s b but I’m not sure but can somebody help me
Answers
Answer:
B
Step-by-step explanation:
The small triangle is half the size of the entire triangle. 27/2 = 13.5
Solve for z. -2 (52 - 4) +62 = -4
Answers
Answer:
if you learned, PEMDAS then it be easier! soo I'll help.
Step-by-step explanation:
-2 ( 52- 4 ) is 48. 48 + 62= 110
if that's wrong, then I'm sorry!
1
62
48
——
110
Please help me in it! It's very difficult, i'm in 6th and I still don't understand this. Please, help me in this!!!
Answers
Answer:
40
Step-by-step explanation:
Answer:
40
Step-by-step explanation:
the lines on the very end are your subtrct the the lowest one from the highest one
Consider an insulated uniform metal rod of length a with exposed ends and with thermal diffusivity 1. Suppose that at t = 0 the temperature profile is 1 0 (x,0) = 10 + sin 3x + 20 sin 5x = 2 sin 7x, but then the ends are held in ice at 0° C. When t is large, the temperature profile is closely approximated by a sinusoidal function of x whose amplitude is decaying to 0. What is the angular frequency of that sinusoidal function? (Hint: Start with the general solution to the heat equation with boundary conditions, and then match it to the given initial condition.)
Answers
The angular frequency of the sinusoidal function that approximates the temperature profile when t is large is 7π.
The general solution to the heat equation with boundary conditions is u(x,t) = A sin(kx) e^(-kt) + B cos(kx) e^(-kt), where k is the wavenumber and t is time. The wavenumber is related to the angular frequency by k = 2π/a, where a is the length of the rod. In this case, k = 7π/a. Therefore, the angular frequency is 7π.
The amplitude of the sinusoidal function will decay to 0 as t approaches infinity. This is because the exponential term e^(-kt) will decrease as t increases.
The initial condition u(x,0) = 10 + sin 3x + 20 sin 5x + 2 sin 7x can be matched to the general solution by setting A = 10, B = 0, k = 3, and k = 5.
The boundary conditions u(0,t) = u(a,t) = 0 can be satisfied by setting A sin(3a) e^(-kta) + B cos(3a) e^(-kta) = 0 and A sin(5a) e^(-kta) + B cos(5a) e^(-kta) = 0. These equations can be solved to find A = 0 and B = 0.
The solution u(x,t) = 0 is a sinusoidal function of x whose amplitude is decaying to 0. The angular frequency of this function is k = 2π/a = 7π.
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Find the value of a and b when x = 10
5x2
2
2x²(x - 5)
10x
Answers
Step-by-step explanation:
If x=10
2(10)²(10-5)
200 × 5
=1000
10x=10(10)
=100
if 7 is 100 % how much is 2 in %
Answers
28.571429% that is 2
hope this helped
Answer:
50% Is the answer
I NEED HELP ASAP!!!!!!
Answers
Answer:
I give bases example the triangle pyramid
In deciding whether to set up a new manufacturing plant, com- pany analysts have determined that a linear function is a reason- able estimation for the total cost C(x) in dollars of producing x items. They estimate the cost of producing 10,000 items as $547,500 and the cost of producing 50,000 items as $737,500. (a) Find a formula for C(x). (b) Find the total cost of producing 100,000 items. (c) Find the marginal cost of the items to be produced in this plant.
Answers
The formula for C(x) is `C(x) = 4.75x + 500,000`.
The total cost of producing 100,000 items is $5,250,000.
The marginal cost of the items to be produced in this plant is $4.75.
Given, Company analysts have determined that a linear function is a reasonable estimation for the total cost C(x) in dollars of producing x items. They estimate the cost of producing 10,000 items as $547,500 and the cost of producing 50,000 items as $737,500.
(a) Find a formula for C(x)
For the given data, let C(x) be the cost of producing x items, we have the two points (10,000, 547,500) and (50,000, 737,500).
We have to find the slope of the line passing through these points.
slope of the line `
m = (y2 - y1) / (x2 - x1)`m = (737,500 - 547,500) / (50,000 - 10,000)m = 190,000 / 40,000m = 4.75
Formula for C(x) can be found by using the slope-intercept form of the equation of a line.
C(x) = mx + b
We know, m = 4.75
Using the point (10,000, 547,500), we get
547,500 = 4.75 (10,000) + b.b = 547,500 - 47,500
b = 500,000
Therefore, the formula for C(x) is `
C(x) = 4.75x + 500,000`
So, the formula for C(x) is `C(x) = 4.75x + 500,000`.
(b) Find the total cost of producing 100,000 items.
Total cost of producing 100,000 items is C(100,000).
C(x) = 4.75x + 500,000
C(100,000) = 4.75 (100,000) + 500,000= 4,750,000 + 500,000= 5,250,000
Therefore, the total cost of producing 100,000 items is $5,250,000.
(c) Find the marginal cost of the items to be produced in this plant.
Marginal cost is the cost incurred for producing one additional item. It can be found by taking the first derivative of the cost function with respect to x.
C(x) = 4.75x + 500,000 `
=>` `dC(x)/dx = 4.75`
The marginal cost of the items to be produced in this plant is $4.75.
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Do social recommendations increase statectiveness? Astudy of the video viewers compared wwers who arnved at an advertising video tra pariatrand by two abarcaron noviewers who won by web Browong Data whited on whether the virtud.comly call band being studerende The results on We Trust you to recommandations?
Answers
Yes, based on the information, it should be noted that social recommendations can increase ad effectiveness.
How to explain the informationThe study you mentioned found that viewers who arrived at an advertising video through social media recommendations were more likely to correctly recall the brand being advertised than viewers who arrived by browsing. This is because social recommendations come from people we trust, and we are more likely to be influenced by their opinions.
