A blimp, suspended in the air at a height of 500 feet, lies directly over a line from a sports stadium to a planetarium. If the angle of depression from the blimp to the stadium is $32^{\circ}$ and from the blimp to the planetarium is $25^{\circ}$, find the distance between the sports stadium and the planetarium. The distance between the sports stadium and the planetarium is feet. (Round to two decimal places as needed.) an example Get more help. Clear all

Step 1 ：Given that the height of the blimp is \(h = 500\) feet, the angle of depression to the stadium is \(\theta_{stadium} = 32^\circ\) and to the planetarium is \(\theta_{planetarium} = 25^\circ\).

Step 2 ：Convert these angles to radians: \(\theta_{stadium} = 0.5585053606381855\) radians and \(\theta_{planetarium} = 0.4363323129985824\) radians.

Step 3 ：Using the tangent of the angles of depression, we can find the distances from the blimp to the stadium and the planetarium. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the height of the blimp forms the opposite side, and the distances from the blimp to the stadium and the planetarium form the adjacent sides.

Step 4 ：Calculate the distance from the blimp to the stadium: \(d_{stadium} = h / \tan(\theta_{stadium}) = 800.1672645205252\) feet.

Step 5 ：Calculate the distance from the blimp to the planetarium: \(d_{planetarium} = h / \tan(\theta_{planetarium}) = 1072.2534602547794\) feet.

Step 6 ：Subtract the distance to the stadium from the distance to the planetarium to find the distance between the stadium and the planetarium: \(d = d_{planetarium} - d_{stadium} = 272.08619573425415\) feet.

Step 7 ：Round the final answer to two decimal places: \(\boxed{272.09}\) feet.