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ozzi
2 months ago
9

Find the average value of the function f(x, y, z) = 5x2z 5y2z over the region enclosed by the paraboloid z = 4 − x2 − y2 and the

plane z = 0.
Mathematics
1 answer:
Leona [12.6K]2 months ago
3 0
The paraboloid intersects the x-y plane when x²+y²=9, defining a circle with a radius of 3, centered at the origin.

<span>Utilizing cylindrical coordinates (r,θ,z), the paraboloid transforms into z = 9−r² and f = 5r²z. </span>

<span>If F represents the average of f over the area 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>Thus, F = 10935π/8 ÷ 81π/2 = 135/4</span>
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F(x) = -4x(2)+ 12x – 9
AnnZ [12381]

For this context, we examine the function:

f (x) = - 4x ^ 2 + 12x-9

y = 0 presented as:

-4x ^ 2 + 12x-9 = 0

The definition of the discriminant in a quadratic equation is provided by:

a = -4\\b = 12\\c = -9

Sentences correspond to the types of roots: Different real roots, equal real roots, or distinct complex roots

d = b ^ 2-4 (a) (c)

d> 0:Upon substituting the provided values, we arrive at

d = 0:

d

This indicates that there are two equal real roots.

To discover the intersections along the x-axis, we apply the quadratic formula:d = (12) ^ 2-4 (-4) (- 9)\\d = 144-144

d = 0

Plugging in the values yields: y = 0:

-4x ^ 2 + 12x-9 = 0

The intersection on the x-axis is

x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}

x = \frac {-12 \pm \sqrt {12 ^ 2-4 (-4) (- 9)}} {2 (-4)}\\x = \frac {-12 \pm \sqrt {144-144}} {- 8}\\x = \frac {-12 \pm0} {- 8}\\x = \frac {-12} {- 8}\\x = \frac {3} {2}

(\frac {3} {2}, 0)

5 0
1 month ago
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Robby had 4 4/9 bags of pet food.All of the bags held the same amount of food when they were full.Of Robby's 4 4/9 bags of pet f
Zina [12379]

Respuesta: 1.8 bolsas

Explicación paso a paso:

A partir de la pregunta, considerando que Robby tiene 4 4/9 bolsas de comida para mascotas, 2/5 eran comida para perros.

2/5 de 4 4/9 = comida para perros.

Convierte 4 4/9 a fracción impropia= 40/9.

2/5 de 40/9 implica 2/5 × 40/9.

= 16/9.

= 1.78 o 1.8.

Espero que esto sea útil, por favor marca como la mejor respuesta.

4 0
3 months ago
Suppose A is a 5x7 matrix. How many pivot columns must A have if its columns span R^5​? ​Why?
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The correct answer is "Option B." There seems to be an error with the options provided; however, the appropriate choice is detailed in the attached file. If the column of the matrix and the span of A are both equal to R^5, then A must have a pivot in each row, hence resulting in five pivot columns that confirm choice B as accurate.
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2 months ago
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You can determine your result by adding +7 to every value.
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