Using polar/cylindrical coordinates, we take
x = r cos(t )
y = r sin(t )
z = z
so that
dV = dx dy dz = r dr dt dz
The solid is given by the set
D = {(x, y, z) : x ² + y ² ≤ 1 and 0 ≤ z ≤ 4 - 2x - y}
or in cylindrical coordinates,
D = {(r, t, z) : 0 ≤ r ≤ 1, 0 ≤ t ≤ 2π, and 0 ≤ z ≤ 4 - 2r cos(t ) - r sin(t )}
Then the volume of the solid is
[tex]\displaystyle \iiint_D \mathrm dV = \iiint_Dr\,\mathrm dr\,\mathrm dt\,\mathrm dz \\\\ = \int_0^{2\pi} \int_0^1 \int_0^{4-2r\cos(t)-r\sin(t)} r\,\mathrm dz\,\mathrm dr\,\mathrm d t \\\\ = \int_0^{2\pi}\int_0^1 (4r-2r^2\cos(t)-r^2\sin(t))\,\mathrm dr\,\mathrm dt \\\\ = \int_0^{2\pi} \left(2 - \frac23 \cos(t) - \frac13 \sin(t)\right) \,\mathrm dt \\\\ = \boxed{4\pi}[/tex]
The volume of the given solid below the plane 2x + y + z = 4 and above the disk x2 + y2 ≤ 1 is 4π.
What is the volume of the solid in polar coordinates?The volume of the solid in polar coordinates can be calculated by the double integral in polar coordinates.
[tex]V = \iint_R f(x,y) \,dA.\\[/tex]
Similarly, by polar coordinates,
x = r cost
y = r sint
z = z
we get, dV = dx dy dz = r dr dt dz
The solid is given
D = {(x, y, z) : x ² + y ² ≤ 1 and 0 ≤ z ≤ 4 - 2x - y}
or in polar coordinates,
D = {(r, t, z) : 0 ≤ r ≤ 1, 0 ≤ t ≤ 2π, and 0 ≤ z ≤ 4 - 2r cos(t ) - r sin(t )}
The volume of the solid is
[tex]\rm \int \int\int\ dV = \int \int\int\ rdr dt dz\\ \rm = \int\limits^{2\pi}_0\int\limits^1_0 \int\limits^{4-2rcost-rsint}_0 rdr dt dz\\\rm =\int\limits^{2\pi}_0\int\limits^1_0 {4r-2r^{2} cost-r^{2} sint} dr dt\\\rm =\int\limits^{2\pi}_0( 2-2/3cost-1/3sint) dt\\\rm =4\pi[/tex]
Hence, The volume of the given solid below the plane 2x + y + z = 4 and above the disk x2 + y2 ≤ 1 is 4π.
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An isosceles triangle has two sides of equal lenght. The length of the third side is 1 ft less than twice the length of one of the equal sides find the length of each side when the perimeter is 23 ft, answer as a comma separated list
Answer: An isosceles triangle has two sides of equal length. The length of one of the equal sides is two more than 3 times the length of the third side. If the perimeter is 46m, find the length of each side.
Step-by-step explanation:
Hope this helps!
Answer:
Step-by-step explanation:
The perimeter is equal to the lengths of all the sides.
Let x equal the two sides that are equal to each other.
Then the 3rd side is 2x - 1 which is 2 times the length of the equal side minus 1.
So the perimeter 23 = x + x + (2x-1)
Now solve for x
23 = x + x + 2x - 1
add 1 to both sides
24 = x + x + 2x
24 = 4x
x = 6
The two equal sides would be 6 feet each
The other side would be 2(6) - 1 = 11
The perimeter is 6 + 6 + 11 = 23
f(x) = x² + 2ax + a² - a
g(x) = 4 - x²
Let variable a be a real number. Find the range of values of a such that the equation f(x) = g(x) has two different solutions.
***The answer should be -2 < a < 4, but I don't know why.
Step-by-step explanation:
f(x) = x² + 2ax + a² - a
g(x) = 4 - x²
f(x) = g(x)
x² + 2ax + a² - a = 4 - x²
2x²+2ax+a²-a -4 =0
the equation has two different solutions
=> (2a)² - 4(2)(a²-a-4) > 0
4a²-8a²+8a +32 >0
-4a²+8a+32 > 0
a²-2a-8 >0
(a-4)(a+2) < 0
the answer should be
-2 < a < 4
can you help me pwease i give brainliest
Answer:
I believe your answer should be B.) I hope this helps! :D
13.On many multiple-choice tests, 1 point is given for each correct
answer and 0.25 point is taken away for every wrong answer.
(This is to discourage guessing.) Answers that are left blank
do not affect the score. Gloria scored 62 on a test with 100
questions. Let C be the number of questions Gloria correctly
answered and let W be the number of wrong answers she had.
