To prove that removing the root of a k-th order binomial tree results in k binomial trees of smaller orders, we will use mathematical induction.
Base case: For k = 1, removing the root of a 1st order binomial tree results in zero smaller binomial trees, which is true.
Inductive hypothesis: Assume that removing the root of a k-th order binomial tree results in k binomial trees of smaller orders, for some positive integer k.
Inductive step: We want to prove that removing the root of a (k+1)-th order binomial tree results in (k+1) binomial trees of smaller orders.
By definition, a (k+1)-th order binomial tree, denoted by B(k+1), is formed by joining two k-th order binomial trees, denoted by Bk, with one of them as the leftmost child of the root of the other. Let us remove the root of B(k+1) to obtain a new binomial tree, denoted by B'.
Since B(k+1) is formed by joining two k-th order binomial trees, we can apply the inductive hypothesis to each of them to obtain that removing their roots results in k smaller binomial trees each. Let these smaller binomial trees be denoted by B1, B2, ..., Bk and C1, C2, ..., Ck,
respectively.
Now, consider B' and observe that it is a k-th order binomial tree, since it has k children, each of which is a smaller binomial tree. Moreover, B' is obtained from B(k+1) by removing the root, which is the parent of the two k-th order binomial trees Bk and Ck. Thus, we have decomposed B' into k smaller binomial trees, namely B1, B2, ..., Bk and C1, C2, ..., Ck.
Therefore, we have shown that removing the root of a (k+1)-th order binomial tree results in (k+1) binomial trees of smaller orders, which completes the induction.
Learn more about binomial tree here: brainly.com/question/30025883
#SPJ4
in hypothesis testing, the alternative hypothesis is . a. the hypothesis tentatively assumed true in the hypothesis-testing procedure b. the hypothesis concluded to be true if the null hypothesis is rejected c. the maximum probability of a type i error d. the maximum probability of a type ii error
Answer:
taking your points
Step-by-step explanation:
A snail moves at a speed of 2 feet per minute. How many yards will the snail have moved in one hour? The snail has moved a total of
3
yards
The snail moves a distance of 40 yards in one hour as it moves at a speed of 2 feet per minute.
We can start by converting the speed of the snail from feet per minute to yards per hour, since the distance moved is given in yards per hour.
There are 60 minutes in an hour, so the snail's speed in yards per hour is:
2 feet per minute * 60 minutes per hour * (1/3) yards per foot = 40 yards per hour
Hence,
The snail moves a distance of 40 yards in one hour.
Since we are given that the snail has moved a total of 3 yards, we can find the amount of time it took by dividing the distance by the rate:
time = distance / rate
time = 3 yards / 40 yards per hour
time = 0.075 hours or 4.5 minutes
Therefore, the snail will move a distance of 3 yards after 4.5 minutes and distance of 40 yards in one hour.
Learn more about distance here brainly.com/question/12759408
#SPJ4
Amy used her first 2 tokens at Glimmer Arcade to play a game of Roll-and-Score. Then she
played her favorite game, Balloon Bouncer, over and over until she ran out of tokens. Balloon
Bouncer costs 3 tokens per game, and Amy started with a bucket of 35 game tokens.
Which equation can Amy use to find how many games of Balloon Bouncer,
she played
Answer:
11
Step-by-step explanation:
Let's call the number of games of Balloon Bouncer that Amy played as "x".
Amy started with 35 tokens and used 2 for Roll-and-Score, so she had 35 - 2 = 33 tokens left.
Each game of Balloon Bouncer costs 3 tokens, so the total number of tokens used for Balloon Bouncer games is 3x.
Therefore, the equation to find the number of games Amy played is:
33 = 3x
Solving for x, we get:
x = 11
So Amy played 11 games of Balloon Bouncer.
Answer: 3g + 3 = 35
Step-by-step explanation:
write the equation for the given graphs. How do we find the "a" value?
