Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x2, x = y2; about y = 1
Answers
D? y
= VE
Now we find the points of intersection first: 27?
-
VE
2 = 0, 1 so when x is O y is also 0
and when x is 1, y is also 1
Hence the points of intersection are (0,0) and (1,1)
Given it is rotated about y
=1
So using disk method, the volume is given by:
The volume of the solid obtained by rotating the region bounded by the given curves about [tex]y = 1[/tex] is [tex]\frac{11\pi}{30} [/tex] cubic units.
Procedure - Determination of the volume of a solid of revolutionLet [tex]f(x) = 1[/tex], [tex]g(x) = x^{1/2}[/tex] and [tex]h(x) = x^{2}[/tex], both [tex]g(x) [/tex] and [tex]h(x)[/tex] have the following intersection points: [tex](x_{1}, y_{1}) = (0, 0)[/tex] and [tex](x_{2}, y_{2}) = (1,1)[/tex]. This means that the interval of x-coordinates for the solid of revolution is [tex]x \in [0, 1][/tex].
The integral formula for the volume of the solid of revolution ([tex]V[/tex]), in cubic units, is:
[tex]V = \pi \int\limits^{1}_{0} {[h(x)-f(x)]^{2}} \, dx - \pi \int\limits^{1}_{0} {[g(x)-f(x)]^{2}} \, dx [/tex] (1)
After substituting on each function and simplifying terms we get the following integral:
[tex]V = \pi \int\limits^{1}_{0} {(x^{2}-1)^{2}} \, dx - \pi \int\limits^{1}_{0} {(x^{1/2}-1)^{2}} \, dx [/tex]
[tex]V = \pi\left[\int\limits^{1}_{0} {x^{4}} \, dx - 2\int\limits^{1}_{0} {x^{2}} \, dx -\int\limits^{1}_{0} {x} \, dx + 2\int\limits^{1}_{0} {x^{1/2}} \, dx \right][/tex]
And the volume of the solid of revolution is:
[tex]V = \pi \cdot \left(\frac{1}{5}-\frac{2}{3} -\frac{1}{2}+\frac{4}{3} \right)[/tex]
[tex]V = \frac{11\pi}{30} [/tex]
The volume of the solid obtained by rotating the region bounded by the given curves about [tex]y = 1[/tex] is [tex]\frac{11\pi}{30} [/tex] cubic units. [tex]\blacksquare[/tex]
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Related Questions
Peter has 18 oranges, 27 pears and 12 bananas. He wants to make fruit baskets with
the same number of each fruit in each basket. What is the greatest number of fruit
baskets he can make? How many of each type of fruit will be in the baskets?
Answers
Answer:
3 baskets and 6 oranges, 9 pears, 4 bananas
Step-by-step explanation:
the common denominator is 3
Below is an excerpt from the home loan contract Kyle signed with his bank.
“The sum of the principal of the loan and all applicable interest will be divided into 360 monthly payments according to the amortization schedule.
The Buyer shall make each monthly payment by the first day of the month it is due. A payment not received by the first day of the month it is due will be considered delinquent. A payment not received by the fifteenth day of the month it is due will be subject to a late charge of $75.00. If a payment is not received by the last day of the month it is due, the Buyer and transaction shall be reported to the appropriate credit agencies.
In the event of delinquency, the Buyer may divide any payment not received into not more than three equal installments to be included on the next consecutive monthly payments. Notification of the Buyer’s intent to do so must be made by written letter or phone call to the phone number and address listed below within one business week of delinquency.”
Due to some changes at his job and the resulting loss of income, Kyle will not be able to make his monthly payment by the first of the month. He does expect to be able to make it by the 9th of the month. According to the contract, what does Kyle need to do?
a.
Other than pay the monthly payment, Kyle does not need to do anything.
b.
Notify the bank to let them know that his payment is going to be eight days late
c.
Divide this payment into three payments to be included in future monthly payments.
d.
Include an extra $75.00 with his payment to cover the late fee that will be assessed.
Answers
9514 1404 393
Answer:
a. Other than pay the monthly payment, Kyle does not need to do anything.
Step-by-step explanation:
If Kyle plans to make a payment by the 15th of the month, he need do nothing, except make the payment.
If he plans to divide the payment into 3 parts, he needs to contact the bank within a week of the first of the month. If he plans to make a payment after the 15th, he needs to add $75 to it for the late charge.
Answer:
i would think it would be [A Other than pay the monthly payment, Kyle does not need to do anything.]