Give three possible pairs of values of C and W.
b. Write an equation that describes all possible solutions.
c. Graph all possible solutions.
a.
Simplify (5 1/3) exponential 3
[tex]\rm\purple{\overbrace{\underbrace{\tt\color{black}{\:\:\:\:\:\:\:\:Answer:\:\:\:\:\:\:\:\:\:}}}}[/tex]
125/3
[tex]\large\red{ChaEunWoo2009}[/tex]
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A copy machine makes 32 copies per minute. How many copies does it make in 4 minutes and 15 seconds?
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Answer:
136
Step-by-step explanation:
15 seconds is 1/4 minute. We can find the number of copies by multiplying the rate by time:
(32 copies/min)(4.25 min) = 136 copies
The machine can make 136 copies in 4 minutes 15 seconds.
6g = -86; g= -16
Need help
Answer:
6g = -86
Divide each term in 6 g = − 86 by 6 and simplify.
Exact Form: g = -43/3
Decimal Form: g = −14.¯3
Mixed Number Form:
g = −14 1/3
—————————————————————
g= -16
Vertical line at x = −16
Write 41 and 4 as a mixed number
Answer:
10 and 1 fourth
Step-by-step explanation:
41 over 4 = 41/4 so you divide 41 by 4 which is 10 R1 so its 10 wholes, 1 and 4
You and your friends Jose and Sarah all live on the same street. You know that Jose lives 7 blocks away from you. Jose says that Sarah lives 6 blocks from him, but he didn't say whether she lives closer to you than he does or further. Let represent the number of blocks from you that Sarah might live.
Answer:
farthur
Step-by-step explanation:
What is the average rate of change of g(x)=4x+5 ?
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Answer:
4
Step-by-step explanation:
The equation is in slope-intercept form:
y = mx +b
where m is the slope (average rate of change).
In the given equation, m = 4.
The average rate of change is 4.
What is the rate of change of the function y=x-2
Answer:
The rate of change is 1.
The equation of a line and y-intercept form is
y = mx + b, where m = the slope of the line.
and our given equation y = x - 2 the slope of the line is 1 in other words y = 1x - 2
a. is the value of the expression (1/4 ^-3)^-2 greater than 1 equal to 1 or less than 1
b. if the value of the expression is greater than 1 show how you can change one sign to make the value less than 1, show how you can change one sign to make the value greater than 1 if the value is equal to 1 show how you can make one change to make the value not equal to 1
please I need an answer quick ._.
A.
[tex]((1/4)^{-3})^{-2} = (1/4)^{(-3)*(-2)}=(1/4)^6<1[/tex]So it is less than 1
B.
We can change -3 to 3 or -2 to 2.
This will result in:
[tex](1/4)^{-6}=4^6 >1[/tex](a) The value of given expression "((1/4)⁻³)⁻²" is less than 1, and (b) To make value greater than 1, change second exponent to positive; to make it equal to 1, change first exponent to 0.
Part (a) : Starting from the innermost exponent, (1/4)⁻³ simplifies to 4³, which is 64.
So, ((1/4)⁻³)⁻² becomes (64)⁻², which is 1/4096. Since 1/4096 is less than 1, the value of the expression is less than 1.
Part (b) : To change the value to greater than 1:
Change the second exponent to a positive value, making it ((1/4)⁻³)²,
This simplifies to (4³)², which is 4096.
To change the value to equal 1:
Change the first exponent to 0, making it ((1/4)⁰)⁻². This simplifies to (1)⁻², which is 1.
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WATTPAD SUGGESTIONS? :)
Answer:
Step-by-step explanation: about a boy who fell for a girl that was twice his age and yeah something about that :)
Find the degree of the monomial -1.57k^2
Answer:
2
Step-by-step explanation:
Degree of the monomial is the highest power in the monomial.
Highest power in the monomial - 1.57k^2 is 2.
So, the degree of the monomial - 1.57k^2 is 2.
explain how to calculate a unit rate such as 300 miles per hour 4 hour
my brain is not working today i need help
Answer:
75
Step-by-step explanation:
divide 4 by 300 and then you will get the answer of 75. I'm so sorry if this is wrong
Three
consecutive
inirgers
17
The length of a flat screen TV is 5 inches less than 3 times its width. If the perimeter of the TV is
150 inches, find the dimensions.
Find the slope of (3,-2) and (4,4)
Answer:
0
Step-by-step explanation:
if ryan has $1127 left after spending 1/5 of his monthly salary for rent and 1/7 of his monthly salary for his utility bill, what was his monthly salary
Answer:
$1715
Step-by-step explanation:
1/5 + 1/ 7 = 12/ 35
So, he is left with 23/35 of his salary
if 23/35 of his salary is 1127
his whole salary is 1715
ingredients: f(6)-20 = (?)cups of sugar. use f(x) = x² - 2x - 3
Answer:-18 mm sugar
Step-by-step explanation:
Thanks me later
simplify each expression and state the number of terms in the final answer.
