The area enclosed by the graphs of y = 1/x, y = 1, and x = 3 is rotated about the line y = -1. Find the volume and show steps.
Answers
Answer:
[tex]A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\approx9.8524[/tex]
Step-by-step explanation:
Use the Washer Method [tex]A=\pi\displaystyle \int\limits_{a}^{b}{\bigr[R(x)^2-r(x)^2\bigr] \, dx[/tex] where [tex]R(x)[/tex] is the outer radius and [tex]r(x)[/tex] is the inner radius.
If we sketch out the graph, we see that [tex]y=1[/tex] intersects points [tex](1,1)[/tex] and [tex](3,1)[/tex], which will be our bounds of integration.
Here, our outer radius will be [tex]R(x)=(-1-1)=-2[/tex] and our inner radius will be [tex]r(x)=-1-\frac{1}{x}[/tex].
Thus, we can compute the integral and find the volume:
[tex]A=\pi\displaystyle\int\limits^{3}_{1} {(-2)^2-\biggr(-1-\frac{1}{x}\biggr)^2 } \, dx\\ \\A=\pi\displaystyle\int\limits^{3}_{1} {4-\biggr(1+\frac{2}{x}+\frac{1}{x^2} \biggr) } \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {4-1-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {3-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\biggr[3x-2\ln(|x|)+\frac{1}{x}\biggr]\Biggr|_{1}^{3}\\[/tex]
[tex]A=\pi\displaystyle\biggr[\biggr(3(3)-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3(1)-2\ln(|1|)+\frac{1}{1}\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(9-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3+1\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(\frac{28}{3}-2\ln(|3|)\biggr)-\biggr(4\biggr)\biggr]\\A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\\A\approx9.8524[/tex]
In conclusion, the volume of the solid of revolution will be about 9.8524 cubic units. See the attached graph for a helpful visual!
Related Questions
If 2 cups = 1 pint, how many cups of liquid are in a 11-pint container?
Group of answer choices
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Convert 12 yards 3 feet and 8 inches into inches
Answers
Answer:
476 inches. i dont know how to do an explanation i just looked it up lol
Answer:
476 inches
Step-by-step explanation:
First convert 12 yards to 36 feet. We know this is the conversion because there are 3 feet in every foot, and 12 times 3 is 36.
We add 36 feet to 3 feet and we get 39 feet. There are 12 inches in a foot, so we do 12 times 39 feet and we get 468 inches.
The we add 468 inches to 8 inches and finally get 476 inches.
What is the equation of a hyperbola with a = 1 and c = 9? Assume that the transverse axis is horizontal.
Answers
Answer:
Step-by-step explanation:
The general formula for a horizontal hyperbola :
[tex]\frac{x^{2} }{a^{2} } -\frac{y^{2} }{b^{2} } =1[/tex]
Given: a = 1 we know a² = 1 ,and that c = 9 so we know c² = 9² = 81
We also know that the relation between a, b, c for a hyperbola is c²= a²+b²
c²= a²+b², substitute what we know
81 = 1 +b², subtract 1 from both sides of the equation
80 = b²
The equation of our hyperbola is:
[tex]\frac{x^{2} }{1 } -\frac{y^{2} }{80 } =1[/tex] or [tex]x^{2} -\frac{y^{2} }{80} =1[/tex]
What is the value of x?
Answers
Answer:
Step-by-step explanation:
All the three sides are equal. So, it is an equilateral triangle.
Each angle in an equilateral triangle = 60°
7x + 4 = 60
7x = 60 -4
7x = 56
x = 56/7
x = 8
How can you tell that (496 + 77 + 189) x 10 is twice as large as (496 + 77 +189) x 5 without doing complicated calculations?
Answers
Answer:
Because 10 is twice as large as 5.
Step-by-step explanation:
An Eiffel Tower mural is 2 meters high and 0.77 meters wide. If the actual tower is 125 meters wide, how tall is it to the nearest meter?
Answers
If the actual tower is 125 meters wide, then the height will be approximately 325m
Similarity ratio of shapesThe ratio of the similar sides of the shape is equal to a constant.
If an Eiffel Tower mural is 2 meters high and 0.77 meters wide the actual tower is 125 meters wide, then the new height will be expressed as;
2/0.77 = H/125
Cross multiply
0.77H = 2 * 125
0.77H = 250
H = 250/0.77
H = 325m
This means that if the actual tower is 125 meters wide, then the height will be approximately 325m
Learn more on similar shapes here: https://brainly.com/question/2644832
Brand X rice sells for $70 per kg. Brand Y rice sells for $20 per 400 g. Both brands have similar qualities 8 . Which brand is the better buy? Why do you think so?
Answers
Answer:
Brand Y would have a better deal.
