Find the width of the boats?

Question

A waterway in a theme park has a semicircular cross section with diameter 11 ft. The boats that are going to be used in this waterway have rectangular cross sections and are found to submerge 1 ft into the water. If the waterway is to be filled with water 4.5 ft deep, what is the maximum possible width of the boats?

Answer 1

Need to draw an upside down semicircle. Put the origin at the top middle of the semicircle. Draw a horizontal line down from the top for the surface of the water. Then draw another horizontal line below that for the bottom of the boat touching the sides of the semicircle.

The total depth of the semicircular waterway is the radius of the circle, or 11/2 =5.5 ft. If the depth of the water is 4.5 ft, then the surface of the water is 1 foot below the top of the waterway (5.5-4.5). If the bottom of the boat is submerged another foot, then that is 2 feet below the top of the waterway. So the bottom boat will just touch the sides of the waterway when y = -2 feet.

The equation for the semicircle is x²+y² = r². r = 5.5 and y = -2. Need to solve for x and then multiply by 2 to get the total width of the boat.

x = √(5.5²-(-2)²) ≈ 5.123

2x ≈ 10.247 feet