1) 4m+3j, 2 terms.
2) -11x+6y+9, 2 terms
3) [tex]11g^5+6g^4-4gh\\[/tex], 2 terms
Ross sold b boxes of cookies, and gabby sold 35 boxes of cookies. Together they sold a total of 69 boxes of cookies
Question 3 of 10 Classify the following triangle. Check all that apply. 70° 15.8 8 80" 30 15 O A. Equilateral B. Obtuse C. Isosceles D. Right E. Acute F. Scalene
Answer:
the answer F or D
Step-by-step explanation:
how to solve this derivative.
f(x)={ln(3x)}^2x
with steps...
Rewrite f(x) as a nested exponential-logarithm expression :
[tex]\left(\ln(3x)\right)^{2x} = \exp\left(\ln\left(\ln(3x)\right)^{2x}\right)[/tex]
(where [tex]\exp(x) = e^x[/tex])
One of the properties of logarithms lets us drop the exponent as a coefficient:
[tex]\exp\left(\ln\left(\ln(3x)\right)^{2x}\right) = \exp\left(2x\ln\left(\ln(3x)\right)\right)[/tex]
Now, by the chain rule, we have
[tex]f(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \implies \\\\ f'(x) = \left(\exp\left(2x\ln\left(\ln(3x)\right)\right)\right)' \\\\ f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left(2x\ln\left(\ln(3x)\right)\right)'[/tex]
By the product rule,
[tex]f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left( (2x)' \ln\left(\ln(3x)\right) + 2x\left(\ln\left(\ln(3x)\right)\right)' \right) \\\\ f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left( 2\ln\left(\ln(3x)\right) + 2x\left(\ln\left(\ln(3x)\right)\right)' \right)[/tex]
By the chain rule again,
[tex]f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left(2\ln\left(\ln(3x)\right) + 2x \cdot \dfrac1{\ln(3x)} \cdot \left(\ln(3x)\right)' \right) \\\\ f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left(2\ln\left(\ln(3x)\right) + \dfrac{2x}{\ln(3x)} \cdot \dfrac1{3x} \cdot (3x)' \right) \\\\ f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left(2\ln\left(\ln(3x)\right) + \dfrac{2}{3\ln(3x)} \cdot 3 \right) \\\\ f'(x) = \exp\left(2x\ln\left(\ln(3x)\right)\right) \cdot \left(2\ln\left(\ln(3x)\right) + \dfrac{2}{\ln(3x)} \right)[/tex]
Then simplify this to
[tex]f'(x) = \boxed{\left(\ln(3x)\right)^{2x} \left(2\ln\left(\ln(3x)\right) + \dfrac{2}{\ln(3x)} \right)}[/tex]
When x is a nonzero integer, the sum of −10 + x is sometimes negative and sometimes positive. negaive or positive, explain your answer.
Answer: if the x is negative the answer will be positive but if -10 is the only negative it would be negative. The way I remember is if their is a even amount of negatives is will be positive if theirs a odd amount it would be negative . Hope this helps :)
Step-by-step explanation:
If a 4-pound roast takes 150 minutes to cook, how long should a five-pound roast take
Answer:
A 5-pound roast would take about 187.5 minutes to cook.
Divide 150 minutes by 4 pounds and you get 37.5 minutes. This would be how long it takes to cook each pound. So, you multiply 5 pounds by 37.5 and you get 187.5 minutes for a 5-pound roast to cook.
451 mi/ 7.3 hr. Round to the appropriate number of significant digits.
Answer:
5
Step-by-step explanation:
Rewrite the fraction 1/6 as a division expression with the same numbers.
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Answer:
1 ÷ 6
Step-by-step explanation:
1/6 is a division expression. It means "one divided by six."
The slash (/) can be replaced by the "divided by" symbol (÷), if you like.
1/6 = 1 ÷ 6 . . . . means "one divided by six"
What is the value of n in the proportion below?
n/28 = 4/7
•1
•12
•14
•16
Answer:
n = 16
Step-by-step explanation:
Given the proportion, n/28 = 4/7:
Cross multiply the two fractions:
7n = 28 × 4
7n = 112
Divide both sides by 7 to solve for n:
7n/7 = 112/7
n = 16
Therefore, the value of n = 16.
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¿A qué velocidad debe circular un auto de carreras para
recorrer 50km en un cuarto de hora?
Answer:
creo que la respuesta es 3,333.33 mph
Step-by-stepStep-by-step explanation: