Let X and Y be independent continuous random variables with PDFs fx,and fy, respectively, and let Z X+Y (a) Show that far (zlx) = fyG-x). (b) Assume that X and Y are exponentially distributed with parameter λ Find the conditional PDF of X, given that Z - z. (c) Assume that X and Y are normal random variables with mean zero and variances a2 1, and a2 2. respectively. Find the conditional PDF of X, given that Z-z. 7.
Answers
a. This is equal to \(\(f_Y(z-x)\)\), which proves the desired result.
b. The normalized conditional PDF is:
\(\[f_{X|Z}(z|x) = \frac{\lambda e^{-\lambda (z-x)}}{\lambda e^{\lambda z} \cdot \frac{1}{\lambda}} = e^{-\lambda x}\]\)
c. The normalized conditional PDF is:
\(\[f_{X|Z}(z|x) = \frac{1}{\sqrt{2\pi\sigma_2^2}} e^{-\frac{(z-x)²}{2\sigma_2^2}}\]\)
What is probability?Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is.
(a) To show that \(\(f_{X|Z}(z|x) = f_{Y}(z-x)\)\), we can use the definition of conditional probability:
\(\[f_{X|Z}(z|x) = \frac{f_{X,Z}(x,z)}{f_Z(z)}\]\)
Since X and Y are independent, the joint probability density function (PDF) can be expressed as the product of their individual PDFs:
\(\[f_{X,Z}(x,z) = f_X(x) \cdot f_Y(z-x)\]\)
The PDF of the sum of independent random variables is the convolution of their individual PDFs:
\(\[f_Z(z) = \int f_X(x) \cdot f_Y(z-x) \, dx\]\)
Substituting these expressions into the conditional probability formula, we have:
\(\[f_{X|Z}(z|x) = \frac{f_X(x) \cdot f_Y(z-x)}{\int f_X(x) \cdot f_Y(z-x) \, dx}\]\)
Simplifying, we get:
\(\[f_{X|Z}(z|x) = \frac{f_Y(z-x)}{\int f_Y(z-x) \, dx}\]\)
This is equal to \(\(f_Y(z-x)\)\), which proves the desired result.
(b) If X and Y are exponentially distributed with parameter λ, their PDFs are given by:
\(\[f_X(x) = \lambda e^{-\lambda x}\]\)
\(\[f_Y(y) = \lambda e^{-\lambda y}\]\)
To find the conditional PDF of X given Z = z, we can use the result from part (a):
\(\[f_{X|Z}(z|x) = f_Y(z-x)\]\)
Substituting the PDFs of X and Y, we have:
\(\[f_{X|Z}(z|x) = \lambda e^{-\lambda (z-x)}\]\)
To normalize this PDF, we need to compute the integral of \(\(f_{X|Z}(z|x)\)\) over its support:
\(\[\int_{-\infty}^{\infty} f_{X|Z}(z|x) \, dx = \int_{-\infty}^{\infty} \lambda e^{-\lambda (z-x)} \, dx\]\)
Simplifying, we get:
\(\[\int_{-\infty}^{\infty} f_{X|Z}(z|x) \, dx = \lambda e^{\lambda z} \int_{-\infty}^{\infty} e^{\lambda x} \, dx\]\)
The integral on the right-hand side is the Laplace transform of the exponential function, which evaluates to:
\(\[\int_{-\infty}^{\infty} e^{\lambda x} \, dx = \frac{1}{\lambda}\]\)
Therefore, the normalized conditional PDF is:
\(\[f_{X|Z}(z|x) = \frac{\lambda e^{-\lambda (z-x)}}{\lambda e^{\lambda z} \cdot \frac{1}{\lambda}} = e^{-\lambda x}\]\)
This is the PDF of an exponential distribution with parameter λ, which means that given Z = z, the conditional distribution of X is still exponential with the same parameter.
(c) If X and Y are normally distributed with mean zero and variances σ₁² and σ₂², respectively, their PDFs are given by:
\(\[f_X(x) = \frac{1}{\sqrt{2\pi\sigma_1^2}} e^{-\frac{x^2}{2\sigma_1^2}}\]\)
\(\[f_Y(y) = \frac{1}{\sqrt{2\pi\sigma_2^2}} e^{-\frac{y^2}{2\sigma_2^2}}\]\)
To find the conditional PDF of X given Z = z, we can use the result from part (a):
\(\[f_{X|Z}(z|x) = f_Y(z-x)\]\)
Substituting the PDFs of X and Y, we have:
\(\[f_{X|Z}(z|x) = \frac{1}{\sqrt{2\pi\sigma_2^2}} e^{-\frac{(z-x)^2}{2\sigma_2^2}}\]\)
To normalize this PDF, we need to compute the integral of \(\(f_{X|Z}(z|x)\)\) over its support:
\(\[\int_{-\infty}^{\infty} f_{X|Z}(z|x) \, dx = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi\sigma_2^2}} e^{-\frac{(z-x)^2}{2\sigma_2^2}} \, dx\]\)
This integral can be recognized as the PDF of a normal distribution with mean z and variance σ₂². Therefore, the normalized conditional PDF is:
\(\[f_{X|Z}(z|x) = \frac{1}{\sqrt{2\pi\sigma_2^2}} e^{-\frac{(z-x)²}{2\sigma_2^2}}\]\)
This is the PDF of a normal distribution with mean z and variance σ₂², which means that given Z = z, the conditional distribution of X is also normal with the same mean and variance.
Learn more about probability on:
https://brainly.com/question/13604758
#SPJ4
Related Questions
What code would you use to assign a vector of the odd numbers between 20 and 30 to an object called ""stats""?.
Answers
To assign a vector of the odd numbers between 20 and 30 to an object called "stats" in a programming language like R, you can use the following code:
R stats <- seq(21, 29, 2)
- The `seq()` function is used to generate a sequence of numbers.
- The first argument, 21, specifies the starting point.
- The second argument, 29, specifies the ending point.
- The third argument, 2, specifies the step size.
- By setting the step size to 2, the `seq()` function generates a sequence of odd numbers.
- The resulting sequence of odd numbers between 20 and 30 is assigned to the object "stats" using the assignment operator `<-`.
After executing this code, the vector of odd numbers (21, 23, 25, 27, 29) will be stored in the "stats" object for further use in your program.
learn more about sequence here:
https://brainly.com/question/30262438
#SPJ11
Maggy needs to ship watches for her boss. Maggy has 50 watches and 6 boxes. Each box must contain an equal number of
watches. How many watches does she have left over?
Answers
Answer:
2
Step-by-step explanation:
took usatest prep
thank you
What value of x makes the equation -4+12x=8 true?
Answers
Answer:
1
Step-by-step explanation:
a car uses 30 gallons of gasoline for a trip of 750 miles. How many gallons would be used on a trip of 450 miles
Answers
Answer:
The answer is 18
Step-by-step explanation:
Given;A car uses 30 gallons of gasoline for a trip of 750 miles.
To Find;Gallons would be used on a trip of 450 miles.
Now, Gasoline used for one mile
30 ÷ 750 = 0.04
So,
0.04 × 450 = 18
Thus, 18 gallons would be used on a trip of 450 miles.
-TheUnknownScientist 72
what is 4(5x + 3) 4(5x + 3)
Answers
Answer:
what math is this???????
Next step is 16(5x+3)^2
So solution is 400x^2+480x+144
How many points of intersection are there between the graphs of the hyperbola and ellipse?
3x2 – 4y2 + 4x – 8y +4= 0
3x2 + y2 + 4x – 3y +4= 0
A.0
B. 1
C. 3
D.4
Answers
The point of intersection is the point where two or more functions meet.
The graphs of the hyperbola and ellipse have 0 point of intersection
The given parameters are:
\(\mathbf{3x^2 - 4y^2 + 4x - 8y + 4 = 0}\)
\(\mathbf{3x^2 + y^2 + 4x - 3y + 4 = 0}\)
To determine the points of intersection, we simply equate both equations.
So, we have:
\(\mathbf{3x^2 + y^2 + 4x - 3y + 4 = 3x^2 - 4y^2 + 4x - 8y + 4 }\)
Cancel out the common terms
\(\mathbf{y^2 - 3y = - 4y^2 - 8y }\)
Collect like terms
\(\mathbf{y^2 +4y^2= 3y - 8y }\)
\(\mathbf{5y^2= - 5y }\)
Divide through by 5
\(\mathbf{y^2= - y}\)
Divide through by y
\(\mathbf{y= -1}\)
The system of equation does not give room to calculate the x-coordinate at the x-axis.
