At the top of their 50 foot tall sales building, there is a 10 feet tall sign that attracts the attention of pedestrians. The company president expressed to Professor Schmidt that to obtain the best view of the sign, pedestrians were standing about fifty feet from the building and trampling the company's lawn because the side walk is 60 feet from the building. He seemed unaffected when Professor Schimdt expressed that it is better that the pedestrians wander onto the lawn rather than into the street to get the best view. Since the sign is badly worn and since it has attracted a good deal of business for Almaject, the company president wants to replace it. But he wants to replace it with a sign for which the ideal viewing distance is 60 feet from the building. He was so pleased with Professor Schmidt's team of consultants who provided a valuable service to Almaject on aircraft tracking issues that he is asking for their service again. For an observer, the angle between the lines of sight from the observer's eye to the top and the bottom of the sign is a measure of the observer's view of the sign, and the best view occurs when this angle is largest possible. Assuming that the observer's eyes are 5 ft off the ground (so that the observer is approximately 5'3"), you can determine that the ideal viewing distance from the building is approximately 49.7 feet if the sign were 10 feet tall. You are to determine the height of the new sign placed at the top of the building so that the best view occurs when the observer is 60 feet from the building. Be sure to justify that you have maximized the appropriate function. Here is a word of caution. Let h be the height of the new sign, and let x be the distance that the observer stands from the building. Avoid destroying a maximization problem by letting x=60 too soon. In addition, the company is contemplating putting in a walkway with a bench a distance of 35 feet from the base of the building. The company would also like to know how tall the sign should need to be so that the best view of the new sign is at this distance from the building.
