Answers
Answer:
2 people should go
Step-by-step explanation:
if more than 2 people go it would be over $500 for food, hope this helps and stay positive :)
Related Questions
True or false: if the truth is x, and y and z are conflicting claims, it must be true that either y≠x or z≠x, or neither?
Answers
Tt is also possible neither y nor z is x.
Now, According to the question;
The x and y-axis are two important lines of the coordinate plane. The x-axis is a horizontal number line and the y-axis is a vertical number line. These two axes intersect perpendicularly to form the coordinate plane. The x-axis is also called the abscissa and the y-axis is called the ordinate.
If x is true, a non true statement cannot be the same as it.
Now, In symbolic logic, = doesn't mean two statements have the same value—it means they are the same literal statement.
So, if x is true, and y and z contradict, at least one of y and z is false, and that false statement cannot possibly be x.
Hence, it is also possible neither y nor z is x.
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The table shows data for attendance at a music festival based on the number of bands performing.
This year’s festival will feature 10 bands. The local TV station reports that the expected attendance is 100,000.
Is 100,000 attendees for 10 bands a reasonable prediction?
No, there is no recognizable pattern in the data from which to make a prediction.
No, the pattern is exponential. The number should be 600,000.
Yes, the pattern is quadratic. The second differences are a constant 1.
Yes, the pattern is linear. The average rate of change is about 10,000.
Answers
It should be noted that 100,000 attendees for 10 bands a reasonable prediction as D. Yes, the pattern is linear. The average rate of change is about 10,000.
How to explain the linear relationshipA straight-line link between two variables is referred to statistically as a linear relationship (or linear association).
A positive slope indicates a positive linear relationship, meaning that as one increases, the other also increases.
From the information, the year’s festival will feature 10 bands and the local TV station reports that the expected attendance is 100,000. The average rate will be:
= 100000 / 10
= 10000
The correct option is D.
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Find the centroid of the triangle whose vertices are A(2, 6), B(4, 9), and C(6,15).
Answers
As a result, the triangle's centroid at vertices A(2, 6), B(4, 9), and C(6, 15) is (4, 10).
Given the vertices of a triangle, how do you get its centroid? The center of the thing is represented by the centroid.The centroid of a triangle is the location where the triangle's three medians cross.The junction of all three medians is another definition of it.A line connecting a side's midway to the triangle's opposite vertex is called the median.C = [(y1 + y2 + y3)/ 3, (x1 + x2 + x3)/ 3]. The triangle's centroid is indicated with the letter C.The x-coordinates of a triangle's three vertices are x1, x2, and x3.The y-coordinates for a triangle's vertices are y1, y2, and y3. As a result, the triangle's centroid at vertices A(2, 6), B(4, 9), and C(6, 15) is (4, 10).To learn more about triangle's centroid refer
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Please answer the question :)
Answers
According to the information in the graph, it can be established that the profit for 600 people would be $3,013.96
How to calculate the profit for 600 people?To calculate the profit for 600 people we must take into account the information in the graph. In this case we must establish how many people participated in the survey and how much was the total profit of the store with these customers.
11 + 8 + 9 + 19 + 3 = 5087.98 + 66.45 + 32.50 + 34.50 + 29.95 = 251.08Now we must make a rule of three to establish how much the profit would be in a group of 600 buyers.
50 - $251.08600-x600 * 251.08 / 50 = 3,012.96According to the above, the profit of the store with a group of 600 buyers would be 3,012.96
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Two students in Mr. Huber's class, Emmy and Warren, have been assigned a workbook to complete at their own pace. They get together at Emmy's house after school to complete as many pages as they can. Emmy has already completed 27 pages and will continue working at a rate of 13 pages per hour. Warren has completed 33 pages and can work at a rate of 10 pages per hour. Eventually, the two students will be working on the same page. How long will that take? How many pages will each of them have completed?
Answers
The time taken by both of them to work on the same page is 2 hours. The number of pages written by Emmy is 53 and the number of pages written by Warren is 53.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that Emmy has already completed 27 pages and will continue working at a rate of 13 pages per hour. Warren has completed 33 pages and can work at a rate of 10 pages per hour.