First, social recommendations are more personalized. They are based on the interests of the person who is making the recommendation, so they are more likely to be relevant to the person who is receiving the recommendation. Second, social recommendations are more credible. We trust the opinions of our friends and family, so we are more likely to believe their recommendations. Third, social recommendations are more timely. They are shared in real time, so they are more likely to be relevant to the current moment.
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Cool-down Melanie and Kala each started solving equation 2 for x. 1 (7x – 6) = 6x – 10 The result of Melanie's first step was: 3.5x = 6 = 6x - 10 The result of Kala's first step was: 7x - 6= 12x - 20 One of them made an error. Who was it, and what was the error?
I need to find who made the error and what was the error
Answers
Answer:
Kala's first step.
Step-by-step explanation:
Solving linear equations.
Step 1: Simplify each side, if needed. Step 2: Use Add./Sub. Properties to move the variable term to one side and all other terms to the other side. Step 3: Use Mult./Div. Step 4: Check your answer. I find this is the quickest and easiest way to approach linear equations. Example 6: Solve for the variable.But Kala did not do that. Instead, she skipped the first stages and moved on ahead, making her equation invalid.
The temperature is 0 degrees, every hr it's dropping 3 degrees the temperature is -6 degrees at a certain time. How mush did the temperature decrease, how many hrs did it take to be at -6 degrees
Answers
Answer:
i think 2
Step-by-step explanation:
Is 7n+6 equal to 13n?
Answers
Answer:
No
Step-by-step explanation:
Say n was equal to 2
7(2) + 6 =20
13(2) = 26
11.Tell whether each situation can be represented by a negative number, 0, or a positive number. Negative Number 0 Positive Number Situation 1: A football team's first play resulted in a loss of 15 yards. Situation 2: A store marks up the price of a calculator $5.20. Situation 3: Nina withdrew $50 from her bank account. Situation 4: A porpoise is swimming at sea level. Situation 5: Kylie scored 2 goals in yesterday's soccer game.
Answers
Answer:
negative
positive
negative
zero
positive
Step-by-step explanation:
Situation 1: A football team's first play resulted in a loss of 15 yards.
negative
Situation 2: A store marks up the price of a calculator $5.20.
positive
Situation 3: Nina withdrew $50 from her bank account.
negative
Situation 4: A porpoise is swimming at sea level.
zero
Situation 5: Kylie scored 2 goals in yesterday's soccer game.
positive
Write all your steps leading to the answers.
A process X(t) is given by X(t)= Acosω_0t+Bsinω_0t, where A and B are independent random variables with E{A}=E{B}=0 and σ^2_A=σ^3_B=1. ω_0, is a constant. Find E{X(t)} and R(t_1, t_2).
Answers
The expected value of X(t), E{X(t)}, is 0, indicating that on average, the process X(t) fluctuates around the zero mean. The autocovariance function R(t₁, t₂) is given by R(t₁, t₂) = e^(-ω₀|t₁-t₂|), which signifies that the covariance between X(t₁) and X(t₂) decays exponentially with the difference in time values |t₁-t₂|.
To find E{X(t)}, we need to calculate the expected value of the given process X(t) = Acos(ω₀t) + Bsin(ω₀t), where A and B are independent random variables with mean 0.
E{X(t)} = E{Acos(ω₀t) + Bsin(ω₀t)}
Since E{A} = E{B} = 0, the expected value of each term is 0.
E{X(t)} = E{Acos(ω₀t)} + E{Bsin(ω₀t)}
= 0 + 0
= 0
Therefore, E{X(t)} = 0.
To find R(t₁, t₂), the autocovariance function of X(t), we need to calculate the covariance between X(t₁) and X(t₂).
R(t₁, t₂) = Cov[X(t₁), X(t₂)]
Since A and B are independent random variables with σ²_A = σ²_B = 1, the covariance term becomes:
R(t₁, t₂) = Cov[Acos(ω₀t₁) + Bsin(ω₀t₁), Acos(ω₀t₂) + Bsin(ω₀t₂)]
Using trigonometric identities, we can simplify this expression:
R(t₁, t₂) = Cov[Acos(ω₀t₁), Acos(ω₀t₂)] + Cov[Bsin(ω₀t₁), Bsin(ω₀t₂)] + Cov[Acos(ω₀t₁), Bsin(ω₀t₂)] + Cov[Bsin(ω₀t₁), Acos(ω₀t₂)]
Since A and B are independent, the covariance terms involving them are 0:
R(t₁, t₂) = Cov[Acos(ω₀t₁), Acos(ω₀t₂)] + Cov[Bsin(ω₀t₁), Bsin(ω₀t₂)]
Using trigonometric identities again, we can simplify further:
R(t₁, t₂) = cos(ω₀t₁)cos(ω₀t₂)Cov[A,A] + sin(ω₀t₁)sin(ω₀t₂)Cov[B,B]
Since Cov[A,A] = Var[A] = σ²_A = 1 and Cov[B,B] = Var[B] = σ²_B = 1, the expression becomes:
R(t₁, t₂) = cos(ω₀t₁)cos(ω₀t₂) + sin(ω₀t₁)sin(ω₀t₂)
= cos(ω₀(t₁ - t₂))
Therefore, R(t₁, t₂) = e^(-ω₀|t₁-t₂|).
The expected value of X(t), E{X(t)}, is 0, indicating that on average, the process X(t) fluctuates around the zero mean. The autocovariance function R(t₁, t₂) is given by R(t₁, t₂) = e^(-ω₀|t₁-t₂|), which signifies that the covariance between X(t₁) and X(t₂) decays exponentially with the difference in time values |t₁-t₂|.
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