How do I solve this? I know it's one sheet and I have its base formula here:
What/where is capital A, B, and Z in this equation? I have a and b but no idea the rest of the parts. How do I start? Thank you.
This is what I got but it's wrong:
$\left(\frac{x}{10}\right)^2+\left(\frac{y}{4}\right)^2=\left(\frac{z}{\sqrt{7}}\right)^2+1$
On
The illustration has given semi- major/minor axes $(A,B)$ values at height of $Z=7$. You have to see it.
You can find the only remaining unknown $c$ by plugging in its value into the the equation of the elliptic hyperboloid of 1 sheet:
$$ \dfrac{A^2}{a^2}+ \dfrac{B^2}{b^2}= \dfrac{7^2}{c^2} +1. $$
Write the equation $$ \left(\frac{x}{10}\right)^2+\left(\frac{y}{4}\right)^2=\left(\frac{7}{c}\right)^2+1 $$ then divide everything by the RHS, so that the new RHS becomes $1.$ Now you have an ellipsis and you have to extract the semi-axes and put them equal to $20$ and $8,$ respectively. $$ \frac{x^2}{10^2\left[\left(\frac{7}{c}\right)^2+1\right]}+\frac{y^2}{4^2\left[\left(\frac{7}{c}\right)^2+1\right]}=1 $$ then $$ 10^2\left[\left(\frac{7}{c}\right)^2+1\right]=A^2=20^2\\ \left(\frac{7}{c}\right)^2+1=2^2\\ \left(\frac{7}{c}\right)^2=3\\ \left(\frac{c}{7}\right)^2=\frac{1}{3}\\ c^2=\frac{7^2}{3}\\ c=\pm\frac{7}{\sqrt{3}}\\ $$ where you can take the positive sign, because $c$ only appear squared in the final equation.
You obtain the same if you use $B$, so the equation is $$ \left(\frac{x}{10}\right)^2+\left(\frac{y}{4}\right)^2=3\left(\frac{z}{7}\right)^2+1 $$