We need to find the inverse of the function f(x) = 1/x. Therefore, h'(3) = -1/[tex]3^2[/tex] = -1/9. So, h'(3) = -1/9.
In the equation expressing the function, swap out f(x) with y. Swap out x and y. To put it another way, swap out every x for a y and vice versa.
An inverse in mathematics is a function that is used to another function.
Calculate y using a solution.
To find the inverse, we switch the x and y variables and solve for y:
x = 1/y
y = 1/x
So the inverse function of f(x) = 1/x is h(x) = 1/x.
Now, we need to find h'(3), the derivative of h(x) at x = 3.
h(x) = 1/x, so using the power rule of differentiation, we get:
h'(x) = -1/[tex]x^2[/tex]
Therefore, h'(3) = -[tex]1/3^2[/tex] = -1/9.
So, h'(3) = -1/9.
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Correct Question:
If h is the inverse function of f and if f(x) = 1/x, then h'(3) =
Pita has 12 coins in her bag.
There are three £1 coins and nine 50p coins.
She takes 3 coins out of the bag at random.
What is the probability that she takes out exactly £2.50?
Suppose these data show the number of gallons of gasoline sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks.
Week Sales (1,000s
of gallons)
1 17
2 22
3 19
4 24
5 19
6 16
7 21
8 19
9 23
10 21
11 16
12 22
(a)
Compute four-week and five-week moving averages for the time series.
Week Time Series
Value 4-Week
Moving
Average
Forecast 5-Week
Moving
Average
Forecast
1 17 2 22 3 19 4 24 5 19 6 16 7 21 8 19 9 23 10 21 11 16 12 22 (b)
Compute the MSE for the four-week moving average forecasts. (Round your answer to two decimal places.)
Compute the MSE for the five-week moving average forecasts. (Round your answer to two decimal places.)
(c)
What appears to be the best number of weeks of past data (three, four, or five) to use in the moving average computation? MSE for the three-week moving average is 11.15.
Three weeks appears to be best, because the three-week moving average provides the smallest MSE.Three weeks appears to be best, because the three-week moving average provides the largest MSE. Four weeks appears to be best, because the four-week moving average provides the smallest MSE.Five weeks appears to be best, because the five-week moving average provides the smallest MSE.None appear better than the others, because they all provide the same MSE.
The five-week moving average has a slightly lower MSE than the four-week moving average, suggesting that it may be a better choice for forecasting.
Using the given data, we can calculate the four-week and five-week moving averages as shown in the table below:
Week Sales (1,000s of gallons) 4-Week Moving Average Forecast 5-Week Moving Average Forecast
1 17 - -
2 22 - -
3 19 - -
4 24 20.5 -
5 19 21.0 20.2
6 16 21.0 20.6
7 21 20.0 20.2
8 19 19.8 20.0
9 23 19.8 20.2
10 21 20.5 20.6
11 16 21.0 20.6
12 22 20.5 20.6
To compute the Mean Squared Error (MSE) for the four-week moving average forecasts, we need to calculate the difference between the actual sales and the four-week moving average forecast for each week, square these differences, and then take the average of the squared differences.
Similarly, to compute the MSE for the five-week moving average forecasts, we need to calculate the difference between the actual sales and the five-week moving average forecast for each week, square these differences, and then take the average of the squared differences.
Using the given data and the formulas for MSE, we can calculate the MSE for the four-week and five-week moving average forecasts as follows:
MSE for four-week moving average forecasts = 6.58
MSE for five-week moving average forecasts = 6.32
The MSE for the four-week moving average forecasts is 6.58, while the MSE for the five-week moving average forecasts is 6.32. The MSE measures the average squared difference between the actual sales and the forecasted sales, so a lower MSE indicates a better forecast. In this case, the five-week moving average has a slightly lower MSE than the four-week moving average, suggesting that it may be a better choice for forecasting.
Ultimately, the choice of the best number of weeks to use in the moving average computation depends on the specific needs of the business or decision-maker.
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Fernando is typing 70 words in 4 minutes. How long will it take him to type 350 words? How many words can he type in 6 minutes?
Answer: it will take 20 min to type 350 words
105 words in 6 min
Step-by-step explanation:
Answer:
It will take 20 minutes to type 350 words.
In 6 minutes, 105 words can be typed.
Step-by-step explanation:
To find the time taken to type 350 words, divide 4 by 70 and then multiply it by 350.
[tex]\sf \text{Time taken to type 1 word = $\dfrac{4}{70} $}\\\\\text{Time taken to type 350 words = $\dfrac{4}{70}*350$}[/tex]
= 20 minutes
To find the number of words to be typed in 6 minutes, first find how many he can type in 1 minute.
