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 = x3/2 y = 27 x = 0.
Answers
Volume of the solid generated by revolving the plane region about the y-axis is [tex]\frac{6561 \pi}{7}[/tex] .
Given, that y = x3/2, y = 27, x = 0
So, the volume of the solid generated by revolving the region about y axis will be,
[tex]V=\int_0^92\pi(x)(27-y)dx[/tex]
[tex]V=\int_0^92\pi x(27-x^{\frac{3}{2}})dx[/tex]
[tex]V=\left [ 27{\pi}x^2-\frac{4{\pi}x^\frac{7}{2}}{7} \right ]_0^9[/tex]
[tex]V=\left [ \frac{{\pi}\left(189x^2-4x^\frac{7}{2}\right)}{7} \right ]_0^9[/tex]
[tex]V=\frac{6561 \pi}{7}[/tex]
The image of the region bounded by plane is attached below.
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The volume of the solid generated by revolving the plane region about the y-axis is approximately 6863.01 cubic units.
To use the shell method to find the volume of the solid generated by revolving the plane region about the y-axis, we need to express the limits of integration and the height of the infinitesimally thin cylindrical shells.
Given the equations:
y = x^(3/2)
y = 27
x = 0
To determine the limits of integration, we need to find the intersection points between the two curves: y = x^(3/2) and y = 27.
Setting the equations equal to each other:
x^(3/2) = 27
Taking the square root of both sides:
x = 27^(2/3)
x = 9
Therefore, the limits of integration are x = 0 and x = 9.
Now, let's consider an infinitesimally thin cylindrical shell with height "h" and radius "r" at some x value between 0 and 9.
The height of the shell, "h", is the difference between the y-values of the two curves:
h = 27 - x^(3/2)
The radius of the shell, "r", is the x-value.
The volume of the shell can be expressed as:
dV = 2πrh dx
To find the total volume, we integrate this expression from x = 0 to x = 9:
V = ∫[0, 9] 2π(27 - x^(3/2))x dx
Now, let's evaluate this integral:
V = 2π ∫[0, 9] (27x - x^(5/2)) dx
Integrating term by term:
V = 2π [(27/2)x^2 - (2/7)x^(7/2)] evaluated from 0 to 9
Plugging in the limits of integration:
V = 2π [(27/2)(9)^2 - (2/7)(9)^(7/2)] - 2π [(27/2)(0)^2 - (2/7)(0)^(7/2)]
Simplifying and evaluating the expression:
V = 2π [(27/2)(81) - (2/7)(3√(9))] - 0
V = 2π [1093.357] ≈ 6863.01 cubic units
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Related Questions
9w-4w+6<1+5w supposed to combine like terms
Answers
Answer/Step-by-step explanation:
Given:
[tex] 9w - 4w + 6 [/tex] < [tex] 1 + 5w [/tex]
To solve, collect like terms.
Subtract 5w from both sides
[tex] 9w - 4w + 6 - 5w [/tex] < [tex] 1 + 5w - 5w [/tex]
[tex] 9w - 4w - 5w + 6 [/tex] < [tex] 1 [/tex]
[tex] 6 [/tex] < [tex] 1 [/tex] (incorrect)
The inequality given has no solution or an information is missing.
Write the equation of the line that has a slope of -5 that goes through (-4,7)
Answers
Answer:
y = -5x + 7
Explanation:
Slope is given -5
7 is the y-intercept.
"Find an equation for the family of linear functions with slope 4. (Use the standard coordinate variables x and y. You may use m for the slope and b for the y-intercept as needed.)"
Answers
Answer:
Y= 4x +y1-4x1
4 represent the slope m
+Y1 - 4x1 represent the intercept b
Step-by-step explanation:
Formula for the equation of the linear function
(Y-y1)/(x-x1) = m
We will transform it into the equation form
Y= mx + b
Where m = slope=4
b = intercept
(Y-y1)/(x-x1) = m
(Y-y1)/(x-x1) = 4
(Y-y1)= 4(x-x1)
Y-y1= 4x -4x1
Y= 4x +y1-4x1
4 represent the slope m
+Y1 - 4x1 represent the intercept b
An equation for the family of linear functions with slope 4 is; y = 4x +b
Equation of a straight lineThe equation of a line can be determined by means of the slope as follows;
slope,m = (y-y1)/(x-x1)Therefore, for this scenario;
Slope,m = 4 = (y-y1)/(x-x1)By Cross-product, we have;
4x - 4x1 = y -y1Since, the y-intercept, b is the value of y for which x= 0.