Hence, the graphs of the hyperbola and ellipse have 0 point of intersection
Read more about points of intersection at:
https://brainly.com/question/13373561
Answer: 0
Explanation: Took the test
A water well drilling rig can dig -48 feet after one full day (24 hours) of continuous use. Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
Answers
we have the next information
48 ft in 24 hours
x ft in 15 hours
x is the height of the drill after 15 hours
x can be calculated if we multiply 15 by 48 and then we divide the result between 24
\(\begin{gathered} x=\frac{15\cdot48}{24} \\ \end{gathered}\)\(\begin{gathered} x=\frac{720}{24} \\ x=30 \end{gathered}\)the height that was drill after 15 hours was 30 ft
Allison can jog 3 blocks in the same amount of time it takes her brother, Blake, to jog 2 blocks. If they keep jogging at the same rate, how many blocks will Blake jog if Allison jogs 18 blocks?
Answers
Answer:
Blake will jog 12 blocks
Step-by-step explanation:
Allison : Blake = 3 : 2
If they keep jogging at the same rate, how many blocks will Blake jog if Allison jogs 18 blocks?
Allison : Blake = 18 : x
Equate the ratios
3 : 2 = 18 : x
3/2 = 18/x
Cross multiply
3 * x = 2 * 18
3x = 36
Divide both sides by 3
x = 36 / 3
= 12
x = 12
Blake will jog 12 blocks
Answer:
12 blocks
Step-by-step explanation: she ran 12 blocks
A machine cuts a strip of carpet into two pieces. The length of the smaller piece is 5 m greater than 1/4 the length of the larger piece if the length of the smaller piece is 12 m the length of the big piece is.
Answers
Let length of the big piece be x.
It is told that the length of the small piece of the carpet is 5m greater (+5) than 1/4 the length of the larger piece (1/4 × x)
Putting it all together in an equation we can calculate the length of the big piece:
12 m (length of the smaller piece) = \(\frac{1}{4}\) × x (length of larger piece) + 5m
Solving the equation for x we get,
12m - 5m = \(\frac{1}{4}\) x
7m × 4 = x
28m = x
Therefore, the length of the larger piece is:
x = 28m
Read more about solving word problems with algebraic equations:
https://brainly.com/question/4541471
https://brainly.com/question/13818690
Doughnut Muffins
cups flour 1 cup oil
teaspoons baking powder cups sugar
teaspoons salt 3 eggs
teaspoons nutmeg cups milk
teaspoon cinnamon
Combine flour, baking powder, salt, nutmeg, and cinnamon in a bowl. In another bowl, combine oil, sugar, egg, and milk. Add liquid ingredients to dry ingredients; stir only to combine. Bake in paper-lined muffin tins at 350° for 20 to 25 minutes.
Donna wants to make a batch of doughnut muffins that only yields 12 muffins. Modify the recipe for her, making sure to include new measurements of every ingredient.
Answers
Answer:
1 3/4 cups of flour
1/3 cup oil
1 1/2 teaspoons baking powder
3/4 cups sugar
1/2 teaspoons salt
1 egg
3/4 cups milk
1/4 teaspoon cinnamon
I hope this is right!
Step-by-step explanation:
Looks like I can update my answer!
Okay...the recipe is...
Doughnut muffins
Yields: 36 muffins
5 1/4 cups of flour
1 cup oil
4 1/2 teaspoons baking powder
2 1/4 cups sugar
1 1/2 teaspoons salt
3 eggs
2 1/4 cups milk
3/4 teaspoon cinnamon
Alright, it looks as though what you need to do here first is to turn everything into improper fractions. This would result in:
21/4 cups of flour
1/1 cup oil
9/2 teaspoons baking powder
9/4 cups sugar
3/2 teaspoons salt
3/1 eggs
9/4 cups milk
3/4 teaspoon cinnamon
Then, divide everything by three...
7/4 cups of flour
1/3 cup oil
3/2 teaspoons baking powder
3/4 cups sugar
1/2 teaspoons salt
1/1 egg
3/4 cups milk
1/4 teaspoon cinnamon
...and simplify.