Number of words typed in 4 minutes = 70 words
[tex]\sf \text{Number of word typed in 1 minute = $\dfrac{70}{4}$}\\\\\text{Number of word typed in 6 minute = $\dfrac{70}{4}*6$}[/tex]
= 105 words
Which equation represents this graph
The exponential function that represents the graph is given as follow:
y = 2^(x - 1) + 2.
How to define an exponential function?An exponential function has the definition presented as follows:
y = ab^x.
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The function in this problem has a horizontal asymptote at y = 2, hence:
y = ab^x + 2.
When x increases by one, y is multiplied by two, hence the parameters a and b can given as follows:
a = 1, b = 2.
The function is translated one unit right, hence it is defined as follows:
y = 2^(x - 1) + 2.
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Eva and Aiden own competing taxicab companies. Both cab companies charge a one-time pickup fee for every ride, as well as a charge for each mile traveled. Eva charges a $3 pickup fee and $1.20 per mile. The table below represents what Aiden's company charges.
Based on their unit rates, Aiden Company charges more per mile and fixed fee than Eva Company.
What is the unit rate?The unit rate is the ratio of one value compared to another.
The unit rate (also known as the slope or the constant rate of proportionality) is the quotient of two quantities.
Eva's Taxicab Company:Fixed pickup fee per ride = $3
Variable fee per mile = $1.20
Aiden's Taxicab Company:Slope (unit rate) = Rise/Run = $1.40 ($18 - $11) / (10 - 5)
Variable fee per mile = $1.40 ($7 ÷ 5)
Fixed pickup fee per ride = $4 ($11 - $1.4(5)
Thus, while Eva charges a fixed cost of $3 for every ride and $1.20 per mile, Aiden charges a fixed cost of $4 for every ride and $1.40 per mile, thereby charging more overall.
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Question Completion:Which company charges more?
Find the solutions using the Zero Product Property:
The solution is, the solutions using the Zero Product Property: is x = 7 and -2.
The expression to be solved is:
x² - 5x - 14 = 0
we know that,
The zero product property states that the solution to this equation is the values of each term equals to 0.
now, we have,
x² - 5x - 14 = 0
or, x² - 7x + 2x - 14 = 0
or, (x-7) (x + 2) = 0
so, using the Zero Product Property:
we get,
(x-7) = 0
or,
(x + 2) = 0
so, we have,
x = 7 or, x = -2
The answers are 7 and -2.
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Define a relation - by a-b a mod 4 = b mod 4. Find the equivalence class of - Be sure to start with at least 3 ellipses, 2 negative numbers, 2 positive numbers, and 3 ellipses like {. .., -2,-1,0, 1,
The relation "a-b a mod 4 = b mod 4" means that for any two numbers a and b, if their difference is divisible by 4, then they belong to the same equivalence class. To find the equivalence class of -, we need to find all the numbers that have the same modulus as - when divided by 4.
We can start by listing out some numbers with the same modulus as -. For example, we have {-9, -5, -1, 3, 7, ...}, since these numbers are all congruent to -1 mod 4. Similarly, we have {0, 4, 8, 12, ...} for numbers that are congruent to 0 mod 4, and {1, 5, 9, 13, ...} for numbers that are congruent to 1 mod 4.
Therefore, the equivalence class of - is {-9, -5, -1, 3, 7, ...}, which contains all the negative numbers that are congruent to -1 mod 4.
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Which answers describe the shape below? Check all that apply.
A. Parallelogram
B. Rectangle
C. Square
D. Rhombus
E. Trapezoid
Answer:
A
Step-by-step explanation:
Answer the question. Please!!!
★ Area of Semicircle:-
we have given Radius of Semicircle is 5.6 cm .
➺ Area = ½ π r²
➺ Area = ½ × 22/7 × 5.6²
➺ Area = ½ × 22/7 × 5.6 × 5.6
➺ Area = (22/2×7) × 5.6 × 5.6
➺ Area = 22/14 × 5.6 × 5.6
➺ Area = 11/7 × 5.6 × 5.6
➺ Area = (11 × 5.6 × 5.6/7)
➺ Area = (61.6 × 5.6/7)
➺ Area = (61.6 × 5.6/7)
➺ Area = 344.96/7
➺ Area = 49.28 cm
★ Perimeter of Semicircle:-
Radius = 5.6 ( given)➺ Perimeter = πr + 2r
➺ Perimeter = 22/7 × 5.6 + 2 × 5.6
➺ Perimeter =( 22× 5.6/7 ) + 2 × 5.6
➺ Perimeter =123.2/7 + 2 × 5.6
➺ Perimeter =123.2/7 + 11.2
➺ Perimeter =123.2 + 78.4 / 7
➺ Perimeter =201.6/7
➺ Perimeter =28.8 cm
★ Therefore:-
Area of Semicircle = 49.28 cmPerimeter of Semicircle = 28.8 cmStep-by-step explanation:
the area of a circle is
pi×r²
and of a half-circle (= half of a circle)
pi×r²/2
the area here is therefore
pi×5.6²/2 = pi×31.36/2= 15.68pi = 49.26017281... cm²
the perimeter is the sum of half of the circle's circumference plus the diameter (2×radius).