The time taken to reach the same page is,
13t + 27 = 33 + 10t
13t - 10t = 33 - 27
3t = 6
t = 2 hours
The number of pages written by Emmy is,
N = 13t + 27
N = ( 13 x 2) + 27 = 53
The number of pages written by Warren is,
N = 33 + 10t
N = 33 + 20
N = 53
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What is the Slope of (8,6) and (16,11)
Answers
Answer:
5/8
Step-by-step explanation:
[tex]\textrm{Slope } = $\dfrac{y_2 - y_1}{x_2 - x_1}$[/tex]
where[tex](x_1, y_1) \;\; (x_2, y_2)[/tex] are two points on the graph
The given points on the graph are (8,6) and (16,11)
Difference in y values, y2 - y1 = 11 - 6 = 5
Difference in x values,: x2 - x1 = 16 - 8 = 8
Slope = 5/8
Solve the inequality 4+p<11
Answers
Answer:
p<11-4
p<7
Step-by-step explanation:
we have 4+p<11 where we send 4 in another side and we get p<11-4 =p<7
A zoo feeds its monkeys a mixture of fruits and biscuits. The diagram shows the ratio of fruit mass to biscuit mass.
A tape diagram with 2 tapes of unequal lengths. The first tape has 7 equal parts. A curved bracket above the first tape is labeled Fruit mass. The second tape has 5 equal parts of the same size as in the first tape. A curved bracket below the second tape is labeled Biscuit mass.
The table shows the mass, in grams, of fruit and biscuits that the zoo fed two of the monkeys.
Based on the ratio, complete the missing values in the table.
Answers
Note that the missing values in the table are:
1) A - 28 : 20
2) B: - 63 : 49.
Note that this is a test of ratios and proportionality.
What is the rationale for the above answer?Note that the first tape comprises 7 boxes for the Fruit Mass and
the second tape 5 boxes for the Buiscuit mass.
So we know that the the figures represent Lowest Common Multiples when we are presented with the table.
From the table, we see that Fruitmass has a related number which is 28. Immediately we can see the relationship.
7 x 4 = 28
This means that the missing number can be arrived at by saying:
5 x 4 = 20
Repeat this same logic for the missing number in B to obtain: 63: 49.
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The lifetime of the timing belt of a certain make of cars is normally distributed with mean 125,000 miles and standard deviation 10,000 miles.
a) Find the probability that a timing belt lasts until the car runs 140,000 miles.
b) The auto maker recommends that owners have the timing belt replaced when the mileage reaches 90,000 miles. What is the probability that the timing belt fails before the car reaches the manufacturer’s recommended mileage?
c) An owner of this type of car wants to take a chance and replace the timing belt at the 1st percentile of the distribution. What should be the mileage of the car when he has the timing belt replaced?
Answers
The probability that a timing belt lasts until the car runs 140,000 miles is 0.9332.
a) The probability that a timing belt lasts until the car runs 140,000 miles can be found using the Z-score formula. The Z-score is calculated as (140,000 - 125,000)/10,000 = 1.5. The probability can be found using the Z-score table or by using a calculator, and is equal to 0.9332.
b) The probability that the timing belt fails before the car reaches the manufacturer’s recommended mileage of 90,000 miles can be found using the Z-score formula. The Z-score is calculated as (90,000 - 125,000)/10,000 = -2.5. The probability can be found using the Z-score table or by using a calculator, and is equal to 0.0062.
c) The 1st percentile of a normal distribution is the Z-score corresponding to the cumulative probability of 0.01. The Z-score for the 1st percentile is -2.33. The mileage of the car when the timing belt should be replaced can be found using the Z-score formula. The mileage is equal to 125,000 - (2.33 * 10,000) = 110,700 miles.
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Equation: &-
Solution:
,. © Justify Conclusions Seventy-five percent, or 15, of the students in Emily's homeroom class are going on a field trip. Two thirds, or 12, of the students in Santiago's homeroom class are going on the field trip. Which class has more students? Justify your answer.
Answers
The class that has more students, given the percentage of students in both classes is Emily's homeroom tab .
How to find the class with more student ?To find the class with the higher number of students, you need to find the total number of students in these classes.
The number of students in Emily's homeroom class is:
= Number of students / Percentage of students
= 15 / 75 %
= 20 students
Students in Santiago's homeroom class is :
= 12 / ( 2 / 3 )
= 12 x 3 / 2
= 18 students
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If you know Emma Waton peech "HeforShe" what ele could I analye other than her tone in voice , word choice and body language ?
