Question

Calculate the volume of the solid, bounded by the surfaces: z=4x2 +4y2; z=x2+y2; z=4.

Answer 1

For any given "z", each surface represents a circle:\[z=4x^2+4y^2\rightarrow R_1=\sqrt{\frac{ z }{ 4 }}=\frac{ 1 }{ 2 }\sqrt{z}\]
\[z=x^2+y^2\rightarrow R_2=\sqrt{z}\]
These circles are centered in x=0 and y=0 and radii are R1 and R2
The area between the 2 circles is:\[A=\pi(R_2^2-R_1^2)=\pi(z-\frac{ z }{ 4})=\frac{ 3 }{ 4 }\pi z\]
This is the inner area between the two circles for any given z. In order to get de dV when moving in z-axis you get:\[dV=Adz \rightarrow dV=\frac{ 3 }{ 4 }\pi zdz\] and the total volume is:
\[V=\int\limits_{z=0}^{z=4}Adz=\frac{ 3 }{ 4}\pi \int\limits_{z=0}^{z=4}zdz=\frac{ 3 }{ 4}\pi \left[ \frac{ z^2 }{ 2} \right]_0^4=\frac{ 3 }{ 4 }\pi·8=6 \pi\]

Answer 2

70% of the counties of the particular state received some rain on monday, and 65% of the counties received some rain on tuesday. no rain fell either day in 25% of the counties. what % of the counties received some rain on monday and tuesday? a. 45.5% b. 40% c. 12.5% d. 60% Remove I'm Feeling Lucky »
70mm

Answer 3

If 25% of the counties received no rain, it means 75% of the counties received some rain. Those that received rain on monday amount to 70% and thoses that received rain on tuesday amount to a 65%. The total (70%+65%=135%) exceeds 75% (the counties that received some rain) in 60%, which is exactly the percentage of counties that have received some rain on monday and tuesday

Answer 4

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