the circumference of a circle is
2×pi×r
and half of that is
2×pi×r/2 = pi×r
in our case that is
pi×5.6 = 17.59291886... cm
the full perimeter is then
17.59291886... + 2×5.6 = 28.79291886... cm
About 34% of physicians in the U.S. have been sued for malpractice. We select infinitely many
samples of 100 physicians and create a sampling distribution of the sample proportions. What is
the probability that more than 40% of 100 randomly selected physicians were sued?
a.About 1%
b.About 10%
c.About 40%
d.About 18%
The probability that more than 40% of 100 randomly selected physicians were sued is about 10%. Therefore, the answer is b. About 10%.
To determine the probability that more than 40% of 100 randomly selected physicians were sued, we need to find the mean and standard deviation of the sampling distribution and then use the z-score to find the probability.
1. Find the mean (µ) and standard deviation (σ) of the sampling distribution:
µ = p = 0.34 (the proportion of physicians sued for malpractice)
q = 1 - p = 0.66 (the proportion of physicians not sued for malpractice)
n = 100 (sample size)
[tex]Standard deviation (σ) = \sqrt{\frac{pq}{n} } = \sqrt{\frac{(0.34)(0.66)}{100} } = 0.047[/tex]
2. Calculate the z-score for the desired proportion (40% or 0.40):
[tex]z = \frac{X-µ}{σ} = \frac{0.40-0.34}{0.047} = 1.28[/tex]
3. Use a z-table or calculator to find the probability associated with the z-score:
P(Z > 1.28) =0.100 (rounded to three decimal places)
The probability that more than 40% of 100 randomly selected physicians were sued is about 10%. Therefore, the answer is b. About 10%.
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the total sales (in thousands) of a video game are given by , where 89, 45, and is the number of months since the release of the game. find and . use these results to estimate the total sales after 11 months. do not compute the total sales after 11 months. round to the nearest hundredth (2 decimal places). approximately video games after 11 months
The estimated total sales after 11 months is approximately 235.54 thousand video games. To find and in the given equation for total sales, The equation is: total sales = 89 + 45ln(number of months since release) We can see that the coefficient of the natural logarithm function is 45.
So, we have: 45 = k where k is the growth rate of the video game sales. Now, to estimate the total sales after 11 months, we need to substitute 11 for in the equation: total sales = 89 + 45ln(11) Using a calculator, we get: total sales ≈ 235.54 Rounding to the nearest hundredth, we get: total sales ≈ 235.54 thousand.
So, the estimated total sales after 11 months is approximately 235.54 thousand video games.
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Question 4: ( 6 + 8+ 6 marks) a. Divide:x3-27/9 - x2 : x2+3x+9/ x2+9x+18
b. Solve: √3x + 2-2√x=0 c. Solve: 3x7 - 24 x4=0
a. The division of (x³ - 27/9 - x²) by (x² + 3x + 9/x² + 9x + 18) is x - 3.
b. The solution to the equation √3x + 2 - 2√x = 0 is 1/3.
c. The solution to the equation 3x⁷ - 24x⁴ = 0 is 0 or 2√2/3.
For part (a), we first factorize the denominator and simplify the numerator. Then, we use long division to divide the numerator by the denominator, resulting in a quotient and a remainder.
(x³ - 27/9 - x²) / (x² + 3x + 9/x² + 9x + 18)= x - 3For part (b), we can simplify the equation by squaring both sides, rearranging, and then substituting y = √x. This results in a quadratic equation, which can be easily solved.
√3x + 2 - 2√x = 0 x = 1/3For part (c), we factorize the equation by taking out the common factor of 3x⁴. This results in a simpler equation, which can be solved by setting each factor equal to zero.
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The y-intercept of a function always occurs where y is equal to zero true or false
The y-intercept of a function always occurs where y is equal to zero is false.
Not where y is equal to zero, but where the value of x equals zero, is where a function's y-intercept is found. The y-intercept is the location where the function and y-axis cross. The value of y is equal to the y-coordinate of the intersection point at this time, while the value of x is zero.
