Example 1
Completion requirements
| Example 1 |
When a liquid contained in a cylinder is rotated about a vertical axis, the cross section of its surface forms a parabola. Suppose a cylindrical tank of water with a diameter of \(200 \thinspace \rm{cm}\) is rotated such that the surface at the centre of the tank is \(4 \thinspace \rm{cm}\) lower than the surface at the edge of the tank. Write an inequality that represents all the locations of water for a cross section of this tank of water.
No point of reference is given, so choose one that will be easy to work with.
Let the vertex be \((0, 0)\). Another point on the parabola is \((100, 4)\).
\(\begin{align}
y &= a\left( {x - 0} \right)^2 + 0 \\
y &= ax^2 \\
4 &= a\left( {100} \right)^2 \\
0.000\thinspace 4 &= a \\
\end{align}\)
An equation for the boundary is \(y = 0.000 \thinspace 4x^2\). Use a test point to determine the direction of the inequality sign. The point \((0, –1)\) is part of the solution region.
There is water at the boundary line, so the inequality \(y \le 0.000 \thinspace 4x^2\) represents the solution region. However, the possible \(x\)-values should be limited to the dimensions of the tank. The height of the tank is not given, so no restriction will be placed on \(y\).
\(y \le 0.000 \thinspace 4x^2, \thinspace - 100 \le x \le 100\)
Let the vertex be \((0, 0)\). Another point on the parabola is \((100, 4)\).
\(\begin{align}
y &= a\left( {x - 0} \right)^2 + 0 \\
y &= ax^2 \\
4 &= a\left( {100} \right)^2 \\
0.000\thinspace 4 &= a \\
\end{align}\)
An equation for the boundary is \(y = 0.000 \thinspace 4x^2\). Use a test point to determine the direction of the inequality sign. The point \((0, –1)\) is part of the solution region.
| Left Side | Right Side |
|---|---|
|
\[\begin{array}{r} y \\ -1 \end{array}\] |
\[\begin{array}{l} 0.000 \thinspace 4x^2 \\ 0.000 \thinspace 4(0)^2 \\ 0 \end{array}\] |
| \(\hspace{25pt}\)LS \(\lt\) RS | |
There is water at the boundary line, so the inequality \(y \le 0.000 \thinspace 4x^2\) represents the solution region. However, the possible \(x\)-values should be limited to the dimensions of the tank. The height of the tank is not given, so no restriction will be placed on \(y\).
\(y \le 0.000 \thinspace 4x^2, \thinspace - 100 \le x \le 100\)