Answers
Women Emma Watson spoke intelligently, significantly, and powerfully about gender inequity and how to combat it.
She did this in order to begin the HeForShe campaign, which tries to recruit men and boys to the feminist movement's struggle for gender parity.
The purpose of this campaign is to encourage as many men and boys to support gender equality as they can. The terrible reality that "feminism" is all too frequently associated with "man hate," effectively stopping the progress toward attaining gender equality, was brought to light by Emma Watson in her powerful statement.
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What is the letter name of the point (-3, 4)?
Answers
Answer: R
Step-by-step explanation: you are gonna go left 3 and up 4.
x=-3 and y=4
find the rules for the composite functions f ∘ g and g ∘ f.
Answers
The composite functions f ∘ g and g ∘ f is defined as follows:
f ∘ g (x) = f(g(x))
g ∘ f (x) = g(f(x))
In other words, f ∘ g is the composition of two functions f and g, such that the output of g is used as the input to f. Similarly, g ∘ f is the composition of two functions g and f, such that the output of f is used as the input to g.
It's important to note that the order in which the functions are composed matters, as f ∘ g is not necessarily equal to g ∘ f. In general, the composite function of two functions is not commutative, meaning that the order in which the functions are composed affects the result.
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a one-sample z-test for a population proportion will be conducted using a simple random sample selected without replacement from a population. which of the following is a check for independence? A
npo 10 and n(1-P) 10 for sample size n and population proportion P-
Each sample proportion value is less than or equal to 0.5.
B
c
The sample size is more than 10 times the population size.
D
The population size is more than 10 times the sample size.
E
The population distribution is approximately normal.
Answers
The check for independence in a one- sample is
The population size is further than 10 times the sample size.
The Correct option is D.
What is Z- test?
A Z- test is a statistical thesis test used to determine whether a sample mean significantly differs from a given population mean when the population standard divagation is known.
For a one- sample z- test for a population proportion, the sample is named without relief from a finite population.
In order for the test to be valid, it's necessary to insure that the sample is small relative to the population size.
still, it can be considered large enough to satisfy the independence supposition, If the population size is further than 10 times the sample size. This supposition is necessary for the validity of statistical conclusion in this environment.
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2x1+48-5x48+96-23= blank please helppppppppp
Answers
Answer:
-117
Step-by-step explanation:
2x1+48-5x48+96-23=-117
First multiplication
2×1=2
5×48=240 ->
2+48-240+96-23
2+48=50
50-240=-190
-190+96=-94
-94-23=-117
Hope that helped! Let me know if you have any further questions.
Compare the key feature (slope and y-intercept) of each linear function g with those of the graph of f(x)= 2x + 3. Describe how the value of k affects the key features.
Answers
23. g(x)=3(2x+3) has a slope of 6 and y-intercept of 9.
24. g(x)=2(0.5x) +3 has a slope of 1 and y-intercept of 3.
25. g(x)=1(2x+3) has a slope of 2 and y-intercept of 3.
26. g(x)=2(1)x+3 has a slope of 2 and y-intercept of 3.
How did we get the values?23. g(x)=3(2x+3) can be written as g(x)=6x + 9. This shows that the slope of the function is 6 and the y-intercept is 9.
24. g(x)=2(0.5x) +3 can be written as g(x)=x + 3. This shows that the slope of the function is 1 and the y-intercept is 3.
25. g(x)=1(2x+3) can be written as g(x)=2x + 3. This shows that the slope of the function is 2 and the y-intercept is 3.
26. g(x)=2(1)x + 3 can be written as g(x)=2x + 3. This shows that the slope of the function is 2 and the y-intercept is 3.
The value of k in the linear function g(x)=k(mx+b) affects the slope of the function by multiplying the slope of the original function m by k. The y-intercept of the function remains unchanged.
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Use the given function to find the following.
x + 10
5
f (x) =
=
-1
f-¹ (x).
Find
-1
Find f (ƒ-¹ (5)).
