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SpyIntel [72]
3 years ago
6

Use cylindrical coordinates to evaluate the triple integral ∭ where E is the solid bounded by the circular paraboloid z = 9 - 16

(x^2 + y^2) and the xy -plane.
Mathematics
1 answer:
4vir4ik [10]3 years ago
4 0

Answer:

\mathbf{\iiint_E  E \sqrt{x^2+y^2} \ dV =\dfrac{81 \  \pi}{80}}

Step-by-step explanation:

The Cylindrical coordinates are:

x = rcosθ, y = rsinθ and z = z

From the question, on the xy-plane;

9 -16 (x^2 + y^2) = 0 \\ \\  16 (x^2 + y^2)  = 9 \\ \\  x^2+y^2 = \dfrac{9}{16}

x^2+y^2 = (\dfrac{3}{4})^2

where:

0 ≤ r ≤ \dfrac{3}{4} and 0 ≤ θ ≤ 2π

∴

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} \int ^{9-16r^2}_{0} \ r \times rdzdrd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} r^2 z|^{z= 9-16r^2}_{z=0}  \ \ \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0} r^2 ( 9-16r^2})  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0} \int ^{\dfrac{3}{4}}_{0}  ( 9r^2-16r^4})  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( \dfrac{9r^3}{3}-\dfrac{16r^5}{5}})|^{\dfrac{3}{4}}_{0}  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( 3r^3}-\dfrac{16r^5}{5}})|^{\dfrac{3}{4}}_{0}  \ drd \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV = \int^{2 \pi}_{0}   ( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}}) d \theta

\iiint_E  E \sqrt{x^2+y^2} \ dV =( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}}) \theta |^{2 \pi}_{0}

\iiint_E  E \sqrt{x^2+y^2} \ dV =( 3(\dfrac{3}{4})^3}-\dfrac{16(\dfrac{3}{4})^5}{5}})2 \pi

\iiint_E  E \sqrt{x^2+y^2} \ dV =(\dfrac{81}{64}}-\dfrac{243}{320}})2 \pi

\iiint_E  E \sqrt{x^2+y^2} \ dV =(\dfrac{81}{160}})2 \pi

\mathbf{\iiint_E  E \sqrt{x^2+y^2} \ dV =\dfrac{81 \  \pi}{80}}

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Uses 10 4/5 oz of icing for every 1/4 of a cake what is the unit rate in ounces of icing for cake
Anton [14]
1 whole cake= 1/4 + 1/4 + 1/4 + 1/4
1/4= 0.25

So we multiply 10 4/5oz per 1/4 cake by four to find out how many ounces are needed for 1 whole cake.

T= total ounces for 1 whole cake
T= (1 ÷ cake portion)(# ounces)
T= (1/0.25)(10 4/5)
change mixed fraction to improper
T= (4)(54/5)
multiply 4 by the numerator
T= (4*54)/5
T= 216/5 ounces for whole cake

T= 43 1/5 ounces for whole cake

Hope this helps! :)
4 0
3 years ago
PLEASE HELP ASAP!! Ben worked 6 hours at x dollars per hour. Julie worked 9 hours and earned twice as much as Ben per hour. Phil
krok68 [10]

Answer:

A

Step-by-step explanation:

Given that Ben worked 6 hours at x dollars per hour. That is,

Ben earns = 6x

Julie worked 9 hours and earned twice as much as Ben per hour. That is,

Julie earns = 9 × 2x = 18x

Phil worked 16 hours and earned 3 dollars an hour less than Julie

That is,

Phil earns = 16 × ( 2x - 3 )

= 32x - 48

Total wages = 6x + 18x + 32x - 48

Total wages = 56x - 48

Option A is therefore the best option which is the answer to the question.

4 0
3 years ago
What is the solution to the equation 9(x − 2) = 27
ehidna [41]

9(x − 2) = 27

x=5

5 is your final answer

4 0
3 years ago
What pair of fractions is 0.8 between on a number line
madam [21]
0.8  has one decimal.. so... let's make it a fraction first, by using one zero with a one as denominator.

\bf 0.\underline{8}\implies \cfrac{08}{1\underline{0}}\impliedby \textit{one decimal, one zero, no dot}
\\\\\\
\cfrac{8}{10}\implies \cfrac{4}{5}
\qquad \qquad \qquad 
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is between any of those, among many other fractions as well.
3 0
3 years ago
Again I need help, sorry but thank you!
musickatia [10]

Answer:

B)13 pints

Step-by-step explanation:

SO first we find the unit rate

We know that in 90 minutes (hour and a half), the water leaked 6 pints form 2;30 to 4

Now So we can divide to find unit rate

6/90=1/15

So 1 pint for every 15 minutes

And now we want to know the amount it takes from 2:30 to 6:30 which is 4 hours

40 hours is 240 minutes

SO we can divide now

240/15=16

SO 16 pints leaked but the question asked for remaining water in container

So 29-16 is 13

So B)13

3 0
3 years ago
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