Question

find h as indicated in the figure. h = (round to the nearest integer as needed.)

Explanation:

Step1: Set up tangent - ratio equations

Let the base of the larger right - triangle be $x$. For the larger right - triangle with height $330 + h$ and angle $23.8^{\circ}$, $\tan(23.8^{\circ})=\frac{330 + h}{x}$, so $x=\frac{330 + h}{\tan(23.8^{\circ})}$. For the smaller right - triangle with height $330$ and angle $40.4^{\circ}$, $\tan(40.4^{\circ})=\frac{330}{x}$, so $x = \frac{330}{\tan(40.4^{\circ})}$.

Step2: Equate the two expressions for $x$

Since $\frac{330 + h}{\tan(23.8^{\circ})}=\frac{330}{\tan(40.4^{\circ})}$, we can cross - multiply to get $(330 + h)\tan(40.4^{\circ})=330\tan(23.8^{\circ})$.

Step3: Expand and solve for $h$

Expand: $330\tan(40.4^{\circ})+h\tan(40.4^{\circ})=330\tan(23.8^{\circ})$. Then $h\tan(40.4^{\circ})=330\tan(23.8^{\circ})-330\tan(40.4^{\circ})$. So $h=\frac{330(\tan(23.8^{\circ})-\tan(40.4^{\circ}))}{\tan(40.4^{\circ})}$.
We know that $\tan(23.8^{\circ})\approx0.445$ and $\tan(40.4^{\circ})\approx0.859$.
$h=\frac{330(0.445 - 0.859)}{0.859}=\frac{330\times(- 0.414)}{0.859}=\frac{-136.62}{0.859}\approx - 159$. But height cannot be negative in this context. The correct setup should be $\frac{330 + h}{\tan(40.4^{\circ})}=\frac{330}{\tan(23.8^{\circ})}$.
Cross - multiplying gives $(330 + h)\tan(23.8^{\circ})=330\tan(40.4^{\circ})$.
Expand: $330\tan(23.8^{\circ})+h\tan(23.8^{\circ})=330\tan(40.4^{\circ})$.
$h\tan(23.8^{\circ})=330\tan(40.4^{\circ})-330\tan(23.8^{\circ})$.
$h=\frac{330(\tan(40.4^{\circ})-\tan(23.8^{\circ}))}{\tan(23.8^{\circ})}$.
$h=\frac{330(0.859 - 0.445)}{0.445}=\frac{330\times0.414}{0.445}=\frac{136.62}{0.445}\approx307$.

Answer:

$307$