Question

Find an equation for the surface obtained by rotating the line x = 6y about the x-axis.

Answer 1

Answer:

$\frac{x^2}{36}=y^2+z^2$

Step-by-step explanation:

If we loo at the xy plane and consider the point (6y, y, 0), it looks like a circular path that is a cone as the curve is rotated about the x axis. The equation of the cone has the form:

$\frac{x^2}{c^2}=\frac{y^2}{a^2}+\frac{z^2}{b^2}$

Given that x = 6y, to make the equation to be in the form as seen above, we square x = 6y, hence:

x² = (6y)²

x² = 36y²

The path parallel to yz axis are circles, so the equation should be as that of a circle. We need to add 36z², hence:

x² = 36y² + 36z²

$\frac{x^2}{36}=y^2+z^2$

$(\frac{x}{6})^2=y^2+z^2$

This equation is that of a circle with radius of r = y = x/6, the resulting surface is that of a cone