The y-intercept, or value of y when x is equal to zero, is represented by the symbol b in the equation for a straight line, y = mx + b, where m denotes the slope and b the y-intercept. Because the y-intercept depends on the value of b, it does not follow that if the value of y is zero, it also means that the y-intercept is zero.
In conclusion, a function's y-intercept is the value of y when x is equal to zero and is located where the function meets the y-axis.
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Test the claim that for the adult population of one town, the mean annual salary is given by µ=$30,000. Sample data are summarized as n=17, x(bar)=$22,298 and s=$14,200. Use a significance level of α=0. 5. Assume that a simple random sample has been selected from a normally distribted population
After testing the claim, the required t-statistic value will come out to be approximately -2.235.
it is given that,
Population mean annual salary is μ=$30000
Sample size is n=17
Sample mean annual salary is ¯x=$22298
Sample standard deviation of the salaries is s=$14200
Level of significance is α=0.05
To test the assertion that the mean annual salary for the adult population of one town is $30000, one must determine the test statistic.
The issue is determining whether the adult population of one town makes a mean annual wage of $30,000 or not. It shows that $30000 is taken as the mean annual salary under the null hypothesis. The alternative hypothesis, however, contends that the mean annual salary is not $30000.
The alternative hypothesis and the null are thus:
H0:μ=$30000
H0:μ≠$30000
Regarding the question, it has a small sample size and there is no known population standard deviation.
Consequently, is the proper test statistic as t-statistic.
The test statistic is determined as: assuming the null hypothesis is correct.
[tex]t= \frac{¯x−μ}{\frac{s}{√n} } \\ = \frac{22298 - 30000}{ \frac{14200}{ \sqrt{17} \\} } \\ = \frac{ - 7702 \sqrt{17} }{14200} \\ = - 2.236349[/tex]
or we can take the nearest decimals and it'll be -2.236. Thus, the value of the required t-statistic is approximately -2.236.
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At a local Brownsville play production, 420 tickets were sold. The ticket prices varied on the seating arrangements and cost $8, $10, or $12. The total income from ticket sales reached $3920. If the combined number of $8 and $10 priced tickets sold was 5 times the number of $12 tickets sold, how many tickets of each type were sold?
Answer:
Number of $ 8 priced tickets = 210
Number of $10 priced tickets = 140
Number of $ 12 priced tickets = 70
Step-by-step explanation:
Framing and solving equations with three variables:
Let the number of $ 8 priced tickets = x
Let the number of $10 priced tickets = y
Let the number of $ 12 priced tickets = z
Total number of tickets = 420
x + y + z = 420 --------------(i)
Total income = $ 3920
8x + 10y + 12z = 3920 ----------------(ii)
Combined number of $8 and $10 priced tickets= 5 * the number of $12 priced tickets
x + y = 5z
x + y - 5z = 0 -------------------(iii)
(i) x + y + z = 420
(iii) x + y - 5z = 0
- - + + {Subtract (iii) from (i)}
6z = 420
z = 420÷ 6
[tex]\sf \boxed{\bf z = 70}[/tex]
(ii) 8x + 10y + 12z = 3920
(iii)*8 8x + 8y - 40z = 0
- - + - {Now subtract}
2y + 52z = 3920 -----------------(iv)
Substitute z = 70 in the above equation and we will get the value of 'y',
2y + 52*70 = 3920
2y + 3640 = 3920
2y = 3920 - 3640
2y = 280
y = 280 ÷ 2
[tex]\sf \boxed{\bf y = 140}[/tex]
substitute z = 70 & y = 140 in equation (i) and we can get the value of 'x',
x + 140 + 70 = 420
x + 210 = 420
x = 420 - 210
[tex]\sf \boxed{x = 210}[/tex]
Number of $ 8 priced tickets = 210
Number of $10 priced tickets = 140
Number of $ 12 priced tickets = 70
Find the area of the composite figure. In neccesary, round your answer to the nearest hundredth.
Answer:
(1/2)(3)(4) + (1/2)π(2^2)
= 6 + 2π square kilometers
= 12.28 square kilometers
What is the total surface area, in square centimeters, of the pyramid that Susan will paint.
The surface area of the pyramid is 186 cm².
Given that the net diagram of a square pyramid, we need to find the surface area of the same,
SA = a² + 2al
a = side length, l = height
SA = 6² + 2×6×12.5
= 186 cm²
Hence, the surface area of the pyramid is 186 cm².
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One factor of the function f(x) = x^3 − 9x^2 + 20x − 12 is (x − 6). Describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. Show your work and include all intercepts in your answer.
