Question

Determine which value best approximates the volume of the solid between the \(x y\) -plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(f(x, y)=\sqrt{x^{2}+y^{2}}\) \(R:\) circle bounded by \(x^{2}+y^{2}=9\) (a) 50 (b) 500 (c) -500 (d) 5 (e) 5000

Short answer

The best approximation of the volume of the solid is (a) 50.

Step-by-step solution

Understanding the function

The function \(f(x, y)=\sqrt{x^{2}+y^{2}}\) describes a cone with its peak at the origin (0,0) and extending infinitely in the positive z-axis. In the xy-plane, the area covered by the function extends from the origin out to the circle described by \(x^{2}+y^{2}=9\), which has a radius of 3.

Visualizing and approximating the volume

This cone's base is the circle described by \(x^{2}+y^{2}=9\) and the height of the cone is equal to the radius of the circle, which is 3. Using reasoning based on the formula of the volume of a cone even without performing any calculations, the volume of the cone would be approximately 1/3 of the volume of a cylinder with the same radius and height, as the base area (\(\pi r^{2}\)) of the cone is identical to that of the cylinder and the height is also the same.

Evaluating the options

Comparing the formula for the volume of a cone and the given answer options, one can see that all values except (a) 50 are much larger than 1/3 of the volume of a cylinder with radius 3 and height 3 (which is an easy calculation of \(9\pi\), about 28). The next smallest option (d) 5 is even smaller. Of the given options, 50 is the best approximation of the volume of the cone as described by the function and region \(R\).