Step-by-step explanation:
Step 1. There are 1,000 grams in 1 kilogram. Knowing this, you can see that it would be $70 for 1,000 grams of rice and $20 for 400 grams of rice.
Step 2. You would have to do 70 divided by 1000 (0.07) and 20 divided by 400 (0.05) to see which has the better deal.
Step 3. Brand X is 7 cents per grain and Brand Y is 5 cents per grain, making Brand Y the better deal. I hope that this helps! :)
A cube has an edge of 2 feet. The edge is increasing at the rate of 4 feet per minute. Express the volume of the cube as a function of m, the number of minutes elapsed.
Hint: Remember that the volume of a cube is the cube(third power) of the length of a side.
v(m)=???? feet^3
Answers
Answer:
V(m) = [tex](4m+2)^3[/tex]
Step-by-step explanation:
The edge of the cube is increasing by 4 feet every minute
V(m) = [tex](4m + 2)^3[/tex]
Two is the starting sidelength of the cube, and 4m is how many feet the side of the cube has grown, depending on the number of minutes that have passed
Samantha is a waitress and earns a weekly paycheck of $350 plus any tips she earns thought out the week . One week . Samantha noticed she earned about $12 tips each hour . Let x represent the number of hours Samantha works during a week where her average hourly earnings is $22
Write a rational equation that models this situation
Answers
A home builder needs to cover the rectangular floor of a closet with a carpet the closet is 2 yards long and 2/3 of a yard wide how much carpet will the home builder need to cover the closet?
Answers
Your Answer is 4 Yards
Step-by-step explanation:
2/3 yards is 2 feet
2 yards is 6 feet
2 times 6 feet is 12 foot e.g. (2 x 6 = 12)
12 feet is 4 yards.
Formula (L x W = A)
L = length
W = Width
A = Area
Hope this helps! Please let me know if you need more help or think my answer is incorrect. Brainliest would be MUCH appreciated. Have a wonderful day!
Help picture below problem 10
Answers
Solution :-
Here, we have been given that lines f and g are parallel. Thus, the angle measuring 135° and ∠2 are vertically opposite angles.
And we know that vertically opposite angles measure same. Thus,
∠2 = 135°
And,
∠2 + ∠6 = 180° ( Co - interior angles sum up to 180° )
135° + ∠6 = 180°
∠6 = 180° - 135°
∠6 = 45°
Hope that helps :)
{(4x+2y=6),(-2x+2y=18):} Solve by using the elimination method
Please help and show me how you did it
Answers
Answer:
[tex]\displaystyle [-2, 7][/tex]
Step-by-step explanation:
You want to eradicate one pair of variables so they are set to zero. It does not matter which pair is selected:
[tex]\displaystyle \left \{ {{4x + 2y = 6} \atop {-2x + 2y = 18}} \right.[/tex]
{½[4x + 2y = 6]
{−2x + 2y = 18
>> [tex]\displaystyle \left \{ {{2x + y = 3} \atop {-2x + 2y = 18}} \right. \\ \\ \frac{3y}{3} = \frac{21}{3} \\ \\ \\ y = 7[/tex]
Plug this back into the system to get the x-value of [tex]\displaystyle -2.[/tex]
I am joyous to assist you at any time.
Joy's class voted on whether or not to do science fair projects in groups. Out of 24 votes, 18
were in favor of working in groups. What percentage of the votes were in favor of working in
groups
Write your answer using a percent sign (%).
Submit
Answers
Answer:
75%.
Step-by-step explanation:
That would be (18/24) * 100
= 3/4 * 100
= 75%
I well give u brainliest
What are the x- and y-intercepts for the linear equation 2x – 4y = 8 ?
Question 6 options:
(4,0) and (0,-2)
(2,0) and (0,4)
(-2,0) and (4,0)
(-2,0) and (0,4)
Answers
The answer is a. (4,0) and (0,-2)
Help to get the answer
Answers
Hope this helps!
find the square root of 576 and do it in factorizing method and your answer is going to be in index form
Answers
Answer:
24.
Step-by-step explanation:
Solution,
By Prime Factorisation method,
576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
576 = (2)² × (2)² × (2)² × (3)²
Now,
By Taking square roots on both the sides,
√576 = √ (2)² × (2)² × (2)² × (3)²
√576 = 2 × 2 × 2 × 3
√576 = 24.
FIRST CORRECT GETS BRAINLIEST
Answers
Answer:
First choice.
Step-by-step explanation:
cos 11π/6 = cos π/6 = √3/2
So cos 11π/12 = √(1 + cos 11π/6) / √2
= √(1 + √3/2) / √2
= √(2 + √3) / 2.
Find the actual area of the triangle below.
Can someone pls help?