(no spaces)
Answers
The composite function f (ƒ-¹ (5)) is 5
How to determine the composite functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = (x + 10)/5
Also, we have
f-¹ (x) = 5x - 10
The composite function f (ƒ-¹ (5)) is calculated as
f (ƒ-¹ (x)) = (5x - 10 + 10)/5
Evaluate the difference
f (ƒ-¹ (x)) = 5x/5
So, we have
f (ƒ-¹ (x)) = x
Substitute 5 for x
f (ƒ-¹ (5)) = 5
Hence, the solution is 5
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Complete question
Use the given function to find the following.
f(x) = (x + 10)/5
f-¹ (x) = 5x - 10
Find f (ƒ-¹ (5)). (no spaces)
Liam’s class has 30 students each either 11 or 12 years old the sum of their ages is 352 years how many of each age are in his class 
Answers
Answer: There are 8 11-year-old students and 22 12-year-old students in Liam's class.
Step-by-step explanation:
Let's call the number of 11-year-old students in class x. Then, the number of 12-year-old students in the class would be 30 - x.
The total number of years that the 11-year-old students represent is 11 * x.
The total number of years that the 12-year-old students represent is 12 * (30 - x).
The sum of all the years is 352, so we can set up an equation based on the above information:
11x + 12(30 - x) = 352
Expanding the second term on the left-hand side:
11x + 360 - 12x = 352
Combining like terms:
-x + 360 = 352
Subtracting 360 from both sides:
-x = -8
Dividing both sides by -1:
x = 8
So, there are 8 11-year-old students and 30 - 8 = 22 12-year-old students in Liam's class.
Liquid/Vapor saturation pressure Psat is often represented as a function of temperature by an equation of the form:log10Psat/torr = a - b/ [ t/oC + c ]Here parameters a, b, and c are substance-specific constants. Suppose it is required to represent Psat by the equivalent equation:ln Psat/kPa = A - B/[ T/K + C]Show how the parameters in the two equations are related.
Answers
If we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.
What is the logarithmic equation?
A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first, see whether you can write both sides of the equation as powers of the same number.
The parameters in the two equations are related as follows:
A = log10(Psat/kPa) / ln(10)
B = b * ln(10)
C = c
T = t + 273.15 (to convert from Celsius to Kelvin)
Therefore, the relationship between the parameters in the two equations is:
A = log10(Psat/kPa) / ln(10)
B = b * ln(10)
C = c
T = t + 273.15
Hence, if we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.
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Evander King is ordering a luxury convertible. The base price is $81,975. Options include satellite radio, $475; keyless entry, $555; navigation system, $1,250; and custom paint, $983. The destination charge is $890. The dealer's costs are 90% of the base price and 88% of the options price. The dealer will sell Evander the car for $200 more than the dealer's cost plus a 6% sales tax. What is the total cost of the convertible, including tax?
Answers
Answer: $82,403.28
Step-by-step explanation:
475+555+1250+983 = 3263 (total option price)
81,975 (total base price)
0.9 * 81,975 = 73,777.5 + 200
.88 * 3263 = 2,871.44
1.06 * (73,977.5 + 2871.44 + 890) = 82,403.28 (Rounded to nearest tenth)
elementary statistics, a step-by-step approach, 10th ed., bluman
Answers
The subject is taught in a non-theoretical manner in Al Bluman's Elementary Statistics. Today's world speaks statistics, and Bluman's market-leading Step-by-Step Approach makes it simple to learn and comprehend.
Bluman gives your students all the assistance they need to understand the foundations of statistics and draw that link by assisting them in transitioning from the computational to the conceptual.
Elementary Statistical Methods are the collection, analysis, presentation, and interpretation of data, and probability. The analysis includes descriptive statistics, correlation and regression, confidence intervals, and hypothesis testing.
Thus, The subject is taught in a non-theoretical manner in Al Bluman's Elementary Statistics. Today's world speaks statistics, and Bluman's market-leading Step-by-Step Approach makes it simple to learn and comprehend.
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Let u and v be vectors in R^2. Which of the following is/are true?(select all that apply)[ ] None of the choices is correct.[ ] The set of all vectors which are scalar multiple of a nonzero vector u is a line through u and 0.[ ] The operation of vector addition is not commutative.[ ] A column vector in R^2 is a 1x2 matrix.[ ] The magnitude of a vector cv is c times the magnitude of v, where c is any scalar.
Answers
None of the choices is correct, and a column vector in R^2 is not a 1x2 matrix.
[ ] The set of all vectors which are scalar multiple of a nonzero vector u is a line through u and 0.