The equations of the graphs are y = 1/2|x - 2| + 5 and y = -7|x + 2| - 11
How to determine the equations of the graphsFrom the question, we have the following parameters that can be used in our computation:
The graphs
The lines are absolute function graphs, and they can be represented as
y = a|x - h| + k
Where
Vertex = (h, k)
For the blue line, we have
(h, k) = (2, -5) and (x, y) = (0, -4)
So, we have
-4 = a|0 - 2| - 5
2a = 1
a = 1/2
So, the equation is y = 1/2|x - 2| + 5
For the red line, we have
(h, k) = (-2, 3) and (x, y) = (0, -11)
So, we have
-11 = a|0 + 2| + 3
2a = -14
a = -7
So, the equation is y = -7|x + 2| - 11
Read more about absolute value functions at
https://brainly.com/question/10538556
#SPJ1
Consider the function f(x) = (x - 3)².
a. How could you restrict the domain of f(x) so that its inverse will be a function?
b. Graph f(x) with its restricted domain and then graph its inverse on the same set of axes.
c. Find the equation of the inverse of f(x) with its restricted domain.
The inverse of f(x) has no restriction on the domain.
The graph of the inverse function is given below.
The inverse equation is [tex]f^{-1}[/tex](x) = x² + 3.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(x) = (x - 3)²
The inverse of f(x).
Step 1:
Make f(x) = y
y = (x - 3)²
Step 2:
Interchange x and y
x = (y - 3)²
Step 3:
Solve for y.
square root on both sides.
x² = y - 3
y = x² + 3
So,
[tex]f^{-1}[/tex](x) = x² + 3.
a)
[tex]f^{-1}[/tex](x) = x² + 3
we see that for any values of x, [tex]f^{-1}[/tex](x) is a function.
So,
There is no restriction on the domain.
b)
The graph is given below.
c)
The inverse equation is
[tex]f^{-1}[/tex](x) = x² + 3.
Thus,
The inverse of f(x) has no restriction on the domain.
The inverse equation is [tex]f^{-1}[/tex](x) = x² + 3.
Learn more about functions here:
https://brainly.com/question/28533782
#SPJ1
Four years ago, the sum of ages of father and his son was 27 years. After five years, the ratio of the ages of the son and father will be 2: 7. Find the present ages of father and his son. Ans: 30 yrs, 5 yrs
Step-by-step explanation:
x current age of the father
y current age of the son
first equation (x-4)+(y-4)=27 so x=27+4+4-y=35-y
second equation (y+5) : (x+5)= 2:7
so (y+5)*7=(x+5)*2
let's replace x we found at the first equation with the one at second equation
(y+5)*7=(35-y+5)*2
7y+35=(40-y)*2
7y+35=80-2y
7y+2y=80-35
9y=45
y=5
x=35-y=35-5=30
How does the distance change during the last section of tom’s journey? What does this mean? Explain your reasoning
The distance is decreasing with time meaning that the object is going back to the starting point.
What does it mean when distance decreases with time?When distance decreases with time, it means that the object is moving towards its starting point or moving in the opposite direction from its initial position. In other words, the object is moving backwards or reversing its direction of motion.
Distance is a measure of the total amount of ground covered by an object in motion, while time is a measure of how long it takes the object to cover that distance. If the object is moving towards its starting point, it means that it is covering less and less distance over time, or the distance it covers is decreasing.
Learn more about distance time graph:https://brainly.com/question/19836052
#SPJ1
simplify 3 to the -2 power
Answer: 0.1 decimal 1/9 fraction
Step-by-step explanation:
Answer:
0.1 or 1/9 i think so but I am sure
there 10 cars in a race. in how many ways can three cars finish the race as the 1st, 2nd, and the 3rd places?
There are 10 cars in a race, in 720 ways can three cars finish the race as the 1st, 2nd, and the 3rd places.
An arrangement of particulars in a specific order is appertained to as a permutation. Then, the factors of sets are arranged in a direct or successional order. For case, the set A = ,6 has a permutation of 2, which is. There's no other way to organise the factors of set A, as you can see.
There are 10 cars in the race,
Three cars finish the race in the 1st ,2nd and 3rd order
1st finish the race have 10 chances
P(A) = 10
2nd finish the race have 9 chances
P(B) = 9
3rd finish the race have 8 chances
P(C) = 8
The number in which this can be put,
= P(A) x P(B) x P(C)
= 10*9*8
=720 ways
Therefore, in 720 ways can three cars finish the race as the 1st, 2nd, and the 3rd places.
Learn more about Permutation:
https://brainly.com/question/3933713
#SPJ4
Yen takes her pomeranian kiwii to the dog park once inside the fence kiwii runs across the filed in 10 seconds his path is shown on the graph belowwhat is the average velocity 4 to 8 seconds
The average velocity from 4 to 8 seconds is given as follows:
v = [position at 8 seconds - position at 4 seconds]/4.
