Superlesson
Project 8-3
Advanced Algebra
Chapter 8, Polynomials and Polynomial Functions
Several important formulas relating to ocean exploration involve division of one or more variables by others. To solve a problem involving such variables, you may need to graph a rational function or solve a rational equation.
Part A, Rational Expressions;
Part B, Adding and Subtracting Rational Expressions
1. A SCUBA diver descended at a rate of 10 ft/min to the wreck of the Kenosha off Long Island, New York.
a. How long did it take the diver to reach the ship? Use the greater depth in your calculation.
b. The diver spent 40 minutes exploring the Kenosha, moving at an average rate of 8 ft/min. How far did the diver travel?
c. In order to avoid a potentially fatal condition called the bends, divers must pause at specific depths for extended periods of time as they ascend to the surface. The U.S. Navy recommends that divers take about 33 minutes to ascend after a 40-min stay at the depth of the Kenosha. What is the recommended rate of ascent in ft/min?
d. Find the diver's total distance traveled, total time, and average rate of speed.
e. A diver descends at a rate of 5 ft/min to a shipwreck at a depth of d ft. Over the next 30 minutes the diver explores the ship, traveling a distance equal to 4 times the ship's depth. Finally, the diver ascends to the surface in 60 minutes. Find and simplify the diver's average rate of speed.
f. Is the average speed a rational expression? Explain.
Part E, Making Connections
2. Charles's Law relates the pressure P, volume V, and temperature T of a gas under two sets of conditions:
A killer whale with a volume (V1) of 200 ft3 of oxygen in its lungs rests on the ocean surface. Follow these steps to discover how the oxygen volume changes when the whale dives.
a. Find P1, the air pressure at sea level, in pounds per square inch.
b. At the ocean's surface, where the whale is basking, the temperature is a balmy 27° C. To use Charles's Law, temperature must be expressed on the Kelvin scale. Find the relationship between Kelvin temperature and Celsius temperature. Then find T1, the Kelvin temperature at sea level.
c. The total pressure at any point in the ocean is the sum of the air pressure at sea level plus the water pressure. Water pressure increases by 0.44 lb/in2 for every foot descended. Write an expression for P2, the total pressure at a depth of d feet.
d. Write an expression for T2, the Kelvin temperature at depth d. Assume that temperature drops 0.01 degree for each foot of descent.
e. Use Charles's Law to find a rational expression for V2, the oxygen volume at depth d.
f. Graph volume as a function of depth.
g. What happens to the volume of oxygen as the whale descends?
h. At what depth is the volume of oxygen half of the original volume?
i. What is the volume of oxygen at a depth of 1000 ft?