[ ] The operation of vector addition is not commutative.
[ ] The magnitude of a vector cv is c times the magnitude of v, where c is any scalar.
None of the choices is correct, and a column vector in R^2 is not a 1x2 matrix.
None of the choices is correct. The set of all vectors which are scalar multiple of a nonzero vector u is a line through u and the origin (not just 0). The operation of vector addition is not commutative, meaning that the order of addition matters. The magnitude of a vector cv is c times the magnitude of v, where c is any scalar and v is a vector. A column vector in R^2 is a matrix with two columns and one row (1x2 matrix).
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Lupe's family traveled 5 7 of the distance to her aunt’s house on Saturday. They traveled 1 2 of the remaining distance on Sunday. What fraction of the total distance to her aunt’s house was traveled on Sunday?
Answers
Answer:
3/14
Step-by-step explanation:
They traveled 5/7 of the distance on Saturday and 1/2 of the remaining
distance, 3/7, on Sunday so..
(1/2)(3/7) = 3/14
Hope this helps :)
Answer:Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.
On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance.
Step-by-step explanation:
Lupe's family traveled 5/7 of the total distance to her aunt's house on Saturday, so the remaining distance was 1 - (5/7) = 2/7.
On Sunday, they traveled 1/2 of the remaining distance, which was 2/7, so the total distance traveled on Sunday was 1/2 * 2/7 = 2/14 of the total distance. :D
reaching into the change drawer, sue notices that she only has nickels (worth 5 cents) and dimes (worth 10 cents). from prior experience, she knows that the variance of the number of nickels is 8, the variance of the number of dimes is 10 and the covariance of the number of nickels to the number of dimes is 5. what is the covariance of the number of coins to the value of the coins (in cents)?
Answers
The covariance between the number of coins and the value of the coins is 240.
The covariance between the number of coins (N) and the value of the coins (V) can be calculated as
Cov(N, V) = Cov(n nickels + n dimes, 5n nickels + 10n dimes)
= 5 Cov(n nickels, n nickels) + 10 Cov(n nickels, n dimes) + 5 Cov(n dimes, n nickels) + 10 Cov(n dimes, n dimes)
= 5 * Var(n nickels) + 10 * Cov(n nickels, n dimes) + 5 * Cov(n dimes, n nickels) + 10 * Var(n dimes)
= 5 * 8 + 10 * 5 + 5 * 5 + 10 * 10
= 40 + 50 + 50 + 100
= 240
So the covariance between the number of coins and the value of the coins is 240.
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In Exercises 17-20, find the volume of the solid generated by revolv- ing ` the shaded region about the given axis. 17. About the x-axis 18. About the J-axis x + 2 = 2 1 19. About the y-axis 20. About the x-axis sin x cOS X Lan'
Answers
the volume of the solid generated by revolving the shaded region about the given axis x- axis is: 2π/3
What is shaded region?The difference between the area of the complete polygon and the area of the unshaded portion inside the polygon is the area of the shaded region. There are two methods for the area of the darkened component to appear in polygons. The darkened area can be found either in the polygon's center or on one of its sides.
We first solve for y from the equation:
x+2y=2
or, 2y = 2 - x
or, y = (2 - x) / 2
From the graph we see that the equation is from x=0 to x=2
And it is rotating about the x axis. We can use the Disk method to find the volume of the solid:
= π [tex]\int\limits^2_0 {((2 - x)/2)^2} \, dx[/tex]
= π/4 [4x - 2x² + x³/3]₀²
= 2π/3
so, 2π/3 is the volume of the solid, it is rotating about the x axis.
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The complete question is as follows:
3/5x7/(2x67) yavavac
Answers
Answer:
ar many types the total bar o
Answer:
I'm not sure if this is what you meant but if it's not, put a picture of the equation and I'll edit my answer
[tex]\frac{21}{670}[/tex]
Step-by-step explanation:
Assuming the equation is:
[tex]\frac{3}{5} * \frac{7}{2*67}[/tex]
The first thing you have to do is simplify:
2 * 67 = 134
[tex]\frac{3}{5} * \frac{7}{134}[/tex]
Multiply the numerators:
3 * 7 = 21
Then multiply the denominators:
5 * 134 = 670
Then simplify again:
[tex]\frac{21}{670}[/tex]
vector a⃗ =6.0i^−4.0k^. construct a unit vector that is parallel to
Answers
The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.