What is the relation between velocity, distance and time?Velocity is distance divided by time, hence the following equation is built to model the relationship between these variables:
v = d/t.
The change in the distance from 4 to 8 seconds is given as follows:
position at 8 seconds - position at 4 seconds
The change in the time from 4 to 8 seconds is given as follows:
8 - 4 = 4 seconds.
Hence the velocity is of:
v = [position at 8 seconds - position at 4 seconds]/4.
Missing InformationThe problem is incomplete, hence the general procedure to obtain the average velocity was presented.
More can be learned about the relation between velocity, distance and time at https://brainly.com/question/24316569
#SPJ1
What is the length of the radius of the above sphere if the volume is 14,130 in3?
Solve this equation and you will have the solution:
4 • 3.14 • r3 ÷ 3 = 14,130
Your last step when solving this equation is to find the cube root of a both sides!
The length of the radius of the sphere is 15 in.
What is a sphere?It is a three-dimensional figure and has a diameter and a radius and the volume of a sphere is 4/3πr³.
We have,
Volume of the sphere = 14130 in³
So,
4/3πr³ = 14130 _____(1)
Solve for r.
4/3 x 3.14 x r³ = 14130
keep r on one side and the rest on the other side.
r³ = 14130 x 3 / 4 x 3.14
Cube root on both sides.
r³ = 3375
∛r³ = ∛3375
[ 15 x 15 x 15 = 3375 ]
So,
r = 15
Thus,
The radius of the sphere is 15 in.
Learn more about sphere here:
https://brainly.com/question/12390313
#SPJ9
what is the meaning of square?
Answer:
Step-by-step explanation:
A square is a 4 sided polygon in which all of its side lengths are equal and the diagonals are also be
a manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 420 gram setting. is there sufficient evidence at the 0.01 level that the bags are underfilled? based on a 37 bag sample, the manufacturer fails to reject the null hypothesis. what is the conclusion?
The conclusion is that the bag filling machine appears to be working correctly at this setting.
Based on the information provided, the manufacturer conducted a hypothesis test to determine if the bags are underfilled at the 420 gram setting. The null hypothesis (H0) is that the bags are not underfilled, while the alternative hypothesis (H1) is that the bags are underfilled.
Since the manufacturer failed to reject the null hypothesis at the 0.01 level of significance, this means that there is not enough evidence to support the claim that the bags are underfilled at the 420 gram setting. In other words, the bag filling machine appears to be working correctly at this setting.
It's important to note that the conclusion may change if the sample size or the level of significance is different. Additionally, other factors such as the accuracy of the measuring device used or the consistency of the banana chips themselves may also impact the results.
To learn more about machine here:
https://brainly.com/question/23878728
#SPJ4
What is the frequency of the function f(z)?
f(z)=-2 sin (2) +3
Express the answer in fraction form
Enter your answer in the box.
I don’t understand this can someone help
The ordered pair (2,10) means that the cost to feed 200 animals is of $10,000.
How to interpret the ordered pair?The general format of an ordered pair is given as follows:
(x,y).
The meaning of an ordered pair (x,y) is that y = f(x), meaning that the numeric value of the function at the value of x is of y.
The variables for this problem are given as follows:
x -> number of animals, in hundreds, hence x = 2 means that there are 200 animals.y -> cost to feed the animals, in thousands, hence y = 10 means that the cost is of $10,000.More can be learned about ordered pairs at brainly.com/question/1528681
#SPJ1
QuestionIntermediate value theorem states that if a function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the intervalATrueBFalse
The given statement about the intermediate value theorem is false
The given statement is
"Intermediate value theorem states that if a function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval"
But according to Intermediate value theorem "if a continuous function f with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval"
So if we takes discontinuous function the given statement will be wrong
Therefore, the statement is false
Learn more about intermediate value theorem here
brainly.com/question/29712240
#SPJ4
Find your Average daily balance
Scenario 2: You have a credit card with 25% APR. You have a classic car/truck you
are restoring and decide to buy some things with your new credit card. At the
beginning of your You buy new wheels and tires for 1500 dollars You don't buy
anything else for 14 days. However, your battery dies and your brake calipers start
leaking so you spend another 500 dollars. You don't buy anything else for the rest
of the month. You also don't make any payments. Your billing cycle is 30 days.
Follow the steps to calculate how much interest will be charged at the end of the
billing cycle.
The Average daily balance of the given scenario is $2083.
