To write the given equation for the hyperboloid of one sheet in standard form, let's start with the given equation:

$5x^2 + 2y^2 - 10z^2 - 10 = 0$

Step-by-Step Simplification:

1. Move the constant term to the other side: $5x^2 + 2y^2 - 10z^2 = 10$

2. Divide the entire equation by 10 to normalize the coefficients: $\frac{5x^2}{10} + \frac{2y^2}{10} - \frac{10z^2}{10} = \frac{10}{10}$

   Simplifying each term: $\frac{x^2}{2} + \frac{y^2}{5} - z^2 = 1$

The standard form of the hyperboloid of one sheet is:

$\frac{x^2}{2} + \frac{y^2}{5} - z^2 = 1$

This is the required equation in standard form.

Would you like more details or have any further questions? Here are five related questions to deepen understanding:

1. What are the key characteristics of a hyperboloid of one sheet?
2. How does the standard form of a hyperboloid differ from that of an ellipsoid?
3. Can you describe the axis of symmetry for this hyperboloid?
4. How would the equation change if it described a hyperboloid of two sheets?
5. What would happen to the shape if the sign in front of $z^2$ were positive?

Tip:

A hyperboloid of one sheet has two asymptotic directions and looks somewhat like a "saddle," curving inwards along one axis and outwards along the others.