We investigate geometric sets determined by implicit equations.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

1 Geometry disguised as algebra

Consider the equations:
\begin{align*} x+y &= 6\\ x^2+y^2+z^2&=18 \end{align*}

Find a solution to these equations.

Guess-and-check is not a bad method for this problem.
Explain what the equations
\begin{align*} x+y &= 6\\ x^2+y^2+z^2&=18 \end{align*}

describe from a geometric point of view.

Use geometry to explain why the equations
\begin{align*} x+y &= 6\\ x^2+y^2+z^2&=18 \end{align*}

have exactly one solution where \(x\), \(y\), and \(z\) are real numbers.

2 Thinking about a tetrahedron

Sketch the triangle in \(\R ^3\) whose vertices are the intersections of the plane
\[ 20x + 15y + 12z = 60 \]
and the coordinate axes.
Compute the volume of the tetrahedron (triangular-based pyramid) of in \(\R ^3\) bounded by the planes \(x=0\), \(y=0\), \(z=0\), and the plane \(20x + 15y + 12z = 60\).
Recall that the volume of a cone (pyramid!) is given by:
\[ V = \left (\frac {1}{3}\right ) (\text {area of base})(\text {height}) \]
Compute the area of the triangle in \(\R ^3\) whose vertices are the intersections of the plane \(20x + 15y + 12z = 60\) and the coordinate axes.
For now, use your old friend:
\[ A = \frac {bh}{2} \]
and use calculus to minimize the distance between the line connecting \((3,0,0)\) to \((0,4,0)\), and the point \((0,0,5)\).