Chapter 8, Problem 33T

Solution

To calculate: The equations of that model the water’s surface with the glass upright and after the glass is tilted.

The glass is upright formula is $x^2+y^2=2.25$ When it is tilted formula is $\frac{x^2}{9}+\frac{y^2}{2.25}=1$ .

Given information:

The goal of this task is to find one equation that describes the water surface when glass is upright and when glass is tilted.

Radius of cylindrical water glass $1.5$

  $\text{inches}$ .

Calculation:

First when glass is upright and radius $r=1.5$ it is a circle with a center at $(0,0)$ so the equation is

  $\begin{array}{r}\hfill x^2+y^2=r^2\\ \hfill x^2+y^2=2.25\end{array}$

Now when glass is tilted it is in the shape of an ellipse and if make that major axis is horizontal then

minor axis is

  $3\to b=1.5$

major axis is

  $6\to a=3$

Now substitute the values of $a$ and $b$ into the standard formula for an ellipse with the major horizontal axis.

  $\begin{array}{r}\hfill \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\\ \hfill \frac{x^2}{9}+\frac{y^2}{2.25}=1\end{array}$

When the glass is upright formula is $x^2+y^2=2.25$ When it is tilted formula is $\frac{x^2}{9}+\frac{y^2}{2.25}=1$