Question

\(x = t\,cost,y = t,z = t\,sin t, t \ge 0\).

Short answer

The parametric equations \(x = t cost,y = t\), and \(z = t sin t\) matches with the graph II.

Step-by-step solution

Parametric Equation.

Consider the parametric equations\(x = t\,cost,y = t,\)and\(z = t\,sin t\), where\(t \ge 0\).

Square and add the parametric equations of\(x = t\,cost,\)and\(z = t\,sint\).

That is, calculate the value of the expression \({x^2} + {z^2}\).

\({x^2} + {z^2} = {(t cost)^2} + {(t sint)^2}\)

\( = {t^2} co{s^2}t + {t^2}si{n^2}t\)

\( = {t^2} \left( {co{s^2}t + si{n^2}t} \right)\)

\( = {t^2} \)

Substitute\(y = t\)in the above computed equation as\({x^2} + {z^2} = {y^2}\).

The equation\({x^2} + {z^2} = {y^2}\)represents that the circular cone, spiraling along the\(y\)-axis for\(y \ge 0\)which is away from the\(xz\)plane and the circular spiral increases as\(t\)increase.

Thus, The parametric equations \(x = t cost,y = t\), and \(z = t sin t\) match with the graph II.\(\)