Step-by-step explanation:
The population in millions of a city t years after 1990 is given by the equation p(t) = 2.9+0.08t in this function
Answers
Answer:
The answer would be 2.9 million is the population of the city in 1990 and 0.08 million is the increase per year in the population.
Hope it help and if it its correct please give brainliest
Stay Safe And Healthy
Thank You
what is the diameter of a circle that has the circumference of 400
Answers
Answer:
400/π
Step-by-step explanation:
Well, we know that the formula for circumference is πd. (d is the diameter) Then, we set up the equation 'πd = 400'. Now, simplifying for d, we get d = 200/π.
Answer:
400/π
Step-by-step explanation:
Simply the expression
-2.4f + 0.4f -15-3
Answers
Answer:-2.4f + 0.4f -15-3Step-by-step explanation:
Answer:
-1.8f - 19
Step-by-step explanation:
Jamal had $25.00. He spent $9.00. Which decimal represents the fraction of the $25.00 that
Jamal spent?
A. 0.64
B. 0.36
C. 04
D. 06
Answers
Answer:
B. 0.36
Step-by-step explanation:
9/25 = x/100
9x100= 900
900 divided by 25 will be 36/100, so 0.36
passes through (2,-3) with a slope of -9/2
Answers
Answer:
y = — 9/2x + 6
Step-by-step explanation:
m= —9/2
y — y1 = m (x — x1)
y + 3 = —9/2 (x — 2)
y = — 9/2x + 6
If
x
=
4
and
y
=
7
, evaluate the following expression:
2
(
4
x
+
3
y
)
Answers
Answer:
expression.
2(4x+3y)
2(4(4)+3(7))
2(16+21)
2( 37)
74
Help!!! Show work!!!!!
Answers
Answer:
1. (-2,-5) (-1,-1) (0,3) (1,7) (2,11)
2. (-8,-4) (-4,-3) (0,-2) (4,-1) (8,0)
3. (-4,3) (-2,2) (0,1) (2,0) (4,-1)
4. (-2,-1) (-1,2) (0,5) (1,8) (2,11)
Step-by-step explanation:
1. Slope = (11-7)/(2-1) = 4/1 = 4/1
y-intercept: 7 - (4)(1) = 3
Start in the vertical line (y) on 3, 4 up and 1 to the right (you need to points to trace a line)
2. Slope = (0- - 2)/(8-0) = 1/4
y - intercept = -2-(1/4)(0) = -2
Start on -2 vertical line, 1 up and 4 to the right.
3. Slope = (1-0)/(0-2 = -1/2
y-Intercept: 0 - (-1/2)(2)=1
Start on 1 vertical line then 1 down and two to the right (the slope is negative, the line goes downhill)
4. Slope: (8-1)/(1-0) = 7/1
y-intercept = 1 - (7)(0) = 1
Start on 1 vertical line (y) , seven up and 1 to the right.
Please help! Steps if possible: If n=31, ¯x (x-bar)=41, and s=12, construct a confidence interval at a 90% confidence level. Assume the data came from a normally distributed population.
Give your answers to one decimal place.
decimal place.
________ < μ < _______
Answers
Answer: [tex]\boldsymbol{37.5} < \mu < \boldsymbol{44.5}[/tex]
===========================================================
Explanation:
At a 90% confidence level, the z critical value is roughly z = 1.645. Use a reference table or a calculator to determine this. Despite not knowing what sigma is, we can use a z interval here because n > 30. If the sample size n was smaller than 30, then we'd have to use a T distribution instead.
We'll plug that z value, along with the other given values, into the formula for the lower boundary L of the confidence interval.
[tex]L = \text{lower boundary}\\\\L = \overline{x} - z*\frac{s}{\sqrt{n}}\\\\L = 41 - 1.645*\frac{12}{\sqrt{31}}\\\\L = 41 - 3.545\\\\L = 37.455\\\\L = 37.5\\\\[/tex]
This value is approximate.
Do the same for the upper boundary as well
[tex]U = \text{upper boundary}\\\\U = \overline{x} + z*\frac{s}{\sqrt{n}}\\\\U = 41 + 1.645*\frac{12}{\sqrt{31}}\\\\U = 41 + 3.545\\\\U = 44.545\\\\U = 44.5\\\\[/tex]
This value is also approximate.
