Question

find the length x to the nearest whole number. (do not round until the final answer. then round to the nearest whole number.)

Explanation:

Step1: Set up tangent - ratio equations

Let the height of the part of the triangle with the $62^{\circ}$ angle be $h$. The height of the whole triangle is $460$. For the smaller right - triangle with the $62^{\circ}$ angle, $\tan(62^{\circ})=\frac{h}{x}$, so $h = x\tan(62^{\circ})$. For the larger right - triangle with the $31^{\circ}$ angle, $\tan(31^{\circ})=\frac{460}{x}$, so $460=x\tan(31^{\circ})+h$.

Step2: Substitute $h$ into the second equation

Substitute $h = x\tan(62^{\circ})$ into $460=x\tan(31^{\circ})+h$. We get $460=x\tan(31^{\circ})+x\tan(62^{\circ})$.
Factor out $x$: $460 = x(\tan(31^{\circ})+\tan(62^{\circ}))$.

Step3: Solve for $x$

We know that $\tan(31^{\circ})\approx0.6009$ and $\tan(62^{\circ})\approx1.8807$.
Then $x=\frac{460}{\tan(31^{\circ})+\tan(62^{\circ})}=\frac{460}{0.6009 + 1.8807}=\frac{460}{2.4816}$.
$x\approx185$.

Answer:

$185$