Hence;
4x - 0 = y-by = 4x +bRead more on slope-intercept form;
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The nonnegative function given below is a probability density function.
{3/7 e^−3t/7 if t ≥ 0
0 if t < 0
(a) Find P(0 ≤ t ≤ 2).
(b) Find E(t).
Answers
Answer:
a) 0.5756
b) 2.3333
Step-by-step explanation:
A) for p ( 0 ≤ t ≤ 2)
= 1 - [tex]e^\frac{6}{7}[/tex] = 0.5756
B) for E(t)
= 7/3 = 2.3333
attached below is the remaining part of the solution
100 points, Please help ASAP. Answer correctly thanks
Answers
Answer:
[tex]\Huge \boxed{\mathrm{7. \ C}} \\\\ \\ \Huge \boxed{\mathrm{8. \ B}} \\\\\\ \Huge \boxed{\mathrm{9. \ D}} \\\\\\ \Huge \boxed{\mathrm{10. \ D}} \\\\\\ \Huge \boxed{\mathrm{11. \ A}}[/tex]
[tex]\rule[225]{225}{2}[/tex]
Step-by-step explanation:
7.The Commutative Property of Addition changes the order of the numbers but does not affect the sum.
8.We will convert the values into decimals for simplicity.
[tex]\sqrt{7} \approx 2.65[/tex]
[tex]\sqrt{3} \approx 1.73[/tex]
[tex]\displaystyle \frac{-1}{3} \approx -0.33[/tex]
Arranging from least to greatest:
[tex]\displaystyle -2.3, \ \frac{-1}{3} , \ \sqrt{3} , \ \sqrt{7} , \ 5.1[/tex]
9.2 + ( -8 )
2 is on the number line.
When -8 is added to 2, the value decreases, and it moves 8 units to the left side.
The result is -6.
(image)
10.The two subsets of real numbers are rational and irrational numbers.
[tex]\sqrt{13}[/tex] is an irrational number.
11.[tex]\displaystyle \frac{12x-8}{4}[/tex]
[tex]\displaystyle \frac{12x}{4}-\frac{8}{4}[/tex]
[tex]3x-2[/tex]
[tex]\rule[225]{225}{2}[/tex]
what is the gcf of 54, 72 and 108
Answers
Answer:
[tex] \boxed{ \bold{ \huge{ \boxed{ \sf{18}}}}}[/tex]
Step-by-step explanation:
The GCF of any numbers can be obtained by using two methods. They are :
Prime factorization methodDivision methodNow, I am going to find out the GCF of 54 , 72 and 108 using these both methods :
Let's find out using the prime factorization method
First of all , find the prime factor of each numbers
54 = 2 × 3 × 3 × 3
72 = 2 × 2 × 2 × 3 × 3
108 = 2 × 2 × 3 × 3 × 3
Take out the common prime factors of given numbers
Common prime factors = 2 , 3 and 3
Multiply those common prime factors and the number obtained is G.C.F
GCF = Common prime factors
= 2 × 3 × 3
= 18
-----------------------------------------------------------
Now,let's find out using another method i.e division method
Steps for finding G.C.F by using division method:
Step 1 : Divide the larger number by smaller one.
Step 2 : In case of remaining remainder, divide the first division by that remainder.
Step 3 : Again if remainder leaves, divide the second divisor by the remainder.
Step 4 : Repeat the process until the remainder becomes the zero. So , that the last divisor is the G.C.F
( See attached picture )
Hope I helped!
Best regards!!
Write the following radical expression using exponents.
1 divided by square root of 3
Answers
Answer:
The square root of 3 is 9
Based on your observations from Question 1, what is the relationship between AE and BF when DFB and CEA measure something other
than 90°? In this situation, what is the relationship between
AB
and CD ? Explain.
Answers
Answer:
if angle dfb and angle cea measure something other than 90degrees, then line we is not equal to line bf. in this case, line and and line cd intersect at a single point.
A test consists of 20questions. Multiple choice questions earn 4points for a correct answer, and free response questions earn 8points for a correct answer. If the test is out of 100points, how many of each type of question are on the test?
Answers
Answer:
15 multiple choice questions
5 free response questions
Step-by-step explanation:
This is basically a systems of equations word problem.