What is the amount of a fixed period payment to be done to pay off the loan amount which is increasing compoundly?Suppose we're given that:
P_v = Present value of the loan amount
r = rate of interest (compound interest) per period of time
n = number of period of time in which loan is to be paid off
P = A fixed period payment needed to be done for paying off the loan in n number of periods.
Then, we get:
[tex]P = \dfrac{r\times P_v}{1 - (1 + r)^{-n}}[/tex]
Given;
APR= 25%
Cost of wheels and tires= $1500
Time=14days
Cost of brake calipers and leaking= $500
Time=30days
Now,
P= 25x2000/1-(1+25)^-30
P=50000/24
P=2083
Therefore, the monthly balance of loan will be $2083.
Learn more about loan payment here: MONTHLY PAYMENT
https://brainly.com/question/4234016
#SPJ9
In any set of positive integers the replacement of a set of three 2s with a set of two threes will not change the sum
The statement is incorrect. The replacement of three 2s with two threes will change the sum.
For example, consider the set {2, 2, 2}. The sum of this set is 6. However, if we replace three 2s with two threes, the new set becomes {3, 3}, and its sum is 6, which is different from the original sum. Hence, replacing a set of three 2s with a set of two threes does change the sum. An integer is a whole number that can be positive, negative, or zero. Integers are numbers that don't have any fractional or decimal component. Some examples of integers are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, and so on. The set of integers is denoted by the symbol "Z", which stands for Zahlen, the German word for "numbers". Integers are an important mathematical concept used in many branches of mathematics and science, including algebra, number theory, and computer science.
To learn more about integers click here
brainly.com/question/1768254
#SPJ4
Complete Question
In any set of positive integers the replacement of a set of three 2s with a set of two threes will not change the sum" State whether the statement was correct or incorrect.
the sun is shining and a spherical snowball of volume 260 ft3 is melting at a rate of 11 cubic feet per hour. as it melts, it remains spherical. at what rate is the radius changing after 6.5 hours?
If the spherical snowball is melting , then the rate at which the radius changing after 6.5 hours is 0.208 ft/h .
A spherical snowball has volume = 260 ft³ that is melting at a rate of 11 cubic feet per hour, while remaining spherical.
We have to find the rate at which the radius is changing after 6.5 hours.
So , let "V" = volume of snowball, "r" = radius of snowball, and "t" = time.
Volume Of Sphere is V = (4/3)πr³ and dV/dt = -11 ft³/h ;
So, dr/dt when t = 6.5 hours. is dV/dt = (dV/dr)×(dr/dt) ;
derivative of volume formula with respect to "r", we get:
⇒ dV/dr = 4πr² ;
Substituting the given values in chain rule equation, we get:
⇒ -11 = (4/3)πr² × dr/dt ;
⇒ dr/dt = -11/((4/3)πr²)
For rate of change of the radius when t = 6.5 hours, we find the radius at that time.
The initial volume of the snowball = 260 ft³.
After 6.5 hours of melting at a rate of 11 ft³/h, the remaining volume will be: V = 260 - (11 × 6.5) = 188.5 ft³ ;
So , V = (4/3)πr³
⇒ 188.5 = (4/3)πr³
⇒ r³ = (188.5 × 3 / (4π)) ;
≈ 3.55ft
Now we substitute r = 3.55 feet
⇒ dr/dt = -11/[(4/3)π(3.55)²] ;
⇒ dr/dt ≈ -0.208 ft/h ;
Therefore , the rate at which radius is changing after 6.5 hours is approximately 0.208 feet per hour and negative sign represents that radius is decreasing .
Learn more about Rate Of Change here
https://brainly.com/question/28891125
#SPJ4
SOMEBODY PLEASE HELP ME ASAP I WILL LOVE YOU FOREVER
By determining the value of x when y = 0, (x, 0), and the value of y when x = 0, respectively, the x-intercept and y-intercept are determined (0, y).
What is linear equation?A two-variable equation with a line-shaped graph is referred to as a linear equation. A collection of coordinate plane points that are all the solutions to the linear equation make up the graph. If all of the variables have true numerical values, the equation can be graphed by first drawing a sufficient number of points to identify a pattern, then connecting those points to include all of the points.
A linear equation is represented in standard form as
Ax+By=C,A,B≠0
The equation for y must first be solved before you may graph a linear equation in its usual form.
