1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
LUCKY_DIMON [66]
3 years ago
5

Find the average value of the function f(x, y, z) = 5x2z 5y2z over the region enclosed by the paraboloid z = 9 − x2 − y2 and the

plane z = 0.
Mathematics
1 answer:
Katyanochek1 [597]3 years ago
4 0
The paraboloid meets the x-y plane when x²+y²=9. A circle of radius 3, centre origin. 

<span>Use cylindrical coordinates (r,θ,z) so paraboloid becomes z = 9−r² and f = 5r²z. </span>

<span>If F is the mean of f over the region R then F ∫ (R)dV = ∫ (R)fdV </span>

<span>∫ (R)dV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] rdrdθdz </span>

<span>= ∫∫ [θ=0,2π, r=0,3] r(9−r²)drdθ = ∫ [θ=0,2π] { (9/2)3² − (1/4)3⁴} dθ = 81π/2 </span>


<span>∫ (R)fdV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] 5r²z.rdrdθdz </span>

<span>= 5∫∫ [θ=0,2π, r=0,3] ½r³{ (9−r²)² − 0 } drdθ </span>

<span>= (5/2)∫∫ [θ=0,2π, r=0,3] { 81r³ − 18r⁵ + r⁷} drdθ </span>

<span>= (5/2)∫ [θ=0,2π] { (81/4)3⁴− (3)3⁶+ (1/8)3⁸} dθ = 10935π/8 </span>

<span>∴ F = 10935π/8 ÷ 81π/2 = 135/4</span>
You might be interested in
Consider △RST and △RYX.
astra-53 [7]

Answer:

B. StartFraction R Y Over R S EndFraction = StartFraction R X Over R T EndFraction = StartFraction X Y Over T S EndFraction

i.e \frac{RY}{RS} = \frac{RX}{RT} = \frac{XY}{TS}

Step-by-step explanation:

Two or more shape or figures are similar when their sides and angles can be compared appropriately.

In the given figure, ΔRXY is within ΔRST. Since the two triangles are similar, then their length of sides can be compared in the form of required ratios.

So that by comparison,

\frac{RY}{RS} = \frac{RX}{RT} = \frac{XY}{TS}

Therefore, the correct option to the question is B.

8 0
2 years ago
Read 2 more answers
A wood frame costs $1.50. Use a property to write an equivalent expression for the cost of a wood frame.
podryga [215]
Let the width of the frame = x
Let the cost of a linear foot of the frame = y
Let the inside dimension of the length of the frame = a
Let the inside dimension of the width of the frame = b.

The total length of the frame = 2*a + 4x
The total width of the frame = 2b

The total amount of the frame in feet is (2a + 4x + 2b)*y = Cost
If this is incorrect please leave a message on my home page.
6 0
3 years ago
2 3/4 x 8 = blank/4 x 8/1
Alex

Answer: 11

Step-by-step explanation:

2 3/4 × 8 = p/4 × 8/1

Let p be the missing number

Convert the mixed fraction 2 3/4 to proper fraction, multiply 4 x 2, then add 3= 11/4

11/4 × 8 = p/4 ×8

p= 11

I hope this helps.

5 0
3 years ago
Read 2 more answers
A plane that carries mail makes a round trip each day from Chicago to New York. It makes 2 intermediate stops on the way to New
Afina-wow [57]

Answer:

Step-by-step explanation:

7(d)

3 0
3 years ago
What is the square root of 69
Rasek [7]
The square root of 69 is 8.30.because 8.30*8.30 is 69
6 0
3 years ago
Read 2 more answers
Other questions:
  • This is how zirconium appears in the periodic table. Rounded to the nearest whole number, how many electrons are in an atom of z
    12·2 answers
  • Write an equation in slope-intercept form for the line shown.
    8·1 answer
  • In a normal distribution, what is true about the mean, median and the mode?​
    14·1 answer
  • The number of days it takes a theater crew to set up a stage for a musical varies inversely as the number of
    5·1 answer
  • Find the exact value of x
    7·2 answers
  • Write into a single logarithm log b (X+1)+ log b (X-8)
    9·1 answer
  • Finding values or<br> k(x) = x - 2<br> h(x) = x2 +1<br> (h + k)(2) =
    7·1 answer
  • 12x 3000 this is for u my bud answer it quickly
    13·2 answers
  • Match the following pair of angles with their respective definitions.<br> Pls help
    7·2 answers
  • Please Help! I will give you the brainiest and a lot of points!
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!