1 3/4 cups of flour
1/3 cup oil
1 1/2 teaspoons baking powder
3/4 cups sugar
1/2 teaspoons salt
1 egg
3/4 cups milk
1/4 teaspoon cinnamon
There you go!
The discrete random variable X is the number of students that show up for Professor Adam's office hours on Monday afternoons. The table below shows the probability distribution for X. What is the probability that fewer than 2 students come to office hours on any given Monday? X Р(Х) 0 40 1 30 2 .20 3 .10 Total 1.00 0.50 0.40 0.70 0.30
Answers
The probability that fewer than 2 students come to office hours on any given Monday is 0.70.
How we find the probability?To find the probability that fewer than 2 students come to office hours on any given Monday, we need to calculate the sum of the probabilities of X=0 and X=1.
P(X < 2) = P(X = 0) + P(X = 1)
= 0.40 + 0.30
= 0.70
From the given probability distribution, we can see that the probability of X=0 is 0.40 and the probability of X=1 is 0.30. These represent the probabilities of no students or one student showing up for office hours, respectively.
To find the probability that fewer than 2 students come to office hours on any given Monday, we need to add these probabilities together since X can only take on integer values.
Therefore, P(X < 2) = P(X = 0) + P(X = 1) = 0.40 + 0.30 = 0.70.
This means that there is a 70% chance that either no students or one student will show up for office hours on any given Monday.
Learn more about Probability
brainly.com/question/16484393
#SPJ11
Archie bought a shirt on sale that was 20% less than the original price. The original price was $5 more than the sale price. What was the original price?
Answers
The original price of the shirt was $6.25
In this question, we have been given Archie bought a shirt on sale that was 20% less than the original price. The original price was $5 more than the sale price.
We need to find the original price.
Let the original price of the shirt be x and the sale price of the shirt is y
20x/100 = y
x/5 = y
x = 5y
Now the second equation that we get is
x = y + 5
5y - y = 5
4y = 5
y = 5/4
Now putting the value of y in the first equation we get
x = 5y
= 5 (5/4)
= 25/4
= 6 1/4
= $6.25
Then we can say that the original price of the shirt was $6.25
Therefore, the original price of the shirt was $6.25
Learn more about an equation here:
https://brainly.com/question/649785
#SPJ1
Which equation is the inverse of 2(x - 2)2 = 8(7 + y)?
-
=
O -2(x - 2)2 = -8(7 + y)
O y-1x2-x-6
O y=-2+28+4x
y = 2+28+4x
Answers
The inverse of an equation is the opposite of the equation
The inverse of the function is \(y =2 \pm \sqrt{28 + 4x\)
How to determine the inverse of the functionThe equation is given as:
\(2(x - 2)^2 = 8(7 + y)\)
Divide both sides of the equation by 2
\((x - 2)^2 = 4(7 + y)\)
Open the bracket
\((x - 2)^2 = 28 + 4y\)
Take the square roots of both sides
\(x - 2 = \pm\sqrt{28 + 4y\)
Add 2 to both sides
\(x =2 \pm \sqrt{28 + 4y\)
Swap the occurrences of x and y
\(y =2 \pm \sqrt{28 + 4x\)
Hence, the inverse of the function is \(y =2 \pm \sqrt{28 + 4x\)
Read more about inverse functions at:
https://brainly.com/question/8120556
the manager of a supermarket tracked the amount of time needed for customers to be served by the cashier. after checking with his statistics professor, he concluded that the checkout times are exponentially distributed with a mean of 5.5 minutes. what propotion of customers require more than 12 minutes to check out?
Answers
Approximately 0.357 or 35.7% of customers require more than 12 minutes to check out.
Since the checkout times are exponentially distributed with a mean of 5.5 minutes, we can use the exponential distribution formula to find the probability that a customer will take more than 12 minutes to check out:
P(X > 12) = 1 - P(X ≤ 12)
where X is the checkout time.
To find P(X ≤ 12), we can use the cumulative distribution function (CDF) of the exponential distribution, which is:
F(x) = 1 - e^(-λx)
where λ is the rate parameter of the distribution. For an exponential distribution with mean μ, the rate parameter λ is equal to 1/μ.