The x-intercept is found by finding the value of x when y = 0, (x, 0), and the y-intercept is found by finding the value of y when x = 0, (0, y).
To know more about linear equations visit:
brainly.com/question/14323743
#SPJ1
If you get paid $13 per hours and work 8 hours and 45 minutes how much money did you earn? round your answer to the nearest hundred?
Answer:
113.75
Step-by-step explanation:
13$ x 8.75 h (45 minutes is 3/4 of an hour) = $113.75
keeping in mind that an hour has 60 minutes, so you're really getting paid $13 per 60 minutes, so now, 8 hours and 45 minutes is just 8*60 + 45 = 525 minutes.
[tex]\begin{array}{ccll} \$&minutes\\ \cline{1-2} 13 & 60\\ x& 525 \end{array} \implies \cfrac{13}{x}~~=~~\cfrac{60}{525} \\\\\\ \cfrac{ 13 }{ x } ~~=~~ \cfrac{ 4 }{ 35 }\implies 455=4x\implies \cfrac{455}{4}=x\implies 113.75=x[/tex]
Two different investment companies offer college savings plans, one at 8. 4% compounded continuously and the other at 8. 6% compounded quarterly. Which is the better investment?
The 8.4% compounded quarterly is the better investment because it gives a good return.
Compound interest is “interest-on-interest”, or the ability of a financial instrument to generate earnings on its earnings. Compound interest, can be calculated using the formula :
[tex]A = P(1 + \frac{r}{n} )^{nt}[/tex]
where A = final amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods elapsed
(i)S = P
S = P(1.0854) (1.0854)
AP(1) = 1.0846 - 1
AP(1)= 0.08546 or 8.546%
(ii)r = 0.084/4
AP(2) = (1 + 0.084/4)⁴ -1
AP(2) = 1.08688 - 1
AP(2) = 8.668%
The 8.4% compounded quarterly investment is therefore preferable.
To learn more about compound interest
https://brainly.com/question/30661956
#SPJ4
43. given a standard deck of 52 playing cards, where half (clubs and spades) are black and half (hearts and diamonds) are red, what is the probability of picking three black cards in a row if the cards are not replaced?
The probability of picking three black cards in a row, without replacement, from a standard deck of 52 playing cards is 2/17, or approximately 0.1176 (rounded to four decimal places).
The probability of picking a black card on the first draw = 26/52 = 1/2
(since there are 26 black cards in the deck)
The probability of picking a black card on the second draw, given that a one was picked on the first draw and the card is not replaced = 25/51,
(since there are now only 25 black cards left out)
The probability of picking a black card on the third draw, given that the ones picked on the first two draws are not replaced = 24/50 = 12/25,
(since there are now only 24 black cards left out)
To pick three black cards in a row, we need to multiply the probabilities of each individual draw = (1/2) * (25/51) * (12/25) = 6/51
Simplifying this fraction, we get = 6/51 = 2/17
Learn more about Probabality and Deck of cards from the link below:
brainly.com/question/12602921
#SPJ4
Which statements describe the graph of 2 8/9>x
There is an open circle on ²9.
21909.
The graph contains the point 0
,8
There is a closed circle on ²ğ.
The arrow points to the left.
The arrow points to the right.
2x on a number line? Select three options.
The inequality 26/9 > x represents a line with an open circle at 26/9 going toward positive infinity represented by an arrow.
What are inequalities and their types?Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
Given, An inequality [tex]2\frac{8}{9} > x[/tex].
26/9 > x.
Now, As the inequality is strictly greater than 26/9 it would be represented by a line open circled at 26/9 and going towards +∞.
learn more about inequalities here :
https://brainly.com/question/30228778
#SPJ9
can someone help please
Answer: x=3, y=-1
Step-by-step explanation:
2x+y=5
2y=2x-8
In equation 1, y=5-2x.
Substitute this value in equation 2.
2(5-2x)=2x-8
10-4x=2x-8
Thus,
4x+2x=10+8 (collecting terms)
6x=18
x=3
Now, we said earlier that y=5-2x.
Therefore, y=5-2(3) [because x=3]
y=5-6
y=-1
Hope this helps :]
pins each cost £1.10
1000 pins are sold
35% of the pins are donated to charity
the rest are sold: 60% are sold for £3
40% are sold are sold £5 for 2
work out the percentage profit the company should make
The profit is £1,370. Then the percentage profit of the company will be 124.54%.
