June 25, 2008
SMDEP calculus
Group 1: According to Boyle’s Law, if the temperature of a confined gas is held fixed, then the product of the pressure P and the volume V is a constant. Suppose that, for a certain gas, PV = 800, where P is measured in pounds per square inch and V is measured in cubic inches. (a) Find the average rate of change of P as V increases from 200 in^3 to 250 in^3. (b) Express V as a function of P and show that the instantaneous rate of change of V with respect to P is inversely proportional to the square of P. (c) What is the instantaneous rate of change of V with respect to P when P is 4 pounds per square inch?
Group 2: Water is flowing at a constant rate into a spherical tank. Let V(t) be the volume of water in the tank and H(t) be the height of the water in the tank at time t. (a) What are the meanings of V’(t) and H’(t)? Are these derivatives positive, negative, or zero? (b) Is V’’(t) positive, negative, or zero? Explain. (c) Let t1, t2, and t3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H’’(t1), H’’(t2), and H’’(t3) positive, negative, or zero? Why?
Group 3: A car is traveling at night along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100 meters west and 100 meters north of the origin and travels in an easterly direction. There is a statue located 100 meters east and 50 meters north of the origin. At what point on the highway will the car’s headlights illuminate the statue?
Group 4: Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. (a) If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why the result is true.
Group 5: A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/sec. Find the rate at which the area within the circle is increasing after (a) 1 second, (b) 3 seconds, and (c) 5 seconds. What can you conclude?
Group 6: If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V = 5000(t – t/40)^2, for 0 ≤ t ≤ 40. Find the rate at which water is draining from the tank after (a) 5 minutes, (b) 10 minutes, (c) 20 minutes, and (d) 40 minutes. At what time is the water flowing out the fastest? The slowest? Summarize your findings.
Group 7: When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.4 = C, where C is a constant. Suppose that at a certain instant the volume is 400 cm^3 and the pressure is 80 kPa and decreasing at a rate of 10 kPa/min. At what rate is the volume increasing at this instant?
Group 8: Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W^(2/3), where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L^2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species’ brain growing when the average length was 18 cm?