We are given the function, [tex]\underline{f(x)=x^3-9x^2+20x-12}[/tex], and are asked to find the x and y intercepts of the function.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
What is an intercept?
An intercept is where the graph of a function cross either the x or y axis. The x-intercept(s) crosses the x-axis and the y-intercept(s) crosses the y-axis.
How do find the x-intercept(s)?
To find the x-intercepts let y in your function equal zero, then solve for x.
How do find the y-intercept(s)?
To find the y-intercepts let x in your function equal zero, then solve for y.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Refer to the attached image for the rest.
find the z-value needed to calculate one-sided confidence bounds for the given confidence level. (round your answer to two decimal places.) a 81% confidence bound
To find the z-value needed to calculate one-sided confidence bounds for an 81% confidence level, we first need to determine the area under the normal distribution curve to the left of the confidence level. Since we are looking for one-sided confidence bound, we only need to consider the area to the left of the mean.
Using a standard normal distribution table or calculator, we can find that the area to the left of the mean for an 81% confidence level is 0.905.
Next, we need to find the corresponding z-value for this area. We can use the inverse normal distribution function to do this.
z = invNorm(0.905)
Using a calculator or a table, we can find that the z-value for an area of 0.905 is approximately 1.37.
Therefore, the z-value needed to calculate one-sided confidence bounds for an 81% confidence level is 1.37 (rounded to two decimal places).
The z-value needed to calculate a one-sided confidence bound with an 81% confidence level.
1. First, since it's one-sided confidence bound, we need to find the area under the standard normal curve that corresponds to 81% confidence. This means the area to the left of the z-value will be 0.81.
2. Now, to find the z-value, we can use a z-table or an online calculator that provides the z-value corresponding to the cumulative probability. In this case, the cumulative probability is 0.81.
3. Using a z-table or an online calculator, we find that the z-value corresponding to a cumulative probability of 0.81 is approximately 0.88.
So, the z-value needed to calculate a one-sided 81% confidence bound is 0.88, rounded to two decimal places.
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2. (10p) There are two points P1(1,2,2) and P2(-1,1,0) in Cartesian coordinate system. For position vectors R1 and R2, solve following problems. (1) Cross product of Ri and R2 (2) Inner angle between R1 and R2 (3) Area of a triangle OP1P2 (4) Circumference of the triangle OP.P2
1) the cross product of R1 and R2 is -4i + 2j + 3k.
2) the inner angle between R1 and R2 is given by:56.35 degrees
3) the area of the triangle OP1P2 is (1/2) sqrt(29).
4)the circumference of the triangle OP1P2 is: C
(1) Cross product of R1 and R2:
The cross product of two vectors R1 and R2 is given by:
R1 × R2 = (R1yR2z - R1zR2y)i - (R1xR2z - R1zR2x)j + (R1xR2y - R1yR2x)k
Substituting the values of R1 and R2, we get:
R1 × R2 = (2×0 - 2×1)i - (1×0 - (-1)×2)j + (1×1 - 2×(-1))k
= -4i + 2j + 3k
Therefore, the cross product of R1 and R2 is -4i + 2j + 3k.
(2) Inner angle between R1 and R2:
The inner angle between two vectors R1 and R2 is given by:
cos θ = (R1 · R2) / (|R1||R2|)
where R1 · R2 is the dot product of R1 and R2, and |R1| and |R2| are the magnitudes of R1 and R2, respectively.
Substituting the values of R1 and R2, we get:
R1 · R2 = 1×(-1) + 2×1 + 2×0 = -1 + 2 = 1
|R1| = sqrt(1^2 + 2^2 + 2^2) = sqrt(9) = 3
|R2| = sqrt((-1)^2 + 1^2 + 0^2) = sqrt(2)
Therefore, the inner angle between R1 and R2 is given by:
cos θ = 1 / (3sqrt(2))
θ = cos^(-1) (1 / (3sqrt(2)))
θ ≈ 56.35 degrees
(3) Area of a triangle OP1P2:
Let R = R2 - R1 be the vector connecting P1 to P2. Then the area of the triangle OP1P2 is given by:
A = (1/2) |R1 × R2|
= (1/2) |(-4i + 2j + 3k)|
= (1/2) sqrt((-4)^2 + 2^2 + 3^2)
= (1/2) sqrt(29)
Therefore, the area of the triangle OP1P2 is (1/2) sqrt(29).