We have 90% confidence that the population mean [tex]\mu[/tex] (greek letter mu) is somewhere between L = 37.5 and U = 44.5
Therefore, we would write the 90% confidence interval as [tex]\boldsymbol{37.5} < \mu < \boldsymbol{44.5}[/tex]
This is the same as writing the confidence interval in the form [tex](\boldsymbol{37.5} , \boldsymbol{44.5})[/tex]. Both forms are two different ways to say the same thing. The first form being [tex]L < \mu < U[/tex] while the second form is [tex](L,U)[/tex].
Side note: The value 3.545 calculated earlier is the approximate margin of error.
Determine the length of the model.
-60 ft-
1 in. = 8 ft
=
Answers
Length of the model = 11.6 inch
An electrician must use 15 miles of electrical cable between electric poles. How many yards of cable will be used?
13,200
26,400
39,600
79,200
158,400
Answers
Answer:
B) 26,400
An electrician must use 15 miles of electrical cable between electric poles. How many yards of cable will be used?
13,200
26,400
39,600
79,200
158,400
26400 yards of cable is used.
the answer is B) 26,400
What is unit conversion?
Unit conversion is the process of changing a physical quantity from one unit of measurement to another. For example, converting kilometers to miles, Celsius to Fahrenheit, or liters to gallons. The process involves multiplying or dividing the original measurement by a conversion factor, which is a ratio of equivalent units
To convert miles to yards, we need to multiply by the conversion factor of 1 mile = 1760 yards.
15 miles * 1760 yards/mile = 26,400 yards
So, the electrician will use 26,400 yards of cable.
Hence, 26400 yards of cable is used.
the answer is B) 26,400
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x^3-3x^2+x-2/10x^4-14x^3-10x^2+6x-10
Answers
Answer:
The quotient is 10x+16
The remainder is 28x^2+10x+22
Plz help!! Given: DE is a midsegment of ABC. Show all work.
Find the value of x and y.
And What is the length of AC? !Please show all work!
Incase photo is blurry
AE-y+6
EC-x-3
DE-x+2
BC-24
Answers
Answer:
AC=14
X=10
Y=1
Hoped this helped!
Step-by-step explanation:
DE is half of BC, therefore x=10. AE and EC have to be the same length, which is 7. SO, y=1 and x is 10.
A food store is making 60 jars of vegetables. Each jar will get 7.83 ounces of peas and carrots
combined. The food store has 384 ounces of peas for all the jars, and the carrots will be divided
equally among the jars. How many ounces of carrots will go into each jar?
Answers
Answer:
1.43 ounces
Step-by-step explanation:
384 / 60 = 6.4
7.83 - 6.4 = 1.43
write the fraction as a decimal use bar notatin if necessary
-2 5/22
Answers
Answer:
1.13
Step-by-step explanation:
to turn a fraction to a decimal divide the denomitator by the numerator [tex]\frac{-25}{22}[/tex] can also be written as -25÷22 which equals 1.13 or 1.136 with a line over it
Hope this helps!
For constant term: f(a,b,g(t)), pattern term: f(a,Y,X) when pattern matching we can use the following assignment Y=b and X=g(t) to make the two terms match.
Note: A variable in the pattern is allowed to match anything in the other structure.
What do the uppercase letters X,Y represent?
Answers
Answer:
fuctions and constants
Step-by-step explanation:
please mark as brainliest if it helped :D
Solve the equation for B. PLEASE SHOW YOUR WORK.
2(B + 3.5 - B = -3(6 - 2B)
Answers
9514 1404 393
Answer:
B = 5
Step-by-step explanation:
The given equation has mismatched parentheses. We assume the intent is ...
2(b +3.5) -b = -3(6 -2b)
Eliminating parentheses, we have ...
2b +7 -b = -18 +6b
25 = 5b . . . . . . . . . . . add 18-b to both sides and collect terms
b = 5 . . . . . . . . . . . . divide both sides by 5
a car travels at 56 km per hour find the time it takes to travel 300 metres give your answer in seconds correct to the nearest second
Answers
1km = 1000 m , thus 56km = 56000 m
1hr = 60 mins
1min = 60 secs, therefore 60 mins = 60*60 seconds = 3600 seconds = 1hr
so then, we know the car travels 56000 meters in 3600 secs, how many seconds will that be for 300 meters?