So, in our case, λ = 1/5.5 = 0.1818, and we can calculate P(X ≤ 12) as:
P(X ≤ 12) = F(12) = 1 - e^(-0.1818 × 12) ≈ 0.643
Therefore, the probability that a customer will take more than 12 minutes to check out is:
P(X > 12) = 1 - P(X ≤ 12) ≈ 1 - 0.643 ≈ 0.357
To learn more about statistics click on,
https://brainly.com/question/15291758
#SPJ4
Help needed ASAP will give you BRAINLIEST and 5 stars
Answers
Answer:
y-x
Step-by-step explanation:
Answer:
Y=-x
Step-by-step explanation:
Because any other ones have stuff they should.
8.
For which set of data is the median greater than the mean?
A.
B.
C.
D.
{8, 12, 14, 14, 20}
{8, 12, 14, 14, 22}
{9, 13, 14, 15, 19}
(9, 13, 14, 15, 20}
Answers
Answer:
B
Step-by-step explanation:
For a set of data to have a median that is greater than the mean, it must have an odd number of values and be skewed to the right. This means that the majority of the values in the set must be smaller than the median, with a few larger values on the right side of the distribution.
Out of the given options, the only set of data that satisfies these conditions is (B) {8, 12, 14, 14, 22}. This set has an odd number of values, and the median of 14 is greater than the mean of 14.4. The other options have either an even number of values, or are not skewed to the right.
Therefore, the correct answer is (B) {8, 12, 14, 14, 22}.
The range of cells in column D and rows 1 to 10 is represented as:
1:10 D1:D10 D:D
Answers
Answer:
how?
Step-by-step explanation:
I didnt get question
The degrees of freedom for a chi-square test applied to a cross-tabulation with 5 rows and 2 columns is equal to:
Answers
The degrees of freedom for a chi-square test applied to a cross-tabulation with 5 rows and 2 columns is equal to 6
How to determine the degrees of freedom applied to a cross-tabulation ?The given parameters are:
Rows = 5
Column = 2
The degrees of freedom is then calculated as:
df = Rows + Column - 1
Substitute the known values in the above equation
df = 5 + 2 - 1
Evaluate the expression
df = 6
Hence, the degrees of freedom for a chi-square test applied to a cross-tabulation with 5 rows and 2 columns is equal to 6
Read more about degrees of freedom at
https://brainly.com/question/17305237
#SPJ1
x-5=√x + 1
solve and check for extraneous solutions
Answers
Hope this helps!
Geometry. Math nation section 3
Answers
∠g and ∠h are complementary angles and ∠g and ∠h are acute angles are true statements from the given information
Two angles are given.
∠g = (2x-90)°
∠h = (180-2x)°
We have to find the statement which is true about the angles g and h.
If both angles are greater than zero.
Complementary angles add up to 90 degrees
i.e., ∠g and ∠h are complementary if ∠g + ∠h = 90°.
Substituting the given values:
∠g + ∠h
= (2x-90)° + (180-2x)° = 90°
Thus, ∠g and ∠h are complementary angles.
and both the angles are less than 90 degrees so we can tell that angles ∠g and ∠h are acute.
So the statement ∠g and ∠h are acute angles is also true
Hence, ∠g and ∠h are complementary angles and ∠g and ∠h are acute angles are true statements
To learn more on Angles click:
https://brainly.com/question/28451077
#SPJ1
draw the graph y= 3 - 1/2x
Answers
To graph this by hand, first put your equation in y intercept form
y = mx + b
m = slope
b = y intercept
y = -1/2x + 3
Start with y intercept. Plot your y intercept of 3 on the y axis. This coordinate would be (0, 3) on the graph.
From there, use slope of -1/2 ( y/x ) to plot your mixture point. From y intercept of 3, move down -1 and right +2. Plot your point. Do this again, -1 down and right +2. Plot your point. Do this again so you have enough points to construct a straight line.
* note: sometimes you need points going the other direction for a straight line. You can do the inverse of -1/2 and instead do 1/-2 which is the same slope. Go up +1 and left -2. I did this in green on the attachment.
14.2 cm correct 1 decimal place
Answers
The 14.2 cm correct 1 decimal place is 14.2 after rounding to the nearest 0.1 or the tenth place.