Answers
Answer:
200 meters2
Step-by-step explanation:
4 * 5 = 20
20 * 20 ÷ 2 = 200
How many students were in the sample?
Math Test Scores
5 0 4
6 5 7 8
7 5 8
8 2 4 5 5 5 6 7 8 9
9 0 0 0 0 4 6 7 9
5 0 = 50
29
5
24
Answers
LMNO is a parallelogram, with and . Which statements are true about parallelogram LMNO? Select three options.
x = 11
m∠L = 22°
m∠M = 111°
m∠N = 59°
m∠O = 121°
Answers
Answer:
the answers 1, 4, and 5.
m∠N = 59° and m∠O = 121°.
Option 3 and 4 are correct.
What is a parallelogram?A parallelogram is a geometric object with sides that are parallel to one another in two dimensions. It is a form of polygon with four sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length. A parallelogram has adjacent angles that add up to 180 degrees. You must have studied a variety of 2D shapes and sizes in geometry, including circles, squares, rectangles, rhombuses, etc. Each of these forms has a unique set of characteristics.
As per the given data:
LMNO is a parallelogram
The sum of adjacent angles in a parallelogram is always equal to 180°
11x + 6x - 7 = 180
17x = 187
x = 11
m∠L = m∠N {opposite angles in a parallelogram are equal}
m∠N = 6x - 7 = 6(11) - 7 m∠N
m∠L = 59°
m∠M = m∠O {opposite angles in a parallelogram are equal}
m∠M = 11x = 11(11) = 121°
m∠O = 121°
Hence, m∠N = 59° and m∠O = 121°.
To learn more about parallelogram, click:
brainly.com/question/11220936
#SPJ5
a manufacturer wishes to recieve $4800 for a batch of items. when 1200 items were found to be defective, the selling cost per item increased by $0.20. how many items were manufactured.
Answers
Answer:
Abput 12000 items were produced yes and i think it is nice that yk like yes
Step-by-step explanation:
What is the mean of the set of data below?
Answers
Answer:
Mean = 8
Step-by-step explanation:
add all the numbers
2+3+5+6+8+11+11+12+14 = 72
then divide by how many number there are which is 9
72 / 9 = 8
I don’t get it because I don’t have a calculator
Answers
I don’t see a picture or what you need help on.
what is the answer
how do i do it
Answers
Answer:
-8.9° C
Step-by-step explanation:
Question:
Doing a scientific experiment the temperature of a container decreased by 35.7° C at a constant rate over 8 1/2 hours.
What value represents the change in the temperature of the container each hour?
Answer:
-8.9° C
Help picture below problem 6
Answers
Answer:
option B is the correct answer
Step-by-step explanation:
mark me brainliest
27 is 0.5% of what number
Answers
Answer:
54
Step-by-step explanation:
27 is half of what number?
27 x 2 = 54
:)
Answer:
135
Step-by-step explanation:
27 x 0.5 = 135
Hope it helps!!!!!
The radius of the half dollar in centimeters is shown.
Which of the following is closest to the area of the face of the
half dollar?
A. 12 cm2
B. 15 cm2
C. 28 cm2
D. 735 cm2
Answers
Step-by-step explanation:
12 is your answer
22/7 × 1.95× 1.95
you will get 11.9
round off to get
12
bai
Solve. 9 m 3 − 27 m 2 − 252 m = 0
Answers
Answer:
m = 0
m = 7
m = -4
Step-by-step explanation:
[tex]\sf 9m^3-27m^2-252m=0[/tex]
Divide by 9:
[tex]\sf \implies m^3-3m^2-28m=0[/tex]
Factor out common term m:
[tex]\sf \implies m(m^2-3m-28)=0[/tex]
Factor expression in parentheses:
[tex]\sf \implies m(m-7)(m+4)=0[/tex]
Therefore,
[tex]\sf \implies m=0[/tex]
[tex]\sf \implies m-7=0 \implies m=7[/tex]
[tex]\sf \implies m+4=0 \implies m=-4[/tex]
how do I do it? I didn't understand the math.
Answers
Answer:
please try do the rest by yourself..Of you didn't get the answer you can ask me through comment..
Marketing analysis determined 55% of females between the ages of 25 to 34 years old search for green technology and practice being green, as compared to 33% of men in the same age group. What is the probability that a randomly selected man between the age of 25 and 34 does not search for green technology
Answers
According to the given percentages, it is found that there is a 67% probability that a randomly selected man between the age of 25 and 34 does not search for green technology.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
We have that 33% of men search for green technology, hence, there is a 100 - 33 = 67% probability that a randomly selected man between the age of 25 and 34 does not search for green technology.
More can be learned about probabilities at https://brainly.com/question/14398287