[tex]\begin{array}{ccll} meters&seconds\\ \cline{1-2} 56000&3600\\ 300&x \end{array}\implies \cfrac{56000}{300}=\cfrac{3600}{x}\implies \cfrac{560}{3}=\cfrac{3600}{x} \implies 560x=10800 \\\\\\ x = \cfrac{10800}{560}\implies x = \cfrac{135}{7}\implies x \approx 19.286\implies \stackrel{\textit{rounded up}}{x = 19}[/tex]
if 1 dollar equals 0.72p what is the exchange rate from pounds to dollars
Answers
Answer:
1 pound = 1.389 dollars ( to nearest thousandth)
Step-by-step explanation:
1 pound = 1/0.72 = 1.388888...
0.72p is equal to 72/100 pounds.
On the left is dollars and the right is the pounds
1 = 72/100
Multiply both sides by 100 as we want the right side to equal 1
100 = 72
Divide both sides by 72 to get 1 (pound) on the right
100/72 = 1
This means that the answer is:
1 pound is equal 100/72 or 1.38888888…. Dollars
find measurement 1 and 4
Answers
Answer:
the info they gave us is to suppose that angle 6 is = 126°
step 1:
We know that angle 8 = angle 6 = 126°
Why? — when a circle (360°) is split by 2 lines that intersect, there will be 2 equal pairs of angles formed
step 2:
angle 2 = angle 8 = 126°
Why? — they’re alternate angles
step 3:
angle 4 = angle 2
Why? — when a circle (360°) is split by 2 lines that intersect, there will be 2 equal pairs of angles formed
step 4:
angle 1 = 180-126 (angle 4) = 54°
Why? — total angles in a straight line = 180°
Key Term:
Alternate angles:
Alternate angles are angles that occur on opposite sides of the transversal line and have the same size. Alternate angles are equal: We can often spot interior alternate angles by drawing a Z shape: There are two different types of alternate angles, alternate interior angles and alternate exterior angles.
a rectangle is made by joining two squares adjacent to each other as shown below. if one squared has an area of 144 squared centimeters what is the area of the rectangle?
Answers
Answer:
Step-by-step explanation:
2(144) = 288 cm²
Complete the table for an object that goes 3/4 mile in 6 minutes .
Answers
Let be an angle in quadrant I such that sin +10 Find the exact values of sec 0 and tan 0.
Answers
Answer:
just look at picture's That's the answer
1 3/10 x 18 please show work and answer
Answers
Answer:
Step-by-step explanation:
1 3/10 x 18
(1 x 10 +3)/10 = 13/10
13/10 x 18 = (13 x 18) / 10 = 234/10 = (234/2) / (10/2) = 117/5 = 23 2/5 (a fraction is a division)
The answer to the expression showing the product of 1 3/10 x 18 is 23 2/5.
Given is an expression showing the product of 1 3/10 x 18, we need to solve it,
To multiply a mixed number by a whole number, you can convert the mixed number to an improper fraction and then multiply the fractions.
Here's how you can solve the problem:
Step 1: Convert the mixed number to an improper fraction:
1 3/10 = (1 × 10 + 3) / 10 = 13/10
Step 2: Multiply the fractions:
13/10 x 18/1 = (13 x 18) / (10 x 1) = 234 / 10
Step 3: Simplify the fraction, if possible:
234/10 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2:
234/10 = (234 ÷ 2) / (10 ÷ 2) = 117/5
Therefore, 1 3/10 x 18 = 117/5.
Alternatively, you can express the answer as a mixed number by dividing the numerator by the denominator:
117 ÷ 5 = 23 with a remainder of 2
So, the final answer is 23 2/5.
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g A pipe of negligible diameter is to be carried horizontally around a corner from a hallway 8 ft wide into a hallway 4 ft wide. What is the longest length that the pipe can be
Answers
The length of the longest pipe that can be carried horizontally is given by the minimum value of the function of the length of the pipe.
The longest length that the pipe can be is approximately 16.648 feet.
Reasons:
The length of the pipe, L
Width of the first hallway = 8 ft.
The width of the second hallway = 4 ft.
We have;
(L - a)·sin(θ) = 8
a·cos(θ) = 4
Therefore;
L - a = 8·csc(θ)
a = 4·sec(θ)
L(θ) = 8·csc(θ) + 4·sec(θ)
At the minimum length, we have;
L'(θ) = 4·sec(θ)·tan(θ) - 8·csc(θ)·cot(θ) = 0
4·sec(θ)·tan(θ) = 8·csc(θ)·cot(θ)
Therefore;
[tex]\displaystyle \frac{sec(\theta) \cdot tan(\theta)}{csc(\theta) \cdot cot(\theta)} = \frac{sin(\theta) \cdot tan(\theta) \cdot tan(\theta) }{cos(\theta) } =tan^3(\theta) = \frac{8}{4}[/tex]
[tex]\displaystyle tan(\theta) =\sqrt[3]{\frac{8}{4} } ={\sqrt[3]{2} }[/tex]
θ = arctan([tex]\sqrt[3]{2}[/tex]) ≈ 51.56°
a = 4×sec(arctan([tex]\sqrt[3]{2}[/tex]))
a ≈ 6.434
L = 8·csc(θ) + a = 8 × csc(arctan([tex]\sqrt[3]{2}[/tex])) + 4×sec(arctan([tex]\sqrt[3]{2}[/tex])) ≈ 16.648
The longest length that the pipe can be, L ≈ 16.648 ft.