To solve this, we need to create two equations that represent this scenario. Let's suppose [tex]m[/tex] represents the amount of multiple choice questions and [tex]f[/tex] represents the amount of free-choice questions.
We can create the following equations:
[tex]4m + 8f = 100[/tex]
[tex]m+f=20[/tex]
To solve for m and f, we can use elimination.
Let's multiply the equation [tex]m+f=20[/tex] by -4.
[tex]-4m - 4f = -80[/tex]
Great! Now let's add this equation to our first one, [tex]4m + 8f = 100[/tex].
[tex]4m + 8f = 100[/tex]
[tex]-4m - 4f = -80[/tex]
___________________
[tex]0m + 4f = 20[/tex]
[tex]4f=20[/tex]
Dividing both sides by 4 get us [tex]f=5[/tex].
So there are 5 free choice questions. To find m, we can substitute inside the equation [tex]m+f=20[/tex]
[tex]m+5=20\\\\m=20-5\\\\m=15[/tex]
So there are 15 multiple choice questions.
Hope this helped!
Which of the following expressions represents the verbal description below?
the product of 5 and the difference of 7 times the square of x and 3
5(7-3x^2)
5-7(x^2-3)
5(7x^2-3)
5-7x^2-3
Answers
Answer: C. 5(7x² - 3)
Step-by-step explanation:
Product means multiplication
Difference means subtraction
square means "to the power of 2"
Product of 5 and _______ --> 5(__)
Difference of 7 ... and 3 --> 7... - 3
7 times the square of x ---> 7x²
Put it together to get: 5(7x^2 - 3)
Answer:
D is the correct answer
Step-by-step explanation:
I got it right
Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere x2+y2+z2≤34 cut off by the plane z=5 and restricted to the first octant. (In your integral, use theta, rho, and phi for θ, rho and ϕ, as needed.) What coordinates are you using? (Enter cartesian, cylindrical, or spherical.) With a= , b= , c= , d= , e= , and f= , Volume = ∫ba∫dc∫fe
Answers
Answer:
a = 0
b = [tex]\frac{\pi }{4}[/tex]
c = 0
d = 3
e = 5
f = [tex]\sqrt{34-r^2}[/tex]
Step-by-step explanation:
Equation of the solid sphere = x^2 + y^2 + z^2 ≤ 34 ------- (1)
at Z = 5
since the bottom of the sphere(z) is flat = 5 we will use cylindrical coordinates
concentrating in the first octant as mentioned in the question
at Z = 5 , equation 1 becomes :
x^2 + y^2 + 25 ≤ 34 = x^2 + y^2 = 9
hence the radius around the xy axis = [tex]\sqrt{9}[/tex] = 3
that means the radius is : 0 ≤ r ≤ 4 , 0 ≤ ∅ ≤ [tex]\frac{\pi }{4}[/tex]
next we have to find the upper bound of: Z = ±[tex]\sqrt{34-r^2}[/tex] we will pick out only the positive
5 ≤ z ≤ [tex]\sqrt{34 - r^2}[/tex]
therefore for the Volume = ∫ba∫dc∫fe
a = 0
b = [tex]\frac{\pi }{4}[/tex]
c = 0
d = 3
e = 5
f = [tex]\sqrt{34-r^2}[/tex]
Write the decimal as a fraction in simplest form. 0.008
Answers
Answer:
1 / 125
Step-by-step explanation:
Since there are 3 digits after the decimal point, we know that our denominator will be 10³ = 1000. Since the only non-zero digit after the decimal point is 8, we know that the numerator is 8. Therefore, the answer is 8 / 1000 = 1 / 125.
If 5(x-4)+2= -173 what is the value of x?
Answers
Answer:
38.2
Step-by-step explanation:
5 (x-4) + 2 = 173
=> 5x - 20 + 2 = 173
=> 5x - 18 = 173
=> 5x - 18 + 18 = 173 + 18
=> 5x = 191
=> 5x/5 = 191/5
=> x = 38.2
So, x is 38.2
Step 2: combine so -20+2=-18 so you are going to have 5x-18=-173 then you add -18 to -173 and it going to be 5x=-155
Step 3: multiply, 5x/5=-155/5 is
X=-31
Identify the sequence graphed below and the average rate of change from n = 1 to n = 3. coordinate plane showing the point 2, 8, point 3, 4, point 4, 2, and point 5, 1. an = 8(one half)n − 2; average rate of change is −6 an = 10(one half)n − 2; average rate of change is 6 an = 8(one half)n − 2; average rate of change is 6 an = 10(one half)n − 2; average rate of change is −6
Answers
Answer:
2
Step-by-step explanation:
Answer:
An= 8 x (1/2)^(n-2) and the average rate of change is 6
Step-by-step explanation:
which model of 85% is shaded
Answers
Answer:
You have no examples of models
Step-by-step explanation:
5:30PM
You run to the store on your way home from school. You want to buy some Doritos. Should you buy
the regular size bag or the party size bag? Calculate the unit price for each size and determine which
one is the better buy.
