Question

A rotating liquid forms a surface in the form of a paraboloid. The surface is 2m
deep at the centre and 10m across. Obtain an equation of the surface.

Answer 1

Answer on Question #78890 – Math – Analytic Geometry

Question

A rotating liquid forms a surface in the form of a paraboloid. The surface is 2m deep at the centre and 10m across. Obtain an equation of the surface.

Solution

The paraboloid which has radius $a$ at height $h$ is then given parametrically by

$$x(u, \theta) = a \sqrt{u / h} \cos \theta$$ $$y(u, \theta) = a \sqrt{u / h} \sin \theta$$ $$z(u, \theta) = u$$ $$x^2 + y^2 = a^2 \left(\frac{u}{h}\right) \cos^2 \theta + a^2 \left(\frac{u}{h}\right) \sin^2 \theta = a^2 \left(\frac{u}{h}\right)$$

The equation of the surface

$$\frac{x^2}{a^2} + \frac{y^2}{a^2} = \frac{z}{h}$$ $$\frac{x^2}{2^2} + \frac{y^2}{2^2} = \frac{z}{10}$$ $$\frac{x^2}{4} + \frac{y^2}{4} = \frac{z}{10}$$

Answer: $\frac{x^2}{4} + \frac{y^2}{4} = \frac{z}{10}$

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