(4) Circumference of the triangle OP1P2:
Let a, b, and c be the side lengths of the triangle OP1P2 opposite to the points O, P1, and P2, respectively. Then the circumference of the triangle is given by:
C = a + b + c
To find the length of side c, we can use the distance formula:
c = |R2 - R1| = sqrt((-1 - 1)^2 + (1 - 2)^2 + (0 - 2)^2) = sqrt(18)
To find the length of sides a and b, we can use the fact that the triangle isisosceles (since the angles at P1 and P2 are equal), so a = b:
a = b = |P1 - O| = sqrt(1^2 + 2^2 + 2^2) = sqrt(9) = 3
Therefore, the circumference of the triangle OP1P2 is:
C
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If there where 340 donuts, and sixth graders ate 25% of the donuts, how many donuts did sixth graders eat?
The sixth graders ate 85 donuts.
C. Arman added up all the water he drank over the 14 days and realized it was exactly 26 quarts. If he redistributed all the water so he drank exactly the same amount every day, about how many quarts would he drink each day? Check one.
A about 1 1/4 quarts
B about 2 1/4 quarts
C about 3 quarts
D about 1 7/8 quarts
differential Solve the following simultaneous Dx = axt by Dy = a'xt b'y
The general solution to the system of differential equations is:
x =
To solve the simultaneous differential equations:
Dx = axt
Dy = a'xt + b'y
We can use the method of integrating factors to solve the second equation.
Let v = exp(∫b'dt) be the integrating factor. Then we can multiply both sides of the second equation by v:
vDy = va'xt + vb'y
Notice that the left-hand side is the product rule of the derivative of vy with respect to t. So we can rewrite the equation as:
D(vy) = va'xt
Integrating both sides with respect to t, we get:
vy = exp(∫va'dt) ∫va'xt exp(-∫va'dt) dt + C
where C is a constant of integration.
Now, let's differentiate the first equation with respect to t:
D(Dx) = D(axt)
D²x = aDx + ax
Substituting Dx into the above equation, we get:
D²x = a²xt + ax
Notice that this is a linear homogeneous differential equation of the form:
D²x - ax = a²xt
which can be solved using the method of undetermined coefficients. We guess a particular solution of the form xp = bt, where b is a constant to be determined. Substituting xp into the above equation, we get:
D²(bt) - abt = a²xt
bD²t - abt = a²xt
Solving for b, we get:
b = a²/(a² - a)
Therefore, the general solution to the first equation is:
x = c₁e^t + c₂e^(-at) + a²t/(a² - a)
where c₁ and c₂ are constants of integration.
Now, let's substitute x into the equation for vy:
vy = exp(∫va'dt) ∫va'xt exp(-∫va'dt) dt + C
vy = exp(∫va'dt) ∫va'(c₁e^t + c₂e^(-at) + a²t/(a² - a)) exp(-∫va'dt) dt + C
vy = exp(∫va'dt) [c₁∫va'e^t exp(-∫va'dt) dt + c₂∫va'e^(-at) exp(-∫va'dt) dt + a²/(a² - a)∫va't exp(-∫va'dt) dt] + C
vy = exp(∫va'dt) [c₁e^(∫va'dt) + c₂e^(-a∫va'dt) + a²/(a² - a) ∫va't exp(-∫va'dt) dt] + C
where C is another constant of integration.
We can differentiate vy with respect to t to obtain y:
y = (1/v) D(vy)
y = (1/v) D(exp(∫b'dt) [c₁e^(∫va'dt) + c₂e^(-a∫va'dt) + a²/(a² - a) ∫va't exp(-∫va'dt) dt] + C)
y = exp(-∫b'dt) [c₁va'e^(∫va'dt) - c₂va'e^(-a∫va'dt) + a²/(a² - a) va't] + C'
where C' is another constant of integration.
Therefore, the general solution to the system of differential equations is:
x =
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if you invest $2500 in an account, what is the balance in the account and the amount of interest after 4 years if you earn a) 1.7% interest annually? , b) 1.5% compounded monthly? , c) 1.2 compounded daily? , d) 0.7% compounded continuously?
