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Modeling Exponential Decay

5.12.1 Modeling Exponential Decay

5.12.1 • Modeling Exponential Decay

Warm Up

Jada received a gift of $180. In the first week, she spent a third of the gift money. She continues spending a third of what is left each week thereafter.

1.

Which equation best represents the amount of gift money g g , in dollars, she has after t t weeks?

a. g = 180 − 1 3 t g = 180 − 1 3 t

b. g = 180 · ( 1 3 ) t g = 180 · ( 1 3 ) t

c. g = 1 3 · 180 t g = 1 3 · 180 t

d. g = 180 · ( 2 3 ) t g = 180 · ( 2 3 ) t

Compare your answer:

g = 180 · ( 2 3 ) t g = 180 · ( 2 3 ) t

2.

Explain your reasoning.

Compare your answer:

  • Jada's initial gift is $180. Then, she spends a third 1 3 1 3 of the whole amount each week. This means that she has 2 3 2 3 left after each week because 1 − 1 3 = 2 3 1 − 1 3 = 2 3 (the whole amount minus a third).
  • When writing the equation, a = 180 a = 180 because that is the starting amount she was given. Then, because she spends 1 3 1 3 each week, that means only 2 3 2 3 is leftover. so, b = 2 3 b = 2 3 . The equation that results is g = 180 · ( 23 ) t g = 180 · ( 23 ) t .