1. Calculate magnitude of a⃗:
a⃗ = 6.0i^−4.0k^
|a⃗| = √(6.0^2+4.0^2) = √74
2. Construct unit vector, u⃗= (6/√74)i^−(4/√74)k^
u⃗ = (6/√74)i^−(4/√74)k^
3. The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.
The unit vector that is parallel to a⃗ is (6/√74)i^−(4/√74)k^.
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The graph of a line passes through the points (0, 0.5) and (6, 5). A second i
has the equation 5y = 7-6x. What is the solution to the system of
equations?
Answers
The solution to the system of equations is (0.839, 1.129).
What is a graph ?
Graph can be defined as visual representation of the data through the given ordered pairs .
Given ,
The graph of a line passes through the points (0, 0.5) and (6, 5). A second i has the equation 5y = 7-6x.
The equation of the first line can be found using the two given points, using the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the values from the two points, we have:
m = (5 - 0.5) / (6 - 0) = 4.5 / 6 = 0.75
So the equation of the first line is:
y = 0.75x + 0.5
To find the solution to the system of equations, we need to find the point where the two lines intersect, which means finding the values of x and y that satisfy both equations. To do this, we can substitute the equation of the first line into the second equation:
5y = 7 - 6x
0.75x + 0.5 = 7 - 6x
Solving for x, we get:
7.75x = 6.5
x = 6.5 / 7.75 = 0.839
Substituting x back into either equation to find y, we get:
y = 0.75x + 0.5 = 0.75 * 0.839 + 0.5 = 0.629 + 0.5 = 1.129
So the solution to the system of equations is (0.839, 1.129).
Therefore, The solution to the system of equations is (0.839, 1.129).
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find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 2e−x, y = 2, x = 4; about y = 4
Answers
The volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.
The volume of the solid obtained by rotating the region bounded by the given curves about the line y=4 is given by
V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx
We can calculate the integral using integration by parts.
Let u = 4-2[tex]e^{-x}[/tex]
⇒ du = 2d[tex]e^{-x}[/tex]
and dv = 2dx
⇒ v = x
So,
V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx
= 2π (x (4-2[tex]e^{-x}[/tex])² - 0 (4-2[tex]e^{-x}[/tex]})² - 2∫⁴₀ x d (4-2[tex]e^{-x}[/tex]))
= 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)
Using integration by parts again
Let u = 4-2[tex]e^{-x}[/tex] ⇒ du = -2d[tex]e^{-x}[/tex]
and dv = xdx ⇒ v = x²/2
So,
V = 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)
= 2π (2×4³ - 2×2∫⁴₀ x² [tex]e^{-x}[/tex] dx)
Again we can calculate the integral using integration by parts.
Let u = x² ⇒ du = 2xdx
and dv = [tex]e^{-x}[/tex] dx ⇒ v = -[tex]e^{-x}[/tex]
So,
V = 2π (2×4³ - 2×2∫⁴₀4 x^2 [tex]e^{-x}[/tex]} dx)
= 2π (2 × 4³-2 × 2 × ([tex]-e^{-4}[/tex] + 2∫⁴₀ [tex]e^{-x}[/tex]dx))
= 2π (2×4³ - 2×2 (1+4))
= 2π (2×4³ - 16)
= 2π (128-16)
= 256π
Therefore, the volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.
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prove the following equalities using mathematical induction: a. ∑ ― 1 = 1 ( 1 i(i 1)) = 1 ― 1/, ≥ 2.
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We can prove the equality using mathematical induction. We assume it holds for n and show it holds for n+1. Then, we add n+1 to both sides of the equation, proving the equality holds for n+1.
To prove the equality using mathematical induction, we must first assume it holds for n. Then, we can add n+1 to both sides of the equation:
∑ ― 1 = 1 ( 1 i(i 1)) + (n + 1) = 1 ― 1/ + (n + 1)
The right side of the equation is equal to 1 ― 1/ + n + 1, which simplifies to 1 ― 1/(n+1), proving the equality holds for n+1. Therefore, the equality holds for all n ≥ 2, and we have proven the equation using mathematical induction. Note: The expression in the equation should be n(n+1)/2, not 1/i(i+1).
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