Find the volume v of the described solid s. the base of s is an elliptical region with boundary curve 4x2 + 9y2 = 36. cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.
Answers
defines an ellipse centered at [tex](0,0)[/tex] with semi-major axis length 3 and semi-minor axis length 2. The semi-major axis lies on the [tex]x[/tex]-axis. So if cross sections are taken perpendicular to the [tex]x[/tex]-axis, any such triangular section will have a base that is determined by the vertical distance between the lower and upper halves of the ellipse. That is, any cross section taken at [tex]x=x_0[/tex] will have a base of length
[tex]\dfrac{x^2}9+\dfrac{y^2}4=1\implies y=\pm\dfrac23\sqrt{9-x^2}[/tex]
[tex]\implies \text{base}=\dfrac23\sqrt{9-{x_0}^2}-\left(-\dfrac23\sqrt{9-{x_0}^2}\right)=\dfrac43\sqrt{9-{x_0}^2}[/tex]
I've attached a graphic of what a sample section would look like.
Any such isosceles triangle will have a hypotenuse that occurs in a [tex]\sqrt2:1[/tex] ratio with either of the remaining legs. So if the hypotenuse is [tex]\dfrac43\sqrt{9-{x_0}^2}[/tex], then either leg will have length [tex]\dfrac4{3\sqrt2}\sqrt{9-{x_0}^2}[/tex].
Now the legs form a similar triangle with the height of the triangle, where the legs of the larger triangle section are the hypotenuses and the height is one of the legs. This means the height of the triangular section is [tex]\dfrac4{3(\sqrt2)^2}\sqrt{9-{x_0}^2}=\dfrac23\sqrt{9-{x_0}^2}[/tex].
Finally, [tex]x_0[/tex] can be chosen from any value in [tex]-3\le x_0\le3[/tex]. We're now ready to set up the integral to find the volume of the solid. The volume is the sum of the infinitely many triangular sections' areas, which are
[tex]\dfrac12\left(\dfrac43\sqrt{9-{x_0}^2}\right)\left(\dfrac23\sqrt{9-{x_0}^2}\right)=\dfrac49(9-{x_0}^2)[/tex]
and so the volume would be
[tex]\displaystyle\int_{x=-3}^{x=3}\frac49(9-x^2)\,\mathrm dx[/tex]
[tex]=\left(4x-\dfrac4{27}x^3\right)\bigg|_{x=-3}^{x=3}[/tex]
[tex]=16[/tex]
The volume [tex]\( V \)[/tex] of the solid [tex]\( S \)[/tex] is 32 cubic units.
To find the volume [tex]\( V \)[/tex] of the solid [tex]\( S \),[/tex] where the base is an elliptical region defined by the equation [tex]\( 4x^2 + 9y^2 = 36 \),[/tex] and the cross-sections perpendicular to the [tex]\( x \)[/tex]-axis are isosceles right triangles with their hypotenuse lying on the base, we proceed as follows:
The equation [tex]\( 4x^2 + 9y^2 = 36 \)[/tex] represents an ellipse centered at the origin with semi-major axis [tex]\( \sqrt{9} = 3 \)[/tex] along the [tex]\( y \)[/tex]-axis and semi-minor axis [tex]\( \sqrt{4} = 2 \)[/tex] along the [tex]\( x \)[/tex]-axis.
Each cross-section perpendicular to the [tex]\( x \)[/tex]-axis is an isosceles right triangle with its hypotenuse on the elliptical base. The height [tex]\( h(x) \)[/tex] of each triangle at a given [tex]\( x \)[/tex] is determined by the elliptical equation.
