Geometric sets

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We investigate geometric sets determined by implicit equations.

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

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.

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 $(0,0,5)$ .