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>
In response to the earlier inquiry, the diagram depicts horizontal lines along with a vertical segment; FB measures 3. If C is defined as 3, then the coordinates for point D would be (-6,-4). I trust this response has clarified your question.
Response:
a. 55 cars
b. 25 cars
Detailed explanation:
Let’s denote the quantity of cars with stereo systems as N(ss), those with air conditioning as N(ac), and those with sunroofs as N(sr).
We find that:
N(ss) = 30
N(ac) = 30
N(sr) = 40
N(ss and ac and sr) = 15
N(at least two) = 30
a.
To calculate how many cars possess at least one feature (N(at least one) or N(ss or ac or sr)), we apply:
N(ss or ac or sr) = N(ss) + N(ac) + N(sr) - N(ss and ac) - N(ss and sr) - N(ac and sr) + N(ss and ac and sr)
N(ss or ac or sr) = 30 + 30 + 40 - (N(at least two) + 2*N(ss and ac and sr)) + 15
Substituting, we find N(ss or ac or sr) = 30 + 30 + 40 - (30 + 2*15) + 15 = 55
b.
For those cars that have exactly one feature, we have:
N(only one) = N(at least one) - N(at least two)
N(only one) = 55 - 30 = 25
We can simplify by reducing factors between the 5 and the 50, as well as between the 42 and the 3.