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The longest length of the pipe is an illustration of trigonometry functions
The longest length of the pipe is 16.64 ft
The widths of the hallways are given as:
[tex]\mathbf{w_1 = 8}[/tex]
[tex]\mathbf{w_2 = 4}[/tex]
See attachment for the diagram that represents the scenario
From the diagram, we have:
[tex]\mathbf{sin(\theta) = \frac 4x}[/tex]
[tex]\mathbf{cos(\theta) = \frac 8y}[/tex]
Make x and y the subject
[tex]\mathbf{x = \frac 4{sin(\theta)}}[/tex]
[tex]\mathbf{y = \frac 8{cos(\theta)}}[/tex]
Where, the total length (L) of the pipe is:
[tex]\mathbf{L = x + y}[/tex]
So, we have:
[tex]\mathbf{L = \frac 4{sin(\theta)} + \frac 8{cos(\theta)}}[/tex]
Differentiate
[tex]\mathbf{L' = -\frac{4cos(\theta)}{sin^2(\theta)} + \frac{8sin(\theta)}{cos^2(\theta)}}[/tex]
Set to 0
[tex]\mathbf{-\frac{4cos(\theta)}{sin^2(\theta)} + \frac{8sin(\theta)}{cos^2(\theta)} = 0}[/tex]
Rewrite as:
[tex]\mathbf{\frac{4cos(\theta)}{sin^2(\theta)} = \frac{8sin(\theta)}{cos^2(\theta)}}[/tex]
Divide through by 4
[tex]\mathbf{\frac{cos(\theta)}{sin^2(\theta)} = \frac{2sin(\theta)}{cos^2(\theta)}}[/tex]
Cross multiply
[tex]\mathbf{2sin^3(\theta) = cos^3(\theta)}[/tex]
Divide both sides by [tex]\mathbf{2cos^3(\theta)}[/tex]
[tex]\mathbf{tan^3(\theta) = \frac 12}[/tex]
Take cube roots of both sides
[tex]\mathbf{tan(\theta) = \sqrt[3]{\frac 12}}[/tex]
[tex]\mathbf{tan(\theta) = 0.7937}[/tex]
Take arc tan of both sides
[tex]\mathbf{\theta = tan^{-1}(0.7937)}[/tex]
[tex]\mathbf{\theta = 38.4}[/tex]
Substitute [tex]\mathbf{\theta = 38.4}[/tex] in [tex]\mathbf{L = \frac 4{sin(\theta)} + \frac 8{cos(\theta)}}[/tex]
[tex]\mathbf{L = \frac 4{sin(38.4)} + \frac 8{cos(38.4)}}[/tex]
Evaluate
[tex]\mathbf{L = 6.43+ 10.21}}[/tex]
[tex]\mathbf{L = 16.64}}[/tex]
Hence, the longest length of the pipe is 16.64 ft
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Celia's scores in the 5 quizzes Math are 18,12,15,19. What is her average score in Math?
d. 18
a. 15
c. 17
b. 16
Answers
Step-by-step explanation:
(18+12+15+19)÷5
64÷5
15
Explanation
(18+12+15+19)/4
64/4
16
Sara has only nickels in her hand, and David has exactly the same
number of dimes and no other coins. Together, they have a total of $1.05. How many nickels does Sara have? How many dimes does David have? Show your work
using numbers, labeled sketches, or words.
Answers
David: Dimes: 7 dimes
What is the property of this math problem please help
Answers
Answer:
distributive property
Step-by-step explanation:
Because distributive property formula is a(b+c) = ab + ac
meaning that w(6+5) = 6w + 5w
Change the following numbers to decimals and round to the nearest hundredth
5/9
Answers
Answer:
Answer: 5/9 as a decimal is 0.556.
5/9 ≈ 0.56 rounded to the nearest hundredth, good luck