6. Bo and Li are standing 325 m apart,watching a hot air balloon above hem. Bo measures the angle of elevation to the balloon to be 54 degrees measures the angles of elevation to the balloon to be 38 degrees.How is each person from the balloon , to the nearest metre. Find the height balloon, to the nearest metre.

## Step 1: Visualizing the Problem<br />### We have two observers, Bo and Li, a certain distance apart, looking at a hot air balloon. This forms a triangle. We know the distance between Bo and Li, and the angles of elevation from each of them to the balloon.<br /><br />## Step 2: Setting up the Triangle<br />### Let B be Bo's position, L be Li's position, and H be the point on the ground directly below the balloon. Let the balloon's position be A. We have triangle BLH with BL = 325m. Let x be the distance BH. Then LH = 325 - x. Angle ABH = 54° and angle ALH = 38°. Let h be the height of the balloon, AH.<br /><br />## Step 3: Using Trigonometry<br />### In triangle ABH, $\tan(54°) = \frac{h}{x}$. In triangle ALH, $\tan(38°) = \frac{h}{325-x}$.<br /><br />## Step 4: Solving for x<br />### From the equations in Step 3, we have $h = x\tan(54°)$ and $h = (325-x)\tan(38°)$. Therefore, $x\tan(54°) = (325-x)\tan(38°)$. This simplifies to $x(\tan(54°) + \tan(38°)) = 325\tan(38°)$. So, $x = \frac{325\tan(38°)}{\tan(54°) + \tan(38°)}$.<br /><br />## Step 5: Calculating x<br />### Using a calculator, $\tan(54°) \approx 1.376$ and $\tan(38°) \approx 0.784$. Therefore, $x \approx \frac{325 \times 0.784}{1.376 + 0.784} \approx \frac{254.8}{2.16} \approx 118.2$ m.<br /><br />## Step 6: Calculating h<br />### Now we can find h using $h = x\tan(54°)$. $h \approx 118.2 \times 1.376 \approx 162.6$ m.<br /><br />## Step 7: Calculating Distances to the Balloon<br />### Distance from Bo to the balloon (AB) is $\sqrt{x^2 + h^2} \approx \sqrt{118.2^2 + 162.6^2} \approx \sqrt{13971.24 + 26438.76} \approx \sqrt{40400} \approx 201$ m.<br />### Distance from Li to the balloon (AL) is $\sqrt{(325-x)^2 + h^2} \approx \sqrt{(325-118.2)^2 + 162.6^2} \approx \sqrt{206.8^2 + 162.6^2} \approx \sqrt{42778.24 + 26438.76} \approx \sqrt{69217} \approx 263$ m.