6-14 More Exponential Functions as Mathematical Models WK #19 Date. r n. b. How many customers will Paul have after 1 year?

Transcription

1 Alg2H 6-14 More Exponential Functions as Mathematical Models WK #19 Date Continuously Compounded Interest: A = Pe rt Natural Growth & Decay: y = ne kt (k is positive for growth & negative for decay) Interest Compounded n times per year: A P 1 nt r n Appreciation/Depreciation: V n = P(1 + r) n depreciation) (r is positive for appreciation, negative for 1. As word of mouth spreads about Paul s new business, the number of customers grows exponentially as the weeks go by. Paul started his business with 12 customers, after 8 weeks he has 78 customers. a. Write an equation for this function. b. How many customers will Paul have after 1 year? c. Paul decides that he will have to take on a partner when he gets 200 customers. After how many weeks will he have to take on a partner? 2. Mary deposits $600 into a bank account that pays 6.8% interest (compounded continuously). How long will it take her to reach $1400? 3. Bacteria grow in salt water in umber from 68 to 347 in 5 hours. Find k for the growth formula.

2 4. The class of 2000 invested $1200 at 5% interest (compounded continuously) at the beginning of their freshman year (year 0) to purchase a gift for the Math department. How much will they have at the end of their senior year (year 4) to spend on the present? 5. If you deposit $10,000 into an account that pays 8.5% interest compounded quarterly, how long will it take to reach $17,000? 6. For a radioactive substance the constant k is How long will it take 500 grams of the substance to reach its half-life? 7. The following data shows the increasing cost of certain goods from 1980 to 1985: Year Cost a. Is the cost exponentially related to time? b. If the cost continues to rise according to the same pattern, how much will the goods cost in 2003? (Provide the equation that you used to determine the answer.) c. When will the goods cost $1000?

3 8. A hard-boiled egg has a temperature of 98 C. If it is put into a sink that maintains a temperature of 18 C, its temperature x minutes later is given by Y = e-0.28x. Stephanie needs her egg to be at 30 C for decorating. How long should she leave it in the sink? 9. Phil bought a house for $200,000 in He wants to make a profit of $100,000 when he sells the house. If the house appreciates 7% per year, when should Phil sell his house? 10. Write an exponential equation to fit the data below. How many items will be in the supermarket in 2010?

4 11. After you pour a cup of coffee, it cools off in such a way that the difference between the coffee temperature and the room temperature decreases exponentially with time. This model is called Newton s Law of Cooling. Suppose that you pour a cup of coffee. Three minutes later, you measure its temperature and find that it is 85 C. Five minutes after the first reading, you find that it has cooled to 72 C. The room is at 20 C. Let c = number of degrees of coffee temperature. Let D = number of degrees difference (coffee minus room). Let t = number of minutes since first temperature reading. a. Show that you understand the definitions of D and t by writing the given information as two ordered pairs, (t, D). b. Write the particular equation expression D in terms of t. c. What was t when the coffee was first poured? Substitute this value of t into the equation to find D when the coffee was first poured. Based on your answer, what was the temperature, c, of the coffee when it was first poured? d. Assume that coffee is drinkable when its temperature is at least 55 C. For how long after it was poured will the coffee be drinkable? 12.

5 The pressure of the air in the Earth s atmosphere decreases exponentially with altitude above the surface of the Earth. The pressure at the Earth s surface (sea level) is about 14.7 pounds per square inch (psi) and the pressure at 2000 feet is approximately 13.5 psi. a. Write the particular equation expressing pressure in terms of altitude. b. Predict the pressure at i. Mexico City (altitude 7500 feet) ii. Mount Everest (altitude 29,000 feet) iii. Where U-2 spy planes fly (80,000 feet) iv. The edge of space (defined by NASA to be 50 miles up) c. Airplane cabins are pressurized because human blood at body temperature will boil if the pressure drops below 0.9 psi. At what altitude would your blood start to boil if you were in an unpressurized airplane? d. Recently it has been discovered that the Mediterranean Sea dried up about 6 million years ago, leaving a valley 10,000 feet below sea level. What would the air pressure have been at the bottom of this valley?