Justin is riding in a hot air balloon, directly over Ben, who is standing on the ground. Justin spots Patti on the ground at an angle of depression of 27 degrees. The balloon rises 35m. Now the angle of depression to Patti is 36 degrees. How far is Patti from Ben?
h = initial balloon height , d = Patti/Ben distance
h / d = tan(27º) ... h / tan(27º) = d
(h + 35) / d = tan(36º) ... (h + 35) / tan(36º) = d
(h + 35) tan(27º) = h tan(36º) ... 35 tan(27º) = h [tan(36º) - tan(27º)]
solve for h , then substitute back to find d
To find the distance between Patti and Ben, we can use trigonometry. Let's denote the distance between Patti and Ben as "x" (in meters).
First, let's analyze the situation when Justin spots Patti at an angle of depression of 27 degrees. This forms a right-angled triangle between Patti, Justin, and a point directly below Justin (let's call it point A). The side opposite the angle of depression is the difference in height between Justin and point A, which is 35 meters because the balloon rises 35 meters. The side adjacent to the angle of depression is the distance between Justin and Patti, which is "x".
Using trigonometry, we can write:
tan(27 degrees) = opposite / adjacent
tan(27 degrees) = 35m / x
To find "x", we can rearrange the equation:
x = 35m / tan(27 degrees)
Now, let's analyze the situation when the balloon rises 35 meters and the angle of depression to Patti becomes 36 degrees. In this case, the right-angled triangle is formed between Patti, Justin (who is now higher), and a point directly below Justin (let's call it point B). The side opposite the angle of depression is still x (the distance between Patti and Ben), and the side adjacent to the angle of depression is the difference in height between Justin (at point B) and point A (which is still 35 meters).
Using trigonometry, we can write:
tan(36 degrees) = opposite / adjacent
tan(36 degrees) = x / 35m
To find "x", we can rearrange the equation:
x = 35m * tan(36 degrees)
Now, we can substitute the value of "x" from the first equation into the second equation to find the distance between Patti and Ben:
x = 35m * tan(36 degrees)
x = 35m * 0.7265
x ≈ 25.42 meters
Therefore, Patti is approximately 25.42 meters away from Ben.