The balance and interest earned for each option are:
a) Balance ≈ $2801.97, Interest = $301.97b) Balance ≈ $2804.63, Interest = $304.63c) Balance ≈ $2806.54, Interest = $306.54d) Balance ≈ $2809.60, Interest = $309.60How to solve for the balance and the interesta) a) 1.7% interest annually?If you earn 1.7% interest annually, the balance in the account after 4 years would be:
Balance = $2500 × (1 + 0.017)^4 ≈ $2801.97
The interest earned would be the difference between the balance and the initial investment:
Interest = $2801.97 - $2500 = $301.97
b) 1.5% compounded monthly?If you earn 1.5% interest compounded monthly, we need to first calculate the monthly interest rate:
Monthly rate = 1.5% / 12 = 0.125%
The balance in the account after 4 years would be:
Balance = $2500 × (1 + 0.00125)^48 ≈ $2804.63
The interest earned would be the difference between the balance and the initial investment:
Interest = $2804.63 - $2500 = $304.63
c) 1.2 compounded daily?If you earn 1.2% interest compounded daily, we need to first calculate the daily interest rate:
Daily rate = 1.2% / 365 = 0.003288%
The balance in the account after 4 years would be:
Balance = $2500 × (1 + 0.00003288)^1460 ≈ $2806.54
The interest earned would be the difference between the balance and the initial investment:
Interest = $2806.54 - $2500 = $306.54
d) 0.7% compounded continuously?If you earn 0.7% interest compounded continuously, we use the formula:
Balance = $2500 × e^(0.007 × 4) ≈ $2809.60
The interest earned would be the difference between the balance and the initial investment:
Interest = $2809.60 - $2500 = $309.60
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A football game in 2?011 had an attendance of 213,070 fans. The headline in the newspaper the next day read, About 200,000 people attend the big game! Is the newspapers estimate reasonable
The newspaper's estimate is not a reasonable approximation of the actual attendance of the football game.
The newspaper's estimate of "about 200,000 people" attending the football game with an actual attendance of 213,070 is not a very accurate estimate.
The newspaper's estimate is an approximation and rounded off to the nearest hundred thousand, which is a difference of 13,070 people. This is a large discrepancy and represents an error of more than 6% of the actual attendance.
It is important to note that rounding off numbers is a common practice, but it should be done with care and precision. In this case, rounding off to the nearest hundred thousand leads to a significant difference and makes the estimate quite unreliable.
Therefore, the newspaper's estimate is not a reasonable approximation of the actual attendance of the football game.
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1. Let X₁,..., Xy be independent random variables. Prove the following statements:
(a) If for each i = 1,2..., N one has P(X; <6) ≤6 for all 6 € (0, 1), then
n
P(ΣIXI0.
i=l
Hint: you may want to prove that EIe-ax,1I≤2/1, 1 > 0.
(b) If for each i = 1,..., N one has P(X; <6) ≥d for some 8 € (0, 1), then
n
P[ΣIxiI
i=l
The assumption that P(Xi < 6) ≥ d for some 8 € (0, 1), we can show that Var(Xi) ≤ 6^2 - (6d)^
(a) To prove that P(ΣIXI0 for all t > 0, we can use Markov's inequality, which states that for any non-negative random variable Y and any positive constant a, we have:
P(Y ≥ a) ≤ E(Y)/a
Let Y = e^(tΣIXi) and a = e^t. Then we have:
P(ΣIXi ≥ t) = P(e^(tΣIXi) ≥ e^t) ≤ E(e^(tΣIXi))/e^t
Now, we need to show that E(e^(tΣIXi)) ≤ e^(t^2/2). To do this, we can use the fact that for any independent random variables Y1, Y2, ..., Yn, we have:
E(e^(t(Y1+Y2+...+Yn))) = E(e^(tY1)) E(e^(tY2)) ... E(e^(tYn))
Uszng this formula and the assumption that P(Xi < 6) ≤ 6 for all 6 € (0, 1), we get:
E(e^(tXi)) = ∫₀^₆ e^(tx) fXi(x) dx ≤ ∫₀^₆ e^(6t) fXi(x) dx = e^(6t) E(Xi)
Therefore, we have:
E(e^(tΣIXi)) = E(e^(tX1) e^(tX2) ... e^(tXn)) ≤ E(e^(6t)X1) E(e^(6t)X2) ... E(e^(6t)Xn) = (E(X1) e^(6t))^(n)
Since Xi is non-negative, we have E(Xi) = ∫₀^₆ fXi(x) dx ≤ 1, so we get:
E(e^(tΣIXi)) ≤ (e^(6t))^n = e^(6nt)
Finally, substituting this inequality into the earlier expression, we get:
P(ΣIXi ≥ t) ≤ E(e^(tΣIXi))/e^t ≤ (e^(6nt))/e^t = e^(6n-1)t
Since this inequality holds for all t > 0, we have:
P(ΣIXi ≥ 0) = lim t→0 P(ΣIXi ≥ t) ≤ lim t→0 e^(6n-1)t = 1
Therefore, we have shown that P(ΣIXi ≥ 0, as required.
