The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

A)9?
B)81?
C)18?
D)24?
E)36?

Question

The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

A)9?
B)81?
C)18?
D)24?
E)36?

Quizplus · Accepted Answer

The equation $z^2 + r^2 = 9$ describes a cylinder in cylindrical coordinates with radius 3 and extending infinitely in both the positive and negative z-directions. However, since it's bounded, it implies a cylinder of finite height. To find the volume of this solid, we recognize that the equation also represents a circle of radius 3 in the r-z plane, which when revolved around the z-axis forms a cylinder. The volume of a cylinder is given by $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height. In this case, the radius $r = 3$, and the height $h$ can be inferred from the equation as the range of $z$ values that satisfy the equation, which is from $-3$ to $3$, giving a height of $6$. Thus, the volume is $V = \pi (3)^2 (6) = 54\pi$. Since none of the given options are in terms of $\pi$, we must calculate the numerical value: $54\pi \approx 169.646$, which does not directly match any of the options. However, recognizing that the question might have intended to ask for the volume without specifying the need for $\pi$, and considering the closest options provided, the correct interpretation of the volume of the solid described by the equation in terms of the options given would be to understand the solid's geometry correctly and calculate its volume based on the standard formula for the volume of a cylinder. The error in my initial calculation step was in not recognizing that the options provided suggest a simplification or a different interpretation of the volume calculation, possibly assuming a unit other than what was calculated or an error in the interpretation of the equation's description of the solid's shape. The correct approach involves recognizing the solid as a sphere (due to the equation's form in cylindrical coordinates), not a cylinder, and calculating its volume accordingly.

The Volume of the Solid Bounded By z2+r2=9z ^ { 2 } + r ^ { 2 } = 9z2+r2=9

The Volume of the Solid Bounded By $z ^ { 2 } + r ^ { 2 } = 9$