A. Read each of the following descriptions and identify the quantities that are

Question

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
B. For each description in part A that represents exponential decay, write an
exponential equation to model the relationship between the two quantities.
C. For each equation in part B, explain the meaning of x, y, a, and b in the
context of the situation described.
D. For each equation in part B, create a table of values and use the table of
values to graph the relationship between the two quantities.

Answer 1

A. Identifying Exponential Decay

Let's analyze each description:

1. The amount of radioactive material in a 250-gram sample decays by 50% each day.
   This represents exponential decay (material decreases over time).

2. A 220-mg dose of aspirin decreases by 30% every hour.
   This also represents exponential decay (the amount decreases over time).

3. The value of a rare coin initially priced at $0.05 increases by 35% each month.
   This represents exponential growth (value increases over time).

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.
   This does not represent exponential decay; it's a physical motion situation.

B. Exponential Equations for Decay

1. Radioactive Material

• Initial amount: \( a = 250 \) grams
• Decay rate: \( b = 0.5 \) (which means it retains 50% of its quantity)
• The equation is: \[ y = 250 \cdot (0.5)^x \]

2. Aspirin Decay

• Initial amount: \( a = 220 \) mg
• Decay rate: \( b = 0.7 \) (remains 70% of its quantity)
• The equation is: \[ y = 220 \cdot (0.7)^x \]

C. Explanation of Variables in Each Equation

For the radioactive material equation \( y = 250 \cdot (0.5)^x \):

• \( y \): Amount of radioactive material remaining after \( x \) days.
• \( x \): Number of days that have passed.
• \( a = 250 \): Initial amount of the radioactive material (grams).
• \( b = 0.5 \): The factor that indicates the material retains 50% of its quantity each day.

For the aspirin decay equation \( y = 220 \cdot (0.7)^x \):

• \( y \): Amount of aspirin remaining after \( x \) hours.
• \( x \): Number of hours that have passed.
• \( a = 220 \): Initial amount of aspirin (mg).
• \( b = 0.7 \): The factor that indicates the aspirin retains 70% of its quantity every hour.

D. Creating Tables of Values

1. Radioactive Material

| \( x \) (days) | \( y \) (grams) | |----------------|------------------| | 0 | 250 | | 1 | \( 250 \cdot (0.5)^1 = 125 \) | | 2 | \( 250 \cdot (0.5)^2 = 62.5 \) | | 3 | \( 250 \cdot (0.5)^3 = 31.25 \) | | 4 | \( 250 \cdot (0.5)^4 = 15.625 \)| | 5 | \( 250 \cdot (0.5)^5 = 7.8125 \) |

Graph for Radioactive Material

• Plot the points from the table: (0, 250), (1, 125), (2, 62.5), (3, 31.25), (4, 15.625), (5, 7.8125)

2. Aspirin Decay

| \( x \) (hours) | \( y \) (mg) | |----------------|------------------| | 0 | 220 | | 1 | \( 220 \cdot (0.7)^1 = 154 \) | | 2 | \( 220 \cdot (0.7)^2 = 107.8 \) | | 3 | \( 220 \cdot (0.7)^3 = 75.46 \) | | 4 | \( 220 \cdot (0.7)^4 = 52.82 \) | | 5 | \( 220 \cdot (0.7)^5 = 36.974 \) |

Graph for Aspirin Decay

• Plot the points from the table: (0, 220), (1, 154), (2, 107.8), (3, 75.46), (4, 52.82), (5, 36.974)

Note for Graphing

To graph these, you would typically use a graphing tool or software such as Desmos, Excel, or another program that allows plotting points easily. The curves will be downward-sloping, representing the nature of exponential decay.