For a fixed [tex]\( x \),[/tex] the corresponding [tex]\( y \)[/tex] values on the ellipse satisfy [tex]\( 4x^2 + 9y^2 = 36 \).[/tex] Solving for [tex]\( y \)[/tex], we get:
[tex]\[ y = \frac{2}{3} \sqrt{36 - 4x^2} \][/tex]
The height of the triangle is [tex]\( \frac{2}{3} \sqrt{36 - 4x^2} \).[/tex]
To find the volume [tex]\( V \)[/tex] of the solid [tex]\( S \),[/tex] integrate the area of each triangular cross-section along the [tex]\( x \)[/tex]-axis from [tex]\( x = -3 \) to \( x = 3 \):[/tex]
[tex]\[ V = \int_{-3}^{3} \text{Area of triangle at } x \, dx \][/tex]
The area of each triangle is [tex]\( \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2h(x) \cdot h(x) = h(x)^2 \).[/tex]
Thus,
[tex]\[ V = \int_{-3}^{3} h(x)^2 \, dx = \int_{-3}^{3} \left( \frac{2}{3} \sqrt{36 - 4x^2} \right)^2 \, dx \][/tex]
[tex]\[ V = \int_{-3}^{3} \frac{4}{9} (36 - 4x^2) \, dx \][/tex]
[tex]\[ V = \frac{4}{9} \int_{-3}^{3} (36 - 4x^2) \, dx \][/tex]
[tex]\[ V = \frac{4}{9} \left[ 36x - \frac{4x^3}{3} \right]_{-3}^{3} \][/tex]
Solving further,
[tex]\[ V = \frac{4}{9} \left[ \left( 36 \cdot 3 - \frac{4 \cdot 27}{3} \right) - \left( 36 \cdot (-3) - \frac{4 \cdot (-27)}{3} \right) \right] \][/tex]
[tex]\[ V = \frac{4}{9} \left[ (108 - 36) - (-108 + 36) \right] \][/tex]
[tex]\[ V = \frac{4}{9} \left[ 72 \right] \][/tex]
[tex]\[ V = \frac{4 \cdot 72}{9} \][/tex]
[tex]\[ V = 32 \][/tex]
Therefore, the volume [tex]\( V \)[/tex] of the solid [tex]\( S \)[/tex] is 32 cubic units.
Related Questions
What is the solution to the equation 0.8m − 4 = −0.8m?
Answers
0.8m + 0.8m = 4
1.6m = 4
m = 4 /1.6
m = 2.5
Each of a certain type of tire on a 4 wheel car lasts exactly 10,000 miles. How many spare tires of this same variety must carried so that a trip of 15,000 miles can just be completed.
Answers
Tire A, B, C, D (and spare is E F G)
first 5,000 miles you use tire A B C D
second 5000 miles you use tire B C D E
third 5000 miles you use A E F G
Each tires is used no more than twice (10,000 miles).
today there are 3,659 thousand elementary and secondary teachers employed in a certain country. this number is expected to increase to 3,856 thousand by next decade. what is the percent increase?
Answers
= (3856 - 3659) / 3659...* 100
= 197 / 3659....* 100
= 0.0538 * 100
= 5.38% <==
PLEASE HELP !!! HURRY!! Given m || n, find x.
Answers
4x+35 + 5x-8 = 180
9x +27 =180
9x =153
x=153/9 = 17
x = 17
palace help me solve these two equations 14, 15
Answers
check the picture below... so.. is just a semi-circle and a rectangle, nothing fancy, get their area, sum them up.
15)
so... the voter turnout, is referring to, what's the percentage that voted, which year has the highest percentage... well, we simply check the percentage of the value provided from the total eligible voters, keeping in mind that, all eligible voters are the 100%
so.. for 1944
[tex]\bf \begin{array}{ccllll} amount&\%\\ ------&----\\ 86,607,000&100\\ 47,976,670&x\\ \end{array}\implies \cfrac{86,607,000}{47,976,670}=\cfrac{100}{x} \\\\\\ x=\cfrac{100\cdot 47,976,670}{86,607,000}\implies x\approx 55.39583405498[/tex]
for 1984
[tex]\bf \begin{array}{ccllll} amount&\%\\ -----&-----\\ 167,727,000&100\\ 92,659,000&x \end{array}\implies \cfrac{167,727,000}{92,659,000}=\cfrac{100}{x} \\\\\\ x=\cfrac{100\cdot 92,659,000}{167,727,000}\implies x\approx 55.2439380660239[/tex]
Describe the relationship between the similarity ratio of two triangles and the ratio of their areas.