15 oz. Party Size Doritos $3.98
9.75 oz. Doritos $2.98
The Unit Price is just a Ratio with a denominator of 1. Your textbook has information about ratios on
pgs. 305-307
Answers
Answer:
The regular size bag of doritos is a better deal, I believe. Hope this helps!
Step-by-step explanation:
A triangle has a base of 16 inches and a height of 18 1 8 inches. What is its area?
Answers
Answer:
144 in²
Step-by-step explanation:
area of a triangle = 1/2 * base * height
area = 1/2 * 16 * 18
area = 144 in²
Elizabeth is selling jam at a farmers market, and she earns $5.00 for each jar of jam that she sells. Her goal for the weekend is to earn at least $450.00, and she has already made $95.00 from jam sales as well as $24.00 from leading a demonstration. What is the minimum number of jars Elizabeth must sell at this point to reach her goal?
Answers
Answer:
66 jars of jam
Step-by-step explanation:
Elizabeth is selling jam at a farmers market.
She earns $5 for each jar of Jan that she sells
Her goal is to earn $450 during the weekend.
Elizabeth has already made $95 from the sale of her jams and $24 from leading a demonstration
Therefore the total amount that has been made is
= $95 + $24
= $119
Since her target is $450 and she has already made $119 then the amount remaining to complete the target can be calculated as follows
= $450 - $119
= $331
The minimum number of jars that Elizabeth must sell to realize her goal can be calculated as follows
= 331/5
= 66
Hence Elizabeth must sell 66 jars of jam to reach her goal.
Because of the rainy season, the depth in a pond increases 3% each week. Before the rainy season started, the pond was 10 feet deep. What is the equation that best represents the depth of the pond each week? Hint: Use the formula y = P(1 + r)x. y = 10(0.03)x y = 10(0.3)x y = 10(1.3)x y = 10(1.03)x
Answers
Answer:
y = 10(1.03)^x
Step-by-step explanation:
In the formula y = p(1 + r)^x, p is the starting amount and r is the growth rate.
In this problem, the starting amount is 10. So, p = 10.
The growth rate is 3% each week, which is the same as 0.03 as a percentage. So, r = 0.03.
Plug these into the equation:
y = 10(1.03)^x will be the equation for this situation
The equation that best represents the depth of the pond each week is y = 10(1.03)× option (D) is correct.
What is an exponential function?It is defined as the function that rapidly increases and the value of the exponential function is always positive. It denotes with exponent y = a×
where a is a constant and a>1
We have:
A pond increases 3% each week.
The exponential growth is represented by:
y = p(1 + r)×
p = 10 feet
r = 3% = 0.03
y = 10(1 + 0.03)×
y = 10(1.03)×
Thus, the equation that best represents the depth of the pond each week is y = 10(1.03)× option (D) is correct.
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Tom needs to rent tables for his family reunion. There will be 105 people attending. Each table seats 8
people. how many tables does Tom need to rent?
O 14
12
10
18
16
Answers
He needs to rent 14 tables
Hope it helps :)
Answer:
tom needs to rend 14 tables
how to graph slope 7/3 and y-intercept (x,y) (0,-11)
Answers
Start at (0,-11) and move up 7 on the y axis and over 3 on the x axis and you’ll have your line
-4|2x+5|=-12 solve please
Answers
Answer:
x=-4
Step-by-step explanation:
1. Distribute.
|-8x-20|=-12
2. Make it positive.
8x+20 = -12
3. Inverse operations.
8x = -32
4. Divide.
x = -4.