(b) To prove that P(ΣIXi ≥ t) ≥ 1 - ne^(-2t^2/d^2) for all t > 0, we can use Chebyshev's inequality, which states that for any random variable Y with finite mean and variance, we have:
P(|Y - E(Y)| ≥ a) ≤ Var(Y)/a^2
Let Y = ΣIXi and a = t. Then we have:
P(|ΣIXi - E(ΣIXi)| ≥ t) ≤ Var(ΣIXi)/t^2
Now, we need to find an upper bound for Var(ΣIXi). Since the Xi are independent, we have:
Var(ΣIXi) = Var(X1) + Var(X2) + ... + Var(Xn)
Using the assumption that P(Xi < 6) ≥ d for some 8 € (0, 1), we can show that Var(Xi) ≤ 6^2 - (6d)^
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Use General Linear Process to determine the mean function and the autocovariance function of ARC2) given by Xt = ∅1X't-1- ∅2X't-2 +et
The GLP's mean function is (t) = (1 + 2), and the GLP's autocovariance function is γ(h) = ∅1² γ(h-1) + ∅2² γ(h-2) - ∅1∅2 γ(h-2), where γ(0) = σ² / (1 - ∅1² - ∅2²).
What is function?A function connects an input with an output. It is analogous to a machine with an input and an output. And the output is somehow related to the input. The standard manner of writing a function is f(x) "f(x) =... "
To use the General Linear Process approach, we first express the given AR(2) model in the following form:
Xt = ∅1Xt-1 - ∅2Xt-2 + et
where et is a white noise process with zero mean and variance σ².
The mean function of this GLP is given by:
μ(t) = E[Xt] = E[∅1Xt-1 - ∅2Xt-2 + et] = ∅1E[Xt-1] - ∅2E[Xt-2] + E[et]
Since et is a white noise process with zero mean, we have E[et] = 0. Also, by assuming that the process is stationary, we have E[Xt-1] = E[Xt-2] = μ. Therefore, the mean function of the GLP is:
μ(t) = μ(∅1 + ∅2)
The autocovariance function of this GLP is given by:
γ(h) = cov(Xt, Xt-h) = cov(∅1Xt-1 - ∅2Xt-2 + et, ∅1Xt-1-h - ∅2Xt-2-h + e(t-h))
Note that et and e(t-h) are uncorrelated since the white noise process is uncorrelated at different time points. Also, we assume that the process is stationary, so that the autocovariance function only depends on the time lag h. Using the properties of covariance, we have:
γ(h) = ∅1² γ(h-1) + ∅2² γ(h-2) - ∅1∅2 γ(h-2)
where γ(0) = Var[Xt] = σ² / (1 - ∅1² - ∅2²).
Therefore, the mean function of the GLP is μ(t) = μ(∅1 + ∅2), and the autocovariance function of the GLP is γ(h) = ∅1² γ(h-1) + ∅2² γ(h-2) - ∅1∅2 γ(h-2), where γ(0) = σ² / (1 - ∅1² - ∅2²).
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For the scale model of an airplane Jamie is building, 4 feet is proportional to 6 inches. If the length of the airplane Jamie is modeling is 20 feet, what will be the length of his model ?
Figure these out ……….
Answer:
o
Step-by-step explanation:
(a) Prove by contradiction: If the sum of two primes is prime, then one of the primes must be 2.
You may assume that every integer is either even or odd, but never both.
(b) Prove by contradiction: Suppose n is an integer that is divisi- ble by 4. Then n + 2 is not divisible by 4.
[tex]$m-k=\frac{1}{2}$[/tex], which contradicts the assumption that m and k are integers. Hence, our assumption that [tex]$n+2$[/tex] is divisible by 4 is false.
what is algebra?Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.
(a) Suppose that the sum of two primes, [tex]$p_1$[/tex] and [tex]$p_2$[/tex], is prime and neither [tex]$p_1$[/tex] nor [tex]$p_2$[/tex] is 2. Since [tex]$p_1$[/tex] and [tex]$p_2$[/tex] are both odd primes, they must be of the form [tex]$p_1=2k_1+1$[/tex] and [tex]$p_2=2k_2+1$[/tex] for some integers [tex]$k_1$[/tex] and [tex]$k_2$[/tex]. Therefore, their sum can be written as:
[tex]$p_1+p_2=2k_1+1+2k_2+1=2(k_1+k_2)+2=2(k_1+k_2+1)$[/tex]
Since [tex]$k_1+k_2+1$[/tex] is an integer, [tex]$p_1+p_2$[/tex] is even and greater than 2, and therefore cannot be prime, contradicting our assumption. Therefore, one of the primes must be 2.
(b) Suppose, for the sake of contradiction, that n is divisible by 4 and n+2 is also divisible by 4. Then we can write:
n=4k for some integer k,
n+2=4m for some integer m.
Subtracting the first equation from the second, we get:
2=4(m-k)
Therefore, [tex]$m-k=\frac{1}{2}$[/tex], which contradicts the assumption that m and k are integers. Hence, our assumption that n+2 is divisible by 4 is false.
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