A company designs spinning toys using the family of functions $y = cx\sqrt{4 - x^{2}}$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches. (a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$ for $c = 6$. (b) It is known that, for $y = cx\sqrt{4 - x^{2}}$, $\frac{dy}{dx} = \frac{c(4 - 2x^{2})}{\sqrt{4 - x^{2}}}$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy? (c) For another spinning toy, the volume is $2\pi$ cubic inches. What is the value of $c$ for this spinning toy?
A company designs spinning toys using the family of functions $y = cx\sqrt{4 - x^{2}}$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.\\
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$ for $c = 6$.\\
(b) It is known that, for $y = cx\sqrt{4 - x^{2}}$, $\frac{dy}{dx} = \frac{c(4 - 2x^{2})}{\sqrt{4 - x^{2}}}$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?\\
(c) For another spinning toy, the volume is $2\pi$ cubic inches. What is the value of $c$ for this spinning toy?