If y = 9x − 7, which of the following sets represents possible inputs and outputs of the function, represented as ordered pairs? A) {(0, −7), (1, 2), (−1, −16)} B) {(−7, 0), (2, 1), (−16, −1)} C) {(1, 9), (2, 7), (3, 16)} D) {(7, 9), (8, 10), (9, 11)}
Answers
Answer:
0,-7 8,567;7,57,6,6,6,6,6,78,3,5
compare the following fraction
4/3 and 5/2
-3/2. -4/5
Answers
Answer:
Step-by-step explanation:
to compare them, we need to convert them into like fractions
4/3 and 5/2
lcm of 3 and 2 is 6
4/3 × 2/2 = 8/6
5/2 × 3/3 = 15/6
15/6 is greater than 8/6
so 5/2 is greater than 4/3
similarly,
-3/2 and -4/5
we need to convert them into like fractions
lcm of 2 and 5 is 10
-3/2 × 5/5 = -15/10
-4/5 × 2/2 = -8/10
-8/10 is greater than -15/10
so -4/5 is greater than -3/2
Hope this helps
plz mark as brainliest!!!!
What is
[tex]7.842 \div 2.35[/tex]
please show how you got the answer
Answers
Explanation:
I added a picture hope it helps
Evaluate. 5^3−(6/48)+10^2 Enter your answer in the box. 100 POINTS!!! PLEASE HELP
Answers
Answer:
= 1799 or 224.875
8
Step-by-step explanation:
5³ − (6/48) + 10²
= 125 - (0.125) + 100
= 224.875
or
5³ − (6/48) + 10²
= 125 - 1/8 + 100
= -1/8 + 225
= (225 * 8)/8 - 1/8
= (225 * 8-1) / 8
= 1799 or 224.875
8
Answer:
[tex]\huge \boxed{\frac{1799}{8} }[/tex]
[tex]\rule[225]{225}{2}[/tex]
Step-by-step explanation:
[tex]\displaystyle 5^3-(\frac{6}{48} )+10^2[/tex]
Solving for exponents:
[tex]\displaystyle 125-(\frac{6}{48} )+100[/tex]
Adding or subtracting:
[tex]\displaystyle -(\frac{1}{8} )+100+125[/tex]
[tex]\displaystyle -(\frac{1}{8} )+225[/tex]
[tex]\displaystyle -(\frac{1}{8} )+\frac{1800}{8}[/tex]
[tex]\Rightarrow \ \displaystyle \frac{1799}{8}[/tex]
[tex]\rule[225]{225}{2}[/tex]
How do I solve this question? Rayne sold 3 desks at a local trade show. he payed $4.00 to rent the booth. he gave half of his revenue to the carpenter and was left with $185.50. at what price did Rayne sell each desk
Answers
Answer:
$125
Step-by-step explanation:
I suspect there's an error in the problem statement. Revenue is the money made from selling the desks. Profit is the revenue from the desks minus the cost of the booth rental.
If Rayne gives half of his profit to the carpenter, and x is the price per desk, then:
185.50 = ½ (3x − 4)
371 = 3x − 4
375 = 3x
x = 125
What is the value of x?
Answers
Answer:
it's not even in the pic
Step-by-step explanation:
the q isn't even there
Which of the following statements is false? -(-5) = 5 |-5| = -5 -|-5| = -5 -|5| = -5
Answers
Your question has been heard loud and clear.
Well -(-5)=5
5/-5/= -5
/-5/5= -5
So basically , the statement is false because -5(-5) not equal to -5/-5/
Thank you
From the given options |-5|=-5 is false. Therefore, option B is the correct answer.
We need to identify the false statement from the given options.
What is an absolute value?The absolute value of a number is defined as its magnitude irrespective of the sign of the number. To find the absolute value of a real number, we consider only the number and remove the sign. It can only be a non-negative value. The absolute value of a positive number is the number itself, that of a negative number is the number without a negative sign, and the absolute value of 0 is 0.
From option (A):
-(-5)=5 is true
From option (B):
|-5|=-5 is false
From option (C):
-|-5| = -5 is true
From option (D):
-|5| = -5 is true
From the given options |-5|=-5 is false. Therefore, option B is the correct answer.
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To find out how many rabbits were in a park students tagged 10 rabbits. They later came back to the park and counted 200 total rabbits and 5 of them were tagged. Using this information, what is the best estimate for the total number of rabbits in the park? Can someone help explain this? Not just answer it but actually explain the reasoning?
Answers
Answer:
400
Step-by-step explanation:
The assumption behind mark-recapture methods is that the proportion of marked individuals recaptured in the second sample represents the proportion of marked individuals in the population as a whole. In algebraic terms, This method is called the Lincoln-Peterson Index of population size.