find h to the nearest tenth.\nthe length of h is approximately \n(round the final answer to the nearest…

Explanation:

Step1: Express (AB) and (BD) in terms of (h)

In right - triangle (ACD), (\tan39^{\circ}=\frac{h}{AD}), so (AD = \frac{h}{\tan39^{\circ}}). In right - triangle (BCD), (\tan67^{\circ}=\frac{h}{BD}), so (BD=\frac{h}{\tan67^{\circ}}). And we know that (AD - BD=343).

Step2: Substitute (AD) and (BD) into the equation (AD - BD = 343)

Substitute (AD=\frac{h}{\tan39^{\circ}}) and (BD = \frac{h}{\tan67^{\circ}}) into (AD - BD=343), we get (h(\frac{1}{\tan39^{\circ}}-\frac{1}{\tan67^{\circ}})=343). Since (\tan39^{\circ}\approx0.80) and (\tan67^{\circ}\approx2.36), then (\frac{1}{\tan39^{\circ}}-\frac{1}{\tan67^{\circ}}=\frac{1}{0.80}-\frac{1}{2.36}\approx1.25 - 0.42 = 0.83).

Step3: Solve for (h)

From (h(\frac{1}{\tan39^{\circ}}-\frac{1}{\tan67^{\circ}})=343), we have (h=\frac{343}{\frac{1}{\tan39^{\circ}}-\frac{1}{\tan67^{\circ}}}). Substitute (\frac{1}{\tan39^{\circ}}-\frac{1}{\tan67^{\circ}}\approx0.83) into the formula, then (h=\frac{343}{0.83}\approx413.3).

Answer:

(413.3)