Answers
[tex]\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}[/tex]
Help Please question in the file.
Answers
py+7=6y+q
what is the answer for this question (linear equations with unknown coefficients)?
Answers
py+7=6y+q
-6y -7 -6y -7
(p-6)y = q-7
divide both sides by p-6
y=(q-7)/(p-6)
Answer:
The required equation is [tex]y =\frac{(q-7)}{(p-6)}[/tex].
Step-by-step explanation:
Consider the provided equation.
[tex]py+7=6y+q[/tex]
We need to simplify the equation and convert it into a linear equation with unknown coefficients.
Subtract 6y from both sides.
[tex]py - 6y + 7 = q[/tex]
Subtract 7 from both sides.
[tex]py - 6y = q - 7[/tex]
Take y common from left side.
[tex]y(p-6) = q-7[/tex]
Divide both sides by p-6.
[tex]y =\frac{(q-7)}{(p-6)}[/tex]
Hence, the required equation is [tex]y =\frac{(q-7)}{(p-6)}[/tex].
A sofa is on sale for 26% off. The sale price is $629 . What is the regular price?
Answers
74% ..............$629
100% ..............?
629 x 100/74 = 850
answer
regular price is $850
The circumference of a circle is 6.28. What is the area of the circle.
Answers
First, let's evaluate the question.
"The circumference of a circle is 6.28. What is the area of a circle?"
Now, let's remember the different formulas for area and circumference.
The circumference is "2•3.14•r", while the area is "3.14•r•r".
We have our circumference, 6.28.
However, we are looking for the area. Since we have the circumference, we need to narrow down to the radius (so we can solve for the area).
Let's set this up as an equation;
C = 2 • 3.14 • r
Plug in the value for our circumference.
6.28 = 2 • 3.14 • r
Multiply 2 by 3.14 and r to simplify the right side of the equation.
2 • 3.14 • r = 6.28 • r = 6.28r
We're now left with:
6.28 = 6.28r
Divide both sides by 6.28 to solve for r.
6.28 / 6.28 = 1
6.28r / 6.28 = r
We are now left with the radius:
R = 1.
Now, we can solve for the area.
Remember our formula for the area.
A = r • r • 3.14.
Plug in 1 for r.
A = 1 • 1 • 3.14
A = 3.14.
Your area is 3.14 units^2.
I hope this helps, and has prepared you for your future problems in relation to this topic!
6.28=3.14*d
6.28/3.14=d
D=2
Area= πr*r
=3.14*1*1
=3.14
If 2m = 4x and 2w = 8x, what is m in terms of w?
Answers
m*ln2=n*ln2 divide both sides by ln2
m=n
One weekend, a newsstand sold twice as many Sunday papers as Friday papers. The Sunday paper costs $1.50, and the Friday paper costs $0.75. How many Sunday papers were sold if the newsstand took in $116.25?
Answers
2f = s
1.50(2f) + 0.75f = 116.25
3f + 0.75f = 116.25
3.75f = 116.25
f = 116.25/3.75
f = 31...31 friday papers
2f = s
2(31) = s
62 = s <=== 62 sunday papers
Answer:
62
Step-by-step explanation:
In a certain company, 74 employees were hired in the last 5 years. If
this represents 43% of the current number of employees, then how
many current employees does the company have?
- list the steps please & thank you!
Answers
74 = .43x. And dividing both sides by .43 you get that the current number of employees is 172.09, or just 172
To find the total number of employees in the company, set up and solve the equation, 0.43 * X = 74. Solving this provides approximately 172 employees
Explanation:In order to answer this question, we need to use the concept of percentages. Here are the steps:
Since we know 74 employees represent 43% of all employees, we set up the equation: 0.43 * X = 74, where X is the total number of employees in the company. Solving the equation for X leads us to the answer. We divide both sides of the equation by 0.43, which gives us X = 74/0.43. Calculating the right side of the equation, we find X is approximately 172, so there are approximately 172 employees in the company. Learn more about Percentages here:https://brainly.com/question/29306119
#SPJ12
Sam is observing the velocity of a car at different times. After two hours, the velocity of the car is 50 km/h. After six hours, the velocity of the car is 54 km/h.
Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the car at different times. Show your work and define the variables used. (5 points)
Part B: How can you graph the equation obtained in Part A for the first seven hours? (5 points)
Answers
[tex]\bf \begin{array}{ccllll} hours(x)&velocity(y)\\ -----&-----\\ 2&50\\ 6&54 \end{array}\\\\ -------------------------------\\\\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 2}}\quad ,&{{ 50}})\quad % (c,d) &({{ 6}}\quad ,&{{ 54}}) \end{array}[/tex]
[tex]\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{54-50}{6-2}\implies \cfrac{4}{4}\implies 1 \\\\\\ % point-slope intercept y-{{ y_1}}={{ m}}(x-{{ x_1}})\implies y-50=1(x-2)\\ \left. \qquad \right. \uparrow\\ \textit{point-slope form} \\\\\\ y-50=x-2\implies \boxed{y-x=48}\impliedby \begin{array}{llll} standard\\ form \end{array}[/tex]
part B)
well, to graph a LINEar equation, since it's just a LINE, you simply need two points to graph a line, and since we know that y - x = 48.... if you want to know what's "y" when x = 7? or 7 hours, well
y - (7) = 48
y = 48 + 7
y = 55
so.. that's the point for the 7th hour, 7, 55, and you can pick any other point above, and graph it away.
What is the future value of $845 a year for seven years at an interest rate of 11.3 percent? $6,683.95 $6,075.69 $8,343.51 $8,001.38 $8,801.91?
Answers
A=p(1+r/100)^n
where:
A=future amount
r=rate=11.3%=0.113
time=7 years
thus the future value of our cash will be:
A=845(1+0.113)^7
A=845(1.113)^7
A=$1,787.82
what is the area of a circle with a radius of 7 inches
Answers
A=3.1416(radius)2 (radius is 7)
A=3.1416(49)
A=153.94
The area is 153.94 inches.
Hope that was helpful :)
Answer is provided in the image attached.
simplify the expression by using a double angle formula
cos^2 4theta-sin^2 4theta
Answers
cos(2x)=cos^2(x)-sin^2(x)
Substitute x=4theta, then
cos(8theta)=cos^2(4theta)-sin^2(4theta)
=>
cos^2(4theta)-sin^2(4theta) = cos(8theta)
Given a positive integer n, assign true to is_prime if n has no factors other than 1 and itself. (remember, m is a factor of n if m divides n evenly.)
Answers
For the hard code for this problem:
if n == 2:
is_prime = True
elif n % 2 == 0:
is_prime = False
else:
is_prime = True
for m in range (3, int (n * 0.5) + 1, 2):
if n % m == 0:
is_prime = False
break.
To add, a high-level programming language that is widely used for general-purpose programming, created by Guido van Rossum and first released in 1991 is called Python.
Below is the solution of assign true to is_prime if n has no factors other than 1 and itself.
Further explanationPython is an interpreted, high-level, general-purpose programming language. Python is created by Guido van Rossum and first released in 1991. Its design philosophy emphasizes code readability. Python is a more productive language than Java because it is an interpretive language which is accompanied by elegant syntax and it makes an excellent choice for scripting and rapid application development in many areas. Python is a general-purpose, versatile and popular because it is concise and easy to read, and it is also a good language to have in any programmer’s stack as it can be used for everything such as web development, software development, data science applications.
Given a positive integer n, assign true to is_prime if n has no factors other than 1 and itself. And remember, m is a factor of n if m divides n evenly. So:
if n == 2:
is_prime = True
elif n % 2 == 0:
is_prime = False
else:
is_prime = True
for m in range (3, int(n**0.5)+1, 2):
if n % m == 0:
is_prime = False
Learn moreLearn more about python https://brainly.com/question/4331067Learn more about python IDLE or similar environment https://brainly.com/question/6280029Learn more about python function https://brainly.com/question/9806744Answer details
Grade: 9
Subject: mathematics
Chapter: python
Keywords: python, integer, factors, python IDLE, is_prime
A pizza parlor sold 38 1/8 pizzas during a dinner hour. If each pizza contained 8 slices, how many slices of pizza were sold?