What is rounding off the number?Rounding is a technique to reduce a large number to a smaller, more approachable figure which is very similar to the actual. Rounding numbers can be achieved in a variety of ways.
It is given that:
14.2 cm correct 1 decimal place
14.2
Rounded to the nearest 0.1 or
the Tenths Place.
Rounded to the nearest tenth place. The 2 in the tenth place rounds down to 2 or stays the same because the digit to the right in the hundredth place is 14.2
Thus, the 14.2 cm correct 1 decimal place is 14.2 after rounding to the nearest 0.1 or the tenth place.
Learn more about the rounding off numbers here:
https://brainly.com/question/13391706
#SPJ1
Help hurry please brainliest and I will add you
Answers
-0.83 would be the strongest out of these options.
It's negative, so we know it'd going down, and it's also much stronger than -0.23.
There were 100 people in a given town: 15 of them were children under 16,10 retirees, 30 people had full-time jobs, and 12 had part-time jobs. There were 10 full-time homemakers, 5 full-time students over age 16 . The remaining people did not have jobs, but all said they would like one. 3 people had not looked actively for work for three months, however." a. Calculate the adult or working age population or population 16 and older b. Calculate out-of-labour force c. Calculate the labour force d. Calculate the participation rajp - Calculate the employment raja
Answers
The labor force is the sum of those with full-time and part-time jobs, while the participation rate is the percentage of the labor force in relation to the adult population. Finally, the employment rate represents the percentage of the labor force that is employed.
(a) To calculate the adult or working age population (16 and older), we subtract the number of children under 16 (15) and retirees (10) from the total population of 100. Therefore, the adult or working age population is 100 - 15 - 10 = 75.
(b) The out-of-labor force refers to individuals who are not working or actively seeking employment. In this case, it includes full-time homemakers (10) and full-time students over age 16 (5). Thus, the out-of-labor force is 10 + 5 = 15.
(c) The labor force consists of those with full-time jobs (30) and part-time jobs (12). Therefore, the labor force is 30 + 12 = 42.
(d) The participation rate is calculated by dividing the labor force (42) by the adult or working age population (75) and multiplying by 100. The participation rate is (42 / 75) * 100 = 56%.
(e) The employment rate is calculated by dividing the number of employed individuals (full-time and part-time) in the labor force (42) by the labor force and multiplying by 100. The employment rate is (42 / 42) * 100 = 100%.
Learn more about percentage here:
https://brainly.com/question/30348137
#SPJ11
This is the result of incorrectly simplifying the dividing the sum of 5 and 35 by 5. CORRECT My Answer This is the result of correctly following the order (2 x 15-9+3)/(5 +35+5) (30-9+3)/(5+7) (30-3)/(12) 27/12 9/4
Answers
The initial incorrect simplification led to an incorrect result of 9 divided by 4 instead of 2.25.
The incorrect simplification of dividing the sum of 5 and 35 by 5 resulted in an erroneous answer. The correct calculation involves following the order of operations. By multiplying 2 and 15 first, subtracting 9, and adding 3, we obtain (30-9+3). Similarly, by adding the numbers in the denominator, we get (5+35+5). Simplifying further, we have (30-3) divided by 12, which equals 27 divided by 12. The final answer is 9 divided by 4, or 2.25. Therefore, the initial incorrect simplification led to an incorrect result of 9 divided by 4 instead of 2.25.
For more information on simplification visit: brainly.com/question/19537848
#SPJ11
Let f(x) be a nonnegative smooth function (smooth means continuously differentiable) over the interval [a, b]. Then, the area of the surface of revolution formed by revolving the graph of y f(x) about the x-axis is given by
S= b∫a πf(x)1√+[f′(x)]^2 dx
Answers
The formula for the surface area of revolution, S, formed by revolving the graph of y = f(x) about the x-axis over the interval [a, b], is given by S = ∫(a to b) 2πf(x) √(1 + [f'(x)]^2) dx.
To calculate the surface area of revolution, we consider the small element of arc length on the graph of y = f(x). The length of this element is given by √(1 + [f'(x)]^2) dx, which is obtained using the Pythagorean theorem in calculus. We can approximate the surface area of revolution by summing up these small lengths over the interval [a, b]. Since the surface area of a revolution is a collection of circular disks, we multiply the length of each element of arc by the circumference of the disk formed by revolving it, which is 2πf(x). Integrating this expression from a to b, we obtain the formula for the surface area of revolution:
S = ∫(a to b) 2πf(x) √(1 + [f'(x)]^2) dx.
