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Question 11

Read each of the following descriptions and identify the quantities that are undergoing exponential decay.
the amount of radioactive material in a 250-gram sample decays by 50% each day
a 220-mg dose of aspirin decreases by 30% every hour
the value of a rare coin initially priced at $0.05 increases by 35% each month
a ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet
Choose one situation in part A that represents exponential decay. Write an equation to model the relationship between the two quantities.
For the situation from part A that you chose, explain the meaning of x, y, a, and b in the context of the situation described.
For the situation from part A that you chose, create a table of values and use the table of values to graph the relationship between the two quantities.

1 answer

To identify the quantities that are undergoing exponential decay from the descriptions provided:

  1. The amount of radioactive material in a 250-gram sample decays by 50% each day. (Exponential decay)
  2. A 220-mg dose of aspirin decreases by 30% every hour. (Exponential decay)
  3. The value of a rare coin initially priced at $0.05 increases by 35% each month. (Not exponential decay; this is exponential growth)
  4. A ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet. (Not exponential decay; this describes a physical motion scenario)

Chosen Situation

Situation: The amount of radioactive material in a 250-gram sample decays by 50% each day.

Equation Modeling Exponential Decay

The general formula for exponential decay can be expressed as:

\[ y = a \cdot b^x \]

where:

  • \( y \) represents the remaining quantity after time \( x \),
  • \( a \) is the initial amount,
  • \( b \) is the decay factor (where \( 0 < b < 1 \)),
  • \( x \) is the number of time periods (days in this case).

For this situation:

  1. Initial amount (a): 250 grams
  2. Decay factor (b): Since the sample decays by 50% each day, the remaining amount is 50%, or \( 0.5 \). So, \( b = 0.5 \).
  3. Time (x): The number of days.

Thus, the equation becomes:

\[ y = 250 \cdot (0.5)^x \]

Explanation of Variables

  • \( x \): Represents the number of days elapsed since the start of the decay (day 0).
  • \( y \): Represents the remaining amount of radioactive material after \( x \) days.
  • \( a \): The initial quantity of radioactive material (250 grams).
  • \( b \): The decay factor (0.5), which indicates that each day, the amount of material is reduced to 50% of what it was the previous day.

Table of Values

Let's create a table for the first 5 days:

| Day (x) | Remaining Amount (y) = 250 * (0.5)^x | |---------|-------------------------------------| | 0 | 250 * (0.5)^0 = 250 grams | | 1 | 250 * (0.5)^1 = 125 grams | | 2 | 250 * (0.5)^2 = 62.5 grams | | 3 | 250 * (0.5)^3 = 31.25 grams | | 4 | 250 * (0.5)^4 = 15.625 grams | | 5 | 250 * (0.5)^5 = 7.8125 grams |

Graphing the Relationship

To graph the relationship, you can plot the values from the table on a coordinate system where the x-axis represents the number of days (x), and the y-axis represents the remaining mass of the radioactive material (y).

  • Points to plot: (0, 250), (1, 125), (2, 62.5), (3, 31.25), (4, 15.625), (5, 7.8125).

The graph will show a continuous curve that decreases rapidly at first and then flattens out, illustrating the nature of exponential decay.