Answers
38*8 = 304
304 + 1 =305 slices were sold
8(38)+8(1/8)
304+1
305 slices.
help find the answer to this question
Answers
18*9/2=81 cm^2
Then subtract the area of the smaller triangle within the larger one (white triangle). It has the same base but a height of only 4.
18*4/2=36 cm^2
81-36=45
Final answer: 45 cm^2
The social studies teacher is giving a test worth 80 points, and containing 34 questions. There are 2 point and 4 point questions on the test. How many of each type of question are on the test? Show your work. A. List the two equations B. Find each type of question
Answers
y is 4 pt questions
x+y=34
2x+4y=80
Multiply the top equation by -2 to use elimination technique
-2x-2y=-68
Then combine the two equations (x is cancelled out)
2y=12
y=6
If there are 6 four point questions, there must be 28 two point questions to get a total of 34 questions.
Suppose that an individual has a body fat percentage of 17.9% and weighs 132 pounds. How many pounds of her weight is made up of fat? Round your answer to the nearest tenth.
Answers
f=2362.8/100
f=23.628
f≈23.6 pounds (to nearest tenth of a pound)
solve for y 6y-8=3y-2
Answers
6y-8 =3y-2
subtract 3y from both sides
3y-8 = -2
add 8 to both sides
3y=6
y=6/3 =2
y = 2
Breanna works in the mall and her hours increase during busy shopping times of the year. Breanna has a plan to save $25 each week for 5 weeks, and then increase that amount by $20 for the next 7 weeks during the busy time, then go back to $25 per week. If Breanna’s plan is to save $540, how many weeks total will she need to save?
a.
11 weeks
b.
16 weeks
c.
22 weeks
d.
23 weeks
Answers
25*5=125 25+20=45 45*7=315 125+315=440 25*4=100 440+100=540 5+7+4=16
B is your answer.
Answer:
Number of weeks Breanna saves are 16 weeks .
Option (b) is correct .
Step-by-step explanation:
As given
Breanna works in the mall and her hours increase during busy shopping times of the year.
Breanna has a plan to save $25 each week for 5 weeks.
Thus
Breanna saves in first 5 weeks = Amount saves per weeks in next 5 weeks × Number of weeks .
= $25 × 5
= $ 125
Increase that amount by $20 for the next 7 weeks during the busy time.
Thus
Amount saves per weeks for next 7 weeks = Amount saves per weeks in first 5 weeks + Increase in amount .
= $25 +$ 20
= $ 45
Breanna saves in next 7 weeks = Amount saves per week× Number of weeks .
Put all the values in the above
= $45 × 7
= $ 315
As given
Breanna saves $25 per week.
If Breanna’s plan is to save $540 .
Let us assume last weeks in which Breanna saves $25 per weeks be x .
Breanna saves in last x weeks = Amount saves for next x weeks × Number of weeks .
= 25 × x
= 25x
Equation becomes
Breanna saves in first 5 weeks + Breanna saves in next 7 weeks + Breanna saves in last x weeks = Total saving amount .
125 + 315 + 25x = 540
440 + 25x = 540
25x = 540 - 440
25x = 100
[tex]x = \frac{100}{25}[/tex]
x = 4 weeks
Thus
Total number of weeks Breanna saves = 5 weeks + 7 weeks + 4 weeks
= 16 weeks
Therefore the number of weeks Breanna saves are 16 weeks .
Option (b) is correct .
John has taken out a loan for college. He started paying off the loan with a first payment of $100. Each month he pays, he wants to pay back 1.1 times the amount he paid the month before. Explain to John how to represent his first 20 payments in sequence notation. Then explain how to find the sum of his first 20 payments, using complete sentences.