What is a profit?A monetary profit, particularly the distinction between the amount gained and the actual cost of purchasing, running, or creating anything.
The profit is given as,
Profit = Selling price - Cost price
The number of pins left after the donation to charity is given as,
⇒ (1 - 0.35) x 1,000
⇒ 0.65 x 1,000
⇒ 650 pins
The selling price when 60% are sold for £3 and 40% are sold for £5 is given as,
Selling price = 0.60 x 650 x £3 + 0.40 x 650 x £5
Selling price = £1,170 + £1,300
Selling price = £2,470
The cost price is given as,
Cost price = £1.10 x 1,000
Cost price = £1,100
The profit is given as,
P = £2,470 - £1,100
P = £1,370
Then the percent of profit is given as,
P = [(£1,370) / £1,100] x 100
P = 124.54%
The profit is £1,370. Then the percentage profit of the company will be 124.54%.
More about the profit is given below.
https://brainly.com/question/15036999
#SPJ9
Is the answer 126.26? Area of compound figures?
Answer:
Step-by-step explanation:
Jonathan rides his bicycle at a velocity of 12 m/s East. He
comes to a hill and over the next 23 seconds, his velocity
decreases to 5 m/s East. What is Jonathan's acceleration?
Jonathan's acceleration is -0.3043 m/s² East, which means he is slowing down.
What is Acceleration?Acceleration is the rate of change of the velocity of an object with respect to time.
Jonathan's initial velocity, u = 12 m/s East
His final velocity, v = 5 m/s East
The time taken, t = 23 seconds
We can use the formula for acceleration:
a = (v - u) / t
Substituting the given values, we get:
a = (5 m/s - 12 m/s) / 23 s
a = -7 m/s / 23 s
a = -0.3043 m/s^2 East
Therefore, Jonathan's acceleration is -0.3043 m/s² East, which means he is slowing down.
To learn more on Acceleration click:
https://brainly.com/question/12550364
#SPJ9
1000, 200, 40, .... Find the 50th term:
PLEASE GIVE BRAINLIEST
thank you and have a good day :)
Answer:
5.6294995342131E-32
Step-by-step explanation:
the common ratio is 1/5, therefore with the equation an = a x r
n-1
we can figure out the answer to this by plugging in the common ratio for r,
and n as the number in the sequence that we want to obtain.
a is simply the first number (1000)
Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis.
y = x^3/2
y = 27
x = 0
Volume of solid generated by revolving the plane region about the y-axis is [tex]\pi \left(\frac{1392}{7}\right)[/tex]
What do you understand by Shell method?The shell method formula is used to find the volume of a solid of revolution, formed by rotating a region about a line that is parallel to one of its sides.
The formula for the shell method is V = 2π* ∫r(x)h(x) dx where V is the volume of the solid, r(x) is the distance between the axis of revolution and the curve being rotated and h(x) is the height of the shell at x.
The shell method can be used when the region is more easily described by an equation in one variable. In contrast, the disk/washer method is used when the region is more easily described by an equation in two variables.
It is a powerful tool for solving many types of problems in calculus, physics and engineering. The shell method formula helps to determine the volume of a solid of revolution, which is an important concept in various fields of study.
Given:
y = [tex]x^\frac{3}{2}[/tex]
y = 27
x = 0
Shell method:
V = [tex]2\pi \int\limits_c^d p(y)h(y) dy[/tex]
Then the volume of a solid is
V = [tex]2\pi \int\limits_0^4x(27 - x^\frac{3}{2})dx[/tex]
= [tex]2\pi \left[ \int\limits_0^4 27x dx - \int\limits_0^4 x^\frac{5}{2}dx\right][/tex]
[tex]= 34\pi \left[\frac{x^2}{2} \right]_0^4 - \frac{4}{7} \pi \left[ x^\frac{7}{2} \right]_0^4[/tex]
[tex]= 34\pi \left(\frac{4^2}{2} \right) - \frac{4\pi}{7}(4)^\frac{7}{2}[/tex]
[tex]= 4\pi \left( \frac{68}{7} - \frac{2^7}{7} \right)[/tex]
[tex]= 4\pi \left( \frac{476 - 128}{7} \right)[/tex]
[tex]4\pi \left( \frac{348}{7} \right)[/tex]
[tex]= \pi \left( \frac{1392}{7} \right)[/tex]
To know more about Shell method visit:
brainly.com/question/30401636
#SPJ1
