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Exploring Exponential Growth: Real-World Situations and Mathematical Models

This study investigates various scenarios illustrating exponential growth and decay, focusing on real-life applications. We explore the growth of bacteria, the depreciation of a car, changes in savings accounts, and more, providing mathematical equations, graphs, and practical domains. Each situation features unique rates of change and initial values, from the doubling of bacteria at midnight to the impact of interest on savings. Through examples like water evaporation and population growth, we reveal the underlying principles of mathematics in everyday life, demonstrating their relevance and importance.

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Exploring Exponential Growth: Real-World Situations and Mathematical Models

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  1. SITUATION 1 Snowfall At midnight, there was 1 bacterium. The number of bacterium doubled every hour GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 2 1 TABLE EQUATION Hour after midnight Number of Bacterium Y = 1(2)x or y = 2x 0 1 X 2 2 1 X 2 2 4 X 2 3 8 SEQUENCE Plot points from graph 1 2 4 8 16 X 2 X 2 X 2 X 2 NEXT-NOW STATEMENT Practical Domain All non-negative numbers Practical Range NEXT=NOW*2; starting at 1 All numbers at least one

  2. SITUATION 2 Amount of Flowers There are 6 flowers in a vase. After each day 1/3 of the flowers are left. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 6 1/3 TABLE EQUATION Y = 6(1/3)x 0 6 1 2 2 .667 = 2/3 3 .222 = 2/9 SEQUENCE .667 = 2/3 .222 = 2/9 .0741 = 2/27 6 2 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *(1/3); Start at 6 nonneg nos. ≤ 6 Practical Range

  3. SITUATION 3 Savings Account Joe has $ 100 in his savings account which earns interest at a rate of 5 % per year. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 100 1.05 TABLE EQUATION y = 100(1.05)x SEQUENCE 110.25 115.76 100 105 121.55 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *(1.05); Start at 100 y ≥ 100 Practical Range

  4. SITUATION 4 Stock Investment Sue has $ 80 that she invests in stock that is growing at a rate of 10 % per year. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 1.1 80 TABLE EQUATION y = 80(1.1)x 0 80 1 88 2 96.80 3 106.48 SEQUENCE 80 117.13 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *(1.1); Start at 80 y ≥ 80 Practical Range

  5. SITUATION Value of a Car Tom has a $ 20,000 car that depreciates at a rate of 15 % per year. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) .85 20,000 TABLE EQUATION y = 20000(.85)x 0 20,000 1 17,000 2 14,450 3 12,282.50 SEQUENCE 20,000 10,440.13 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. nonneg nos. ≤ 20000 NEXT = NOW *(.85); Start at 20000 Practical Range

  6. SITUATION 6 Evaporation of Water Eight ounces of water is in a glass. At the end of each day ¼ of the water is left. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 8 1/4 TABLE EQUATION y = 8(1/4)x 0 8 1 2 2 .5 3 .125 SEQUENCE 8 2 .5 .125 .03125 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. Nonneg. nos. ≤ 8 NEXT = NOW *(1/4); Start at 8 Practical Range

  7. SITUATION 7 Attendance Increase On the opening day of a carnival, 16 people attended. After opening day, attendance increased by 50 %. GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) 1.5 16 TABLE EQUATION y = 16(1.5)x 0 16 1 24 2 36 3 54 SEQUENCE 16 24 36 54 81 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. y ≥ 16 NEXT=NOW*1.5; starting at 16 Practical Range

  8. SITUATION 8 Population of a Village In a certain year, the population of a village is 11, 810. The population grows at a rate of 5 % per year. MULTIPLIER (rate of change) Y-intercept (initial value) 1.05 11,810 TABLE EQUATION y = 11,810(1.05)x 11,810 0 13,672 3 SEQUENCE 11,810 12,400 13,020 13,672 14,353 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *(1.05); Start at 11810 y ≥ 11810 Practical Range

  9. SITUATION 9 Volume of Balloon The volume of a balloon is 2/3 liter. As each gust of helium is added, the balloon triples its volume. GRAPH MULTIPLIER 3 (rate of change) Y-intercept 2/3 (initial value) TABLE EQUATION y = 2/3(3)x 2/3 0 1 2 2 6 3 18 SEQUENCE 2/3 2 6 18 54 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *3; Start at 2/3 y ≥ 2/3 Practical Range

  10. SITUATION 10 The value of an antique vase was $1800 in 2010. Each year, it increased its value by 8.5% MULTIPLIER (rate of change) Y-intercept (initial value) 1.085 1800 TABLE EQUATION y = 1800(1.085)x 0 1800 1 1953 2 2119 3 2299.1 SEQUENCE 1800 1953 2119 2299.1 2494.5 NEXT-NOW STATEMENT Practical Domain Nonneg. nos. NEXT = NOW *1.085; Start at 1800 y ≥ 1800 Practical Range

  11. SITUATION 11 Number of Friends on a Social Media Network Bonnie has been on a social network. When she started, she was chatting with 2 friends. Her number of friends tripled every hour. GRAPH MULTIPLIER (rate of change) 3 Y-intercept (initial value) 2 TABLE EQUATION y = 2(3)x 0 2 1 6 2 18 3 54 SEQUENCE 2 162 NEXT-NOW STATEMENT Practical Domain NEXT = NOW *3; Start at 2 Practical Range

  12. SITUATION 12 ______________________________ GRAPH MULTIPLIER (rate of change) Y-intercept (initial value) TABLE EQUATION SEQUENCE NEXT-NOW STATEMENT Practical Domain Practical Range

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