Answers
Say a certain manufacturing industry has 63.1 thousand jobs in 2008, but is expected to decline at an average annual rate of 1.7 thousand jobs per year from 2008 to 2018. Assuming this holds true, what will be this industry’s percent change from 2008 to 2018? a. 70% b. -27% c. -17% d. -75%
Answers
Using linear function concepts, it is found that the industry’s percent change from 2008 to 2018 will be of:
b. -27%
What is a linear function?A linear function is modeled by:
[tex]y = mx + b[/tex]
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0.In this problem:
The industry had 63.1 thousand jobs in 2008, hence b = 63.1.This number is expected to decline at an average annual rate of 1.7 thousand jobs per year from 2008 to 2018, hence m = -1.7.Thus, the function is:
y = -1.7x + 63.1.
In 2018, which is 10 years after 2008, the value of the function is:
y(10) = -1.7(10) + 63.1 = 46.1.
The percent change is given by the change divided by the initial value, hence:
(46.1 - 63.1)/63.1 = -27%
Hence option b is correct.
More can be learned about linear function concepts at https://brainly.com/question/24808124
The correct answer is b. -27%.
To solve this problem, we need to find the number of jobs in 2018 and then calculate the percent change from 2008 to 2018.
Given information:
- Number of jobs in 2008: 63.1 thousand
- Average annual rate of decline: 1.7 thousand jobs per year
- Time period: 2008 to 2018 (10 years)
Step 1: Find the total decline in jobs from 2008 to 2018.
Total decline in jobs = Average annual rate of decline × Number of years
Total decline in jobs = 1.7 thousand × 10 years = 17 thousand jobs
Step 2: Find the number of jobs in 2018.
Number of jobs in 2018 = Number of jobs in 2008 - Total decline in jobs
Number of jobs in 2018 = 63.1 thousand - 17 thousand = 46.1 thousand jobs
Step 3: Calculate the percent change from 2008 to 2018.
Percent change = (Change in value / Original value) × 100%
Percent change = ((46.1 - 63.1) / 63.1) × 100%
Percent change = (-17 / 63.1) × 100%
Percent change ≈ -27%
Therefore, the industry's percent change from 2008 to 2018 is approximately -27%.
Number of jobs in 2018 = [tex]46.1\text{ thousand} \\[/tex]
Percent change from 2008 to 2018 = [tex]-27\%[/tex]
The radius of the circle whose equation is (x + 4)² + (y - 2)² = 36 is 36 18 6
Answers
(x-h)^2+(y-k)^2+r^2, where (h,k) is the center of the circle and r is the radius
r^2=36
r=√36
r=6
D=6.3cm what is the circumference and area of the circle
Answers
=2×pi×1.6
=10.05
Area= pir^d
=pi × 6.3
=19.79
Question help a robotic rover on mars finds a spherical rock with a diameter of 1313 centimeters. the rover picks up the rock and lifts it 1515 centimeters straight up. the rock has a specific gravity of 5.255.25. the gravitational acceleration on mars is 3.73.7 meters per second squared. if the robot's lifting arm has an efficiency of 3535% and required 1212 seconds to raise the rock 1515 centimeters, how much power (in watts) did the arm use?
Answers
diameter=13 cm/2=0.065m
h=15 cm=0.15m
rho=5.25*1000=5250 kg/m^3
eta=35%=0.35 [efficiency]
t=12 s
g=3.7 m/s^2
mass, m=rho*V=rho*(4pi/3)(r^3)
=5250 kg/m^3 * (4pi/3)(0.065)^3 m^3
=1.922 kg
Work done in lifting h=0.15m
W=mgh
=1.922pi kg * 3.7 m/s^2 * 0.15 m
=1.067pi kg (m/s)^2
=1.067pi J
Average power required, with efficiency eta=0.35
P=(W/t)/eta
=(1.067pi J )/ (12 s) /0.35
=0.254pi J/s
=0.798W (approx.)
Find the value of x
A 41
B 44
C 46
D 36
Answers
Cosine is the adjacent/hypotenuse. In this case, the adjacent is 15 and hypotenuse is 21.
15/21, or like 5/7.
Now put that into a fancy calculator with the [tex]cos^{-1} [/tex] thing or something like that and then you have the answer, or around 44 degrees.