This formula takes into account the variation in the slope of the function f(x) as given by f'(x), ensuring an accurate representation of the surface area of revolution. By evaluating this integral, we can determine the precise surface area for the given function f(x) over the interval [a, b].
Learn more about Pythagorean theorem here:
https://brainly.com/question/14930619
#SPJ11
The following data show the frequency of rainy days in a year less than 0.01 inch 165 days 0.01 -1 inch 90 days 1.01 - 5 inches 60 days 5.01 -10 inches 40 days more than 10 inches 10 days Find the mode.
Answers
The mode of a dataset is the value that appears most frequently. In this case, we need to find the interval of rainfall that occurs most frequently.
From the given data, we can see that the interval "less than 0.01 inch" has the highest frequency with 165 days. Therefore, the mode of this dataset is "less than 0.01 inch"
Effective communication is crucial in all aspects of life, including personal relationships, business, education, and social interactions. Good communication skills allow individuals to express their thoughts and feelings clearly, listen actively, and respond appropriately. In personal relationships, effective communication fosters mutual understanding, trust, and respect.
In the business world, it is essential for building strong relationships with clients, customers, and colleagues, and for achieving goals and objectives. Good communication also plays a vital role in education, where it facilitates the transfer of knowledge and information from teachers to students.
Moreover, effective communication skills enable individuals to engage in social interactions and build meaningful connections with others. Therefore, it is essential to develop good communication skills to succeed in all aspects of life.
Learn more about dataset here:
https://brainly.com/question/26468794
#SPJ11
question 9(multiple choice worth 1 points) (04.01 mc) a watering can dispenses water at the rate of 0.20 gallon per minute. the original volume of water in the can was 8 gallons. which set of ordered pairs shows the volume of water in the can in gallons (y), as a function of time in minutes (x), from the first minute after can starts dispensing water? {(7.8, 1), (7.6, 2), (7.4, 3)} {(1, 7.8), (2, 7.6), (3, 7.4)} {(8.0, 1), (7.8, 2), (7.6, 3)} {(1, 8.0), (2, 7.8), (3, 7.6)}
Answers
The ordered pairs that shows the volume of water in the can in gallons (y), as a function of time in minutes (x), from the first minute after can starts dispensing water: {(1, 7.8), (2, 7.6), (3, 7.4)}
For given question,
A watering can dispenses water at the rate of 0.20 gallon per minute.
The original volume of water in the can was 8 gallons.
We need to find the ordered pairs that shows the volume of water in the can in gallons (y), as a function of time in minutes (x), from the first minute after can starts dispensing water.
The ordered pair is of the form (x, y) means the first coordinate is time in minutes and the second coordinate is the volume of water in the can in gallons.
Initially, for x = 0 minutes, y = 8 gallons
After 1 minute, the amount of water in can is 8.0 - 0.20 = 7.8 gallons.
So, the ordered pair of the form (time, volume of water) are: {(1, 7.8), (2, 7.6), (3, 7.4)}
Therefore, the ordered pairs that shows the volume of water in the can in gallons (y), as a function of time in minutes (x), from the first minute after can starts dispensing water: {(1, 7.8), (2, 7.6), (3, 7.4)}
Learn more about the ordered pair here:
https://brainly.com/question/23922144
#SPJ4
7. What is the sign of c:d, c>0 and d<0?
POSITIVE
O NEGATIVE
ZERO
Answers
Answer:
It would be c>0
Step-by-step explanation:
whoever can figure this our for my ill give u brainlist :)
Answers
Answer:
16 hours
Step-by-step explanation:
Given:
Time Mary spends on math homework per week: 2 2/3 hoursTime Mary spends on art homework per week: 3 1/3 hoursTherefore, time spent on both homeworks over 4 weeks:
= 4 × (math homework + art homework )
\(=4(2 \frac23+1\frac13)\)
\(=4\left(\dfrac83+\dfrac43\right)\)
\(=4\left(\dfrac{12}3\right)\)
\(=4(4)\)
\(=16 \textsf{ hours}\)